数理解析研究所講究録
第 1795 巻 2012 年 1-44

1

On some combinatorial and algebraic properties ofDunkl elements
ANATOL N. KIRILLOV

Research Institute of Mathematical Sciences (RIMS)
Kyoto 606-8502, JapanTo Volodya Korepin, my friend and co-author, on the occasion of his sixtieth birthday,with grateful and admimtion

Abstract
We introduce and study a certain class of nonhomogeneous quadratic algebras
together with the special set of mutually commuting elements inside of each, theso-called Dunkl elements. We describe relations among the Dunkl elements. This
result is a further generalization of similar results obtained in [10],[32] and [21].
As an application we describe explicitly the set of relations among the Gaudin
elements in the group ring of the symmetric group, cf [30].
Yet another objective of our paper is to describe several combinatorial properties of some special elements, the so-called Coxeter element and the longest
element, in the associative quasi-classical Yang-Baxter algebra. In the case of Coxeter element we relate the corresponding reduced polynomials introduced in [40],with the -Grothendieck polynomials for some special permutations. Moreover, we show that the specializationof the -Grothendieck polynomial$\pi_{k}^{(n)}$$\beta$$\beta$$\mathfrak{G}_{\pi_{k}^{(n)}}^{(\beta)}(1)$$\otimes_{(n)}^{(\beta)}(X_{n})$counts the number of k-dissections of a convex $(n+k+1)$ -gon ac-$\pi_{k}$cording to the number of diagonals involved. When the number of diagonals ina k-dissection is the maximal possible, we recover the well-known fact that the
number of k-triangulations of a convex $(n+k+1)$ -gon is equal to the value of acertain Catalan-Hankel determinant, see e.g. [36]. We also show that for a certain
5-parameters family of vexillary permutations, the specialization $x_{i}=1,$,of the corresponding -Schubert polynomials
coincides with some FussNarayana polynomials and their generalizations. We also point out on a conjecturalconnection between the sets of maximal compatible sequences for the permutationand thatfrom one side, and the set of VSASM $(n)$ and thatof CSTCPP(n) correspondingly, from the other, see Comments 3.6 for details.
Finally, in Section 3 we introduce and study a multiparameter generalization of
reduced polynomials introduced in [40], that of Catalan, Narayana and Schrodernumbers.
In the case of the longest element we relate the corresponding reduced polynomial with the Ehrhart polynomial of the Chan-Robbins polytope.
$\forall i\geq 1$$\beta$$\sigma_{n,2n,2,0}$$\sigma_{n,2n+1,2,0}$$\mathfrak{S}_{w}^{(\beta)}(1)$2Introduction
The Dunkl operators have been introduced in the later part of $80$ ’s of the last century
by Charles Dunkl [7], [8] as a powerful mean to study of harmonic and orthogonal
polynomials related with finite Coxeter groups. In the present paper we don need thedefinition of Dunkl operators for arbitrary (finite) Coxeter groups, see e.g. [7], but only
for the special case of the symmetric group .
$t$$S_{n}$.
be the $nng$ of polynomials in variables
Definition 0.1 Let,
opemtors
are
the
Dunkl
(additive)
mtionalThe type
differential-difference
opemtors of the following form
$P_{n}=\mathbb{C}[x_{1}, \ldots.x_{n}]$$x_{1},$$A_{n-1}$$D_{1},$$D_{i}= \lambda\frac{\partial}{\partial x_{i}}+\sum_{j\neq i}\frac{1-s_{ij}}{x_{i}-x_{j}}$Here$s_{ij},$$1\leq i<j\leq n$ ,$\ldots,$$x_{n}$$D_{n}$$\ldots$(0.1),

denotes the exchange (or permutation) opemtor, namely,;$s_{ij}(f)(x_{1}, \ldots, x_{i}, \ldots, x_{j}, \ldots, x_{n})=f(x_{1}, \ldots, x_{j}, \ldots, x_{i}, \ldots, x_{n})$$\frac{\partial}{\partial x}$

stands for the derivative $w.r.t$ . the variable$x_{i}$;$\lambda\in \mathbb{C}$is a parameter.

The key property of the Dunkl operators is the following result.

Theorem 0.1 (C. Dunkl $[7J)$ For any finite Coxeter group $(W, S)$ , where $S=\{s_{1}, \ldots , s_{l}\}$comanddenotes the set of simple reflections, the Dunkl operators$1\leqi,$$j\leql$$D_{i}D_{j}=D_{j}D_{i}$ ,.
mute:$D_{i}$$:=D_{s_{i}}$$D_{j}$$:=D_{s_{j}}$Another fundamental property of the Dunkl operators which finds a wide variety of
applications in the theory of integrable systems, see e.g. [15], is the following statement:
the operator$\sum_{i=1}^{l}(D_{i})^{2}$“essentially” coincides with the Hamiltonian of the rational Calogero-Moser model related to tbe finite Coxeter group $(W, S)$ .

Definition 0.2
level), denoted by(additive) Dunkl opemtor (or the Dunkl operator at critical, , is an opemtor of the form (0.1) with parameter $\lambda=0$ .
$i=1,$$\mathcal{I}hncated$$\mathcal{D}_{i},$For example, the type$l$$\ldots$$A_{n-1}$

rational truncated Dunkl operator has the following form$\mathcal{D}_{i}=\sum_{j\neq i}\frac{1-s_{ij}}{x_{i}-x_{j}}$.

Clearly the truncated Dunkl operators generate a commutative algebra.
The important property of the truncated Dunkl operators is the following result discovered and proved by C.Dunkl [8]; see also [1] for a more recent proof.

3
Theorem 0.2 (C. Dunkl [8], Y.Bazlov [1]) For any finite Coxeter gmup $(W, S)$ the algebm over generated by the truncated Dunkl opemtors
is canonically isomorphic to the coinvariant algebm of the Coxeter gmup $(W, S)$ .
$\mathbb{Q}$$\mathcal{D}_{1},$$\mathcal{D}_{l}$$\ldots,$

Example 0.1 In the case when $W=S_{n}$ is the symmetric group, Theorem 0.2 statesthat the algebra over generated by the truncated Dunkl operators$i=$,
is canonically isomorphic to the cohomology ring of the full flag varietyof$1,$$\ldots$$n$$\mathbb{Q}$.

$\mathcal{D}_{i}=\sum_{j\neq i}\frac{1-s_{i_{J}}}{x_{i}-x_{j}},$$\mathcal{F}l_{n}$type $A_{n-1}$$\mathbb{Q}[\mathcal{D}_{1}, \ldots , \mathcal{D}_{n}]\cong \mathbb{Q}[x_{1}, \ldots , x_{n}]/J_{n}$where$J_{n}$$k\leq n\}$

(0.2)

,

denotes the ideal generated by elementary symmetric polynomials$\{e_{k}(X_{n}),$$1\leq$.

Recall that the symmetric polynomials
generating function$e_{i}(X_{n}),$$i=1,$$\ldots$,$1+ \sum_{i=1}^{n}e_{i}(X_{n})t^{i}=\prod_{i=1}^{n}(1+tx_{i})$$n$, are defined through the,where we set $X_{n}:=$. It is well-known that in the caseisomorphism (0.2) can be defined over the ring of integers Z.
$(x_{1}, \ldots , x_{n})$$W=S_{n}$ ,the$\blacksquare$

Theorem 0.2 by C.Dunkl has raised a number of questions:(A) What is the algebra generated by the truncatedtrigonometric,$\bullet$$\bullet$elliptic,super, matrix, . . .,$(a)$additive Dunkl operators ?
$(b)$$Ruijsenaars-Schneider$-Macdonald operators ?
$(c)$Gaudin operators ?
(B) Describe commutative subalgebra generated by the Jucys-Murphy elements in
the group ring of the symmetric group;
the Hecke algebra;
the Brauer algebra, $BMV$ algebra, . . ..
(C) Does there exist an analogue of Theorem 0.2 forClassical and quantum equivariant cohomology and equivariant K-theory rings ofthe flag varieties ?
Cohomology and K-theory rings of affine flag varieties ?
Diagonal coinvariant algebras of finite Coxeter groups ?
Complex reflection groups ‘!
The present paper is a short Introduction to a few items from Section 5 of [18].
The main purpose of my paper “On some quadratic algebras, II” is to give somepartial answers on the above questions in the case of the symmetric group.
The purpose of the present paper is to draw attention to an interesting class of
nonhomogeneous quadratic algebras closely connected (still mysteriously !) with different
branches of Mathematics such as
Classical and Quantum Schubert and Grothendieck Calculi,
Low dimensional Topology,$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$S_{n}$

4
Classical, Basic and Elliptic Hypergeometric functions,
Algebraic Combinatorics and Graph Theory,
Integrable Systems,

What we try to explain in [18] is that upon passing to a suitable representation of
the quadratic algebra in question, the subjects mentioned above, are a manifestation ofcertain general properties of that quadratic algebra.
From this point of view, we treat the commutative subalgebra generated by the, see
additive (resp. multiplicative) truncated Dunkl elements in the algebra
Definition 2.1, as universal cohomology (resp. universal K-theory) ring of the complete. The classical or quantum cohomology (resp. the classical or quantum
flag variety
K-theory) rings of the flag variety
are certain quotients of the universal $nng$ .
For example, in [20] we have computed relations among the (truncated) Dunkl elements $\{\theta_{i}, i=1, \ldots, n\}$ in the elliptic representation of the algebra $3T_{n}(\beta=0)$ . We
, but
is isomorphic to (yet not
expect that the comlntltative subalgebra.
see [14] the elliptic cohomology ring of the flag variety
Another example from [18]. Consider the algebra $3T_{n}(\beta=0)$ .
One can prove [18] the following identities in the algebra $3T_{n}(\beta=0)$(A) Summation formula$3T_{n}(\beta)$$\mathcal{F}l_{n}$$\mathcal{F}l_{n}$$ot$ ) $tain_{\text{ノ}}\cdot(1$$(1_{t_{\text{ノ}}^{1}}fi_{I1^{1},(1}$$\mathcal{F}l_{n}$

$)$

$n-1$

$n-1$

$\sum$$( \prod$

$j=1$

$j-1$

$n-1$

$u_{b,b+1})u_{1,n}( \prod u_{b,b+1})=\prod u_{a,a+1}$$b=j+1$

$a=1$

$b=1$

(B) Duality transformation formula Let $m\leq n$ , then$n-1$$\sum$ $( \prod$$j=m$$b=j+1$

$j-1$

$m-1$

$n-1$
$u_{b,b+1})$$[ \prod$$u_{a,a+n-1}u_{a,a+n}]u_{m,m+n-1}$

$j=1$

$a=1$

$u_{b,b+1})=$

$b=m$

$a=1$

$\sum[\prod^{m}u_{a,a+n}m-ju_{a+1,a+n}]$$(\square$$( \prod^{n-1}u_{b,b+1})$$b=m$$[ \prod^{j-1}u_{a,a+n-1}u_{a.a+n}]-$$a=1$$\sum_{j=2}^{m}[\prod_{a=j}^{m-1}u_{a,a+n-1}u_{a,a+n}]u_{m,n+m-1}(\prod_{b=m}^{n-1}u_{b,b+1})u_{1,n}$.

One can check that upon passing to the elliptic representation of the algebra $3T_{n}(\beta=$, see [18], Section 5.1.7, or [20] for the definition of elliptic representation, the above
identities (A) and (B) finally end up correspondingly, to be a Summation formula and a),
Duality transformation formula for multiple elliptic hypergeometric series (of typesee e.g. [16] for definition of the latter.
After passing to the so-called $Fay$ representation [18], the identities (A) and (B) becomecorrespondingly to be the Summation formula and Duality transformation formula for
the Riemann theta functions of genus $g>0,$ $[18]$ . These formulas in the case $g\geq 2$
seems to be new.
$0)$

$A_{n-1}$$\blacksquare$5
A few words about the content of the present paper.
In the first section I introduce the so-called dynamical classical Yang-Baxter algebm
as “a natural quadratic algebra” in which the Dunkl elements form a pair-wise commuting
family. It is the study of the algebra generated by the (truncated) Dunkl elements thatis the main objective of our investigation in [18] and the present paper.
In Section 2, see Definition 2.1, we introduce the algebra $3HT_{n}(\beta)$ , which seems to be
the most general (noncommutative) deformation of the (even) Orlik-Solomon algebra,
such that it’s still possible to describe relations among the Dunkl elements, see Theorem 2.1. As an application we describe explicitly a set of relations among the (additive)
Gaudin/Dunkl elements, cf [30].
In Section 3 we describe some combinatorial properties of special elements in theassociative quasi-classical Yang-Baxter algebra 1 $ACYB_{n}$ . The results of Section 3.1,
see Proposition 3.1, items (1) $-(5)$ , are more or less known among the specialists inthe subject, while those of the item (6) seem to be new. Namely, we show that thepolynomial $Q_{n}(x_{ij}=t_{i})$from [40], (6.C8), $(c)$ , essentially coincides with thedeformation [11] of the Lascoux- Sch\"utzenberger Grothendieck polynomial [25] for someparticular permutation.
The results of Proposition 3.1, (6), point out on a deep
connection between reduced forms of monomials in the algebra $ACYB_{n}$ and the Schubertand Grothendieck Calculi. This observation was the starting point for the study ofsome combinatorial properties of certain specializations of the Schubert, -Grothendieck[12] and double -Grothendieck polynomials in Section 3.2. One of the main resultsof Section 3.2 can be stated as follows.
$\beta-$$\beta$$\beta$Theorem 0.3(1) Let $w\in S_{n}$ be a permutation, consider the specialization.
Thenof the -Grothendieck polynomial$:=q,$ $x_{i}=1,$$\forall i\geq 2$,$\mathfrak{G}_{w}^{(\beta)}(X_{n})$$\beta$$\mathcal{R}_{w}(q, \beta+1)$$:=\mathfrak{G}_{w}^{(\beta)}(x_{1}=q, x_{i}=1, \forall i\geq 2)\in \mathbb{N}[q, 1+\beta]$

In other words, the polynomial
For late use we define polynomials$\mathcal{R}_{w}(q, \beta)$.

Let $w\in S_{n}$ be a permutation, consider the specialization
the double -Grothendieck polynomial
.
Then
(2)

$x_{i}$$\emptyset_{w}^{(\beta-1)}(x_{i}:=q, y_{i}:=t, \forall i\geq 1)\in N[q, t, \beta]$$w$$\forall i\geq 1$

, of

.

be a permutation, then$\Re_{v}(1, \beta)=\Re_{1\cross w}(0, \beta)$

Note that$:=q,$ $y_{i}=t,$$\otimes_{w}^{(\beta)}(X_{n}, Y_{n})$$\beta$Let.

has non-negative integer coefficients 2.

$\Re_{w}(q, \beta):=q^{1-w(1)}\mathcal{R}_{w}(q, \beta)$(3)$x_{1}$$\mathcal{R}_{w}(1, \beta)=\mathcal{R}_{w^{-1}}(1, \beta)$

,

but

.

$\mathcal{R}_{w}(t, \beta)\neq \mathcal{R}_{w^{-1}}(t, \beta)$, in the general case.

1
The algebra $ACYB_{n}$ can be treated as “one-half” of the algebra. It appears, see Lemma 3.1,
that the basic relations among the Dunkl elements, which do not mutually commute anymore, are stillvalid.
2For a more general result see Appendix$3T_{n}(\beta)$

6
For the reader convenience we collect some basic definitions and results concerning the-Grothendieck polynomials in Appendix.
, wheredenotes the specialization
Let us observe that$:=1,$
corresponding to permutation .
, of the Schubert polynomial
Therefore, $\Re_{u}(1,1)$ is equal to the number of compatible sequences [4] (or pipe dreams,
see e.g. [36] corresponding to permutation .
$\beta$$\Re_{w}(1,1)=\mathfrak{S}_{w}(1)$$\forall i\geq 1$$x_{i}$$\mathfrak{S}_{w}(1)$$w$$\mathfrak{S}_{w}(X_{n})$

$kl;$

$)$

Problem 0.1
Let $w\in S_{n}$ be a permutation and $:=\ell(u’)$ be its length. Denote by $CS(w)=$$\{a=(a_{1}\leq a_{2}\leq\ldots\leq a_{l})\in N^{l}\}$ the setof compatible sequences $l41$ corresponding to$l$permutation $w$ .
$\underline{Define}$$\bullet$in$a$statistics $r(a)$ on the set of all compatible sequences$CS_{n};= \prod_{w\in S_{n}}CS(w)$such way that$\sum_{a\in CS(w)}q^{a_{1}}\beta^{r(a)}=\mathcal{R}_{w}(q, \beta)$.

Find and investigate a geometric interpretation, combinatorial and algebra-geometmc,
properties of polynomials
the -Schubert polynomial
where for a permutation $w\in S_{n}$ we denoted by$\bullet$$\mathfrak{S}_{w}^{(\beta)}(X_{n})$$\mathfrak{S}_{w}^{(\beta)}(X_{n})$$\beta$defined as follows$\mathfrak{S}_{w}^{(\beta)}(X_{n})=\sum_{a\in CS(w)}\beta^{r(a)}\prod_{i=1}^{l:=\ell(w)}x_{a_{i}}$.

coincides with the Hilbert polynomial of a certain
We expect that polynomial
graded commutative ring naturally associated to permutation .
$\mathfrak{S}_{w}^{(\beta)}(1)$$w$

Remark 0.1 It should be mentioned that, in geneml, the principal specialization$6_{w}^{(\beta-1)}(x_{i}:=q^{i-1}, \forall i\geq 1)$of the $(\beta-1)$ -Gmthendieck polynomial may have negative coefficients.

$\blacksquare$for a special
Our main objective in Section 3.2 is to study the polynomials
class of permutations in the symmetric group @ . Namely, in Section 3.2 we study
for the five parameters familysome combinatorial properties of polynomials$\Re_{u}(q, \beta)$$\infty$$\Re_{\varpi\lambda,\phi}(q, \beta)$

of vexillary permutations$\lambda$$\{\varpi_{\lambda,\phi}\}$which have the shape$:=\lambda_{n,p,b}=(p(n-i+1)+b,$ $i=1,$ $\ldots,$ $n+\overline{1)}$ and flag$\phi:=\phi_{k,r}=(k+r(i-1), i=1, \ldots, n+1)$ .
This class of permutations is notable for many reasons, including that the specialadmits a nice product formula, seeized value of the Schubert polynomial
Theorem 3.6. Moreover, we describe also some interesting connections of polynomials
with the Fuss-Catalan numbers 3 and Fuss-Narayana polynomials, k$\mathfrak{S}_{\varpi_{\lambda,\phi}}(1)$$\Re_{\varpi_{\lambda,\phi}}(q, \beta)$

3We define the (generalized) Fuss-Catalan numbers to be$FC_{n}^{(p)}(b)$of the FUss-Catalan numbers with the p-ballot numbers $Bal_{p}(m, n)$numbers $R_{m}(a, b)$
can be described as follows. Connection
and the Rothe$:= \frac{1+b}{1+b+(n-1)p}(\begin{array}{l}np+bn\end{array})$$:= \frac{n-mp+1}{n+m+1}(\begin{array}{l}n+m+1m\end{array})$$:= \frac{a}{a+bn}(\begin{array}{l}a+bnn\end{array})$$FC_{n}^{(p)}(b)=R_{m}(b+1,p)=Bal_{p-1}(n, (n-1)p+b)$.

7
triangulations and k-dissections of a convex polygon, as well as a connection with two
families of ASM. For example, if $\lambda=(b^{n})$ and $\phi=(k^{n})$ , then the polynomial
defines a-deformation of the number of (descending) plane partitions
sitting in the box. It seems an interesting problem to find an algebra-geometricinterpretation of polynomials
in the general case.
In Section 3.3 we give a partial answer on the question $6.C8(d)$ by R.Stanley [40].
Almost all results in Section 3 state that some two specific sets have the same numberof elements. Our proofs of these results are pure algebraic. It is an interesting problem
to find bijective proofs of results from Section 3 which generalize and extend the bijective
proofs presented in [44], [36], [41] to the case of -Grothendieck polynomials, the Schrodernumbers and k-dissections of a convex $(n+k+1)$ -gon. We are planning to treat andpresent these bijections in (a) separate publication(s).
At the end of Introduction I want to add two remarks.
(a) After a suitable modification of the algebra $3HT_{n}$ , see [22], and the casein[18], one can compute the set of relations among the (additive) Dunkl elements (definedin Section 1, (1.3) . In the case $\beta=0$ and $q_{ij}=q_{i}\delta_{j-i,1},1\leq i<j\leq n$ , whereis the Kronecker delta, the commutative algebra generated by additive Dunkl elements(1.3) appears to be “almost“ isomorphic to the equivariant quantum cohomology ringof the flag variety, see [22] for details. Using the multiplicative version of Dunkl
elements (1.3), one can extend the results from [22] to the case of equivariant quantum
K-theory of the flag variety, see [18].
(b) In fact one can construct an analogue of the algebra $3HT_{n}$ and a commutative
subalgebra inside it, for any graph $\Gamma=(V, E)$ on vertices, possibly with loops andmultiple edges, [18]. We denote this algebra by, and denote by
its nilquotient, which may be considered as a “classical limit of the algebra”.
The case of the complete graph $\Gamma=K_{n}$ reproduces the results of the present paper andthose of [18], i.e. the case of the full flag variety. The case of the complete multipartite gmph
reproduces the analogues of results stated in the present paper
for the full flag variety, see [18] for, in the case of the partial flag variety
details.
We expect that in the case of the complete graph with all edges of multiplicity $m,$, the commutative subalgebra generated by the Dunkl elements in the algebra
is related to the algebra of coinvariants of the diagonal action of the symmetricgroup
on the ring of polynomials.
$(q, \beta)$$\Re_{\varpi_{\lambda,\phi}}(q, \beta)$$b\cross k\cross n$$\Re_{u}(q, \beta)$$\beta$$\blacksquare$$\beta\neq 0$$)$$\delta_{a,b}$$\mathcal{F}l_{n}$$\mathcal{F}l_{n}$$n$$3T_{n}(\Gamma)$$3T_{n}^{(0)}(\Gamma)$$3T_{n}(\Gamma)$$\mathcal{F}l_{n}$$\Gamma=K_{n_{1},\ldots,n_{r}}$$\mathcal{F}l_{n}$$\mathcal{F}_{n_{1},\ldots,n_{r}}$$\Gamma=$$K_{n}^{(m)}$$3T_{n}^{(0)}(\Gamma)$$S_{n}$$\mathbb{Q}[X_{n}^{(1)}, \ldots, X_{n}^{(m)}]$Example 0.2 Take $\Gamma=K_{2,2}$ . The algebm
is genemted by four elements $\{a=$$b=u_{14},$ $c=u_{23},$ $d=u_{24}\}$ subject to the following setof (defining) relations$a^{2}=b^{2}=c^{2}=d^{2}=0$ ,$cb=bc$ , a $d=da$ ,$aba+bab=0=aca+cac$ , $bdb+dbd=0=cdc+dcd$,$3T^{(0)}(\Gamma)$$u_{13},$$\bullet$$\bullet$$abd-bdc-cab+dca=0=acd-bac-cdb+dba$ ,$abca+adbc+badb+brad+cadc+dbcd=0$ .
It not difficult to see that$\bullet$$\iota s$

Hilb $(3T^{(0)}(K_{2,2}), t)=[3]_{t}^{2}[4]_{t}^{2}$ , Hilb $(3T^{(0)}(K_{2,2})^{ab}, t)=(1,4,6,3)$ .

8
its abelization.
Here for any algebm A we denote by
The commutative subalgebra in $3T^{(0)}(K_{2,2})$ , which corresponds to the intersection$:=\theta_{1}+\theta_{2}=(a+b+c+$, is generated by the elements
d and $:=\theta_{1}\theta_{2}=(ac+ca+bd+db+ad+bc)$ . The elements and commute andsatisfy the following relations$A^{ab}$$3T^{(0)}(K_{2,2})\cap Z[\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}]$$)$$c_{1}$$c_{2}$$c_{1}$$c_{2}$$c_{1}^{3}-2c_{1}c_{2}=0$ ,$c_{2}^{2}-c_{1}^{2}c_{2}=0$.

The ring of polynomials
is isomorphic to the cohomology ring $H^{*}(Gr(2,4), Z)$of the Grassmannian variety $Gr(2,4)$ .
, i.e. forThis example is illustrative of the similar results valid for the graphs
and the corresponding
the partial flag varieties [18]. The meaning of the algebra
commutative subalgebra inside it for a general graph , is still unclear.
$Z[c_{1}, c_{2}]$$K_{n_{1},\ldots,n,}$$3T_{n}^{(0)}(\Gamma)$$\Gamma$$\blacksquare$

Conjecture 1

Let $\Gamma=(V, E)$ be a connected subgraph of the complete graph$K_{n}$

on$n$vertices. Then

Hilb $(3T_{n}^{(0)}(\Gamma)^{ab}, t)=t^{|V|-1}T(\Gamma;1+t^{-1},0)$ ,the symbol $T(\Gamma;x, y)$ denotes the Tutte polynomial correwhere for any graphsponding to this graph.
Acknowledgments
I thank Professor Toshiaki Maeno for many years fruitful collaboration. I $m$ also
grateful to Professors Y. Bazlov, A. Buch, I. Burban, B. Feigin, S. Fomin, A. Isaev, M.
Ishikawa, B. Shapiro and Dr. Evgeny Smirnov for fruitful discussions on different stagesof writing [18].
$\Gamma$Dunkl elements

1Let
be the free associative algebra over, .
$:=u_{ii},$ $i=1,$
We set$A_{\eta}$$x_{i}$$\ldots$$Z$

with the set of generators $\{u_{ij}, 1\leq i,j\leq n\}$ .

$n$Definition 1.1 Define (additive) Dunkl elements$\theta_{i},$$i=1,$$\ldots,$$n$, in the algebm$\mathcal{A}_{m}$

to

$be$

$\theta_{i}=x_{i}+\sum_{j\neq l}^{n}J^{=1}u_{ij}$

.

(1.3)

such that
We are interested in to find ‘natural relations” among the generators
the Dunkl elements (1.3) are pair-wise commute. One natural condition which is thecommonly accepted in the theory of integrable systems, is$\iota$$\{u_{ij}\}$$\bullet$(Locality condition)$[x_{i}, x_{j}]=0$,$u_{ij}u_{kl}=u_{kl}u_{ij}$,$if\{i,j\}\cap\{k, l\}=\emptyset$

.

(1.4)

9
Lemma 1.1 Assume that elements$\{u_{ij}\}$satisfy the locality conditions (1.4). Then$u_{ij}+u_{ji}]+[u_{ij},$ $\sum_{k=1}^{n}x_{k}]+\sum_{k\neq i,j}w_{ijk}$,$w_{ijk}=[u_{ij}, u_{ik}+u_{jk}]+[u_{ik}, u_{jk}]+[x_{i}, u_{jk}]+[u_{ik}, x_{j}]+[x_{k}, u_{ij}]$.

$[ \theta_{i}, \theta_{j}]=[x_{i}+\sum_{k\neq i,j}u_{ik},$

where
(1.5)

Therefore in order to ensure that the Dunkl elements form a pair-wise commuting family,
it’s natural to assume that the following conditions hold
(Unitarity)$\bullet$for all $i,j,$$[u_{ij}+u_{ji}, u_{kl}]=0=[u_{ij}+u_{ji}, x_{k}]$

$k,$

$l$

(1.6)

,

i.e. the elements
are central.
(Crossing relations)$?1_{ij}+u_{ji}$$\bullet$$[ \sum_{k=1}^{n}x_{k}, u_{ij}]=0$

for all$i,$ $j$(1.7).

(Dynamical classical Yang-Baxter relations)$\bullet$$[u_{ij}, u_{ik}+u_{jk}]+[u_{ik}, u_{jk}]+[x_{i}, u_{jk}]+[u_{ik}, x_{j}]+[x_{k}, u_{ij}]=0$

,

(1.8)

if $i,j,$ are pair-wise distinct.
We denote by $DCYB_{n}$ the quotient of the algebraby the two-sided ideal generated
by relations (1.4),(1.6),(1.7) and (1.8). Clearly, the Dunkl elements (1.3) generate acommutative subalgebra inside the algebra $CDYB_{n}$ , and the sum
belongs to the center of the algebra $DCYB_{n}$ .
$k$$A_{\eta}$$\sum_{i=1}^{n}\theta_{i}=\sum_{\iota=1}^{n}x_{i}$Example 1.1 (A representation of the algebm $DCYB_{n},$ $cf[9J)$
Given a setof mutually commuting parameters, define, if$i<j$ and set $q_{ij}=q_{ji}$ in the case $i>j$ . Clearly, that if $i<j<k$ , then $q_{ij}q_{jk}=q_{ik}$ .
Letbe a set of variables. Denote by
the corresponding
ring of polynomials. We consider the variable$i=1,$, also as the operator acting
on the ring of polynomials
by multiplication on .
Let
be a transposition. We consider the transposition
also as the operator
which acts on the ringby interchanging and , and fixes all other variables. We
denote by$q_{ij}= \prod_{a=i}^{j-1}q_{a}$$q_{1)}\ldots,$ $q_{n-1}$$z_{1},$$\ldots,$$P_{n}$$z_{n}$$z_{i},$$P_{n}$$:=Z[z_{1}, \ldots , \sim n\vee]$$\ldots,$$n$$\sim i\gamma.$$s_{ij}\in S_{n}$$s_{ij}$$P_{n}$$z_{i}$$z_{j}$$\partial_{ij}=\frac{1-s_{ij}}{z_{i}-z_{j}}$the divided difference operator corresponding to the transposition
operator (cf [9]),$\partial_{(ij)}:=\partial_{i}\cdots\partial_{j-1}\partial_{j}\partial_{j-1}\cdots\partial_{i}$The operators(cf [9])$\partial_{(ij)},$$s_{ij}$. Finally we $defi_{I1}e$if $i<j$ .

$1\leq i<j\leq n$ satisfy (among others) the following set of relations

10
if

,

$j\not\in[i, k]$,$[z_{j}, \partial_{(ik)}]=0$$\bullet$$[\partial_{(ij)}, \partial_{(kt)}]=\delta_{jk}[z_{j}, \partial_{(il)}]+\delta_{il}[\partial_{(kj)}, z_{i}]$Therefore, if we set
a triple $i<j<k$ we will have$u_{ij}=q_{ij}\partial_{(ij)}$,$[ \partial_{(ij)}, \sum_{a=i}^{j}\sim a’\gamma]=0$$\bullet$,, if $i<j$ ,if $i<j$ ,$k<l$ .

and $u_{(ij)}=-u_{(ji)}$ ,if $i>j$ , then for$[u_{ij}, u_{ik}+u_{jk}]+[u_{ik}, u_{jk}]+[_{\sim i},$ $u_{jk}]+[u_{ik,\sim j}\vee]+[z_{k}, u_{jk}]=q_{ij}q_{jk}[\partial_{(ij)}, \partial_{(jk)}]+q_{ik}[\partial\vee]=0$.

and $\{u_{ij}, 1\leq i<J\leq n\}$ define a representation of theThus the elements $i=1,$
algebra $DCYB_{n}$ , and therefore the Dunkl elements$\sim i\prime x,$$\ldots,$$n$$\theta_{i}:=z_{i}+\sum_{j\neq i}u_{ij}=z_{i}-\sum_{j<i}q_{ji}\partial_{(ji)}+\sum_{j>i}q_{ij}\partial_{(ij)}$

form a pairwise commuting family of operators acting on the ring of polynomials, cf [9].
$Z[q_{1}, \ldots, q_{n-1}][z_{1}, \ldots, z_{n}]$$\blacksquare$

Comments 1.1
be the quotient of
the algebraby the two-sided ideal generated by the relations (1.4), (1.7) and (1.8).
Consider elements(Non-unitary dynamical classical Yang-Baxter algebra) Let$\overline{\mathcal{A}_{m}}$$A_{m}$$\theta_{i}=x_{i}+\sum_{a\neq i}u_{ia}$,and$\overline{\theta_{j}}=-x_{j}+\sum_{b\neq j}u_{bj}$.

$1\leq i<j\leq n$ .

Then$[ \theta_{i},\overline{\theta}_{j}]=[\sum_{k=1}^{n}x_{k}, u_{ij}]+\sum_{k\neq i,j}?l;_{ikj}$.

Therefore the elements and commute in the algebra .
, , the relations $w_{ijk}=0$ , assuming
In the case when $x_{i}=0$ for all $i=1,$
that $i,j,$ are all distinct, are well-known as the (non-unitary) classical Yang-Baxter
relations. Note that for a given triple $(i,j, k)$ we have in fact 6 relations. These six. However,relations imply that$\theta_{i}$$A_{n}$$\overline{\theta}_{j}$$\ldots$$n$$k$$[\theta_{i},\overline{\theta_{j}}]=0$$[ \theta_{i}, \theta_{j}]=[\sum_{k\neq i,j}u_{ik},$$u_{ij}+u_{ji}]\neq 0$.

In order to ensure the commutativity relations among the Dunkl elements, i.e.
, one needs to impose on the elements $\{u_{ij}, 1\leq i\neq j\leq n\}$ the “twisted”
for allclassical Yang-Baxter relations, namely$[\theta_{i}, \theta_{j}]=$$0$$i,$ $j$$[u_{ij}+u_{ik}, u_{jk}]+[u_{ik}, u_{ji}]=0$,

if$i,j,$ $k$

are all distinct.

(1.9)

Contrary to the case of non-unitary classical Yang-Baxter relations, it is easy to see thatin the case of twisted classical Yang-Baxter relations, for a given triple $(i, j, k)$ one has
only 3 relations.

11

2

Algebra $3HT_{n}$Consider the twisted classical Yang-Baxter relation
where $i,j,$$[u_{ij}+u_{ia}, u_{ja}]+[u_{ia}, u_{ji}]=0$ ,$k$are distinct.

Having in mind applications of the Dunkl elements, we split the above relation on two
relationsand $u_{jk}u_{ij}=u_{ik}u_{jk}-u_{ji}u_{ik}$ ,$u_{ij}u_{jk}=u_{jk}u_{ik}-u_{ik}u_{ji}$and impose the unitarity constraints$u_{ij}+u_{ji}=\beta$ ,where$\beta$is a central element. Summarizing, we come to the following definition.

Definition 2.1 Define algebm
to be the quotient of the free associative algebm$1\leqi<j\leqn\rangle$,by the set of relations$3T_{n}(\beta)$$\langle$$u_{ij}$$Z[\beta]\bullet$(Locality)$\bullet$if$u_{ij}u_{kl}=u_{kl}u_{ij}$, if $\{i, j\}\cap\{k, l\}=\emptyset$ ,$u_{ij}u_{jk}=u_{ik}u_{ij}+u_{jk}u_{ik}-\beta u_{ik}$ ,$u_{jk}u_{ij}=u_{ij}u_{ik}+u_{ik}u_{jk}-\beta u_{ik}$,$1\leq i<j<k\leq n$ .

For each pair$i<j$ ,

we define element$q_{ij}$$:=u_{ij}^{2}-\beta u_{ij}\in 3T_{n}(\beta)$.

Lemma 2.1
(1) The elements $\{q_{ij}, 1\leq i<j\leq n\}$ satisfy the Kohno-Drinfeld relations(known also as the horizontal four term relations)$q_{ij}q_{kl}=q_{kl}q_{ij}$$[q_{ij}, q_{ik}+q_{jk}]=0$ ,(2),$if\{i,j\}\cap\{k, l\}=\emptyset$ ,$[q_{ij}+q_{ik}, q_{jk}]=0$ ,$ifi<j<k$ .

For a triple $(i<j<k)$ define $u_{ijk};=u_{ij}-u_{ik}+u_{jk}$ . Then$u_{ijk}^{2}=\beta u_{ijk}+q_{ij}+q_{ik}+q_{jk}$(3).

(Deviation from the Yang-Baxter and Coxeter relations)$u_{ij}u_{ik}u_{jk}-u_{jk}u_{ik}u_{ij}=[u_{ik}, q_{ij}]=[q_{jk}, u_{ik}]$$u_{ij}u_{jk}u_{ij}-u_{jk}u_{ij}u_{jk}=q_{ij}u_{ik}-u_{ik}qjk$,.

Comments 2.1 It is easy to see that the horizonta14-term relations listed in Lemma 2.1,
, together with(1), are equivalent to the locality condition among the generators
the commutativity conditions among the Jucys-Murphy elements$\{q_{ij}\}$$d_{i}:= \sum_{j=i+1}^{n}q_{ij}$,$i=2,$$\ldots,$$n$,

namely, $[d_{i}, d_{j}]=0$ . In [18] we describe some properties of a commutative subalgebra
generated by the Jucys-Murphy elements in the Kohno-Drinfeld algebra. It is wellknown that the Jucys-Murphy elements generate a maximal commutative subalgebra in
the group ring of the symmetric group . It is an open problem to describe defining].
relations among the Jucys-Murphy elements in the group$S_{n}$$\overline{ringZ[S}_{n}$$\blacksquare$

12
Finally we introduce the “Hecke quotient” of the algebra$3T_{n}(\beta)$, denoted by $3HT_{n}$ .

Definition 2.2 Define algebm $3HT_{n}$ to be the quotient of the algebm
of relations$q_{ij}q_{kl}=q_{kl}q_{ij}$ ,for all $i,j,$ .
$k,$$3T_{n}(\beta)$by the set$l$

In other words we assume that the all elements $\{q_{ij}, 1 \leq i<j\leq n\}$ are central in. From Lemma 2.1 follows immediately that in the algebra $3HT_{n}$ thethe algebra
satisfy the multiplicative (or quantum) Yang-Baxter relations
elements$3T_{n}(\beta)$$\{u_{ij}\}$$u_{ij}u_{ik}u_{jk}=u_{jk}u_{ik}u_{ij}$ ,if $i<J<k$ .

Therefore one can define multiplicative analogues, we define elements
elements . Namely, to start(2.10)of the Dunkl$1\leq i\leq n$ ,$\Theta_{i},$$wit1_{1}$$\theta_{i}$$h_{ij}$$:=h_{ij}(t)=1+tu_{ij}$ ,We consider
as an element of the algebra
where we assume that all parameters$h_{ij}(t)$$i\neq j$$\overline{3HT_{n}}$$\{\beta, q_{ij}, t, x, y, \ldots\}$.

$:=3HT_{n}\otimes Z[[\beta, q_{ij}^{\pm 1}, t, x, y, \ldots]]$,are central in the algebra $3HT_{n}$ .

Lemma 2.2
(la)
(lb)

$h_{ij}(x)h_{ij}(y)=h_{ij}(x+y+\beta xy)+q_{ij}xy$ ,$h_{ij}(x)h_{ji}(y)=h_{ij}(x-y)+\beta y-q_{ij}xy$ ,if $i<j$ .

if $i<J$ , and therefore theIt follows from (lb) thatelements
are invertible in the algebra $3HT_{n}$ .
(2) $h_{ij}(x)h_{jk}(y)=h_{jk}(y)h_{ik}(x)+h_{ik}(y)h_{ij}(x)-h_{ik}(x+y+\beta xy)$ .
(3) (Multiplicative $Yang-Baxter$ relations)$h_{ij}(t)h_{ji}(t)=1+\beta t-t^{2}q_{ij}$ ,$\{h_{ij}\}$$h_{ij}h_{ik}h_{jk}=h_{jk}h_{ik}h_{ij}$(4),

if $i<j<k$ .

Define multiplicative Dunkl elements (in the algebm$\Theta_{j}$$:= \Theta_{j}(t)=(\prod_{a=j-1}^{1}h_{aj}^{-1})$$( \prod_{a=n}^{j+1}h_{ja})$,$\overline{3HT_{n}}$) as follows$1\leq j\leq n$.

(2.11)

Then the multiplicative Dunkl elements pair-wise commute.

Clearly$\prod_{j=1}^{n}\Theta_{j}=1$,$\Theta_{j}=1+t\theta_{j}+t^{2}($. . . , and$)$$\Theta_{I}\prod_{i\not\in I,j\in I,i<j}(1+t\beta-t^{2}q_{ij})\in 3HT_{n}$.

,Here for a subset $I\subset[1, n]$ we use notation
among
descriptionofrelationsthe multiplicative
Our main result of this Section is a
Dunkl elements.
$\Theta_{I}=\prod_{a\in I}\Theta_{a}$13Theorem 2.1
In the algebra $3HT_{n}$ the following relations hold true
$I \subset[1,n]\sum_{|I|=k}\Theta_{I}i\not\in I,j\in J\prod_{i<j}(1+t\beta-t^{2}q_{ij})=\{\begin{array}{l}nk\end{array}\}$Here$\{\begin{array}{l}nk\end{array}\}$denotes the q-Gaussian polynomial.

Corollary 2.1
Assume thatand

$q_{ij}\neq 0$$u_{ij}^{-1}=q_{ij}^{-1}(u_{ij}-\beta)$

for all $1\leq i<j\leq n$ . Then the all elements
Now define elements
as follows$\{u_{ij}\}$are invertible$\Phi_{i}\in 3HT_{n}$$\Phi_{i}=\{\prod_{a=i-1}^{1}u_{ai}^{-1}\}\{\prod_{a=n}^{i+1}u_{ia}\}$,$i=1,$$\ldots,$$r\iota$.

Then we have(1) (Relationship among$\Theta_{j}$and$\Phi_{j}$)$t^{n-2j+1}\Theta_{j}(t^{-1})|_{t=0}=(-1)^{j}\Phi_{j}$$\underline{(2})$.

The elements $\{\Phi_{i}, 1\leq i\leq n, \}$ genemte a commutative subalgebm in the algebm.
(3) For each $k=1,$
elements
holds$3HT_{n}$$\ldots,$$n$, the following relation in the algebm$3HT_{n}$among the$\{\Phi_{i}\}$$\sum_{I\subset l1,n|i\not\in}\prod_{I,j\in I}(-q_{ij})\Phi_{I}=\beta^{k(n-k)}$$|I|=k$where$\Phi_{I}$$:= \prod_{a\in I}\Phi_{a}$

,

$i<j$

.

In fact the element
admits the following (reduced expression” which is useful for
proofs and applications$\Phi_{i}$$\Phi_{i}=\{\{u_{ij}^{-1}\}\}j\in Ii\in I_{+}?<j\{55_{I}\}j\in I_{+^{i\in}}^{c_{i<j}}\}$.

(2.12)Let us explain notations. For any (totally) ordered set $I=(i_{1}<i_{2}<\ldots< i_{k})$ wedenote by
the set with the opposite order, i.e. $I_{+}=(i_{k}>i_{k-1}>\ldots>i_{1})$ ; if$I_{+}$$I\subset[1, n]$$I$, then $I^{c}=[1, n]\backslash I$ . For any (totally) ordered set$I$

we denote by$5i\in I$

ordered product according to the order of the set .
Note that the total number of terms in the RHS of (2.12) is equal to $k(n-k)>$
$I$

the

14
Finally, from the ”reduced expression” (2.12) for the element$\Phi_{i}$one can see that$. \prod_{\not\in I,g\in I,<J}(-q_{ij})\Phi_{I}=\{5\{ffi_{c}j\in Ii\in I_{+}<J(\beta-u_{ij})\}\}\{ffi_{c}\{5_{j}u_{ij}\}\}:=\overline{\Phi_{I}}\in 3HT_{n}j\in I_{+:_{<}^{\in I}}^{\cdot}$

Therefore the identity$I \subset l1,n|\sum_{|I|=k}\overline{\Phi_{I}}=\beta^{k(n-k)}$is true in the algebra $3HT_{n}$ for arbitrary set of parameters$\{q_{ij}\}$.

Comments 2.2(I)
In fact from our proof of Theorem 2.1 we can deduce more general statement,
namely, consider integers
and such that $1\leq k\leq m\leq n$ . Then
$k$

$m$

$I \subset|1.m|\sum_{|I|=k}\Theta_{I}$$\in l1,m|\backslash Ig\in J\prod_{1<j},(1+t\beta-t^{2}q_{ij})=\{\begin{array}{l}mk\end{array}\}+A\subset l1,n|,B\subset l1,n|\sum_{|A|=|B|=r}u_{A,B}$(2.13),

where, by definition, for two sets $A=$
and $B=(j_{1}, \ldots,j_{r})$ the symbol
is equal to the (ordered) product. Moreover, the elements of the sets$B$and have to satisfy the following conditions:$(i_{1}, \ldots , i_{r})$$\prod_{a=1}^{r}u_{i_{a},j_{a}}$$u_{A,B}$$A$$\bullet$for each $a=1,$$\ldots,$$r$

one has $1\leq i_{a}\leq m<j_{a}\leq n$ , andEven more, if $r=k$ , then sets$\bullet$$A$$B=(j_{1}\leq j_{2}\leq\ldots\leq j_{k})$$k\leq r\leq k(n-k)$ .

and have to satisfy the following additional conditions:and the elements of the set are pair-wise distinct.
$B$

$A$

In the case $\beta=0$ and $r=k$ , i.e. in the case of additive (truncated) Dunkl elements,
the above statement, also known as the quantum Pieri formula, has been stated as
Conjecture in [10], and has been proved later in [32].

Corollary 2.2 $([21J)$, the algebm overIn the case when $\beta=0$ andgenemted, $1\leq i\leq n\}$ is canonically isomorphic
by the multiplicative Dunkl elements
andto the quantum K-theory of the complete flag variety.
of type$Z[q_{1}, \ldots, q_{n-1}]$$q_{ij}=q_{i}\delta_{j-i,1}$$\{\Theta_{i}$$\Theta_{i}^{-1}$$\mathcal{F}l_{n}$$A_{n-1}$

It is still an open problem to describe explicitly the set of monomials
which
appear in the RHS of (2.13) when $r>k$ .
(Truncated Gaudin operators) Let $\{p_{ij}1\leq i\neq j\leq n\}$ be a set of(II)mutually commuting parameters. We assume that parameters
are invertible andsatisfy the Arnold relations$\{u_{A,B}\}$$\{p_{ij}\}$$\frac{1}{p_{ik}}=\frac{1}{p_{ij}}+\frac{1}{p_{jk}}$,$i<j,$For example one can take $p_{ij}=(z_{i}-z_{j})^{-1}$ , where$k$.

$z=(z_{1}, \ldots, z_{n})\in(\mathbb{C}\backslash 0)^{n}$.

15
Definition 2.3 Truncated (mtional) Gaudin opemtor corresponding to the set of pamm, is defined to beeters$\{p_{ij}\}$$G_{i}= \sum_{j\neq i}p_{ij}^{-1}s_{ij}$,$1\leq i\leq n$ ,and , and fixes
wheredenotes the exchange opemtor which switches variables
pammeters.
,
We consider the Gaudin opemtor
as an element of the group ring
$i=1,$
call this element. by Gaudin element and denoted it by.
$x_{i}$$s_{ij}$$x_{\mathcal{J}}$$\{p_{ij}\}$$Z[\{p_{ij}^{\pm 1}\}][S_{n}]$$G_{i}$$G_{i}\in Z[\{p_{ij}^{\pm 1}\}][S_{n}],$$\ldots,$$n$$\theta_{i}^{(n)}$$1\leq i\neq j\leq n$ , define a representation
It is easy to see that the elements$3HT_{n}$of the algebra.
with parameters $\beta=0$ and
Therefore one can consider the (truncated) Gaudin elements as a special case ofthe (truncated) Dunkl elements. Now one can rewrite the relations among the Dunkl
elements, as well as the quantum Pieri formula [10], [32], in terms of the Gaudin elements.
The key observation which allows to rewrite the quantum Pieri formula as a certain
satisfy the
relation among the Gaudin elements is the following one: parameters
Pl\"ucker relations$u_{ij}$$:=p_{ij}^{-1}s_{ij},$$q_{ij}=u_{ij}^{2}=p_{ij}^{2}$$\{p_{i_{J}}^{-1}\}$$\frac{1}{p_{ik}p_{jl}}=\frac{1}{p_{ij}p_{kl}}+\frac{1}{p_{il}p_{jk}}$,if $i<j<k<l$ .

$i=1,$
To describe relations among the Gaudin elements
notation. Let
be a set of invertible parameters as before.
Define polynomials in the variables $h=(h_{1}, \ldots , h_{n})$$\theta_{i}^{(n)},$$\ldots$, $n$ , we need a bit of$\{p_{ij}\}$$G_{m,k,r}( h, \{p_{ij}\})=I\subset\iota 1,n-1|\sum_{|I|=r}\frac{1}{\prod_{i\in I}p_{in}}$$\sum_{J\subset[1,n],|I|+m=|J|+k}(\begin{array}{l}n-|I\cup J|n-m-|I|\end{array})\tilde{h}_{J}$,(2.14)

where$\prod_{k_{a}\in K,l_{a}\in L}p_{k_{a},l_{a}}^{2}$$\prod_{j\in J\backslash (K\cup L)}h_{j}$,$\tilde{h}_{J}=|K|=|L|,K\cap’ L=\emptyset\sum_{K\subset J,L\subset J}$

and summation runs over subsets $K=\{k_{1}, k_{2}<\ldots<k_{r}\}\subset J$ , and $L=\{l_{a}\in J,$ $a=$
, such that $k_{a}<l_{a},$ $1\leq a\leq r$ , and, are pairwise distinct.
$1,$$\ldots,$$r\}$$l_{1},$$l_{r}$$\ldots$(Relations among the Gaudin elements, $[18J,$ $cf[30J)$
Theorem 2.2
Under the assumption that elements $\{p_{ij}, 1\leq i<j\leq n\}$ are invertible, mutuallycommute and satisfy the Arnold relations, one has$\bullet$$\bullet$$G_{m,k,r}(\theta_{1}^{(n)}, \ldots, \theta_{n}^{(n)}, \{p_{ij}\})=0$$G_{0,0,k}(\theta_{1}^{(n)}, \ldots, \theta_{n}^{(n)}, \{p_{ij}\})=e_{k}(d_{2}, \ldots, d_{n})$elements in the group ring

$Z[S_{n}]$, where,$d_{2},$

if $m>k$ ,$\ldots,$of the $symmetr’ic$ gmup

$d_{n}$$S_{n}$.

(2.15)

denote the Jucys-Murphy

16$:=C_{k},$ $k=$
It is well-known that the elementary symmetric polynomials, generate the center of the group ring
, whereas the Gaudin ele, thements. generate a maximal commutative subalgebra. It is well-known, see e.g. [30], that
so-called Bethe subalgebra, in
where. On each -isotypicis the -isotypic component of.
component the value of the central element
is the explicitly known constant
It follows from [30] that the relations (2.15) together with relations$e_{k}(d_{2}, \ldots, d_{n})$$1,$$\ldots,$$n$$Z[\rho_{ij}^{\pm 1}][S_{n}]$$\{\theta_{i}^{(n)}, i=1, \ldots , n\}$$\mathcal{B}(p_{ij})$$Z[\rho_{ij}^{\pm 1}][S_{n}]$$\oplus_{\lambda\vdash n}\mathcal{B}_{\lambda}(\overline{p_{ij}),}$$\mathcal{B}_{\lambda}(p_{ij})$$\mathcal{B}(p_{ij})=$$\lambda$$\lambda$$\mathcal{B}(p_{ij})$$C_{k}$$c_{k}(\lambda)$$G_{0,0.k}(\theta_{1}^{(n)}, \ldots, \theta_{n}^{(n)}, \{p_{ij}\})=c_{k}(\lambda)$,.
defining relations for the algebra
Let us remark that in the definition of the Gaudin elements we can use any set ofmutually commuting, invertible elements
which satisfies the Arnold conditions. For
example, we can takeare$t1_{1}e$$\mathcal{B}_{\lambda}(p_{ij})$$\{p_{ij}\}$$p_{ij}:= \frac{\phi^{-2}(1-q)}{1-\mathscr{S}^{-i}}$,$1\leq i<j\leq n$.

It is not difficult to see that in this case$\lim_{qarrow 0}\frac{\theta_{J}^{(n)}}{p_{1j}}=-d_{j}=-\sum_{a=1}^{j-1}s_{aj}$,denotes the Jucys-Murphy element in the group ringof the symmetricgroup . Basically $hom$ relations (2.15) one can deduce the relations among the JucysMurphy elements
andafter plugging in (2.15) the values
passing to the limit $qarrow 0$ . However the real computations are rather involved.
Finally we note that the multiplicative Dunkl/Gaudin elements
also
generate a maximal commutative subalgebra in the group ring
. Some relations
among the elements
follow from Theorem 2.1, but we don’t know an analogue of
relations (2.13) for the multiplicative Gaudin elements, but see [30].
(III) Shifted Dunkl elements
and
As it was stated in Corollary 2.2, the truncated additive and multiplicative Dunkl
elements in the algebra $3HT_{n}(0)$ generate over the ring of polynomials $Z[q_{1}, \ldots, q_{n-1}]$
correspondingly the quantum cohomology and quantum $K$ –theory rings of the full
flag variety. In order to describdbe the corresponding equivariant theories, we will
introduce the shifted additive and multiplicative Dunkl elements. To start with we need.
at first to introduce an extension of the algebraLetbe a set of mutually commuting elements and$q_{ij}=q_{ji},$$i,j\leq n\}$ be a set of parameters.
where$Z[S_{n}]$$d_{j}$$S_{n}$$d_{2},$$d_{n}$$p_{ij}$$\ldots,$$:= \frac{q^{j-2}(1-q)}{1-q^{g-:}}$$\{\Theta_{i}, 1, \ldots, n\}$$Z[\rho_{ij}^{\pm 1}][S_{n}]$$\{\Theta_{l}\}$$0_{i}$$\mathfrak{D}_{i}$$\mathcal{F}l_{n}$$3HT_{n}(\beta)$$\{z_{1}, \ldots, z_{n}\}$$\{\beta,$$h,$ $t,$$1\leq$to be the semi-direct product of the algeDefinition 2.4
Define algebm
bm
and the ring of polynomials $Z[h, t][z_{1}, \ldots , z_{n}]$ with respect to the cmssing
relations$\overline{3TH_{n}(\beta)}$$3TH_{n}(\beta)$

(1)
(2)

$\sim i\vee u_{kl}=u_{kl}z_{i}$if$i\not\in\{k, l\}$$z_{i}u_{ij}=u_{ij}z_{j}+\beta z_{i}+h,$Now we set as before$h_{ij}$,$z_{j}u_{ij}=u_{ij}z_{i}-\beta z_{i}-h$ ,$:=h_{ij}(t)=1+tu_{ij}$ .

if $1\leq i<j<k\leq n$ .

17Definition 2.5$\bullet$Define shifted additive Dunkl elements to be$0_{i}=z_{i}- \sum_{i<j}u_{ij}+\sum_{i<j}u_{ji}$$\bullet$.

Define shifted multiplicative Dunkl elements to be$\mathfrak{D}_{i}=(\prod_{a=i-1}^{1}h_{ai}^{-1})(1+z_{i})(\prod_{a=n}^{i+1}h_{ia})$.

Lemma 2.3$[\mathfrak{d}_{i}, \mathfrak{d}_{j}]=0$,$[\mathfrak{D}_{i}, \mathfrak{D}_{j}]=0$

for

all

$i_{J}$.

Now we stated an analogue of Theorem 2.1. for shifted multiplicative Dunkl elements.
As a preliminary, for any subset $I\subset[1, n]$ let us set. It is clear that$\mathfrak{D}_{I}=\prod_{a\in I}\mathfrak{D}_{a}$$\mathfrak{D}_{I}i\not\in I,j\in I\prod_{l<j}(1+t\beta-t^{2}q_{ij})\in\overline{3HT_{n}(\beta)}$

Theorem 2.3
In the algebra$\overline{3HT_{n}(\beta)}$.

the following relations hold true

$I \subset[1,n]\sum_{|I|=k}\mathfrak{D}_{I}\iota\not\in I,j\in J\prod_{i<j}(1+t\beta-t^{2}q_{ij})=\{\begin{array}{l}nk\end{array}\}+$$t= \{i_{1},.i_{k}\}\sum_{I\subset[1,.’.n]},\prod_{a=1}^{k}[z_{a}(1+\beta t)^{n-k}+h\frac{(1+\beta t)^{n-k}-(1+\beta t)^{i_{a}-a}}{\beta}]$.

In particular, if $\beta=0$ , we will have

Corollary 2.3 In the algebra$\overline{3HT_{n}(0)}$the following relations hold$I \subset l1,n|\sum_{|I|=k}\mathfrak{D}_{l}?\not\in I,j\in J\prod_{\iota<j}(1-t^{2}q_{ij})=(\begin{array}{l}nk\end{array})+\sum_{I\subset l1,.’.n|I=\{i_{1},.i_{k}\}}.\prod_{a=1}^{k}\prod_{a=1}^{n}(z_{a}+th(n-k-i_{a}+a))$.

One of the main steps in our proof of Theorem 2.3. is the following explicit formula forthe elements.
$\mathfrak{D}_{I}$Lemma 2.4 One has$\overline{\mathfrak{D}_{I}}:=\mathfrak{D}_{I}(1+t\beta-t^{2}q_{ij})=$II (II )$h_{ba}$$\prod_{a\in I}^{\nearrow}((1+z_{a})\prod_{b\not\in I,a<b}^{\searrow}h_{ab})$.

18
Note that if $a<b$ , then $h_{bo}=1+\beta t-u_{ab}$ . Here we have used the symbol$\prod_{b\in I}^{\nearrow}(\prod_{a<b}^{\searrow}a\not\in Ih_{ba})$to denote the following product. At first, for a given $\backslash lelnentb\in I$ let us define theset $I(b)$ $:=\{a\in[1, n]\backslash I, a<b\}$ $:=(a_{1}^{(b)}<\ldots<a_{p}^{(b)})$ for some (depending on ). If
$b$

$p$

then we set$I=(b_{1}<b_{2}\ldots<b_{k})$ ,$\prod_{b\in I}^{\nearrow}(\prod_{a<b}^{\searrow}h_{bo})=\prod_{ja\not\in I=1}^{k}(u_{b_{j},a_{S}}u_{b_{j},a_{s-1}}\cdots u_{b_{j},a_{1}})$.

For example, let us take $n=6$ and $I=(1,3,5)$ . Then$\overline{\mathfrak{D}_{T}}=h_{32}h_{54}h_{52}(1+z_{1})h_{16}h_{14}h_{12}(1+z_{3})h_{36}h_{34}(1+z_{5})h_{56}$.
$\blacksquare$

Combinatorics of associative quasi-classical YangBaxter algebras

3Let$\beta$be a parameter.

Definition 3.1 $([18J)$ The associative quasi-classical Yang-Baxter algebm of weight
denoted by $ACYB_{n}(\beta)$ , is an associative algebm, over the ring of polynomials
genemted by the set of elements $\{x_{ij}, 1\leq i<j\leq n\}$ , subject to the set of relations

if$(a)x_{ij}x_{kl}=x_{kl}x_{ij}$ ,$\{i, j\}\cap\{k, l\}=\emptyset$$(b)x_{ij}x_{jk}=x_{ik}x_{ij}+x_{jk}x_{ik}+\beta x_{ik}$ ,Comments 3.1$\overline{ACYB}_{n}(-\beta)$

The algebra, by the “dual relations”$3T_{n}(\beta)$$\beta$,$Z[\beta]$,,

if $1\leq 1<i<\gamma\leq n$ ,, see Definition 2.1, is the quotient of the algebra$x_{jk}x_{ij}-x_{ij}x_{ik}-x_{ik}x_{jk}+\beta x_{ik}=0$, $i<j<k$ .

$i=1,$, , do not commute in the alThe (truncated) Dunkl elements
gebra $ACYB_{n}(\beta)$ . However a certain version of noncommutative elementary polynomialof degree $k\geq 1$ , still is equal to zero after the substitution of Dunkl elements instead ofvariables, [18]. We state here the corresponding result only “in classical case”, i.e. if$\beta=0$ and $q_{ij}=0$
for all .
asLemma 3.1 ([18]) Define noncommutative elementary polynomial$\theta_{i}=\sum_{j\neq i}x_{ij},$$\ldots$$n$$i,$ $j$$L_{k}(x_{1}, \ldots, x_{n})$follows$L_{k}(x_{1}, \ldots, x_{n})=\sum_{I=(i_{1}<i_{2}<\ldots<i_{k})\subset[1,n]}$$x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}$

.

.
Then
Moreover, if $1\leq k\leq m\leq n$ , then one can show that the value of the noncommutative
formula, seein the algebm $ACYB_{n}(\beta)$ is given by thepolynomial$[32J$.
1$L_{k}(\theta_{1}, \theta_{2}, \ldots, \theta_{n})=0$$L_{k}(\theta_{1}, \ldots , \theta_{m})$

$[$

$0J,$

$Pier\dot{v}$

19
3. 1

Combinatorics of Coxeter element

Consider the “Coxeter element”
which is equal to the ordered product$w;=w_{n}= \prod_{a=1}^{n-1}$of ”simple generators”:. Let us bring the element
to thereduced form in the algebra $ACYB_{n}(\beta)$ , that is, let us consecutively apply the defining
relations $(a)$ andto the element in any order until unable to do so. Denote theresulting (noncommutative) polynomial by. In principal, the polynomial itself
can depend on the order in which the relations $(a)$ andare applied.
$w\in A\overline{CYB}_{n}(\beta)$$w$$x_{a,a+1}$

$(b)$

$w$

$P(x_{ij};\beta)$$(b)$

Proposition 3.1 $(Cf[40J,$ .C5, $(c);[28])$(1) Apart $fmm$ applying the relation $(a)$ (commutativity), the polynomial
does
not depend on the order in which relations $(a)$ and $(b)$ have been applied, and can be written in a unique way as a linear combination:$8$$P(x_{ij};\beta)$$P_{n}(x_{ij}; \beta)=\sum_{s=1}^{n-1}\beta^{n-s-1}$$\prod_{a=1}^{s}x_{i_{a},g_{a}}$,$\sum_{\{i_{a}\}}$

where the second summation runs over all sequences of integers
such that$i_{a}\leq
n-a$
, and
$s-1$ ; moreover, thefor $a=1,$
corresponding sequencecan be defined uniquely by that.
It is clear that the polynomial
also can be written in a unique way as alinear combination of monomials
such that.
$\{i_{a}\}_{a=1}^{s}$$n-1\geq i_{1}\geq i_{2}\geq\ldots\geq i_{s}=1$$\ldots,$$\{j_{a}\}_{a=1}^{n-1}$$\{i_{a}\}_{a=1}^{n-1}$$P(x_{ij};\beta)$$\bullet$$j_{1}\geq j_{2}\ldots\geq j_{s}$$\prod_{a=1}^{s}x_{i_{a},j_{a}}$(2) Denote by $T_{n}(k, r)$ the number of degree

which contain exactly$r$factors of the form

$x_{*,n}$$k$.

monomials in the polynomial $P(x_{ij};\beta)$
Note that $1\leq r\leq k\leq n-1)$ . Then$($

$T_{n}(k, r)= \frac{r}{k}(\begin{array}{ll}n+k-r -2-2n \end{array}) (\begin{array}{l}-n2k-1\end{array})$

In particular,.

and $T_{n}(k, 1)=T(n-2, k-1)$ , where $T(n, k)$ $:=$
is equal to the number of Schroder paths $(i.e$ . consisting of steps $U=(1,1)$ ,$D=(1, -1),$ $H=(2,0)$ and never going below the x-axis) $fmm(0,0)$ to $(2n, 0)$ , having$kU’ s,$ $see[37J$, A088617.
Moreover, $T_{n}(n-1, r)=Tab(n-2, r-1)$ , where Tab $(n, k)$is equal to
the number of standard Young tableaux of the shape $(n, n-k)$ , see $[37J,$ A0097\’o6.
$T_{n}(k, k)=(\begin{array}{l}n-2k-1\end{array})$,$\frac{1}{k+1}(\begin{array}{l}n+kk\end{array})(\begin{array}{l}nk\end{array})$$:= \frac{k+1}{n+1}(\begin{array}{l}2n-kn\end{array})$(3) After the specialization $x_{ij}arrow 1$ the polynomial
polynomial$P_{n}( \beta):=\sum_{k=0}^{n-1}N(n, k)(1+\beta)^{k}$where $N(n, k)$thermore,$:= \frac{1}{n}(\begin{array}{l}nk\end{array})(\begin{array}{l}nk+1\end{array}),$$k=0,$$\ldots,$$P(x_{ij})$

is tmnsformed to the,$n-1$ , stand for the Namyana numbers.

Fur-is the number of ways
to $dmwn-1-d$ diagonals in a convex $(n+2)-gon$ , such that no two diagonals intersect
their interior.
$P_{n}( \beta)=\sum_{d=0}^{n-1}s_{n}(d)\beta^{d}$, where$s_{n}(d)= \frac{1}{n+1}(\begin{array}{l}2n-dn\end{array})(\begin{array}{l}n-1d\end{array})$20
equal to
Therefore, the number of (nonzem) terms in the polynomial, also known as the n-th super-Catalan
the n-th little Schroder number
number, see e.g. $[37J$, A001003.
$P(x_{ij};\beta)i_{j}s$$s_{n}:= \sum_{d=0}^{n-1}s_{n}(d)$(4), and that
After the specializationis tmnsformed to the polynomial$x_{1j}arrow t,$$1\leq J\leq n$$x_{ij}arrow 1$ ,the $polynom\iota alP(x_{ij};\beta)$

if $2\leq i<j\leq n$ ,$P_{n}( \beta, t)=t\sum_{k=1}^{n}(1+\beta)^{n-k}\sum_{\pi}t^{p(\pi)}$ ,where the second summation runs over the set of Dick paths of length $2n$ with exactly
picks (UD-steps), and $p(\pi)$ denotes the number of valleys ( U-steps) that touch upon
the line $x=0$ .
$\pi$

$D$

$k$

is invari ant under the action of anti-involution
see Section 5.1.1 [18] for definitions of and .
(5) The polynomial $P(x_{ij};\beta)$$\phi$$\phi\circ\tau$,$\tau$

Follow $l40J,$ $6.C8,$ $(c)$ , consider the specialization(6)$x_{ij}arrow t_{i}$,$1\leq i<j\leq n$ ,and define $P_{n}(t_{1}, \ldots, t_{n-1};\beta)=P_{n}(x_{ij}=t_{i};\beta)$ .
One can show, ibid, that
$P_{n}(t_{1}, \ldots, t_{n-1};\beta)=\sum\beta^{n-k}t_{i_{1}}\cdots t_{i_{k}}$where the sum runs over all pairs(3.16),$\{(a_{1}, \ldots, a_{k}), (i_{1}, \ldots , i_{k})\in Z\geq 1\cross Z\geq 1\}$andfor all .
Now we are ready to state our main result about polynomials$a_{1}<a_{2}<\ldots<a_{k}$ ,$1\leq i_{1}\leq i_{2}\ldots\leq i_{k}\leq n$such that$j$$i_{j}\leq a_{j}$$P_{n}(t_{1}, \ldots, t_{n};\beta)$Let$\pi\in S_{n}$be the permutation$\pi=(\begin{array}{llllll}1 2 3 \cdots n1 n n -1 \cdots 2\end{array})$.

$\otimes_{w}^{(\beta)}(x_{1}, \ldots, x_{n-1})$

denotes the$\beta$.

$\underline{Then}$$P_{n}(t_{1}, \ldots, t_{n-1};\beta)=(\prod_{i=1}^{n-1}t_{i}^{n-i})\emptyset_{\pi}^{(\beta)}(t_{1}^{-1}, \ldots, t_{n-1}^{-1})$

wheremutation $w\in S_{n},$ $[11]$.
In particular,$1\leq$,-Grothendieck polynomial corresponding to a per-$\emptyset_{\pi}^{(\beta)}(x_{1}=1, \ldots, x_{n-1}=1)=\sum_{k=0}^{n-1}N(n, k)(1+\beta)^{k}$,where $N(n, k)$ denotes the Namyana numbers, see item (3) of Pmposition 3.1.

21$\bullet$$\beta$Note that if $\beta=0$ , then one has, that is the-Grothendieck polynomial at $\beta=0$ , is equal to the Schubert polynomial corresponding$\mathfrak{G}_{w}^{(\beta=0)}(x_{1}, \ldots, x_{n-1})=\mathfrak{S}_{w}(x_{1}, \ldots, x_{n-1})$

to the same permutation$w$. Therefore, if$\pi=(\begin{array}{llllll}1 2 3 \cdots n1 n n -1 \cdots 2\end{array})$, then(3.17)$\mathfrak{S}_{\pi}(x_{1}=1, \ldots , t_{n-1}=1)=C_{n-1}$ ,wheredenotes the $m-$ th Catalan number. Using the formula (3.16) it is not difficult
the following formula for the principal specialization of the Schubert polynomial$C_{m}$$\mathfrak{S}_{\pi}tocheck$$\mathfrak{S}_{\pi}(1, q, \ldots , q^{n-1})=q(\begin{array}{l}n- 13\end{array})C_{n-1}(q)$(3.18),

where $C_{m}(q)$ denotes the Carlitz- Riordan q-analogue of the Catalan numbers, see e.g.
[38]. The formula (3.17) has been proved in [13] using the observation that is a $vex’illar^{v}y$permutation, see [26] for the a definition of the latter. A combinatorial/bijective proofof the formula (3.18) is is due to A.Woo [44].
$\pi$Comments 3.2
The Grothendieck polynomials defined by A. Lascoux and M.-P. Sch\"utzenberger, seee.g. [25], correspond to the case $\beta=-1$ . In this case $P_{n}(-1)=1$ , if $n\geq 0$ , and therefore
the specialization $\otimes_{w}^{(-1)}(x_{1}=1, \ldots, x_{n-1}=1)=1$ for all $w\in S_{n}$ .
$\blacksquare$Multiparameter deformation of Catalan, Narayana andSchroder numbers

3.1.1

Let

be a set of mutually cornmuting pararneters. We define a multiparameter analogue of the associative quasi-classical Yang-Baxter algebra $MACYB_{n}(b)$$b=(\beta_{1}, \ldots, \beta_{n-1})$

as follows.

Definition 3.2 The multipammeter associative quasi-classical Yang-Baxter algebm of
weight , denoted by, is an associative algebm, over the ring of polynomials, genemted by the set of elements $\{x_{ij}, 1\leq i<j\leq n\}$ , subject tothe set of relations$b$$M\overline{ACY}B_{n}(b)$$Z[\beta_{1}, \ldots , \beta_{n-1}]$$(a)x_{ij}x_{kl}=x_{kl}x_{ij}$ ,if$\{i,j\}\cap\{k, l\}=\emptyset$$(b)x_{ij}x_{jk}=x_{ik}x_{ij}+x_{jk}x_{ik}+\beta_{i}x_{ik}$ ,Consider the ”Coxeter element”
product of ”simple generators” :.

if $1\leq 1<i<j\leq n$ ,$w_{n}\in M\overline{ACY}B_{n}(b)$$?l \prime_{n}:=\prod_{a=1}^{n-1}x_{a,a+1}$

which is equal to the ordered.

Now we can use the same method as in [40], 8. $C5,$ $(c)$ , see Section 3.1. to define thereduced form of the Coxeter element. Namely, let us bring the element
to thereduced form in the algebra $MACYB_{n}(\beta)$ , that is, let us consecutively apply the defining$w_{n}$$w_{n}$

22in any order until unable to do so. Denote theto the element
relations $(a)$ and
by $P(x_{ij};b)$ . In principal, the polynomial itself(noncommutative)
polynomial
resultingare applied.
can depend on the order in which the relations $(a)$ and$(b)$$w_{n}$$(b)$

Proposition 3.2 $(Cfl40J, 8.C5, (c);[28])$
Apart from applying the relation $(a)$ (commutativity), the polynomial $P(x_{ij};b)$ does
not depend on the order in which relations $(a)$ and $(b)$ have been applied.
To state our main result of this Section, let us define polynomials$Q(\beta_{1}, \ldots, \beta_{n-1}):=P(x_{ij}=1, \forall i,j;\beta_{1}-1, \beta_{2}-1, \ldots, \beta_{n-1}-1)$.

Example 3.1$Q(\beta_{1}, \beta_{2})=1+2\beta_{1}+\beta_{2}+\beta_{1}^{2}$,$Q(\beta_{1}, \beta_{2}, \beta_{3})=1+2(\beta_{1}+\beta_{2}+\beta_{3})+3\beta_{1}^{2}+\beta_{1}\beta_{2}+\beta_{1}\beta_{3}+\beta_{2}^{2}+\beta_{1}^{3}$,$Q(\beta_{1}, \beta_{2}, \beta_{3}, \beta_{4})=1+7\beta_{1}+\beta_{2}+\beta_{3}+\beta_{1}(9\beta_{1}+3\beta_{3}+5\beta_{4})+\beta_{2}(\beta_{3}+\beta_{4})+\beta_{3}^{2}+$.

$\beta_{1}^{2}(4\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4})+\beta_{1}(\beta_{2}^{2}+\beta_{3}^{2})+\beta_{2}^{3}+\beta_{1}^{4}$

Theorem 3.1
Polynomial$Q(\beta_{1}, \ldots, \beta_{n-1})$

has non-negative integer coefficients.

It follows from [40] and Proposition 3.1 that.

$Q(\beta_{1}, \ldots, \beta_{n-1})|_{\beta_{1}=1,\ldots,\beta_{n-1}=1}=Cat_{n}$and $Q(\beta_{1}+1, \ldots , \beta_{n-1}+1)$ can be considered as a multiPolynomials
parameter deformation of the Catalan and Schroder numbers correspondingly, and theas a multiparameter analogue of Narayana
homogeneous degree part ofpolynomial
expectthethatWe
numbers.
$Q(\beta_{1}, \ldots , \beta_{n-1})$$k$$Q(\beta_{1}, \ldots, \beta_{n-1})$$t_{1}^{n}t_{2}^{n-1}\cdots t_{n}P(x_{ij}=t_{i}^{-1} ; b)$coincides with a multiparameter deformation
Grothendieck polynomial.

$\otimes_{\pi_{k}^{(n)}}^{(b)}(t_{1}, \ldots, t_{n-1})$of the corresponding

3.2

Grothendieck and -Schr\"oder polynomials

3.2.1

Schroder paths and polynomials$q$path of the length is an over diagonal path from $(0,0)$ to
Definition 3.3 A$(n, n)$ with steps 1, $0),$ $(0,1)$ and steps $D=(1,1)$ without steps of typeon the diagonal$x=y$ .
$sCh\mathfrak{X}der$

$n$

$D$

$($

If is a Schroder path, we denote by $d(p)$ the number of the diagonal steps resting on
the path , and by $a(p)$ the number of unit squares located between the path andthe diagonal $x=y$ . For each (unit) diagonal step $D$ of a path we denote by $i(D)$ thex-coordinate of the column which contains the diagonal step $D$ . Finally, define the index$i(p)$ of a pathas the some of the numbers $i(D)$ for all diagonal steps of the path .
$p$

$p$

$p$

$p$

$p$

$p$

23
Definition 3.4 Define q-Schr6der polynomial$S_{n}(q;\beta)$

as follows$S_{n}(q; \beta)=\sum_{p}q^{a(p)+i(p)}\beta^{d(p)}$where the sum runs over the set of all$sCh\mathfrak{X}der$(3.19),

paths of length .
$n$Example 3.2$S_{2}(q;\beta)=1+q+\beta q,$$S_{1}(q;\beta)=1,$$S_{3}(q;\beta)=1+2q+q^{2}+q^{3}+\beta(q+2q^{2}+2q^{3})+\beta^{2}q^{3}$ ,$S_{4}(q;\beta)=1+3q+3q^{2}+3q^{3}+2q^{4}+q^{5}+q^{6}+\beta(q+3q^{2}+5q^{3}+6q^{4}+3q^{5}+3q^{6})+$$\beta^{2}(q^{3}+2q^{4}+3q^{5}+3q^{6})+\beta^{3}q^{6}$ .

Comments 3.3
The q-Schroder polynomials defined by the formula (3.19) are $dif$ ferent from theq-analogue of Schroder polynomials which has been considered in [5]. It seems that there
are no simple connections between the both.

$\blacksquare$

Proposition 3.3(Recurrence relations for -Schr\"oder polynomials)
The
polynomials satisfy the following relations$q$$SCh\mathfrak{X}der$$S_{n+1}(q; \beta)=(1+q^{n}+\beta q^{n})S_{n}(q;\beta)+\sum_{k=1}^{k=n-1}(q^{k}+\beta q^{n-k})S_{k}(q;q^{n-k}\beta)S_{n-k}(q;\beta),$and the initial condition$S_{1}(q;\beta)=1$$(3.20)$.

Note that $P_{n}(\beta)=S_{n}(1;\beta)$ and in particular, the polynomials$P_{n}(\beta)$

recurrence relations

satisfy the following$P_{n+1}( \beta)=(2+\beta)P_{n}(\beta)+(1+\beta)\sum_{k=1}^{n-1}P_{k}(\beta)P_{n-k}(\beta)$(Evaluation of the Schroder
Theorem 3.2
Consider permutation-(3.21).

Hankel Determinant)$\pi_{k}^{(n)}=(\begin{array}{lllllllllll}1 2 \cdots k k +1 k +2 \cdots \cdots n1 2 \cdots k n n -1 \cdots k +1\end{array})$.

Let as before$P_{n}( \beta)=\sum_{j=0}^{n-1}N(n, j)(1+\beta)^{j}$denotes the Narayana-Schroder polynomials.

,$n\geq 1$(3.22),

Then$(1+\beta)(\begin{array}{l}k2\end{array})\otimes_{\pi_{k}^{(n)}}^{(\beta)}(x_{1}=1, \ldots, x_{n-k}=1)=Det|P_{n+k-i-j}(\beta)|_{1\leq i,j\leq k}$

Proof is based on an observation that the permutation
recurrence relations (3.21).

$\pi_{k}^{(n)}$.

(3.23)

is a vexillary one and the24Comments 3.4(1) In the case $\beta=0$ , i.e. in the case of Schubert polynomials Theorem 3.1 has
been proved in [13].
(2) In the cases when $\beta=1$ and $0\leq n-k\leq 2$ , the value of the determinant in theRHS(3.22) is known, see e.g. [5], or M. Ichikawa talk Hankel determinants of Catalan,
Motzkin and Sehroder numbers and its q-analogues, http: $/denjoy.ms.u$-tokyo.ac.jp. One
can check that in the all cases mentioned above, the formula (3.22) gives the same results.
(3) Grothendieck and Narayana polynomials
It follows from the expression (3.22) for the Narayana-Schroder polynomials that
$P_{n}(\beta-1)=\Re_{n}(\beta)$ , where$\mathfrak{N}_{\eta}(\beta):=\sum_{j=0}^{n-1}\frac{1}{n}(\begin{array}{l}nj\end{array})(\begin{array}{ll} nj +1\end{array}) \beta^{j}$,denotes the n-th Narayana polynomial. Therefore, $P_{n}(\beta-1)=\Re_{n}(\beta)$ is a symmetric polynomial in with non-negative integer coefficients. Moreover. the value of thepolynomial $P_{n}(\beta-1)$ at $\beta=1$ is equal to the n-th Catalan number.
It is well-known, see e.g. [42], that the Narayana polynomial
is equal to the( $number$ of peaks of agenerating function of the statistics
path ) $-1$
on the set $Dick_{n}$ of Dick paths of the length $2n$$\beta$$C_{n}$$:= \frac{1}{n+1}(\begin{array}{l}2nn\end{array})$$\mathfrak{N}_{\eta}(\beta)$$Dir,\cdot k$$\pi(\mathfrak{p})=$$\beta^{\pi(\mathfrak{p})}$$\mathfrak{N}_{\eta}(\beta)=\sum_{\mathfrak{p}}$$\mathfrak{p}$.

Moreover, using the Lindstr\"om-Gessel-Viennot lemma see e.g.
http: $//en.wikipedia.org/wiki/Lindstr\ddot{o}m$ -Gessel-Viennot lemma,
one can see that
$DET| \mathfrak{N}_{\eta+k-i-j}(\beta)|_{1\leq i,j\leq k}=\beta(\begin{array}{l}k2\end{array})\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{k})}\beta^{\pi(\mathfrak{p}_{1})+\ldots+\pi(\mathfrak{p}_{k})}$(3.24),

where the sum runs over k-tuple of non-crossing Dick paths
such that thepath starts from the point $(i-1,0)$ and has length $2(n-i+1),$ $i=1,$.
We denote the sum in the $RHS(3.24)$ by. Note thatfor all$(\mathfrak{p}_{1}, \ldots , \mathfrak{p}_{k})$$k$$\mathfrak{p}_{i}$$\ldots,$$\mathfrak{R}_{k-1}^{(k)}(\beta)=1$$\mathfrak{N}_{\eta}^{(k)}(\beta)$$k\geq 2$

.

Thus,

$\mathfrak{N}_{\eta}^{(k)}(\beta)$

is a symmetric polynomial in$\beta$with non-negative integer coefficients,

and.

$\mathfrak{R}_{n}^{(k)}(\beta=1)=C_{n}^{(k)}=\prod_{1\leq i<j\leq n-k+2}\frac{2k+i+j-1}{i+j-1}$

As a corollary we obtain the following statement
Proposition 3.4 Let $n\geq k$ , then.

$\emptyset_{\pi_{k}^{(n)}}^{(\beta-1)}(x_{1}=1, \ldots, x_{n}=1)=\mathfrak{R}_{n}^{(k)}(\beta)$

25
Summarizing, the specialization
is a symmetric polynomial
in with non-negative integer coefficients. and coincides with the generating functionof the statistics
on the set $k-Dick_{n}$ of k-tuple of non-crossing Dick paths.
$\otimes_{(n)}^{(\beta-1)}(x_{1}=1, \ldots, x_{n}=1)$$\pi_{k}$$\beta$$\sum_{i=1}^{k}\pi(\mathfrak{p}_{i})$$(\mathfrak{p}_{1}, \ldots, \mathfrak{p}_{k})$$\blacksquare$

Example 3.3$\otimes_{\pi_{1}^{(5}}^{(\beta)},$It$\iota s$

Take $n=5,$ $k=1$ . Then $\pi_{1}^{(5)}=(15432)$ and one has$(1, q, q^{2}, q^{3})=q^{4}(1,3,3,3,2,1,1)+q^{5}(1,3,5,6,3,3)\beta+q^{7}(1,2,3,3)\beta^{2}+q^{10}\beta^{3}$ .

easy to compute the Carlitz-Riordan q-analogue of the Catalan number$C_{5}$, namely,$C_{5}(q)=(1,3,3,3,2,1,1)$ .

Grothendieck polynomials
and k-dissections
$k\in
N$
Let
and $n\geq k-1$ , be a integer, define a k-dissection of a convex $(n+k+1)-$
gon to be a collection of diagonals in $(n+k+1)$ -gon not containing $(k+1)$ -subset ofpairwise crossing diagonals and such that at least $2(k-1)$ diagonals are coming fromeach vertex of the $(n+k+1)$ -gon in question. One can show that the number of diagonals
in any k-dissectionof a convex $(n+k+1)$ -gon contains at least $(n+k+1)(k-1)$
and at most $n(2k-1)-1$ diagonals. We define the index of a k-dissection to be$i(\mathcal{E})=n(2k-1)-1-\#|\mathcal{E}|$ .
Dnote by(4)$\otimes_{\pi_{k}^{(n)}}^{(\beta)}(x_{1}, \ldots, x_{n})$$\mathcal{E}$$\mathcal{E}$$\mathcal{E}$$\mathcal{T}_{n}^{(k)}(\beta)=\sum_{\mathcal{E}}\beta^{i(\mathcal{E})}$the generating function for the number of k-dissections with a fixed index, where theabove sum runs over the set of all k-dissections of a convex $(n+k+1)$ -gon.

Theorem 3.3$\mathfrak{G}_{\pi_{k}^{(n)}}^{(\beta)}(x_{1}=1, \ldots, x_{n}=1)=\mathcal{T}_{n}^{(k)}(\beta)$.

A k-dissection of a convex $(n+k+1)$ -gon with the maximal number of diagonals(which is equal to $n(2k-1)-1$ ), is called k-triangulation. It is well-known thatthe number of k-triangulations of a convex $(n+k+1)$ -gon is equal to the CatalanHankel number. Explicit bijection between the set of k-triangulations of a convex$(n+k+1)$ -gon and the set of k-tuple of non-crossing Dick paths
such thatthe Dick path connects points $(i-1,0)$ and $(2n-i-1,0)$ , has been constructed in[36], [41].
(5) Polynomials
andLet $w\in S_{n}$ be a permutation andandbe the coressponding-Grothendieck and double -Grothendieck polynomials. We denote by
and by$:=(x_{1}=1,\ldots,x_{n}=1),$$:=(y_{1}=1,\ldots,y_{n}=1)$
the specializationsofthe -Grothendieck polynomials introduced above.
$C_{n-1}^{(k)}$$(\gamma_{1}, \ldots, \gamma_{k})$$\gamma_{i}$$\mathfrak{F}_{w}(\beta),$$\mathfrak{H}_{w}(\beta),$$\mathfrak{G}_{w}^{(\beta)}(X_{n})$$\beta$$\Re_{u}(q;\beta)$$\mathfrak{H}_{w}(q, t;\beta)$$\emptyset_{w}^{(\beta)}(X_{n}, Y_{n})$$6_{w}^{(\beta)}(1)$$\beta$$6_{w}^{(\beta)}(1;1)$$X_{n}$$Y_{n}$$\beta$Theorem 3.4 Let $u;\in S_{n}$ be a permutation.
Then(i) The polynomials
and$\mathfrak{F}_{w}(\beta)$$:=6_{w}^{(\beta-1)}(1)$$\mathfrak{H}_{w}(\beta)$$:=\otimes_{w}^{(\beta-1)}(1;1)$

26
have both non-negative integer coefficients.
(ii) One has$\mathfrak{H}_{w}(\beta)=(1+\beta)^{\ell(w)}S_{w}(\beta^{2})$

(iii)

.

Let $u\in S_{n}$ be a permutation, define polynomials$\mathfrak{H}_{w}(q, t;\beta):=\emptyset_{w}^{(\beta)}(x_{1}=q, x_{2}=q, \ldots, x_{n}=q, y_{1}=t, y_{2}=t, \ldots, y_{n}=t)$

to be the specialization. Then$\{x_{i}=q, y_{i}=t, \forall i\}$,of the double -Grothendieck polynomial$\beta$$\otimes_{w}^{(\beta)}(X_{n}, Y_{n})$$fl_{w}(q, t;\beta)=(q+t+\beta qt)^{\ell(w)}S_{w}((1+\beta q)(1+\beta t))$.

In particular, $\mathfrak{H}_{w}(1,1;\beta)=(2+\beta)^{\ell(w)}S_{w}((1+\beta)^{2})$ .
(iv) Let $w\in S_{n}$ be a $pe7mutation$ , define polynornial$\mathcal{R}_{w}(q;\beta):=\emptyset_{w}^{(\beta-1)}(x_{1}=q, x_{2}=1, x_{3}=1, \ldots)$

to be the specialization.
Then$\{x_{1}=q, x_{i}=1, \forall i\geq 2\}$,of the$(\beta-1)$-Grothendieck polynomial$\mathfrak{G}_{w}^{(\beta-1)}(X_{n})$$\mathcal{R}_{w}(q;\beta)=q^{w(1)-1}\Re_{w}(q;\beta)$

where$\Re_{w}(q;\beta)$

is a polynomial in$\mathbb{R}_{u}(0;\beta=0)=1$

$q$

and

$\beta$,with non-negative integer coefficients, and.
$\blacksquare$

Remark 3.1.
One can show, cf [26], p. 89, that if $w\in S_{n}$ , then$\Re_{w}(q,\beta)=\Re_{w^{-1}}(q,\beta)$
can be violated, and it seems that in
However, the equality,
andgeneral, there are no simple connections between polynomials
if so.
, etc.
From this point we shell use the notation$\mathcal{R}_{w}(1, \beta)=\mathcal{R}_{w^{-1}}(1, \beta)$$\Re_{w}(q, \beta)$$(a_{0}, a_{1}, \ldots , a_{r})_{\beta}$$:= \sum_{j=0}^{r}$$\Re_{w^{-1}}(q, \beta)$$a_{j}\beta^{j}$

Example 3.4
Let us take $w=[1,3,4,6,7,9,10,2,5,8]$ . Then $\Re_{w}(q, \beta)=$
1, 6, 21, 36, 51, 48, $26)_{\beta}+q\beta(6,36,126,216,306,288,156)_{\beta}+$$q^{2}\beta^{3}(20,125,242,403,460,289)_{\beta}+q^{3}\beta^{5}(6,46,114,204,170)_{\beta}$ . Moreover,$\Re_{u}(q, 1)=(189$ , 1134, 1539, $540)_{q}$ .
On the other hand,$w^{-1}=[1,8,2,3,9,4,5,10,6,7]$ , and $\Re_{w^{-1}}(q, \beta)=(1,6,21,36,51,48,26)_{\beta}+$$($$q\beta(1,6,31,56,96,110,78)_{\beta}+q^{2}\beta(1,6,27,58,92,122,120,78)_{\beta}+$$q^{3}\beta(1,6,24,58,92,126,132,102,26)_{\beta}+q^{4}\beta(1,6,22,57,92,127,134,105,44)_{\beta}+$$q^{5}\beta(1,6,21,56,91,126,133,104,50)_{\beta}+q^{6}\beta(1,6,21,56,91,126,133,104,50)_{\beta}$.

Moreover, $\Re_{w^{-1}}(q, 1)=(189,378,504,567,588,588,588)_{q}$ .
Notice that $w=1\cross u$ , where $u=[2,3,5,6,8,9,1,4,7]$ . One can show that’$\Re_{w\cdot\frac{Ontheotherhand}{6)_{\beta}=\Re_{u}(1,\beta)}}(q, \beta)=(,1,6,11,16,11)_{\beta},+q\beta^{2}(10, 20,35,34)_{\beta}+q^{2}\beta^{4}(5,14,26)_{\beta}u^{-1}=[71,2,8,3,4,9,56]and\Re_{u^{-1}}(q,\beta)=(1,6,21,36,51,48,2.$$\blacksquare$27[$\underline{Hereinafter}$

we have used the notation$(a_{0},$ $a_{1},$$\ldots,$$a_{r})_{\beta}$$:= \sum_{j=0}^{r}$$a_{j}\beta^{j}$, etc.]

Problems 3.1(1) Define a bijection be tween monomials of the form
involved in thepolynomial, and dissections of a convex $(n+2)-gon$ by diagonals, such thatno two diagonals intersect their interior.
(2) Describe permutations $uf\in S_{n}$ such that the Grothendieck polynomial
is equal to the ”reduced polynomial “ for a somein the associative qasi-classicalYang-Baxter algebm $ACYB_{n}(\beta)$ .
(3) Study ”reduced polynomial “ corresponding to the monomials$\prod_{a=1}^{s}x_{i_{a},j_{a}}$$P(x_{ij};\beta)$$s$$\emptyset_{w}(t_{1}, \ldots, t_{n})$$\underline{monomial}$$(?$$u_{12}u_{23}\cdots u_{n-1,n}u_{n-2,n-1}\cdots u_{23}u_{12}$,$(u_{12}u_{23}\cdots u_{n-1,n})^{k}$in the algebm $ACYB_{n}(\beta)^{ab}$ .
(4) Construct a bijection between the set of k-dissections of a convex $(n+k+1)-gon$
and “ pipe dreams “ corresponding to the Grothendieck polynomial. As$\emptyset^{(\beta)}(n)(x_{1}, \ldots, x_{n})$$\pi_{k}$for a definition of pipe greams” for $Gr\cdot othe,ndieck$ polyno mials, see

see also [11].

$\cdot$$\cdot$$[2_{(}^{t}iJ,\cdot$

Principal specialization of Grothendieck polynomials, and q-Schroder polynomials

3.2.2

Let
Recall that$\pi_{k}^{(n)}=1^{k}\cross w_{0}^{(n-k)}\in S_{n}$be the vexillary permutation as before, see Theorem 3.1.

$\pi_{k}^{(n)}=(\begin{array}{lllllllll}1 2 \cdots k k k+1 +2 \cdots r\iota 1 2 \cdots k n n-1 \cdots k +1\end{array})$(A)

Principal specialization of the Schubert polynomial.

$\mathfrak{S}_{\pi_{k}^{(n)}}$Note thatis a vexillary permutation of the staircase shape $\lambda=(n-k-1, \ldots, 2,1)$
and has the staircase flag $\phi=(k+1, k+2, \ldots, n-1)$ . It is known, see e.g. [43], [26], thatfor a vexillary permutation $w\in S_{n}$ of the shape and flag,
the corresponding Schubert polynomial
is equal to the multi-Schur polynomial, wheredenotes the flagged set of variables, namely,
and $X_{m}=(x_{1}, \ldots, x_{m})$ . Therefore we can write the folloing determinantal formula forthe principal
cialization of the Schubert polynomial corresponding to the vexillary$\pi_{k}^{(n)}$$\phi=(\phi_{1}, \ldots, \phi_{r}),$$\lambda$$r=\ell(\lambda)$$\mathfrak{S}_{w}(X_{n})$$s_{\lambda}(X_{\phi})$$X_{\phi}=(X_{\phi_{1}}, \ldots , X_{\phi_{r}})$$X_{\phi}$

permutation$sp_{)}e$$\pi_{k}$,$\mathfrak{S}_{\pi_{k}^{(n)}}(1, q, , q^{2}, \ldots)=DET(\{\begin{array}{ll}n -1-i+j k+7-1\end{array}\})_{1\leq i,j\leq n-k}$

where$\{\begin{array}{l}nk\end{array}\}$denotes the q-binomial coefficient.

Let us observe that the Carlitz-Riordan q-analogueof the Catalan number
is equal to the value of the q-Schroder polynomial at $\beta=0$ , namely, $C_{n}(q)=S_{n}(q, 0)$ .
$C_{n}(q)$Lemma 3.2 Let(1)$k,$ $n$$C_{n}$

be integers and $n>k$ , then,

$DET(\{\begin{array}{ll}n -i+:i-1 k+i-1\end{array}\})_{1\leq i,j\leq n-k}=q(\begin{array}{l}n-k3\end{array})C_{n}^{(k)}(q)$

28
(2)

$DET(C_{n+k-i-j}(q))_{1\leq i,j\leq k}=q^{k(k-1)(6n-2k-5)/6}C_{n}^{(k)}(q)$(B) Principal specialization of the Grothendieck polynomial.
$\mathfrak{G}_{\pi_{k}^{(n)}}^{(\beta)}$

Theorem 3.5$q(\begin{array}{l}n- k+13\end{array})-(k-1)(\begin{array}{l}n- k2\end{array})DET|S_{n+k-i-j}(q;q^{i-1}\beta)|_{1\leq i,j\leq k}=$$q^{k(k-1)(4k+1)/6} \prod_{a=1}^{k-1}(1+q^{a-1}\beta)\mathfrak{G}_{\pi_{k}^{(n)}}(1, q, q^{2}, \ldots)$

Corollary 3.1 (1)If $k=n-1$ ,

.

then

$DET|S_{2n-1-i-j}(q;q^{i-1} \beta)|_{1\leq i,j\leq n-1}=q^{(n-1)(n-2)(4n-3)/6}\prod_{a=1}^{n-2}(1+q^{a-1}\beta)^{n-a-1}$,(2)

If $k=n-2$ . then$q^{n-2}DET|S_{2n-2-\iota-j}(q;q^{\iota-1}\beta)|_{1\leq i,j\leq n-2}=$$q^{(n-2)(n-3)(4n-7)/6} \prod_{a=1}^{n-3}(1+q^{a-1}\beta)^{n-a-2}\{\frac{(1+\beta)^{n-1}-1}{\beta}\}$Generalization
Let $n=$.

$\bullet$be a composition of so that $n=n_{1}+\cdots+n_{p}$ . We set$rt^{(0)}=0$ .
$j=1,$,
Now consider the permutation. In
denotes the longest permutation in the symmetric groupwhere
other words,$(n_{1}, \ldots , n_{p})\in N^{p}$$n^{(j)}=n_{1}+\cdots+n_{j},$$n$$\ldots,p,$$w^{(n)}=w_{0^{n)}}^{(1}\cross w_{0^{n)}}^{(2}\cross\cdots\cross w_{0}^{(n_{p})}\in S_{n}$$w_{0}^{(m)}\in S_{m}$$S_{m}$$w^{(n)}=(\begin{array}{lllllllllll}1 2 \cdots n_{1} n^{(2)} \cdots n_{1}+1 \cdots n^{(p-1)} \cdots nn_{1} n_{1}-1 \cdots 1 n_{1}+1 \cdots n^{(2)} \cdots n \cdots n^{(p-1)}+1\end{array})$defined above, one
For the permutation
the Grothendieck polynomial corresponding to$w^{(n)}$$ha^{t}.s$.

the following factorization formula for$w^{(n)},$$[26]$,.

$6_{w^{(n)}}^{(\beta)}=6_{w_{0^{n)}}^{11}}^{(\beta)}\cross\otimes_{1^{n_{1}}\cross w_{0}^{(n)}}^{(\beta)}2\cross\emptyset_{1^{n_{1}+n_{2}}\cross w_{0}^{(n_{3})}}^{(\beta)}\cross\cdots\cross\emptyset_{1^{n_{1}+\ldots n_{\rho-1}}\cross w_{O}^{(n_{p})}}^{(\beta)}$

In particular, if$w^{(n)}=w_{0}^{(n_{1})}\cross w_{0}^{(n_{2})}\cross\cdots\cross w_{0}^{(n_{p})}\in S_{n}$(3.25),of the Grothendieck polynomial corresponding tothen the principal specialization
the permutation , is the product of -Schr\"oder-Hankel polynomials. Finally, we observethat from discussions in Section 3.4, Grothendieck and Narayana polynomials, one
can deduce that$6_{w^{(n)}}^{(\beta)}$

$w$

$q$

$\emptyset_{w^{(n)}}^{(\beta-1)}(x_{1}=1, \ldots, x_{n}=1)=\prod_{j=1}^{p-1}\mathfrak{R}_{n^{(j+1)}}^{(n^{(j)})}(\beta)$

In particular, the polynomial
non-negative integer coefficients.

$6_{w^{(n)}}^{(\beta-1)}(x_{1}, \ldots, x_{n})$.

is a symmetric polynomial in$\beta$with

29
Example 3.5(1) Let us take (non vexillary) permutation $w=2143=s_{1}s_{3}$ . One can check that$6_{w}^{(\beta)}(1,1,1,1)=3+3\beta+\beta^{2}=1+(\beta+1)+(\beta+1)^{2}$ , and $\mathfrak{R}_{4}(\beta)=(1,6,6,1),$$(1,3,1),$. It is easy to see that$\mathfrak{R}_{3}(\beta)=$$\mathfrak{R}_{2}(\beta)=(1,1)$$\beta 6_{w}^{(\beta)}(1,1,1,1)=DET|\begin{array}{ll}\mathfrak{R}_{4}(\beta) \mathfrak{R}_{3}(\beta)\mathfrak{R}_{3}(\beta) \Re_{2}(\beta)\end{array}|$$DET|P_{3}(\beta)P_{4}(\beta)$

On the other hand,.

$P_{2}P_{3\{_{\beta)}^{\beta)}}|=(3,6,4,1)=(3+3\beta+\beta^{2})(1+\beta)$. It is more involved tocheck that.

$q^{5}(1+\beta)6_{w}^{(\beta)}(1, q, q^{2}, q^{3})=DET|\begin{array}{ll}S_{4}(q\cdot\beta) S_{3}(q\cdot\beta)S_{3}(q\cdot q\beta) S_{2}(q.q\beta)\end{array}|$(2) Let us illustrate Theorem 3.3 by a few examples. For the sake of simplicity, we
consider the case $\beta=0$ , i.e. the case of Schubert polynomials. In this case $P_{n}(q;\beta=$$0)=C_{n}(q)$ is equal to the $Carlitz-Rior\overline{danq}$-analogue of Catalan numbers. We arereminded that the q-Catalari-Hankel polynomials are defined as follows$C_{n}^{(k)}(q)=q^{k(1-k)(4k-1)/6}DET|C_{n+k-i-j}(q)|_{1\leq i,j\leq n}$ .

In the case $\beta=0$ the Theorem 3.3 states that if
tationis defined by the use of (3.25), then$n=(n_{1}, \ldots, n_{p})\in \mathbb{N}^{p}$and the permu-$w_{(n)}\in S_{n}$$\mathfrak{S}_{w^{(n)}}(1, q, q^{2}, \ldots)=q^{\Sigma(^{n_{i}})}3C_{n_{1}+n_{2}}^{(n_{1})}(q)\cross C_{n_{1}+n_{2}+n_{3}}^{(n_{1}+n_{2})}(q)\cross C_{n}^{(n-n_{p})}(q)$.

Now let us consider a few examples for $n=6$ .
$\bullet$$\bullet$$n=(1.5),$$n=(2,4)$ ,Note that$\bullet$$\bullet$$\Rightarrow \mathfrak{S}_{w^{(n)}}(1, q, \ldots)=q^{10}C_{6}^{(1)}(q)=C_{5}(q)$$\bullet$.

$\Rightarrow \mathfrak{S}_{w^{(n)}}(1, q, \ldots)=q^{4}C_{6}^{(2)}(q)=DET|\begin{array}{ll}C_{6}(q) C_{5}(q)C_{5}(q) C_{4}(q)\end{array}|$$\mathfrak{S}_{w^{(2,4)}}(1, q, \ldots)=\mathfrak{S}_{w^{(1,1,4)}}(1, q, \ldots)$.

$n=(2,2,2)\Rightarrow \mathfrak{S}_{w^{(n)}}(1, q, \ldots)=C_{4}^{(2)}(q)C_{6}^{(4)}(q)$.

$n=(1,1,4)\Rightarrow \mathfrak{S}_{w^{(n)}}(1, q, \ldots)=q^{4}C_{2}^{(1)}(q)C_{4}^{(2)}(q)=q^{4}C_{4}^{(2)}(q)$the last equality follows from that$\bullet$.

,for all $k\geq 1$ .
$n=(1,2,3)\Rightarrow \mathfrak{S}_{w^{(n)}}(1, q, \ldots)=qC_{3}^{(1)}(q)C_{6}^{(3)}(q)$. On the other hand,$C_{k+1}^{(k)}(q)=1$$n=(3,2,1)\Rightarrow \mathfrak{S}_{w^{(n)}}(1, q, \ldots)=qC_{5}^{(3)}(q)C_{6}^{(5)}(q)=qC_{5}^{(3)}(q)=q(1,1,1,1)$

Note that $C_{k+2}^{(k)}(q)=\{k +11\}$

Exercise.
Let $1\leq k\leq m\leq n$ be integers, $n\geq 2k+1$ .
$w=$$(_{m}1$$m-k+1k$$m-12$$\ldots$

Consider permutation$k+1n$...

$\ldots$

Show that$\mathfrak{S}_{w}(1, q, \ldots)=q^{n(D(w))}C_{n-m+k}^{(m)}(q)$,$n1)\in S_{n}$

.

.

30
where for any permutation $w,$ $n(D(w))= \sum(^{d(w)}:_{2})$ anddenotes the number ofboxes in the i-th column of the (Rothe) diagram $D(w)$ of the permutation , see [26].
p.8.
$d_{i}(w)$$w$$\blacksquare$(C) A determinantal formula for the Grothendieck polynomials$\otimes_{\pi_{k}^{(n}}^{(\beta)}$,

Define polynomials$\Phi_{n}^{(m)}(X_{n})=\sum_{a=m}^{n}e_{a}(X_{n})\beta^{a-m}$,$A_{i,j}(X_{n+k-1})= \frac{1}{(i-j)!}(\frac{\partial}{\partial\beta})^{j-1}\Phi_{k+n-i}^{(n+1-i)}(X_{k+n-i})$if $1\leq i\leq j\leq n$ ,,and

if $1\leq j<i\leq n$ .

$A_{i,j}(X_{k+n-1})= \sum_{a=0}^{i-j-1}e_{n-i-a}(X_{n+k-i})(i- ja -1)$ ,Theorem 3.6$DET|A_{i,j}|_{1\leq i,j\leq n}=\otimes_{\pi_{k+n}^{(k)}}^{(\beta)}(X_{k+n-1})$.

Comments 3.5 One can compute the Grothendieck polynomials for yet another interesting family of permutations. namely, permutations$\sigma_{k}^{(n)}=$$\sigma_{k}^{(n)}=$$(_{1}^{1}$$k-1k-1$$22$$n+kk$$k+1k$$k+1k+2$$\ldots$$s_{k}s_{k+1}\ldots s_{n+k-1}\in S_{n+k}$..

$n+\cdot k.-1$$n+k)=$.

Then$6_{\sigma_{k^{(n)}}}^{(\beta)}(x_{1}, \ldots, x_{n+k})=\sum_{j=0}^{k-1}(\begin{array}{ll}n+j -1j \end{array})e_{n+j}(x_{1}, \ldots, x_{n+k})(1+\beta)^{J}$.

In particular,.

$6_{\sigma_{k}^{(n)}}^{(\beta)}(x_{1}=1, \ldots, x_{n+k}=1)=\sum_{j=0}^{k}(\begin{array}{ll}n+j -1j \end{array}) \beta^{g}$$\blacksquare$

Problems 3.2(1) Give a bijective pmve of Theorem 3.3, $i.e$ . construct a bijection between
the set of k-tuple of mutually non-crossing
pathsof lengths$(n, n-1, \ldots , n-k+1)$ correspondingly, and, wherethe set of pairs
is a k-dissection of a convex $(n+k+1)-gon$ , andis a upper triangle $(0,1)-mat\dot{m}$ of size $(k-1)\cross(k-1)$ ,which is compatible with natuml statistics on both sets.
$SCh\mathfrak{X}der$$\bullet$$\bullet$$m$$(m, \mathcal{T})$$\mathcal{T}$$(\mathfrak{p}_{1}, \ldots , \mathfrak{p}_{k})$31(2) Let $w\in S_{n}$ be a permutation, and $CS(w)$ be the set of compatible sequences
corresponding to , see $e.g$ . $l4l$ .
on the set $CS(w)$ such that
Define statistics$w$$c(\bullet)$$6_{w}^{(\beta-1)}(x_{1}=1, x_{2}=1, \ldots)=\sum_{a\in CS(w)}\beta^{c(a)}$3.2.3

Specialization of Schubert polynomials

Let

$k,$

.

be positive integers andbe non-negative integers such that $r\leq$$p+1$ . It is well-known [26] that in this case there exists a unique vexillary permutation
which has the shape,and the flag
where$\varpi$

$n,$

$r$

$p,$

$:=\varpi_{\lambda,\phi}\in S_{\infty}$$b$$\phi=(\phi_{1}, \ldots , \phi_{n+1})$$\lambda=(\lambda_{1}, \ldots, \lambda_{n+1})$$\phi_{i}=k+1+r(i-1)$ ,$\lambda_{i}=(n-i+1)p+b$ ,$1\leq i\leq n+1$ .

According to a theorem by M.Wachs [43], the Schubert polynomial
following determinantal representation$\mathfrak{S}_{\varpi}(X)$admits the$\mathfrak{S}_{\varpi}(X)=DET(h_{\lambda_{i}-i+j}(X_{\phi_{i}}))_{1\leq i,j\leq n+1}$

Therefore we have$\mathfrak{S}_{\varpi}(1)$$:=\mathfrak{S}_{\varpi}(x_{1}=1, x_{2}=1, \ldots)=$$DET((\begin{array}{l}(n-i+1)p+b-i+j+k+(i-1)rk+(i-1)r\end{array}))_{1\leq i,j\leq n+1}$

We denote the above determinant by $D(n, k, r, b,p)$ .
Theorem 3.7$D(n, k, r, b,p)=$$\prod_{(i,j)\in A_{n,k,r}}\frac{i+b+jp}{i}$$\prod_{(i,j)\in B_{n,k,r}}\frac{(k-i+1)(p+1)+(i+j-1)r+r(b+np)}{k-i+1+(i+j-1)r}$

,

where
$\mathcal{A}_{m,k.r}=\{(i,j)\in Z_{\geq 0}^{2}$$\mathcal{B}_{n,k,r}=\{(i,j)\in Z_{\geq 1}^{2}$$|j\leq n$ ,$j<i\leq k+(r-1)(n-j)\}$ ,$|i+j\leq n+1$ ,$i\neq k+1+rs$ ,$s\in Z_{\geq 0}\}$.

It is convenient to re-wright the above formula for $D(n, k, r, b,p)$ in the following form
$D(n, k, r, b,p)=$$\prod_{j=1}^{n+1}\frac{((n-j+1)p+b+k+(j-1)(r-1))!(n-j+1)!}{(k+(j-1)r)!((n-j+1)(p+1)+b)!}$$\prod_{1\leq i\leq j\leq n}((k-i+1)(p+1)+jr+(np+b)r)$ .

$\cross$

32
The case $r=1$
We consider below some special cases of Theorem 3.5 in the case $r=1$ . To simplify
notation, we set $D(n, k, b, p);=D(n, k, r=1, b,p)$ . Then we can rewrite the above$D(n, k, b,p)=$formula for $D(n, k, r, b,p)$ as follows$(\Rightarrow)$$\prod_{j=1}^{n+1}\frac{((n+k-j+1)(p+1)+b)!((n-j+1)p+b+k)!(j-1)!}{((n-j+1)(p+1)+b)!((k+n-j+1)p+b+k)!(k+j-1)!}$.

Corollary 3.2

If $k\leq n+1$ , then(1)$\prod_{j=1}^{k}$$D(n, k, b,p)=$$(\begin{array}{lll}(n+k +1- j)(p+1)+b n-j +1\end{array})(\begin{array}{l}(k-j)p+b+kj\end{array})\frac{j!(k-j)!(n-j+1)!}{(n+k-j+1)!}$.

In particular,$\bullet$If $k=1$ , then$D(n, 1, b,p)= \frac{1+b}{1+b+(n+1)p}(\begin{array}{l}(p+1)(n+1)+bn+1\end{array}):=F_{n+1}^{(p+1)}(b)$$\frac{1+b}{1+b+(p-1)n,then}\bullet ifk=2(^{pn+b}n)$$whereF_{n}^{p}(b):=,$,

denotes the genemlized Fuss-Catalan number.

$D(n, 2, b,p)= \frac{(2+b)(2+b+p)}{(1+b)(2+b+(n+1)p)(2+b+(n+2)p)}F_{n+1}^{(p+1)}(b)F_{n+2}^{(p+1)}(b)$(2) (R.A. Proctor $[35J)$$\lambda$.

Consider the Young diagmm$:=\lambda_{n,p,b}=\{(i,j)\in Z_{\geq 1}\cross Z_{\geq 1} | 1\leq i\leq n+1,1\leq j\leq(n+1-i)p+b\}$.

For each box $(i,j)\in\lambda$ define the numbers $c(i, j)$ $:=n+1-i+j$ , and$l_{(i,j)}(k)=\{\begin{array}{l}\frac{k+c(p,j)}{c(i,j)}, if j\leq(n+1-i)(p-1)+b,\frac{(p+1)k+c(i,j)}{c(i,j)}, if (n+1-i)(p-1)<j-b\leq(n+1-i)p.\end{array}$Then$D(n, k, b,p)= \prod_{(i,j)\in\lambda}l_{(i,j)}(k)$Therefore, $D(n, k, b,p)\iota s$ a polynomial in(3) If $p=0$ , then$k$.

(3.26)

with mtional coefficients.

,$D(n, k, b, 0)=dimV_{(n+1)^{k}}^{\mathfrak{g}l(b+k)}= \prod_{j=1}^{n+k}(\frac{j+b}{j})^{\min(j,n+k+1-j)}$where for any partition , $\ell(\mu)\leq m$ ,the highest weight . In particular,$\mu$$l^{l}$$V_{\mu}^{\mathfrak{g}t(m)}$denotes the irreducible $g^{[}(m)$ -module with33$\bullet$$D(n, 2, b, 0)= \frac{1}{n+2+b}(\begin{array}{l}n+2+bb\end{array})(\begin{array}{l}n+2+bb+1\end{array})$

is equal to the Narayana number $N(n+b+2, b)$ ;$D(1, k, b, 0)= \frac{(b+k)!(b+k+1)!}{k!b!(k+1)!(b+1)!}:=N(b+k+1, k)$ ,$\bullet$and therefore the number $D(1, k, b, 0)$ counts the number of pairs of non-crossing lattice
paths inside a rectangular of size $(b+1)\cross(k+1)$ , which go from the point 1, $0)$ (resp. fromthat $(0,1))$ to the point $(b+1, k)$ (resp. to that $(b,$ $k+1)$ ), consisting of steps $U=(1,0)$
and $R=(0,1)$ , see [37], A001263, for some list of combinatorial interpretations of the$($

Narayana numbers.

If $p=b=1$ , then(4)

$D(n, k, 1,1)= \prod_{1\leq l<j\leq n+2}\frac{2k+i+j-1}{i+j-1}$.

(5) (R.A. Proctor $[33],[34J)$ If $p=1$ andis odd integer, thenis equal to the dimension of the irreducible representation of the symplectic Lie algebm$Sp(b+2n+1)$ with the highest wnght.
(6) $(Cf[13J)$ Let
be a unique dominant permutation of shapeand$l:= \ell_{n,p,b}=\frac{1}{2}(n+1)(np+2b)$ be its length.
Then
$b$

$D(\gamma\iota, k, b, 1)$$k\omega_{n+1}$$\lambda$$\varpi_{\lambda}$$a\in R(\varpi)\sum_{\lambda}\prod_{i=1}^{\ell}(x+a_{i})=l!B(n, x,p, b)$$:=\lambda_{n,p,b}$.

Here for any permutation of length , we denote by $R(w)$ the set $\{a=(a_{1}, \ldots , a_{l})\}$ of
all reduced decompositions of .
$l$

$w$

$w$

Remark 3.2
It is well-known, see e.g. [35], that the number $D(n, k, b,p)$ is equal to the total
numberof reverse (weak) plane partitions 4 bounded by and contained in
the shape. Finally we recall that the generalized Fuss-Catalan number
counts the number of lattice paths from $(0,0)$ to $(b+r\iota p, n)$ that do not go above theline $x=py$ , see e.g. [24].
$pp^{\lambda_{n,p,b}}(k)$$k$$F_{n+1}^{(p+1)}(b)$$\lambda_{n,b,p}$$\blacksquare$

Theorem 3.8
$1,$

Let

$\varpi_{n,k,p}$

be a unique vexillary permutation of the shapeThen$:=(n,$ $n-$$\lambda_{n.p}$$2,1)p$ and flag $\phi_{n,k}:=(k+1, k+2, \ldots , k+n-1, k+n)$ .
$\ldots,$$\bullet$$\bullet$If$k\geq 2$, then$6_{\varpi_{n,1,p}}^{(\beta-1)}(1)= \sum_{j=1}^{n+1}\frac{1}{n+1}(\begin{array}{ll}n +1 j\end{array}) (\begin{array}{ll}(n +1)pj -1\end{array})\beta^{j-1}$$G_{n,k,p}(\beta)$$:=6_{\varpi_{n,k,p}}^{(\beta-1)}(1)$.

is a polynomial of degree $nk$ in$\beta$,and$Coeff_{[\beta^{nk}]}(G_{n,k,p}(\beta))=D(n, k, 1,p-1,0)$ .

4 Let
be a partition. A reverse plane partition bounded by and shape is a filling of the shapeby the numbers ffom the set $\{0,1, \ldots, d\}$ in such a way that the numbers along columns and rows are
weakly increasing.
$\lambda$$\lambda$$d$$\lambda$

34
is known as the Fuss-Narayana polyThe polynomial.
nomial and can be considered as a t-deformation of the Fuss-Catalan number
counts paths from $(0,0)$ to $(np, 0)$ in the first
Recall that the number
quadrant, consisting of steps $U=(1,1)$ and $D=(1, -p)$ and have peaks (i.e. UD’s),
cf. [37], A108767.
$:=S\mathfrak{N}_{\eta}(t)$$\sum_{j=1}^{n}\frac{1}{n}(\begin{array}{l}nj\end{array})(\begin{array}{l}pnj-1\end{array})t^{j-1}$$FC_{n}^{p}(0)$$\frac{1}{n}(\begin{array}{l}nj\end{array})(\begin{array}{l}pnj-1\end{array})$$J$

For example, take $r\iota=3,$ $k=2,$ $p=3,$ $r=1,$ $b=0$ . Then
$G_{3,2,3}(\beta)=$$\varpi_{3,2,3}=[1,2$ , 12,9,6,3,4,5,7,8, 10, $11]\in S_{12}$ , and(1, 18, 171, 747, 1767, 1995, 1001). Therefore, $G_{3,2,3}(1)=5700=D(3,2,3,0)$ and$Coeff_{[\beta^{6}1}(G_{3,2,3}(\beta))=1001=D(3,2,2,0)$ .

Comments 3.6
It follows from Theorem 3.5 that in the case $r=0$ and $k\geq n$ , one has$D(n, k, 0,p, b)=dimV_{\lambda_{n\rho,b}}^{\mathfrak{g}1(,k+1)}=(1+p)^{(\begin{array}{l}n+12\end{array})}$Now consider the conjugate$\lambda_{n,p,b}$$:=\nu_{n,p,b}$$\nu$$\prod_{j=1}^{n+1}$$\frac{(^{(n-j+1)p+b+k-j+1}k-j+1)}{(\begin{array}{l}(n-j+1)(p+1)+bn-j+1\end{array})}$$:=((n+1)^{b}, n^{p}, (n-1)^{p}, \ldots, 1^{p})$.

of the partition

,

shape
a unique grassmannian permutation
$D(n,
k,
0,p,
b),\overline{that}$is easy to see om the above formula for$:=\sigma_{n,k,p,b^{np+b}}ofthe$$\sigma$$\nu ar$l(1 the flag$[26]$$\psi,$. It$f$$\mathfrak{S}_{\sigma_{n,k,p,b}}(1)=dimV_{\nu_{n,\rho,b}}^{\mathfrak{g}1(k-1)}=$$(1+p)^{(_{2}^{n})} (\begin{array}{l}k+n-lb\end{array})\prod_{j=1}^{n}\frac{(p+1)(n-j+1)}{(n-j+1)(p+1)+b}$$\prod_{j=1}^{n}\frac{(\begin{array}{l}k+j-2(n-j+1)p+b\end{array})}{(^{(n-j+1)(p+1)+b-1}n-j)}$.

After the substitution $k:=np+b+1$ in the above formula we will have.

$\mathfrak{S}_{\sigma_{n.np+b+1,p,b}}(1)=(1+p)^{(_{2}^{n})}\prod_{j=1}^{n}\frac{(_{(n-j+1)p}^{np+b+j-1})}{(\begin{array}{l}j(p+1)-1j-1\end{array})}$In the case $b=0$ some simplifications are happened, namely$\mathfrak{S}_{\sigma_{n,k,p,0}}(1)=(1+p)^{(_{2}^{n})}\prod_{j=1}^{n}\frac{(_{(n-j+1)p}k+j-2)}{(^{(n-j+1)p+n-j}n-j)}$.

Finally we observe that if $k=np+1$ , then$\prod_{j=1}^{n}\frac{(\begin{array}{l}np+j-1(n-j+1)p\end{array})}{(^{(n-j+1)p+n-j}n-j)}=\prod_{j=2}^{n}\frac{(_{(p+1)(j-1)}np+j-1)}{(\begin{array}{l}j(p+1)-1j-1\end{array})}=\frac{\prod_{i=1}^{p}\prod_{j=0}^{n-1}((p+1)j+i)}{\prod_{j=0}^{n(p-1)-1}(n+j)};=A_{n}^{(p)}$,are integers that generalize the numbers of alternating signwhere the numbers, recovered in the case $p=2$ , see [31], [6] for details.
(ASM)
of
size
matrices$A_{n}^{(p)}$$n\cross n$

35
Examples 3.1(1) Let us consider polynomials.
If $n=2$ , then $\sigma_{2,4,2,0}=235614\in S_{6}$ , and $6_{2}(\beta)=(1,2,3)$ $:=1+2\beta+3\beta^{2}$ .
Moreover, $\Re_{\sigma_{2,4,2,0}}(q;\beta)=(1,2)_{\beta}+3q\beta^{2}$ .
If $n=3$ , then $\sigma_{3,6,2,0}=235689147\in S_{9}$ , and 63 $(\beta)=(1,6,21,36,51,48,26)$ .

.
.

$\mathfrak{G}_{n}(\beta)$$:=\mathfrak{G}_{\sigma_{n,2n2,0}}^{(\beta-1)},(1)$Moreover, $\Re_{\sigma_{3,6,2,0}}(q;\beta)=(1,6,11,16,11)_{\beta}+q\beta^{2}(10,20,35,34)_{\beta}+q^{2}\beta^{4}(5,14,26)_{\beta}$ ;.

$\Re_{\sigma_{3,6,2,0}}(q;1)=(45,99,45)_{q}$If $n=4$ , then $\sigma_{4,8,2,0}=[2,3,5,6,8,9,11,12,1.4,7,10]\in S_{12}$ , and(1, 12, 78, 308, 903, 2016, 3528, 4944, 5886, 5696, 4320, 2280, 646).

$\bullet$$\mathfrak{G}_{4}(\beta)=$Moreover, $\Re_{\sigma_{4,8,2,0}}(q;\beta)=(1,12,57_{7}182,392,602,763,730,493,170)_{\beta}+$
21, 126, 476, 1190, 1925, 2626, 2713, 2026, $804)_{\beta}+$
35, 224, 833, 1534, 2446, 2974, 2607, $1254)_{\beta}+q^{3}\beta^{6}(7,54,234,526,909, 1026, 646)_{\beta}$ ;$\Re_{\sigma 4,8,2,0}(q;1)=(3402$ , 11907, 11907, $3402)_{q}=1701(2,7,7,2)_{q}$ .
, etc.][It will be recalled that here we have used the notation
One can show that $deg_{[\beta]}6_{n}(\beta)=n(n-1)$ , and looking on the numbers 3, 26, 646 we$q\beta^{2}($$q^{2}\beta^{4}($$(a_{0},$ $a_{1},$$\ldots,$$a_{r})_{\beta}$$:= \sum_{j=0}^{r}a_{j}\beta^{j}$madeConjecture 2

Let $a(n)$ $:=Coeff[\beta^{n(n-1)}](\otimes_{n}(\beta))$ .

$\underline{Then}$$a(n)=VSAM(n)=OSASM(n)= \prod_{j=1}^{n-1}\frac{(3j+2)(6j+3)!(2j+1)!}{(4j+2)!(4j+3)!}$ ,where
VSASM $(n)$ is the number of alternating sign $2n+1\cross 2n+1$ matrices symmetric
about the vertical axis;
OSASM $(n)$ is the number of $2n\cross 2n$ off-diagonal symmetric alternating sign
matrices.
See [37], A005156, [31] and references therein, for details.
(2)

Let us consider polynomials.
$\sigma_{1,3,2,0}=1342\in
S_{4}$$:=1+2\beta$ .
$n=1$
,
If
then
, and
If $n=2$ , then $\sigma_{2,5,2,0}=1346725\in S_{7}$ , and $\mathfrak{F}_{3}(\beta)=(1,6,11,16,11)$ .
Moreover, $\Re_{\sigma_{2,5,2,0}}(q;\beta)=(1,2,3)_{\beta}+q\beta(4,8,12)_{\beta}+q^{2}\beta^{3}(4,11)_{\beta}$ .
If $n=3$ , then $\sigma_{3,7,2,0}=[1,3,4,6,7,9,10,2,5,8]\in S_{10}$ , and $S_{4}(\beta)=$(1, 12, 57, 182, 392, 602, 763, 730, 493, 170).
Moreover, $\Re_{\sigma_{3,7.2,0}}(q;\beta)=(1,6,21,36,51,48,26)_{\beta}+q\beta(6,36,126,216,306,288,156)_{\beta}$$+q^{2}\beta^{3}(20,125,242,403,460,289)_{\beta}+q^{3}\beta^{5}(6,46,114,204,170)_{\beta}$;$\Re_{\sigma_{3,7,2,0}}(q;1)=(189$ , 1134, 1539, $540)_{q}=27(7,42,57,20)_{q}$ .
If $n=4$ , then $\sigma_{4,9,2,0}=[1,3,4,6,7,9,10,12,13,2,5,8,11]\in S_{13}$ , and $S_{5}(\beta)=$20,
174, 988, 4025, 12516, 31402, 64760, 111510, 162170, 202957, 220200, 202493,(1,153106, 89355, 35972, 7429).
Moreover,$\Re_{\sigma 49,2,0}(q;\beta)=(1,12,78,308,903$ , 2016, 3528, 4944, 5886, 5696, 4320, 2280, $646)_{\beta}+$
8, 96, 624, 2464, 7224, 16128, 28224, 39552, 47088, 45568, 34560, 18240, $5168)_{\beta}+$$ff_{n}(\beta);=6_{\sigma_{n,2n+1,2,0}}^{(\beta-1)}(1)$$\mathfrak{F}_{2}(\beta)=(1,2)$$\bullet$$\bullet$$\bullet$$\bullet$$q\beta$$($$q^{2}\beta^{3}($56. 658, 3220, 11018, 27848, 53135, 78902, 100109, 103436, 84201, 47830,$14467)_{\beta}+$

3656, 728, 3736, 12820, 29788, 50236, 72652, 85444, 78868, 50876, $17204)_{\beta}+$
8, 117, 696, 2724, 7272, 13962, 21240, 24012, 18768,;$\Re_{\sigma}4,9,2.0(q;1)=(30618$ , 244944, 524880, 402408, $96228)_{q}=4374(7,56,120,92,22)_{q}$ .
is a polynomial in of degree , and looking on the
One can show that
170,
2,
11,
numbers7429 we made$q^{3}\beta^{5}($$7429)_{\beta}$$q^{4}\beta^{7}($$n^{2}$$\beta$$\mathfrak{F}_{n}(\beta)$

Conjecture 3 Let $b(n);=Coeff[\beta^{(n-1)^{2}}](S_{n}(\beta))$ .
$b(n)=$ CSTCPP(n). In other words, $b(n)$ is equal to the number of cyclicallysymmetric transpose complement plane partitions in an $2n\cross 2n\cross 2n$ box. This
number is known to be$\underline{Then}$$\prod_{j}^{n-1}\frac{(3j+1)(6j)!(2j)!}{(4j+1)!(4j)!}$,see [37], A051255, [2], p.199.
One has

Proposition 3.5$\Re_{\sigma_{4,2n+1,2,0}}(0;\beta)=\otimes_{n}(\beta)=6_{\sigma_{n.2n,2,0}}^{(\beta-1)}(1)$,.

$\Re_{\sigma_{n,2n,2,0}}(0, \beta)=S_{n}(\beta)=\emptyset_{\sigma_{n,2n+1,2,0}}^{(\beta-1)}(1)$Remarks 1 One can compute the principal specialization of the Schubert polynomial$:=(k, n-k)\in S_{n}$ that interchanges
and $n-k$ ,corresponding to the transposition
and fixes all other elements of 1, .
$k$

$t_{k,n}$

$[$

$n]$

Proposition 3.6$q^{(n-1)(k-1)} \mathfrak{S}_{t_{k.n-k}}(1, q^{-1}, q^{-2}, q^{-3}\ldots.)=\sum_{j=1}^{k}(-1)^{\gamma-1}q^{(_{2}^{j})}\{\begin{array}{l}-1nk-j\end{array}\}$$\{\begin{array}{ll}-2+jn n-k -l\end{array}\}$$\blacksquare$

Exercises.
(1) Let $n\geq 1$ be a positive integer, consider “zig-zag“ permutation$\cdot$$\prime uf=$$(_{1}^{2}$

$43$

...

$2k-12k$$\cdot$$2k+12k+2$. . ...

$2n-12n)\in S_{2n}$ .

Show that$\Re_{w}(q, \beta)=\prod_{k=0}^{n-1}(\frac{1-\beta^{2k}}{1-\beta}+q\beta^{2k})$Let$(k+1)^{m}$ , i.e.
(2)$\sigma_{k,n,m}$.

be grassmannian permutation with shape $\lambda=(n^{m})$ and flag $\phi=$$\sigma_{k,n,m}=(\begin{array}{lllllllllll}1 2 \cdots k k+1 \cdots \cdots k+n k+n+1 \cdots k+n+m1 2 \cdots k k+m +1 \cdots k+m+n k+1 \cdots k+m\end{array})$

Clearly$\sigma_{k+1,n,m}=1\cross\sigma_{k,n,m}$. Show that the coefficient

.

37
Coef

$f_{\beta^{m}}(\Re_{\sigma_{k,n,m}}(1, \beta))$

is equal to the Narayana number $N(k+n+rn, k)$ .

(3) Show that$(a,b,c) \in Z^{3}\sum_{a>0}\{\begin{array}{l}a+b-1b\end{array}\}\{\begin{array}{l}a+c-1c\end{array}\}\{\begin{array}{l}b+cb\end{array}\}=1+\frac{1}{(q;q)^{3}}(\sum_{k\geq 2}(-1)^{k}(\begin{array}{l}k2\end{array})q(\begin{array}{l}k2\end{array}))$

3.2.4
Let

.

Specialization of Grothendieck polynomials

and$2i<n$ be positive integers. Denote byconvex quadrangle having vertices at the points$p,$ $b,$ $n$$i,$$(ip, i)$ ,$(ip, n-i)$ ,$(b+ip, i)$andthe tmpezoid, i.e. a$\mathcal{T}_{p,b,n}^{(i)}$$(b+(n-i)p, n-i)$ .

Definition 3.5 Denote by
the set of lattice path from the point $(ip, i)$ to that$(b+(n-i)p, n-i)$ with east steps $E=(0,1)$ and north steps $N=(1,0)$ , which arelocated inside of the tmpezoid.
is a path, we denote by
the number of peaks, $i.e$ .
If$FC_{b,p,n}^{(i)}$$fl_{p,b,n}^{i)}$$\mathfrak{p}\in FC_{b,p,n}^{(i)}$$p(\mathfrak{p})$$p(\mathfrak{p})=NE(\mathfrak{p})+E_{in}(\mathfrak{p})+N_{end}(\mathfrak{p})$where NE1,$\mathfrak{p}$$(\mathfrak{p})$if the path,is equal to the number of steps NE resting on path$\mathfrak{p}$starts with stepends by the step $N$ andNote that the equality

$0$

$E$

and

$0$

otherwise;$N_{end}(P)$otherwise.

$N_{end}(\mathfrak{p})=1$$\mathfrak{p};E_{in}(\mathfrak{p})$is equal tois equal to 1, if the path

may happened only in the case $b=0$ .

Definition 3.6 Denote by
the set of k-tuples
lattice paths, where for each $i=1,$,.
$FC_{b,p,n}^{(k)}$$\ldots$$k,$$\mathfrak{P}=$$(\mathfrak{p}_{1}, \ldots , \mathfrak{p}_{k})$of non-crossing$\mathfrak{p}_{i}\in FC_{b,p,n}^{(i)}$Let$\beta^{p(\mathfrak{P})}$$FC_{b,p,n}^{(k)}( \beta):=\sum_{;\mathfrak{p}\in FC_{bp,n}^{(k)}}$,denotes the generating function of the statistics$p(\mathfrak{P})$$:= \sum_{i=1}^{k}p(\mathfrak{p}_{i})-k$.

Theorem 3.9 The following equality holds$\emptyset_{\rho_{n,k,p.b}}^{(\beta)}(x_{1}=1, x_{2}=1, \ldots)=FC_{p,b,n+k}^{(k)}(\beta+1)$

3.3.

The “longest element“ and Chan-Robbins polytope

Assume additionally, cf [40], 6. $C8,$ $(d)$ , that the condition $(a)$ in Definition 3.1 isreplaced by that38$(a’):x_{ij}$and$x_{kl}$commute for all $i,j,$

$k$

and .
$l$

to the reduced. Let us bring the element
Consider the element
form, that is, let us consecutively apply the defining relations
to the element
and.
in any order until unable to do so. Denote the resulting polynomial by$(a’)$
andNote that the polynomial itself depends on the order in which the relations
are applied.
We denote by
the specialization $x_{ij}=1$ for all and , of the polynomial$Q_{n}(x_{ij};\beta);yQ_{n}(\beta, t)$ the specialization $x_{ij}=1$ , if $(i,j)\neq(1, n)$ , and $x_{1,n}=t$ , and by.
the specialization$w_{0}$$:= \prod_{1\leq i<j\leq n}x_{ij}$$w_{0}$$(b)$$(a^{l})$$Q_{n}(x_{ij};\beta)$$w_{0}$

$(b)$

$i$

$Q_{n}(\beta)$$Q_{n}(z_{1}, \ldots , z_{i-1})$$j$$x_{ij}=z_{i}$Example 3.6$Q_{3}(\beta)=(2,1)=1+(\beta+1)$ ,$Q_{4}(\beta)=(10,13,4)=1+5(\beta+1)+4(\beta+1)^{2}$ ,$Q_{4}(\beta, t)=t^{4}+t(1+2t^{2}+2t^{3})(\beta+1)+(t+t^{2})^{2}(\beta+1)^{2}$$Q_{4}(z_{1}, z_{2}, z_{3})=z_{1}^{7}z_{2}^{3}z_{3}\emptyset_{[1,5,3,4,2\rceil}^{(\beta)}(z_{1}^{-1}, z_{2}^{-1}, z_{3}^{-1}, z_{1}^{-1})$.

,$Q_{5}(\beta)=(140,336,280,92,9)=1+16(\beta+1)+58(\beta+1)^{2}+56(\beta+1)^{3}+9(\beta+1)^{4}$ ,$Q_{6}(\beta)=1+42(\beta+1)+448(\beta+1)^{2}+1674(\beta+1)^{3}+2364(\beta+1)^{4}+1182(\beta+1)^{5}+169(\beta+1)^{6}$ .

?
What one can say about the polynomial
is equal
It is known, [40], 6. $C8,$ $(d)$ , that the constant term of the polynomial$n\geq
3$ , then. It is not difficult to see that if
to the product of Catalan numbers$deg_{\beta}(Q_{n}(\beta))=2(n-3)$ and.
$Q_{n}(\beta)$$:=Q_{n}(x_{ij};\beta)|_{x_{j}=1,\forall i,j}$$Q_{n}(\beta)$$\prod_{j=1}^{n-1}C_{j}$$Coeff_{[\beta+1]}(Q_{n}(\beta))=2^{n}-1-(\begin{array}{l}n+12\end{array})$

Theorem 3.10

One has$Q_{n}( \beta-1)=(\sum_{m\geq 0}’,(CR_{n+1}, m)\beta^{m})(1-\beta)(\begin{array}{l}n+22\end{array})+1$,

where $CR_{m}$ denotes the Chan-Robbins polytope [3], i.e. the convex polytope given by
the following conditions :$CR_{m}=\{(a_{ij})\in Mat_{m\cross m}(Z_{\geq 0})\}$ such that(1)
(2)$\sum_{i}a_{ij}=1$$a_{ij}=0$ ,,$\sum_{j}a_{ij}=1$

if $j>i+1$ .

.

Here for any integral convex polytope.
points in the set$\mathcal{P}\subset Z^{d}$,$\iota(\mathcal{P}, n)$denotes the number of integer$n\mathcal{P}\cap Z^{d}$Conjecture 4 (A)Let $n\geq 4$ and write$Q_{n}(\beta, t)$(B)$:= \sum_{k=0}^{2n-6}(1+\beta)^{k}c_{k,n}(t)$All roots of the polynomial$Q_{n}(\beta)$,$\underline{then}$$c_{k,n}(t)\in Z_{\geq 0}[t]$belong to the set

Comments 3.7(1) We expect that for each integer $n\geq 2$ the set$\Psi_{n+1}:=\{w\in S_{2n-1}|\mathfrak{S}_{w}(1)=\prod_{j=1}^{n}Cat_{j}\}$.

$\mathbb{R}_{<0}$.

39contains either one or two elements, whereas the set $\{w\in S_{2n-2}|\mathfrak{S}_{w}(1)=\prod_{j=1}^{n}Cat_{j}\}$ isempty. For example. $\Psi_{4}=\{[1,5,3,4,2]\},$ $\Psi_{5}=\{[1,5,7,3,2,6,4], [1,5,4,7,2,6,3]\}$ ,$\overline{\Psi_{6}}=\{w:=[1,3,2,8,6,9,4,5,7], w^{-1}\},$??? .
Question Does there exist a vexillary (grassmannian ?) permutation
such
that
?
For example, ) $=[1,4,5,6,8,3,5,7]\in S_{8}$ is a grassmannian permutation such that, and $\Re_{w}(1, \beta)=(1,9,27,43,38,18,4)$ .
$\Psi_{7}=$

$\}$

$\{$

$w\in S_{\infty}$$\mathfrak{S}_{w}(1)=\prod_{j=1}^{n}Cat_{j}$$u$$\mathfrak{S}_{w}(1)=140$

Remark 3.3 We expect that for $n\geq 5$ there are no permutations.

$w\in S_{\infty}$

such that$Q_{n}(\beta)=\mathfrak{S}_{w}^{(\beta)}(1)$$:= \prod_{j=1}^{n}Cat_{j}$ appear also as the values of the Kostant partition(2) The numbers$A_{n-1}$
on some special vectors. Namely,
function of the type
$C_{n}$$C_{m}=K_{\Phi(1^{n})}(\gamma_{n})$

,

where

$\gamma_{n}=(1,2,3, \ldots, n-1, -(\begin{array}{l}n2\end{array}))$,see e.g. [40], 6. $C10$ , and [17], 173-178. More generally [17], (7,18), (7.25),one has$K_{\Phi(1^{n})}( \gamma_{n,d})=pp^{\delta_{n}}(d)C_{n-1}=\prod_{j=d}^{n+d-2}\frac{1}{2j+1}(\begin{array}{l}n+d+j2j\end{array})$

,

where
$(d+1, d+2, \ldots , d+n-1, -n(2d+n-1)/2)$ , anddenotesthe set of reverse (weak) plane partitions bounded byand contained in the shape, where.
Clearly,
is the n-th$:=(n-1, n-2, \ldots, 1)$$pp^{\delta_{n}}(d)$$\gamma_{n,d}=$$d$$C_{n}$$\delta_{n}$$pp^{\delta_{n}}(1)= \prod_{1\leq i<j\leq n}\frac{i+j+1}{i+j-1}=C_{n}$Catalan number 5.
Conjecture 5
For any permutation $w\in S_{n}$ there exists a graph $\Gamma_{w}=(V, E)$ , possibly with multiple
edges, such that the reduced volume, see e.g. [39] forof the flow polytope
a definition of the former, is equal to.
$?\overline{ol},(\mathcal{F}_{\Gamma_{w}})$$\mathcal{F}_{\Gamma_{w}}$$\mathfrak{S}_{w}(1)$$\blacksquare$

For a family of vexillary permutationsof the shapeand flag $\phi=$$(1,2, \ldots, n-1, n)$ the corresponding graphs
have been constructed in [29], Section 6.
In this case the reduced volume of the flow polytope
is equal to the Fuss-Catalan, cf Corollary 3.2
number$\lambda=p\delta_{n+1}$$w_{n,p}$$\Gamma_{n,p}$$\mathcal{F}_{\Gamma_{n,p}}$$\frac{1}{1+(n+1)p}(^{(n+1)(p+1)}n+1)=\mathfrak{S}_{w_{n,p}}(1)$Problems 3.3(1) Assume additionally to the conditions $(a’)$ and$x_{ij}^{2}=\beta x_{ij}+1$,

above that

if $1\leq i<j\leq n$ .

What one can say about a reduced form of the element
5 For example, if $n=3$ ,$(b)$$w_{0}$

in this case /?

there exist 5 reverse (weak) plane partitions of shape $\delta_{3}=(2,1)$ boundedby 1, namely reverse plane partitions$\{(\begin{array}{ll}0 00 \end{array}),$$(\begin{array}{ll}0 10 \end{array})$,$(\begin{array}{ll}0 l0 \end{array}),$$(\begin{array}{ll}0 l1 \end{array})$,$(\begin{array}{ll}1 11 \end{array})\}$.

40(2) According to a result by S. Matsumoto and J. Novak $[27J$, if $\pi\in S_{n}$ is a per, then the total number of primitive factorizations (seemutation of the cyclic type,denotedby
definition in $[27J)$ of into product of $n-\ell(\lambda)transposition\overline{s,}$is equal to the product of Catalan numbers:$\lambda\vdash n$$PbPrim_{n-\ell(\lambda)}(\lambda)$$\pi$$Prim_{n-\ell(\lambda)}( \lambda)=\prod_{i=1}^{\ell(\lambda)}Cat_{\lambda_{i}-1}$

Recall that the Catalan number
Then$Cat_{n}$$:=C_{n}= \frac{1}{n}(\begin{array}{l}2nn\end{array})$.

. Now take$Q_{n}(1)= \prod_{a=1}^{n}Cat_{a}=Prim_{(_{2}^{n})}(\lambda)$$\lambda=$$(2, 3, \ldots , n+1)$ .

.

Does there exist “a natural bijection between the primitive factorizations and monomials
which appear in the polynomial“$Q_{n}(x_{ij};\beta)9$Grothendieck polynomials

AppendixDefinition Al Let be a parameter. The Id-Coxeter algebra
generated by elements
ciative algebra over the ring of polynomials
subject to the set of relations$|i-j|\geq 2$ ,, if

$IdC_{n}(\beta)$$\beta$$Z[\beta]$$\bullet$$e_{t}e_{j}=e_{j}e_{i}$$\bullet$$e_{i}e_{j}e_{i}=e_{j}e_{i}e_{j}$,if$\langle e_{1},$is an asso$\ldots,$$e_{n-1}\rangle$$|i-j|=1$ ,.
-linear basis of theform aIt is well-known that the elements$w\in
S_{n}$
the productwe denoted by.
algebraHere for a permutation
is any reduced word for a permutation , i.e.
, whereis the length of ).
andbe a set of mutually commuting variables.
,$x_{n}=y,$ $x_{n+1}=z,$Let
We assume that and commute for all values of and . Let us define$\bullet$$e_{i}^{2}=\beta e_{i}$,$1\leq i\leq n-1$

$Z[\beta]$$\{e_{w}, \uparrow n\in S_{n}\}$$IdC_{n}(\beta)$$e_{w}$$\ell=\ell(\prime u;)$$uf=s_{i_{1}}s_{i_{2}}\cdots s_{i\ell}$$x_{1},$ $x_{2},$$u$$x_{n-1},$$\ldots$$w$$(i_{1}, i_{2}, \ldots , i_{\ell})$$e_{i_{1}}e_{i_{2}}\cdots e_{i_{\ell}}\in IdC_{n}(\beta)$$\ldots$$j$$i$$x_{i}$$e_{j}$$h_{i}(x)=1+xe_{i}$ ,and$A_{i}(x)= \prod_{a=n-1}^{i}h_{a}(x)$,$i=1,$$\ldots,$

Lemma Al One has(1) (Addition formula)$h_{i}(x)h_{i}(y)=h_{i}(x\oplus y)$,where we set $(x\oplus y)$ $:=x+y+\beta xy$ ;(2) (Yang-Baxter relation)$h_{i}(x)h_{i+1}(x\oplus y)h_{i}(y)=h_{i+1}(y)h_{i}(x\oplus y)h_{i+1}(x)$Corollary Al.

$n-1$ .

41
(1)
(2)

$[h_{i+1}(x)h_{i}(x), h_{i+1}(y)h_{i}(y)]=0$.

, $n-1$ .
$i=1,2,$The second equality follows from th first one by induction using the Addition formula, whereas the fist equality follows directly froln the Yang-Baxter relation.
Definition A2 (Grothendieck expression)$[A_{i}(x), A_{i}(y)]=0,$$\ldots$$J$$6_{n}(x_{1}, \ldots, x_{n-1})$ $:=A_{1}(x_{1})A_{2}(x_{2})\cdots A_{n-1}(x_{n-1})$

Theorem A ([11]).

The following identity$\mathfrak{G}_{n}(x_{1}, \ldots, x_{n-1})=\sum_{w\in S_{n}}\mathfrak{G}_{w}^{(\beta)}(X_{n-1})e_{w}$holds in the algebra.
Definition A3 We will call polynomial
mial corresponding to a permutation .
Corollary A2$IdC_{n}\otimes Z[x_{1}, \ldots, x_{n-1}]$$\otimes_{w}^{(\beta)}(X_{n-1})$

as the -Grothendieck polyno$\beta$$w$(1) If $\beta=-1$ , the polynomials $6_{w}^{(-1)}(X_{n-1})$ coincide with the Grothendieck polynomials introduced by Lascoux and M.-P. Schutzenberger [25].
(2) The -Grothendieck polynomial
is divisible by(3) For any integer $k\in[1, n-1]$ the polynomial
is apolynomial in the variables and with non-negative integer coefficients.
$x_{1}^{w(1)-1}$$\emptyset_{w}^{(\beta)}(X_{n-1})$$\beta$$\otimes_{w}^{(\beta-1)}(x_{k}=q, x_{a}=1, \forall a\neq k)$$\beta$$q$

Pmof (Sketch)

It is enough to show that the specialized Grothendieck expression
can be written in the algebra $IdC_{n}(\beta-1)\otimes Z[q, \beta]$ asa linear combination of elementswith coefficients which are polynomials inthe variables and with non-negative coefficients. Observe that one can rewrite therelation $e_{k}^{2}=(\beta-1)e_{k}$ in the following form $e_{k}(e_{k}+1)=\beta e_{k}$ . Now, all possible
negative contributions to the expression $6_{n}(x_{k}=q, x_{a}=1, \forall a\neq k)$ can appear only$:=(1+qe_{k})(1+e_{k})^{a}$ . But using the Addition formula one
from products of a form
can see that $(1+qe_{k})(1+e_{k})=1+(1+q\beta)e_{k}$ . It follows by induction on thatis a polynomial in the variables and with non-negative coefficients.
$\otimes_{n}(x_{k}=q, x_{a}=1, \forall a\neq k)$$\{e_{w}\}_{w\in S_{n}}$$\beta$$q$$c_{\sim a}(q)$

$a$

$q$

$c_{a}(q)$$\beta$$\blacksquare$

Definition A4
The double -Grothendieck expression$\beta$$\bullet$$6_{n}(X_{n}, Y_{n})$is defined as follows$6_{n}(X_{n}, Y_{n})=6_{n}(X_{n})6_{n}(-Y_{n})^{-1}\in IdC_{n}(\beta)\otimes Z[X_{n}, Y_{n}]$$\bullet$The double$\beta$-Grothendieck polynomials$\{\otimes_{w}(X_{n}, Y_{n})\}_{w\in S_{n}}$decomposition$\otimes_{n}(X_{n}, Y_{n})=\sum_{w\in S_{n}}\otimes_{w}(X_{n}, Y_{n})e_{w}$of the double -Grothendieck expression in the algebra$\beta$$IdC_{n}(\beta)$.

.

are $(1efi_{I1}ed$ from the42

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