# Noncommutative symmetric functions with matrix parameters (extended abstract)

Alain Lascoux, Jean-Christophe Novelli, and Jean-Yves Thibon

Institut Gaspard Monge, Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France

**Abstract.** We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki.

**Résumé.** Nous définissons de nouvelles familles de fonctions symétriques non-commutatives et de fonctions quasi-symétriques, dépendant de deux matrices de paramètres, et plus généralement, de paramètres associés à des chemins dans un arbre binaire. Pour des spécialisations appropriées, on retrouve les familles à deux vecteurs de Hivert-Lascoux-Thibon et les fonctions de Macdonald non-commutatives de Bergeron-Zabrocki.

**Keywords:** Noncommutative symmetric functions, Quasi-symmetric functions, Macdonald polynomials

## 1 Introduction

The theory of Hall-Littlewood, Jack, and Macdonald polynomials is one of the most interesting subjects in the modern theory of symmetric functions. It is well-known that combinatorial properties of symmetric functions can be explained by lifting them to larger algebras (the so-called combinatorial Hopf algebras), the simplest examples being **Sym** (Noncommutative symmetric functions [3]) and its dual **QSym** (Quasi-symmetric functions [5]).

There have been several attempts to lift Hall-Littlewood and Macdonald polynomials to **Sym** and **QSym** [1, 7, 8, 13, 14]. The analogues defined in [1] were similar to, though different from, those of [8]. These last ones admitted multiple parameters $q_i$ and $t_i$, which however could not be specialized to recover the version of [1].

The aim of this article is to show that many more parameters can be introduced in the definition of such bases. Actually, one can have a pair of $n \times n$ matrices ($Q_n$, $T_n$) for each degree $n$. The main properties established in [1] and [8] remain true in this general context, and one recovers the BZ and HLT polynomials for appropriate specializations of the matrices.

In the last section, another possibility involving quasideterminants is explored.
## 2 Notations

Our notations for noncommutative symmetric functions will be as in [3, 10]. The Hopf algebra of non-commutative symmetric functions is denoted by **Sym**, or by **Sym**(A) if we consider the realization in terms of an auxiliary alphabet. Bases of **Sym**<sub>n</sub> are labelled by compositions I of n. The noncommutative complete and elementary functions are denoted by S<sub>n</sub> and Λ<sub>n</sub>, and the notation S<sup>I</sup> means S<sub>i<sub>1</sub></sub> . . . S<sub>i<sub>r</sub></sub>. The ribbon basis is denoted by R<sub>I</sub>.

The notation $I \vdash n$ means that $I$ is a composition of $n$. The conjugate composition is denoted by $I^{\sim}$. The graded dual of $\mathbf{Sym}$ is $\mathbf{QSym}$ (quasi-symmetric functions). The dual basis of $(S^I)$ is $(M_I)$ (monomial), and that of $(R_I)$ is $(F_I)$. The descent set of $I = (i_1, \dots, i_r)$ is $\mathrm{Des}(I) = \{i_1, i_1 + i_2, \dots, i_r\}$.

Finally, there are two operations on compositions: if $I = (i_1, \dots, i_r)$ and $J = (j_1, \dots, j_s)$, the composition $I \cdot J$ is $(i_1, \dots, i_r, j_1, \dots, j_s)$ and $I \triangleright J$ is $(i_1, \dots, i_r + j_1, \dots, j_s)$.

## 3 Sym<sub>n</sub> as a Grassmann algebra

Since for $n > 0$, $\mathbf{Sym}_n$ has dimension $2^{n-1}$, it can be identified (as a vector space) with a Grassmann algebra on $n-1$ generators $\eta_1, \dots, \eta_{n-1}$ (that is, $\eta_i\eta_j = -\eta_j\eta_i$, so that in particular $\eta_i^2 = 0$). This identification is meaningful, for example, in the context of the representation theory of the 0-Hecke algebras $H_n(0)$ (see [2]).

If $I$ is a composition of $n$ with descent set $D = \{d_1, \dots, d_k\}$, we make the identification

$$R_I \longleftrightarrow \eta_D := \eta_{d_1} \eta_{d_2} \dots \eta_{d_k} \quad (1)$$

For example, $R_{213} \leftrightarrow \eta_2\eta_3$. We then have

$$S^I \longleftrightarrow (1 + \eta_{d_1})(1 + \eta_{d_2})\dots(1 + \eta_{d_k}) \quad (2)$$

and

$$\Lambda^I \longleftrightarrow \prod_{i=1}^{n-1} \theta_i, \quad (3)$$

where $\theta_i = \eta_i$ if $i \notin D$ and $\theta_i = 1 + \eta_i$ otherwise. Other bases have simple expression under this identification, e.g., $\Psi_n$, $\Phi_n$ and Hivert's Hall-Littlewood basis [7].

### 3.1 Structure on the Grassmann algebra

Let $*$ be the anti-involution given by $\eta_i^* = (-1)^i \eta_i$. The Grassmann integral of any function $f$ is defined by

$$\int d\eta f := f_{12\dots n-1}, \quad \text{where} \quad f = \sum_k \sum_{i_1<\dots<i_k} f_{i_1\dots i_k} \eta_{i_1} \dots \eta_{i_k}. \quad (4)$$

We define a bilinear form on $\mathbf{Sym}_n$ by

$$(f,g) = \int d\eta f^* g. \quad (5)$$

Then,

$$(R_I, R_J) = (-1)^{\ell(I)-1} \delta_{I, \bar{J}}, \quad (6)$$

so that this is (up to an unessential sign) the Bergeron-Zabrocki scalar product [1, Eq. (4)].
## 3.2 Factorized elements in the Grassman algebra

Now, for a sequence of parameters $Z = (z_1, \dots, z_{n-1})$, let

$$K_n(Z) = (1 + z_1 \eta_1)(1 + z_2 \eta_2) \dots (1 + z_{n-1} \eta_{n-1}). \qquad (7)$$

We then have

**Lemma 3.1**

$$(K_n(X), K_n(Y)) = \prod_{i=1}^{n-1} (y_i - x_i). \qquad (8)$$

## 3.3 Bases of Sym

We shall investigate bases of $Sym_n$ of the form

$$\tilde{H}_I = K_n(Z_I) = \sum_J \tilde{k}_{IJ} R_J, \qquad (9)$$

where $Z_I$ is a sequence of parameters depending on the composition $I$ of $n$.

The bases defined in [8] and [1] are of the previous form and for both of them, the determinant of the Kostka matrix $K = (\tilde{k}_{IJ})$ is a product of linear factors (as for ordinary Macdonald polynomials). This is explained by the fact that these matrices have the form

$$\begin{pmatrix} A & xA \\ B & yB \end{pmatrix} \qquad (10)$$

where $A$ and $B$ have a similar structure, and so on recursively. Indeed, for such matrices,

**Lemma 3.2** Let $A, B$ be two $m \times m$ matrices. Then,

$$\begin{vmatrix} A & xA \\ B & yB \end{vmatrix} = (y-x)^m \det A \cdot \det B. \qquad (11)$$

## 3.4 Duality

Similarly, the dual vector space $QSym_n = Sym_n^*$ can be identified with a Grassmann algebra on another set of generators $\xi_1, \dots, \xi_{n-1}$. Encoding the fundamental basis $F_I$ of Gessel [5] by

$$\xi_D := \xi_{d_1} \xi_{d_2} \dots \xi_{d_k}, \qquad (12)$$

the usual duality pairing such that the $F_I$ are dual to the $R_I$ is given in this setting by

$$\langle \xi_D, \eta_E \rangle = \delta_{DE}. \qquad (13)$$

Let

$$L_n(Z) = (z_1 - \xi_1) \dots (z_{n-1} - \xi_{n-1}). \qquad (14)$$

Then, as above, we have a factorization identity:

**Lemma 3.3**

$$\langle L_n(X), K_n(Y) \rangle = \prod_{i=1}^{n-1} (x_i - y_i). \qquad (15)$$
# 4 Bases associated with paths in a binary tree

Let $y = \{y_u\}$ be a family of indeterminates indexed by all boolean words of length $\le n-1$. For example, for $n=3$, we have the six parameters $y_0, y_1, y_{00}, y_{01}, y_{10}, y_{11}$.

We can encode a composition $I$ with descent set $D$ by the boolean word $u = (u_1, \dots, u_{n-1})$ such that $u_i = 1$ if $i \in D$ and $u_i = 0$ otherwise.

Let us denote by $u_{m...p}$ the sequence $u_m u_{m+1} \dots u_p$ and define

$$P_I := (1 + y_{u_1} \eta_1)(1 + y_{u_1...2} \eta_2) \dots (1 + y_{u_1...n-1} \eta_{n-1}) \quad (16)$$

or, equivalently,

$$P_I := K_n(Y_I) \quad \text{with} \quad Y_I = (y_{u_1}, y_{u_1...2}, \dots, y_u) := (y_k(I)). \quad (17)$$

Similarly, let

$$Q_I := (y_{w_1} - \xi_1)(y_{w_1...2} - \xi_2) \dots (y_{w_1...n-1} - \xi_{n-1}) =: L_n(Y^I), \quad (18)$$

where $w_{1...k} = u_1 \dots u_{k-1} \overline{u_k}$ where $\overline{u_k} = 1 - u_k$, so that

$$Y^I := (y_{w_1}, y_{w_1...2}, \dots, y_{w_1...n-1}) := (y^k(I)). \quad (19)$$

## 4.1 Kostka matrices

The Kostka matrix is defined as the transpose of the transition matrices from $P_I$ to $R_J$. This matrix is recursively of the form of Eq. (10). Thus, its determinant factors completely. For $n=4$, it is

$$(y_1 - y_0)^4 (y_{01} - y_{00})^2 (y_{11} - y_{10})^2 (y_{001} - y_{000}) (y_{011} - y_{010}) (y_{101} - y_{100}) (y_{111} - y_{110}). \quad (20)$$

**Proposition 4.1** The bases $(P_I)$ and $(Q_I)$ are adjoint to each other, up to normalization:

$$\langle Q_I, P_J \rangle = \langle L_n(Y^I), K_n(Y_J) \rangle = \prod_{k=1}^{n-1} (y^k(I) - y_k(J)), \quad (21)$$

which is indeed zero unless $I = J$.

From this, it is easy to derive a product formula for the basis $P_I$.

**Proposition 4.2** Let $I$ and $J$ be two compositions of respective sizes $n$ and $m$. The product $P_I P_J$ is a sum over an interval of the lattice of compositions

$$P_I P_J = \sum_{K \in [I \triangleright (m), I \cdot (1^m)]} c_{IJ}^K P_K \quad (22)$$

where

$$c_{IJ}^K = \frac{\langle L_{n+m}(Y^K), K_{n+m}(Y_I \cdot 1 \cdot Y_J) \rangle}{\langle Q_K, P_K \rangle}, \quad (23)$$

where $Y_I \cdot 1 \cdot Y_J$ stands for the sequence $(y_1(I), \dots, y_n(I), 1, y_1(J), \dots, y_m(J))$.
Next, if $J$ is a composition of the same integer $n$, form the monomial

$$ \tilde{\mathbf{k}}_{IJ}(Q, T) = \prod_{d \in \text{Des}(J)} z_d(I). \qquad (28) $$

For example, with $I = (4, 1, 2, 1)$ and $J = (2, 1, 1, 2, 2)$, we have $\text{Des}(J) = \{2, 3, 4, 6\}$ and $\tilde{\mathbf{k}}_{IJ} = q_{12}q_{13}t_{14}q_{34}$.

**Definition 5.1** Let $Q = (q_{ij})$ and $T = (t_{ij})$ $(i, j \ge 1)$ be two infinite matrices of commuting indeterminates. For a composition $I$ of $n$, the noncommutative $(Q, T)$-Macdonald polynomial $\tilde{\mathbf{H}}_I(A; Q, T)$ is

$$ \tilde{\mathbf{H}}_I(A; Q, T) = K_n(A; Z(I)) = \sum_{J \vdash n} \tilde{\mathbf{k}}_{IJ}(Q, T) R_J(A). \qquad (29) $$

Note that $\tilde{\mathbf{H}}_I$ depends only on the $q_{ij}$ and $t_{ij}$ with $i + j \le n$.

## 5.2 $(Q, T)$-Kostka matrices

The factorization property of the determinant of the $(Q, T)$-Kostka matrix, which is valid for the usual Macdonald polynomials as well as for the noncommutative analogues of [8] and [1] still holds since the $\tilde{\mathbf{H}}_I$ are specializations of the $P_I$.

**Theorem 5.2** Let $n$ be an integer. Then

$$ \det \mathcal{K}_n = \prod_{i+j \le n} (q_{ij} - t_{ij})^{e(i,j)}, \qquad (30) $$

where $e(i, j) = \binom{i+j-2}{i-1} 2^{n-i-j}$.

## 5.3 Specializations

For appropriate specializations, we recover (up to indexation) the Bergeron-Zabrocki polynomials $\tilde{\mathbf{H}}_I^{BZ}$ of [1] and the multiparameter Macdonald functions $\tilde{\mathbf{H}}_I^{HLT}$ of [8]:

**Proposition 5.3** Let $(q_i)$, $(t_i)$, $i \ge 1$ be two sequences of indeterminates. For a composition $I$ of $n$,

(i) Let $\nu$ be the anti-involution of $\mathbf{Sym}$ defined by $\nu(S_n) = S_n$. Under the specialization $q_{ij} = q_{i+j-1}$, $t_{ij} = t_{n+1-i-j}$, $\tilde{\mathbf{H}}_I(Q, T)$ becomes a multiparameter version of $\nu(\tilde{\mathbf{H}}_I^{BZ})$, to which it reduces under the further specialization $q_i = q^i$ and $t_i = t^i$.

(ii) Under the specialization $q_{ij} = q_j$, $t_{ij} = t_i$, $\tilde{\mathbf{H}}_I(Q, T)$ reduces to $\tilde{\mathbf{H}}_I^{HLT}$.

## 5.4 The quasi-symmetric side

Families of $(Q, T)$-quasi-symmetric functions can now be defined by duality by specialization of the $(Q_I)$ defined in the general case. The dual basis of $(\tilde{\mathbf{H}}_J)$ in $\mathbf{QSym}$ will be denoted by $(\tilde{\mathbf{G}}_I)$. We have

$$ \tilde{\mathbf{G}}_I(X; Q, T) = \sum_J \tilde{\mathbf{g}}_{IJ}(q, t) F_J(X) \qquad (31) $$

where the coefficients are given by the transposed inverse of the Kostka matrix: $(\tilde{\mathbf{g}}_{IJ}) = t(\tilde{\mathbf{k}}_{IJ})^{-1}$.

Let $Z'(I)(Q, T) = Z(I)(T, Q) = Z(\tilde{I}^*)(Q, T)$. Then, thanks to Proposition 4.3 and to the fact that changing the last bit of a binary word amounts to change a $q$ into a $t$, we have
**Proposition 5.4** The inverse of the (Q, T)-Kostka matrix is given by

$$ (K_n^{-1})_{IJ} = (-1)^{\ell(I)-1} \prod_{d \in \text{Des}(I)} z_d'(J) \prod_{p=1}^{n-1} \frac{1}{z_p(J) - z_p'(J)}. \qquad (32) $$

# 6 Multivariate BZ polynomials

In this section, we restrict our attention to the multiparameter version of the Bergeron-Zabrocki polynomials, obtained by setting $q_{ij} = q_{i+j-1}$ and $t_{ij} = t_{n+1-i-j}$ in degree $n$.

## 6.1 Multivariate BZ polynomials

As in the case of the two matrices of parameters, $Q$ and $T$, one can deduce the product in the $\tilde{H}$ basis by some sort of specialization of the general case. However, since $t_{ij}$ specializes to another $t$ where $n$ appears, one has to be a little more cautious to get the correct answer.

**Theorem 6.1** Let $I$ and $J$ be two compositions of respective sizes $p$ and $r$. Let us denote by $K = I \cdot \bar{J} \sim$ and $n = |K| = p + r$. Then

$$ \tilde{H}_I \tilde{H}_J = \frac{(-1)^{\ell(I) + |J|}}{\prod_{k \in \text{Des}(K)} (q_k - t_{n-k})} \sum_{K'} \prod_{k \in \text{Des}(K)} (-1)^{\ell(K)} (z_k(K') - z_k'(K')) \tilde{H}_{K'} \quad (33) $$

where the sum is computed as follows. Let $I'$ and $J'$ be the compositions such that $|I'| = |I|$ and either $K' = I' \cdot J'$, or $K' = I' \triangleright J'$. If $I'$ is not coarser than $I$ or if $J'$ is not finer than $J$, then $\tilde{H}(K')$ does have coefficient 0. Otherwise, $z_k(K') = q_k$ if $k$ is a descent of $K'$ and $t_{n-k}$ otherwise. Finally, $z_k'(K')$ does not depend on $K'$ and is $(Z(I), 1, Z(J))$.

## 6.2 The $\nabla$ operator

The $\nabla$ operator of [1] can be extended by

$$ \nabla \tilde{H}_I = \left( \prod_{d=1}^{n-1} z_d(I) \right) \tilde{H}_I. \qquad (34) $$

Then,

**Proposition 6.2** The action of $\nabla$ on the ribbon basis is given by

$$ \nabla R_I = (-1)^{|I|+\ell(I)} \prod_{d \in \text{Des}(I)} q_d \prod_{d \in \text{Des}(\bar{I})} t_d \sum_{J \ge \bar{I} \sim} \prod_{i \in \text{Des}(I) \cap \text{Des}(J)} (t_i + q_{n-i}) R_J. \quad (35) $$

Note also that if $I = (1^n)$, one has

$$ \nabla \Lambda_n = \sum_{J \vdash n} \prod_{j \in \text{Des}(J)} (q_j + t_{n-j}) R_J = \sum_{J \vdash n} \prod_{j \notin \text{Des}(J)} (q_j + t_{n-j} - 1) \Lambda^J. \quad (36) $$

As a positive sum of ribbons, this is the multigraded characteristic of a projective module of the 0-Hecke algebra. Its dimension is the number of packed words of length $n$ (called preference functions in
[1]). Let us recall that a packed word is a word $w$ over $\{1, 2, \dots\}$ so that if $i > 1$ appears in $w$, then $i - 1$ also appears in $w$. The set of all packed words of size $n$ is denoted by $PW_n$.

Then the multigraded dimension of the previous module is

$$W_n(\mathbf{q}, \mathbf{t}) = \langle \nabla \Lambda_n, F_1^n \rangle = \sum_{w \in PW_n} \phi(w) \qquad (37)$$

where the statistic $\phi(w)$ is obtained as follows.

Let $\sigma_w = \text{std}(\bar{w})$, where $\bar{w}$ denotes the mirror image of $w$. Then

$$\phi(w) = \prod_{i \in \text{Des}(\sigma_w^{-1})} x_i \qquad (38)$$

where $x_i = q_i$ if $w_i^\uparrow = w_{i+1}^\uparrow$ and $x_i = t_{n-i}$ otherwise, where $w^\uparrow$ is the nondecreasing reordering of $w$.

For example, with $w = 22135411$, $\sigma_w = 54368721$, $w^\uparrow = 11122345$, the recoils of $\sigma_w$ are 1, 2, 3, 4, 7, and $\phi(w) = q_1q_2t_5q_4t_1$.

**Theorem 6.3** Denote by $d_I$ the number of permutations $\sigma$ with descent composition $C(\sigma) = I$. Then, for any composition $I$ of $n$,

$$\nabla R_I = (-1)^{|I|+\ell(I)} \theta(\sigma) \sum_{w \in PW_n; \text{ev}(w) \le I} \frac{R_{C(\sigma_w^{-1})}}{d_{C(\sigma_w^{-1})}}, \qquad (39)$$

where $\sigma$ is any permutation such that $C(\sigma^{-1}) = \bar{I}^\sim$, and

$$\theta(\sigma) = \prod_{d \in \text{Des}(\bar{I}^\sim)} t_d. \qquad (40)$$

The behaviour or the multiparameter BZ polynomials with respect to the scalar product

$$[R_I, R_J] := (-1)^{|I|+\ell(I)} \delta_{I, \bar{J}^\sim} \qquad (41)$$

is the natural generalization of [1, Prop. 1.7]:

$$[\tilde{H}_I, \tilde{H}_J] = (-1)^{|I|+\ell(I)} \delta_{I, \bar{J}^\sim} \prod_{i=1}^{n-1} (q_i - t_{n-i}). \qquad (42)$$

# 7 Quasideterminantal bases

## 7.1 Quasideterminants of almost triangular matrices

Quasideterminants [4] are noncommutative analogs of the ratio of a determinant by one of its principal minors. Thus, the quasideterminants of a generic matrix are not polynomials, but complicated rational expressions living in the free field generated by the coefficients. However, for almost triangular matrices,
i.e., such that $a_{ij} = 0$ for $i > j + 1$, all quasideterminants are polynomials, with a simple explicit expression. We shall only need the formula (see [3], Prop.2.6):

$$ \begin{vmatrix} a_{11} & a_{12} & a_{13} & \cdots & \begin{bmatrix} a_{1n} \\ \vdots \\ a_{n1} \end{bmatrix} \\ -1 & a_{22} & a_{23} & \cdots & a_{2n} \\ 0 & -1 & a_{33} & \cdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & a_{n-1n} \\ 0 & \cdots & 0 & -1 & a_{nn} \end{vmatrix} = a_{1n} + \sum_{1 \le j_1 < \cdots < j_k < n} a_{1j_1} a_{j_1+1j_2} a_{j_2+1j_3} \cdots a_{j_k+1n}. \quad (43) $$

Recall that the quasideterminant $|A|_{pq}$ is invariant by scaling the rows of index different from $p$ and the columns of index different from $q$. It is homogeneous of degree 1 with respect to row $p$ and column $q$. Also, the quasideterminant is invariant under permutations of rows and columns.

The quasideterminant (43) coincides with the row-ordered expansion of an ordinary determinant

$$ \mathrm{rdet}(A) := \sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) a_{1\sigma(1)} a_{2\sigma(2)} \cdots a_{n\sigma(n)} \quad (44) $$

which will be denoted as an ordinary determinant in the sequel.

## 7.2 Quasideterminantal bases of Sym

Many interesting families of noncommutative symmetric functions can be expressed as quasi-determinants of the form

$$ H(W, G) = \begin{vmatrix} w_{11}G_1 & w_{12}G_2 & \cdots & w_{1n-1}G_{n-1} & \begin{bmatrix} w_{1n}G_n \\ \vdots \\ w_{2n}G_{n-1} \end{bmatrix} \\ w_{21} & w_{22}G_1 & \cdots & w_{2n-1}G_{n-2} & w_{2n}G_{n-1} \\ 0 & w_{32} & \cdots & w_{3n-3}G_{n-3} & w_{3n}G_{n-2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & w_{nn-1} & w_{nn}G_1 \end{vmatrix} \quad (45) $$

(or of the transposed form), where $G_k$ is some sequence of free generators of Sym, and $W$ an almost-triangular ($w_{ij} = 0$ for $i > j + 1$) scalar matrix. For example, $S_n$ over the $\Lambda^I$ and the $\Psi^I$ (see [3, (37)-(41)]), or over the $\Theta^I$, where $\Theta_n(q) = (1-q)^{-1}S_n((1-q)A)$ (see [10, Eq. (78)]). These examples illustrate relations between sequences of free generators. Quasi-determinantal expressions for some linear bases can be recast in this form as well. For example, the formula for ribbons [3, (50)] can be rewritten as follows. Let $U$ and $V$ be the $n \times n$ almost-triangular matrices

$$ U = \begin{bmatrix} 1 & 1 & \cdots & 1 & 1 \\ -1 & -1 & \cdots & -1 & -1 \\ 0 & -1 & \cdots & -1 & -1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -1 & -1 \end{bmatrix} \quad V = \begin{bmatrix} 1 & 1 & \cdots & 1 & 1 \\ -1 & 0 & \cdots & 0 & 0 \\ 0 & -1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -1 & 0 \end{bmatrix} \quad (46) $$

Given the pair $(U, V)$, define, for each composition $I$ of $n$, a matrix $W(I)$ by

$$ w_{ij}(I) = \begin{cases} u_{ij} & \text{if } i-1 \in \mathrm{Des}(I), \\ v_{ij} & \text{otherwise,} \end{cases} \quad (47) $$
and set

$$H_I(U, V; A) := H(W(I), S(A)). \qquad (48)$$

Then,

$$(-1)^{\ell(I)-1} R_I = H_I(U, V). \qquad (49)$$

Indeed, $H_I(U, V; A)$ is obtained by substituting in (43)

$$a_{j_p+1, j_{p+1}} = \begin{cases} -S_{j_{p+1}-j_p} & \text{if } j_p \in \text{Des}(I), \\ 0 & \text{otherwise.} \end{cases} \qquad (50)$$

This yields

$$S_n + \sum_k \sum_{\{j_1 < \dots < j_k\} \subseteq \text{Des}(I)} S_{j_1} (-S_{j_2-j_1}) \dots (-S_{n-j_k}) \\ = \sum_{\text{Des}(K) \subseteq \text{Des}(I)} (-1)^{\ell(K)-1} S^K = (-1)^{\ell(I)-1} R_I. \qquad (51)$$

For a generic pair of almost-triangular matrices $(U, V)$, the $H_I$ form a basis of $\mathbf{Sym}_n$. Without loss of generality, we may assume that $u_{1j} = v_{1j} = 1$ for all $j$. Then, the transition matrix $M$ expressing the $H_I$ on the $S^J$ where $J = (j_1, \dots, j_p)$ satisfies:

$$M_{J,I} := x_{1j_1-1}x_{1j_2-1}\dots x_{1j_p-1}. \qquad (52)$$

where $x_{ij} = u_{ij}$ if $i-1$ is not a descent of $I$ and $v_{ij}$ otherwise.

As we shall sometimes need different normalizations, we aslo define for arbitrary almost triangular matrices $U, V$

$$H'(W, G) = \text{rdet} \begin{bmatrix} w_{11}G_1 & w_{12}G_2 & \dots & w_{1n-1}G_{n-1} & w_{1n}G_n \\ w_{21} & w_{22}G_1 & \dots & w_{2n-1}G_{n-2} & w_{2n}G_{n-1} \\ 0 & w_{32} & \dots & w_{3n-3}G_{n-3} & w_{3n}G_{n-2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & w_{n,n-1} & w_{nn}G_1 \end{bmatrix} \qquad (53)$$

and

$$H'_I(U, V) = H'(W(I), S(A)). \qquad (54)$$

## 7.3 Expansion on the basis ($S^I$)

For a composition $I = (i_1, \dots, i_r)$ of $n$, let $I^\sharp$ be the integer vector of length $n$ obtained by replacing each entry $k$ of $I$ by the sequence $(k, 0, \dots, 0)$ $(k-1)$ zeros:

$$I^{\sharp} = (i_1 0^{i_1-1} i_2 0^{i_2-1} \dots i_r 0^{i_r-1}). \qquad (55)$$

**Proposition 7.1** *The expansion of $H'(W, S)$ on the $S$-basis is given by*

$$H'(W, S) = \sum_{I \vdash n} \varepsilon(\sigma_I) w_{1\sigma_I(1)} \cdots w_{n\sigma_I(n)} S^I. \qquad (56)$$
## 7.4 Expansion of the basis ($R_I$)

**Proposition 7.2** For $I = (i_1, \dots, i_r)$ be a composition of $n$, denote by $W_I$ the product of diagonal minors of the matrix $W$ taken over the first $i_1$ rows and columns, then the next $i_2$ ones and so on. Then,

$$H'(W, S) = \sum_{I \vdash n} W_I R_I. \qquad (57)$$

## 7.5 Examples

### 7.5.1 A family with factoring coefficients

**Theorem 7.3** Let $U$ and $V$ be defined by

$$u_{ij} = \begin{cases} x^j - y^j & \text{if } i=1, \\ aq_{i-1}x^{j-i+1} - y^{j-i+1} & \text{if } 1 < i < j + 2, \\ 0 & \text{otherwise,} \end{cases} \qquad (58)$$

$$v_{ij} = \begin{cases} x^j - y^j & \text{if } i=1, \\ x^{j-i+1} - bu_{n+1-i}y^{j-i+1} & \text{if } 1 < i < j + 2, \\ 0 & \text{otherwise.} \end{cases} \qquad (59)$$

Then the coefficients $W_J$ of the expansion of $H'_I(U, V)$ on the ribbon basis all factor as products of binomials.

The formula for the coefficient of $R_n$ is simple enough: if one orders the factors of $\det(U)$ and $\det(V)$ as

$$Z_n = (x - aq_1 y, x - aq_2 y, \dots, x - aq_{n-1} y) \qquad (60)$$

and

$$Z'_n = (y - bu_{n-1}x, y - bu_{n-2}x, \dots, y - bu_1x), \qquad (61)$$

then, the coefficient of $R_n$ in $H'_I(U, V)$ is

$$(x - y) \prod_{d \in \text{Des}(I)} z'_d \prod_{e \notin \text{Des}(I)} z_e. \qquad (62)$$

A more careful analysis allows one to compute directly the coefficient of $R_J$ in $H'_I$. For example,

$$\frac{H'_3(U, V)}{(x - y)} = (x - aq_1 y)(x - aq_2 y)R_3 + (x - aq_1 y)(aq_2 x - y)R_{21} \\ + a(x - y)(q_1 x - q_2 y)R_{12} + (aq_1 x - y)(aq_2 x - y)R_{111}. \qquad (63)$$

$$\frac{H'_{21}(U, V)}{(x - y)} = (x - aq_1 y)(bu_1 x - y)R_3 + (x - aq_1 y)(x - bu_1 y)R_{21} \\ + (abq_1 u_1 x - y)(x - y)R_{12} + (aq_1 x - y)(x - bu_1 y)R_{111}. \qquad (64)$$
### 7.5.2 An analogue of the $(1-t)/(1-q)$ transform

Recall that for commutative symmetric functions, the $(1-t)/(1-q)$ transform is defined in terms of the power-sums by

$$p_n\left(\frac{1-t}{1-q}X\right) = \frac{1-t^n}{1-q^n}p_n(X). \qquad (65)$$

With the specialization $x = 1$, $y = t$, $q_i = q^i$, $u_i = 1$, $a = b = 1$, one obtains a basis such that for a hook composition $I = (n-k, 1^k)$, the commutative image of $H'_I(U, V)$ becomes the $(1-t)/(1-q)$ transform of the Schur function $s_{n-k, 1^k}$.

### 7.5.3 An analogue of the Macdonald P-basis

With the specialization $x = 1$, $y = t$, $q_i = q^i$, $u_i = t^i$, $a = b = 1$, one obtains an analogue of the Macdonald P-basis, in the sense that for hook compositions $I = (n-k, 1^k)$, the commutative image of $H'_I$ is proportional to the Macdonald polynomial $P_{n-k, 1^k}(q, t; X)$.

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