数理解析研究所講究録 1429 巻 2005 年 25-42

25

Notes on the $X$ $=M$ $=K$ conjectureMark Shimozono
July 27, 2004
Abstract
This is an expanded version of a talk presented at the 2004 workshop on Combinatorial Aspects of Integrable Systems at the ResearchInstitute of Mathematical Sciences, Kyoto, Japan.
We show that in the large rank limit, for any nonexceptional affine
family of root systems, the one-dimensional sums associated to tensor
modules, have a very simple relationproducts ofship to those of type .
$\mathrm{K}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{v}rightarrow \mathrm{R}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{k}\mathrm{h}\mathrm{i}\mathrm{n}$$A$1

X $=M$ conjecture

1,1

Notationiswhere is a Lie algebra of nonexceptional affine type,
Let
I and
its derived subalgebra and 9 its canonical simple Lie subalgebra. Let
be the vertex sets of the Dynkin diagrams of and 9 respectively,
where $0\in I$ is the distinguished node [9]. Let
be the quantized universal enveloping algebras associated tobe the fundamental weights ofrespectively [10], Let
denotethe dominant weights. For
and-module of highest weight . For
the irreducibleby-module known as thebe the finite-dimensionalletand
is irreducible
Kirillov-Reshetikhin (KR) module [8] [7], Conjecturally.
and has an affine crystal baseof
We warn the reader that unless otherwise stated, we use the opposite
Kashiw ara’s tensor product convention for crystal graphs.
by
We shall use an encoding of dominant weights
assuming
partitions. Using the Dynkin diagram labeling given in (1.1), and $\mathrm{i}=n-1$
and, ,that $a_{i}=0$ for a spin node (that is, $i=n$ for types$\mathfrak{g}’$$\supset \mathfrak{g}’\supset\overline{\mathfrak{g}}$$\mathfrak{g}$$\mathfrak{g}$$\mathfrak{g}$$I\backslash \{0\}$$U_{q}(\mathfrak{g})\supset U_{q}’(\mathfrak{g})\supset U_{q}(\overline{\mathfrak{g}})$$\supset \mathrm{g}’\supset\overline{\mathfrak{g}}$$\mathfrak{g}$$\overline{\mathfrak{g}}$$\{\omega_{i}|\mathrm{i}\in I\backslash \{0\}\}$$\lambda\in P^{+}(\overline{\mathfrak{g}})$$P^{+}(\overline{\mathfrak{g}})=\oplus_{i\in I\backslash \{0\}}\mathbb{Z}\geq 0\omega_{i}$$r\in I\backslash \{0\}$$\lambda$$V^{\lambda}$$U_{q}(\overline{\mathfrak{g}})$$W_{s}^{(r)}$$U_{q}’(\mathfrak{g})$$s\in \mathbb{Z}_{>0}$$W_{s}^{(r)}$$B^{r,s}$$\lambda=\sum_{i\in I\backslash \{0\}}a_{i}\omega_{i}$$B_{n}$$\mathrm{i}$$C_{n}$$D_{n}$

26) we identify the weight A with the partition that has $ai$ columns offorheight for all .
$D_{n}$$i$$\mathrm{i}$Dynkin diagram$\overline{9}n$

m——m
$n-1$
1 2
$n$

$A_{n}$$\mathrm{m}------\ovalbox{\tt\small REJECT}$2$1$$B_{n}$

$n-1$

$n$

$\mathrm{m}------\ovalbox{\tt\small REJECT}$2$1$$C_{n}$

$n-1$

$n$

$\mathrm{m}------ \mathrm{L}^{n}$1$D_{n}$

2$n-1$

Classical decomposition of KR modules

1.2

is generally reducible as a
scribed by [8] [7], has the form
$W_{s}^{(r)}$$U_{q}(\overline{\mathfrak{g}})$-module. This decomposition, pre-$W_{s}^{(r)}\cong V^{s\omega_{\Gamma}}\oplus" \mathrm{c}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{r}\mathrm{e}\mathrm{n}$”We give examples of this decomposition below, using the encoding of ais the rectangle with rows and columns.
weight as a partition. Thus
According to [8] [7] the “children” are obtained by removing certain kindsof “tiles” from this rectangle.
Example 1. Let $r=2$ , $s=3$ , and the rank of 9 large. We use the labelingof affine Dynkin diagrams given in [8] [7]. The classical decompositions ofaarree given as follows.
$r$$s\omega_{r}$$s$$W_{3}^{(2)}$1.

$=A_{n}^{(1)}$$\mathrm{B}$: No children; Remove the empty tile$\emptyset$$W_{3}^{(2)}=\mathfrak{B}$2.

$=D_{n+1}^{(2)}$$\mathfrak{g}$,$A_{2n}^{(2)\dagger}$: Remove$\square$$W_{3}^{(2)}=\mathrm{f}\mathrm{f}\mathrm{l}\oplus \mathrm{f}\mathrm{f}\oplus\infty\oplus(\mathrm{m}\oplus \mathrm{f}\mathrm{f}\mathrm{l}\oplus \mathrm{F}\oplus \mathrm{m}\oplus\Xi\oplus\square \oplus\cdot$

3.

$=C_{n}^{(1)}$$\mathfrak{g}$,$A_{2n}^{(2)}$: Remove$\mathrm{m}$$W_{3}^{(2)}=\mathrm{f}\mathrm{f}\mathrm{l}$$4$.

$=B_{n}^{(1)}$$\mathrm{g}$,$D_{n}^{(1)}$,$A_{2n-1}^{(2)}$: Remove$\oplus \mathrm{F}^{\supset}\oplus\Xi$$\mathrm{B}$$W_{3}^{(2)}=\mathrm{f}\mathrm{f}\mathrm{l}$$\oplus \mathrm{f}\mathrm{f}\mathrm{l}$$\oplus \mathrm{H}$$\oplus\cdot$

27
1.3

The X formula

Consider any finite tensor product of KR modules$W^{L}:=\otimes(W_{s}^{(r)})^{\otimes L_{s}^{(r)}}r,s$-equivariant grading by the energy function $D[12][8][7]$ .
has ais by definition the graded, the one-dimensional sum
For A. It can be describedtoin the restriction ofmultiplicity of. Thisentirely in terms of the combinatorics of the crystal graph ofdefinition makes sense in the cases where the KR modules that occur inand their crystal bases have been constructed.
$W^{L}$$U_{q}(\overline{\mathfrak{g}})$$X_{L,\lambda}(t)$$\in P^{+}(\overline{\mathfrak{g}})$$W^{L}$$V^{\lambda}$$U_{q}(\overline{\mathfrak{g}})$$W^{L}$$W^{L}$

1.4$\mathrm{X}$$=\mathrm{M}$repbe the fermionic fo rmula [8] [7]. It is defined for allLet. It isresenting finite tensor products of KR modules and all A 6$=ML,\lambda(t)$ .
conjectured there that$L$

$M_{L,\lambda}(t)$$\mathrm{r}$$P(\overline{\mathfrak{g}})^{+}$$X_{L,\lambda}(t)$2

The K formula

We now consider the one-dimensional sum for nonexceptional affine alge.
bras in the special case that the rank is large with respect to the data
We have observed that the one-dimensional sums are well-behaved and haveconjectured that they have a simple formula in terms of the one-dimensionalsums of type . This conjectural formula is called the $K$ formula.
$\mathrm{s}$$(L, \lambda)$

$A$

2.1

The large rank limit

has rank . To
be nonexceptional family of affine algebras whereLet
obtain the Dynkin diagram of a nonexceptional affine algebra one attaches
the 0 node somewhere on the left end of one of the classical Dynkin diagrams
followingin (1.1). By examining the fermionic formulas one may prove theresult.
$\overline{\mathfrak{g}}_{n}$$n$$\{\mathfrak{g}_{n}\}$

Proposition 2. [20] For each nonexceptional family
algebras, there is a well-defined large rank limit$M_{L,\lambda}^{F}(t)$$=$$\lim M_{L,\lambda}^{\mathcal{B}n}(t)$$narrow\infty$$\mathcal{F}=\{\mathfrak{g}_{n}\}$of affine28,
called the stable fermionic formula. There are 4 distinct families of. This grouping of affine diagrams depends onlabeled by
the way in which the zero node is attached.
$M^{F}$$\theta$$\in\{\emptyset, 0, \mathrm{m}, \mathrm{B}\}$The grouping is the same as that for the classical decomposition of KR; see section 1.2.
modules$W_{s}^{(r)}$Ubiquity of type

2.2

$A$

for
of the KR module
Kleber [10] observed that the character. The
a nonspin node, should behave like such a character of type-system [8] [7] is a relation among the KR characters, giving an expression
forin terms of , for adjacent to in the Dynkin
diagram of . For a fixed and in large rank, near the Dynkin diagram
always looks locally like that of type An. Therefore such KR characters havethe same relations as those in type A.
$W_{s}^{(r)}$$Q_{s}^{(r)}$$A_{n}^{(1)}$

$r$

$Q$

$(Q_{s}^{(r)})^{2}-Q_{s-1}^{(r)}Q_{s+1}^{(r)}$

$i$

$r$

$r$

$r$

$\overline{\mathfrak{g}}$2.3$Q_{s}^{(i)}$Minimal affinizations

For(containing no spin weight) let
be the associated minimal
affinization [4]. It is an irreducible finite-dimensional
module with-decomposition of the form$\mathcal{T}$$W^{\tau}$$\in P^{+}(\overline{\mathfrak{g}})$$U_{q}’(\mathfrak{g})$$U_{q}(\overline{\mathfrak{g}})$$W^{\tau}\cong V^{\tau}\oplus$

Conjecture 3. For
modules$L$

children.

containing no spin weights, up to filtration, as$W^{L}\cong\oplus_{\tau}X_{L,\tau}^{A}(1)W^{\tau}$$U_{q}’(\mathfrak{g})-$.

In type it is known that
. This agrees with the definition of
as a graded tensor product branching multiplicity$A$$X_{L,\lambda}$$W^{\tau}\cong V^{\tau}$

28
2.4

Decomposition of minimal affinizations

Define the branching coefficients$b_{\tau\lambda}\in \mathbb{Z}\geq 0$by the$W^{\tau}\cong\oplus_{\lambda}b_{\tau\lambda}V^{\lambda}$$U_{q}(\overline{\mathfrak{g}})$decomposition.

is thebe the set of partitions that are tiled by . Explicitly,Let
is the set of all partisingleton set containing just the empty partition,is the set ofis the set of partitions with even row lengths, andtions,
partitions with even column lengths.
$P_{\emptyset}$$\theta$$P\psi$$P_{\square }$$\not\in$$P_{\mathrm{m}}$Conjecture 4. [then$\mathit{3}J$If 9 is in the family$\theta$andcontains no spin weights$\tau$$b_{\tau\lambda}= \sum_{\mu\in P_{\theta}}c_{\lambda\mu}^{\tau}$is the Littlewood-Richardson coefficient or typewheremultiplicity defined by$A_{n}$$c_{\lambda\mu}^{\tau}$tensor product$V^{\lambda}\otimes V^{\mu}\cong\oplus_{\tau}c_{\lambda\mu}^{\tau}V^{\tau}$This conjecture has been proved by Chari for many cases of.
form$\tau$of the$s\omega_{r}[2]$2.5

The K-formula

Comparing the decomposition of$W^{L}$

directly into$V^{\lambda}$$X_{L,\lambda}^{\phi}(1)$$= \sum_{\tau}X_{L,\tau}^{A}(1)\sum_{\mu\in P_{\langle\rangle}}c_{\lambda\mu}^{\tau}$

Inserting a$t$or via$W^{\tau}$, we have.

strategically, we define the $K$ formula$K_{L,\lambda}^{\phi}(t)=t^{|L|-|\lambda|} \sum_{\tau}X_{L,\tau}^{A}(t^{2})\sum_{\mu\in P_{\theta}}c_{\lambda\mu}^{\tau}$.

Originally the $K$ formula was discovered through analogues of creation operators for the symmetric functions given by the large rank limits of classicalcharacters [20].
$t$

2.6

X

$=\mathrm{M}=\mathrm{K}$Conjecture 5. [20] For large rank, X $=M=K$ .

30) and general [13]. A(typeThis was previously known for
special case has independently been conjectured by Lecouvey [14].
Our main result is:$\langle)$$A_{n}^{(1)}$$=\emptyset$$L$

Theorem 6.
$X^{\phi}=K^{\phi}$for$\theta$$\in\{\square ,\subset 0\}$

and

$L$

consisting of tensor factors of the form

The proof is to show bijectively that

$X^{\theta}$$W_{s}^{(1)}$.

satisfies the definition of3

Recording tableaux and the X formula3.1

Crystals$K^{\psi}$.

$B^{1,1}$

is a tensor power of the KR
For simplicity we consider the case th at. In this context we will just regard as the single integer
crystal. We also assume that the rank of the classicalpreviously denoted, we
subalgebra 9 is large, say, $n>L$ . For each partition$:=B^{1,1}$
is drawn in. Its crystal
fix a representative affine family
Figure 1.
$W^{L}$$B^{1,1}$$L$$L_{1}^{(1)}$$n$$\theta$$X_{N}^{(r)}$3.2$\in\{\emptyset,\mathrm{o}_{7}\mathrm{m},\mathrm{B}\}$$B\theta$

Walks in Young’s latticehighestLetbe a classical highest weight vector (that is, aweight vector). Let
be the weight of the first steps in .
Identifying dominant weights with partitions, the path can be encoded,, ,. These sequences should beby the sequence of partitions
regarded as recording tableaux in the language of the Robinson-Schensted
correspondence. The shape of such a tableau is by definition the ending
partition.
We give examples below. Every unbarred (resp. barred) step (resp.
adds (resp. removes) a cell to (resp. from) the r-th row in the partition
diagram. A step leaves the partition unchanged.
$b\in B_{\phi}^{\otimes L}$$U_{q}(\overline{\mathfrak{g}})$$\lambda^{(i)}\in P(\overline{\mathfrak{g}})^{+}$$b$$\mathrm{i}$$b$$\lambda^{(0)}$$\lambda^{(L)}$$\lambda^{(1)}$$\ldots$$r$$\overline{r})$$\emptyset$

1. A standard tableau of shape (4, 1, 1) and type$\square$$\mathrm{m}$$\mathrm{H}^{\supset}$$\mathrm{H}^{\mathrm{I}}$$\emptyset$-path:$\xi^{\square }$112131$\ovalbox{\tt\small REJECT}$

(1)

31

Figure 1: Representative affine families
Usually a standard tableau is written as follow , where the entry isin the r-th row if and only if the i-th element in the path has value .
$\mathrm{i}$$\mathrm{s}$

$r$

(2)

$P$

2. An oscillating tableau of shape $(1, 1)$ and type co (or H) path:
1

1

3

1

2

3. A Motzkin tableau of shape (2) and type$\square$11

3.3$\Pi\supset$El

21$\square$$\overline{3}$

path:

co

$\mathrm{F}$$\mathrm{H}$(3)$\mathrm{B}$$\ovalbox{\tt\small REJECT}$$\mathrm{H}$$\mathrm{F}$$\mathrm{m}$$\square$$\emptyset$

1$\overline{2}$Energy function on paths

is given in [7]; seealso [16]. Under the current assumptions the energy function can be computed as follows. We now reverse the order of the paths (the opposite ofIn general the way to compute the energy function

$D$

32
Kashiwara’s convention for crystal graphs). The subscripts label the gaps
between steps in the path. The subscripts at positions that contribute to
the energy function are underlined.
1. Type : Sum the positions of descents (gaps between steps where thestep on the left is greater than the one on the right).
$\emptyset$$c=1_{1}3_{\underline{2}}1_{3}2_{\underline{4}}1_{5}1$(4)$D_{\emptyset}(c)=2+4=6$

2. Type$\mathrm{m}$: same, with $1<2<3<\cdots<\overline{3}<\overline{2}<\overline{1}$$b=\overline{3}_{\underline{1}}3_{2}\overline{1}_{\underline{3}}2_{\underline{4}}1_{5}1$(5)$D_{\mathrm{m}}(b)=1+3+4=8$

3. Type : same, except a descent of the form$\mathrm{H}$$\overline{1}>1$

is counted twice.

4. Type :$\square$count double(b) Pairs
andcount once for(c) If the rightmost letter is , there is a descent to its right.
(a) Adjacent pairs $x>y$ and$x\emptyset$$\emptyset\emptyset$$x\neq\emptyset$$\emptyset x$$\emptyset$$b=\overline{2}_{1}1_{\underline{2}}\emptyset_{\underline{3}}\overline{1}_{4}2_{5}1_{6}1_{7}===$$D_{\mathrm{I}\supset}(b)=1$

3.4
For

.2+2 $\cdot 1+3$ . 1+4 2+5 $\cdot 2=25$ .
$\cdot$X formula$\phi$

H}$\in\{\emptyset, \square , \coprod\supset,$$X_{L,\lambda}^{\phi}(t)= \sum_{b\in P_{\theta}(L,\lambda)}t^{D_{\phi}(b)}$

where
A.

$P\theta(L, \lambda)$

is the set of classical highest weight vectors in$B_{+}^{L}$of weight

33
3.5

The required bijection

We shall only state it for thethat$\phi$$=\mathrm{m}$case. To prove $X=K$ we must show$\sum$(6)$c_{\lambda\mu}^{\tau}$$X_{L,\lambda}^{\mathrm{m}}(t^{2})=t^{L-|\lambda|} \sum_{\tau}X_{L,\tau}^{\emptyset}(t^{2})$$\mu\in \mathcal{P}\mathrm{m}$$|\mu|=L-|\lambda|$

Therefore we require a bijection$\{\begin{array}{l}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\tau\end{array}\}$(7)$\mathrm{x}LR(\tau;\lambda, \mu)$$\{\begin{array}{l}\mathrm{O}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\lambda\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}L\end{array}\}arrow\mu\in \mathrm{f}\mathrm{f}\mathrm{i}|\tau|=L\cup$$b$$(c, Z)$$\mapsto$. We use theof cardinality
where is an element of a set
is equal to the number of factorizations .
follow ing realization [15]:of any fixed semistandard tableau $P$ of shape , into sem istandard tableauxtheand of shapes A and respectively. The factorization is taken inplactic monoid; see [6].
must
In summary, given the oscillating tableau as input (see (3)), we(1))
construct a standard tableau , either in the form of a tyPe path (see$P$
mustor in the more traditional form, denoted here as (see (2)). Then wewith and tableaux of shapes
also prescribe a factorization of into
A and respectively.
Moreover, the bijection must be grade-preserving:$LR(\tau;\mathrm{A}, \mu)$$Z$$c_{\lambda\mu}^{\tau}$

$T$

$S$

$c_{\lambda\mu}^{\tau}$$\tau$

$T$

$S$

$\mu$

$b$

$\emptyset$

$c$

$P$

$T$

$T\cdot S$

$S$

$\mu$

(8)

2 $D_{\mathrm{m}}(b)=L-|\lambda|+2D_{\emptyset}(c)$by (5) it
Example 7. Let be as in (3); it has shape $\lambda=(1,1)$ , $L=6$ , andcorresponding path
has $D(b)=8$ . We do not say how to find it yet, but the$8=6-2+2\cdot 6$ .
is given in (1); it has $D(c)=6$ by (4). Then (8) becomes 2$b$$\cdot$$c$

4

The VXR map

bijection (7). It
We describe a way to compute the element in the desired
computation ofuses the virtual crystal (VX) construction of [16] and thethe combinatorial -matrix. Thus the name VXR
$c$

$R$

34
Dual crystal

4.1

has a vertex
be the dual crystal graph of $B[10]$ . By definition
Let
$b\in
B$
only
if $f_{i}(c)=b$
if andfor each. The arrows are reversed:
we have (omitting
for , $c\in B$ . For example, for $B_{A}=B^{1,1}$ of typezero arrows)$B^{\vee}$$B^{\vee}$$b^{\vee}$$f_{i}(b^{\vee})=c^{\vee}$$A_{5}^{(1)}$$b$$B_{A}$:$\underline{\Pi^{1}1}arrow\underline{\Pi 2}\mathrm{A}_{3arrow\overline{\cup 4}arrow\overline{\cup 5}arrow\overline{\cup 6}}\underline{\Pi}^{345}$$1^{\vee}arrow\underline{2^{\vee}\prod}12arrow\overline{3^{\vee}\cup}arrow\overline{4^{\vee}\cup}arrow\overline{5^{\vee}\cup}arrow\underline{\prod}3456^{\vee}$$B_{A}^{\vee}:$, one may compute its energy using the usual rule
For an element offor type paths, with the ordering $\ldots<3^{\vee}<2^{\vee}<1^{\vee}$ .
$B_{A}^{\vee\otimes L}$$\emptyset$

4.2

Virtual crystals

For certain aflBne root systems and KR crystals it was shown in [16] thatcrystals of nonsimply laced type could be realized using those of simply laced
type. This is called the virtual crystal construction. For example, the KR
crystal $B_{C}=B^{1,1}$ of type
, can be embedded into the tensor productwhere $B_{A}=B^{1,1}$ is the KR crystal of type
. Let us call thisthe virtual crystal (VX) embedding. Moreover the one-dimensional sum $X$
can be entirely expressed in terms of the crystal of simply-laced type.
For exam, define the embeddings I andby$C_{n}^{(1)}$$A_{2n-1}^{(1)}$$B_{A}\otimes B_{A}^{\vee}$$\Psi’$$\mathrm{p}\mathrm{l}\mathrm{e}$$B_{C}arrow\Psi$$\mathrm{i}$$\overline{\mathrm{i}}$$arrow(2n+1-\mathrm{i})^{\vee}\otimes \mathrm{i}$$\overline{\mathrm{i}}$(9)$arrow \mathrm{i}^{\vee}\otimes(2n+1 -\mathrm{i})$$B_{C}arrow\Psi’$$\mathrm{i}$$B_{A}^{\vee}\otimes B_{A}$$B_{A}\otimes B_{\check{A}}$$arrow \mathrm{i}\otimes(2n+1-\mathrm{i})^{\vee}$(10)$arrow(2n+1-\mathrm{i})\otimes \mathrm{i}^{\vee}$

More generally, there is an embedding$B_{C}^{\otimes L}arrow(B_{A}^{\vee}\otimes B_{A})^{\otimes L}\Psi$$b_{L}\otimes\cdots\otimes$$b_{1}\mapsto\Psi(b_{L})\otimes\cdots\otimes\Psi(b_{1})$defined by the -fold tensor product of the map (9). A similar construction
can be made for .
$L$$\Psi’$

35
The R-matrix

4.3-modules then there
If $B$ and $B’$ are the crystal bases of irreducible$B\otimes B’arrow B’\otimes B$
is a unique isomorphism of affiffiffine crystal graphs $R_{B,B’}$ :called the combinatorial $R$-matrix [12].
$R$
We need an explicit computation of the follow ing -matrix. Here.
has type$U_{q}^{t}(\mathfrak{g})$$B_{A}$$A_{2n-1}^{(1)}$$B_{A}\otimes B_{A}^{\vee}rightarrow B_{\check{A}}\otimes B_{A}R$

if$\mathrm{i}\otimes j^{\vee}rightarrow j^{\vee}\otimes \mathrm{i}$$\mathrm{i}\otimes i^{\vee}rightarrow$$(i\mathrm{f}1)^{\vee}\otimes \mathrm{i}$% 1if$\mathrm{i}\neq j$$\mathrm{i}<2n$$2n\otimes 2n^{\vee}rightarrow 1^{\vee}\otimes 1$

In particular the following diagram commutes:$1\downarrow B_{C}arrow VXB_{A}^{\vee}\otimes B_{A}\downarrow R$(11)$B_{C}arrow VX’B_{A}\otimes B_{A}^{\vee}$

4.4

Local energy function

Given and $B’$ as above, there is a map $H=H_{B,B’}$ :the local energy function. See [12] [7].
$H$ :We also have the value of the local energy function
given in this case by$B\otimes B’arrow \mathbb{Z}$$B$$B_{\check{A}}\otimes B_{A}arrow \mathbb{Z}$$H(x\otimes y)=\{\begin{array}{l}1\mathrm{i}\mathrm{f}x\otimes y=1^{\vee}\otimes 10\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$

Similarly the local energy function $H$ :$B_{A}\otimes B_{A}^{\vee}arrow \mathbb{Z}$

,

(12)

is given by$H(x\otimes y)=\{\begin{array}{l}\mathrm{l}\mathrm{i}\mathrm{f}x\otimes y=2n\otimes 2n^{\vee}0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{e}\end{array}$

4.5

called(13)

VXR map

given $b\in$
We show how to compute the part of the bijection (7) that,, and why the grade-preserving, determines an element
property (8) holds. At this time we do not compute the Littlewood-Richardson.
data$c\in P_{\emptyset}(L, \tau)$$P_{\mathrm{D}}(L, \lambda)$$T\cdot$$S$38
By (11) the follow ing diagram commutes.
—$B\downarrow\Psi C\otimes,L$$B_{C}^{\otimes L}\Psi\downarrow$,$B_{\check{A}}^{\otimes L}\otimes B_{A}^{\otimes L}\mapsto R+(B_{A}^{\vee}\otimes B_{A})^{\otimes L}\overline{R_{0}}(B_{A}\otimes B_{A}^{\vee})^{\otimes L}\overline{R_{-}}B_{A}^{\otimes L}\otimes B_{A}^{\vee\otimes L}$(14)
are compositions of $R$-matrices of the form given,, and
The maps.
. Say it maps
in section 4.3. Consider the composite map
is the one in the desired bijection.
This
is
In the following example $n=3$ . The computation of $R+$ on
shown below. Here we write instead of , etc.
$R_{+}$$R_{-}$$R_{0}$$b\mapsto d\otimes c$$R_{+}\circ\Psi$$c\in B_{A}^{\otimes L}$$\Psi(b)$$1^{\vee}$$\dot{1}$Prom the path one obtains the standard tableau $P$ ; see (2). The
element tells how to make the factorization $P\equiv T\cdot S$ . We prefer tocompute this another way later.
We now prove (8). The main ingredient is the following result.
$c$

$d$

Theorem 8. [16] The virtual crystal embedding$\Psi$

respects energy.

More precisely, in our situation this means that2

$D_{\mathrm{m}}(b)=D_{\emptyset}(d$

&
$($

$c)$

.

be the $R$-matrix and $H$ :be the local energy function [12]. Let
By the definition of $D[7][16]$ we have

Let

$R$

:

$B_{A}^{\vee\otimes L}\otimes B_{A}^{\otimes L}arrow B_{A}^{\otimes L}\otimes B_{A}^{\vee\otimes L}$$d’\otimes$$\mathbb{Z}$$B_{A}^{\vee\otimes L}\otimes B_{A}^{\otimes L}arrow$$c’=R(d\otimes c)\in B_{A}^{\otimes L}\otimes B_{\check{A}}^{\otimes L}$$D(d\otimes c)=D(c)$ $+D(c’)+H(d\otimes c)$.

To prove (8) it is enough to show that
$D(c^{t})=D(c)$$H(d\otimes c)=L-|\lambda|=2$. (the number of barred steps in ).
$b$

(15)
(11)

37
are
To prove (15) we note that the computations of d’ (& andentirely parallel. By the commutativity of the diagram (14) we see thatare just
and in. Since the pairs of factors inreversed, we have the following computation:$c’$$\Psi(b)$$d’\otimes c^{\mathit{1}}=R_{-}\mathrm{o}\Psi’(b)$$d\otimes c$$\Psi’(b)$So $c’=$ 646566. Compare this with $c=$ 131211, In general one can show
theythat and have descents in the same positions, which implies thathave the same energy, proving (15).
$H$ , isTo prove (16), one must use the fact that the desired value of
evaluated at adjacentobtained as the sum of local energy functions
$c$

$c’$

$H_{B_{A}^{\vee}\otimes B_{A}}$$\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}_{\mathrm{C}}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}\mathrm{t}\mathrm{h}\mathrm{e}R- \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{s}B^{\vee}\otimes B_{A}arrow B_{A}\otimes B_{A}^{\vee}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{h}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}R- \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}B_{A}^{\vee\otimes}P_{\otimes B_{A}^{\otimes L}arrow B_{A}^{\otimes L}\otimes}$By (12)matrix is given by
has anthe only contributions to the energy are given by places where one. By studying
application of the local matrix of the form
numberthe above computations we see that such exchanges happen an evenduringof times, and that they occur in symmetric pairs, with one occurrence. One also sees that the number of times thatand the other duringis the number of barred elements in .
such exchanges occur during
Prom these considerations (16) follows.
$B_{A}^{\vee\otimes L}[7][16]$. The latter$R_{-}\mathrm{o}R_{0}\mathrm{o}R\overline{+}^{1}$$R$$1^{\vee}\mathrm{X}1$$R$$\mapsto 2n\otimes 2n^{\vee}$$R_{-}$$R_{+}^{-1}$$b$$R_{+}^{-1}$5

DDF bijection

We now take a completely different approach to the desired

5.1

bijection (7).

DDF

[5]. Let $[L]$The following bijection is due to Delest, Dulucq, and Favreau
be the collection of subsets of $[L]$ of cardinality .
$\{1,2, \ldots, L\}$ and

$=$

$k$

$(_{k}^{[L]})$38
The DDF bijection is between the following sets.

$\ovalbox{\tt\small REJECT}_{\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\lambda}^{\mathrm{O}\mathrm{s}\mathrm{c}\mathrm{i}11\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}L\ovalbox{\tt\small REJECT} \mathrm{t}\mathrm{a}\mathrm{b}1\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}arrow A\in(_{|\lambda|}^{[L]})\cup\ovalbox{\tt\small REJECT}_{\mathrm{a}1\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{t}A}^{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\lambda\ovalbox{\tt\small REJECT} \mathrm{t}\mathrm{a}\mathrm{b}1\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}\mathrm{x}$$b$$\ovalbox{\tt\small REJECT}_{\mathrm{o}\mathrm{n}[L]-A}^{\mathrm{F}\mathrm{i}\mathrm{d}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}1\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\ovalbox{\tt\small REJECT} \mathrm{x}\mathrm{e}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$$(T, I)$$\mapsto$

Let $b=6162\cdots b_{L}$ . Start with$T$

and I both empty. For$\mathrm{i}$from 1 to$L$do:(D1) If $b_{i}=r$ then adjoin$\mathrm{i}$

to

$T$

at row .
$r$(D2) If
then reverse Schensted row insert on
to .
value , say. Add the pair$b_{i}=\overline{r}$Example 9. Let b

The output is$=$$T=\ovalbox{\tt\small REJECT}_{5}^{3}$at row , ejecting the
$r$

$I$

$(\mathrm{i}, a)$

$a$

$T$

112133 as before.

and$I=(24)(16)=(\begin{array}{llll}6 4 2 \mathrm{l}\mathrm{l} 2 4 6\end{array})$

The DDF bijection may be extended to a bijection$\ovalbox{\tt\small REJECT}_{\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\lambda}^{\mathrm{M}\mathrm{o}\mathrm{t}\mathrm{z}\mathrm{k}\mathrm{i}\mathrm{n}}1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}L\ovalbox{\tt\small REJECT} \mathrm{t}\mathrm{a}\mathrm{b}1\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}arrow A\in(_{|\lambda|}^{\{L\mathrm{J}})\cup\ovalbox{\tt\small REJECT}_{\mathrm{a}1\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{t}A}^{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\lambda\ovalbox{\tt\small REJECT} \mathrm{t}\mathrm{a}\mathrm{b}1\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}\mathrm{x}$$b$$\mapsto$$\ovalbox{\tt\small REJECT}_{\mathrm{o}\mathrm{n}:L]-A}^{\mathrm{I}\mathrm{n}\mathrm{v}\mathrm{o}1\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}\ovalbox{\tt\small REJECT}$$(T, I)$

In the extended DDF bijection there is an additional rule.
(D3) If$b_{i}=\emptyset$then add a fixed point ($i\grave{)}$

to .
$I$

.

39
The Burge correspondence

5.2

The Burge correspondence [1] is a bijection$\{\begin{array}{ll}\mathrm{F}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d} \mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e} \mathrm{I}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\end{array}\}arrow\{\begin{array}{l}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{s}\end{array}\}$$I$$\mapsto$$S$

It can be obtained as the restriction of the following bijection.

Involutions$arrow\{\begin{array}{l}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{u}\mathrm{x}\end{array}\}$To compute $I\mapsto S$ , write the involution I as a two-line permutation. Reverse the lower word and row insert it into the empty tableau, obtainings.

Example 10. Let $I=(24)(16)$ in cycle notation. Then in two-line notation
we have$I=(\begin{array}{llll}\mathrm{l} 2 4 66 4 2 1\end{array})$$S$$(\emptysetarrow 1246)=$

Here is another example with $I=(24)(5)(17)$ . We have
$I=(\begin{array}{lllll}\mathrm{l} 2 4 5 77 4 2 5 1\end{array})$$S$$(\emptysetarrow 15247)$ $=$Insert DDF and Burge data
. be the standard tableau obtained by the plactic product ofLet
and $S[6]$ . Then define to be the corresponding type path. This givesfor (7).
us the desired data and

5.3$P\equiv T$

$T$

$S$

$\emptyset$$c$$P\equiv T\cdot$ $S$$c$

Example 11.
$P$$T\cdot S=$

$c=$

Here

$c$

112131is written in the unreversed order

40
Finishing the proof

5.4

One may show that the VXR map and DDF bijection give the sam answer.
The DDF formulation is a composition of bijections and is therefore bijective.
The VXR map was shown to be grade-preserving. This completes the proof
of $X=K$ in the special case that was explained here.
$\mathrm{e}$6Closing remarks

6.1

Level-rank duality

The type[18] [19]$A$

one-dimensional sums satisfy a graded level-rank duality [17]$K_{L,\lambda}^{\emptyset}(t)=t^{||L||}K_{L^{t},\lambda^{t}}^{\emptyset}(t^{-1})$

is obtained
indicates the transpose or conjugate partition of ,where
where $r_{ij}(L)=$by transposing all rectangles in , and$\lambda$$\lambda^{t}$$L$$\sum_{r\geq i}\sum_{s\geq j}L_{s}^{(r)}$$L^{t}$$||L||= \sum_{i,j\geq 1}(_{2}^{r_{\mathrm{i}\dot{g}}(L)})$.

This implies the following identity, where $|L|= \sum_{r,s}rsL_{s}^{(r)}$ :$K_{L^{t},\lambda^{t}}^{\phi^{t}}(t)$$=t^{||L||+|L|-|\lambda|}K_{L,\lambda}^{\phi}(t^{-1})$

The $X=M=K$ conjecture implies that in large rank, types and have
one-dimensional sums which are given by polynomials whose terms occur inthe reverse order from each other. Just from the definitions it is entirely
andshould have any
unclear why the one-dimensional sums of types
relationship with each other.
$B$

$B$

$C$

$C$

The missing case

6.2

Our methods don’t seem to work at all for type H. Even though types coand involve exactly the same kinds of paths, the energy functions for these
two types behave rather differently.
$\Xi$

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42[14] C. Leeouvey, A duality between -multiplicities in tensor products and
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