Pfaffian Formulas and Schur Q-Function Identities

TL;DR

This paper develops Pfaffian analogues of classical matrix formulas and uses them to provide new, clearer proofs of identities related to Schur Q-functions, advancing algebraic combinatorics.

Contribution

It introduces Pfaffian versions of the Cauchy--Binet and minor-summation formulas, offering novel tools for studying Schur Q-functions.

Findings

Pfaffian analogues of classical formulas established

New proofs for identities of Schur Q-functions provided

Enhanced understanding of Pfaffian structures in algebraic combinatorics

Abstract

We establish Pfaffian analogues of the Cauchy--Binet formula and the Ishikawa--Wakayama minor-summation formula. Each of these Pfaffian analogues expresses a sum of products of subpfaffians of two skew-symmetric matrices in terms of a single Pfaffian. By using these Pfaffian formulas we give new transparent proofs to several identities for Schur Q23 pa-functions.

Equations

$$A(\mathbf{x})={\left(\frac{x_j-x_i}{x_j+x_i}\right)}_{1\le i,j\le n},\text{and}D(\mathbf{x})=\prod _{1\le i<j\le n}\frac{x_j-x_i}{x_j+x_i}.$$

$$V_{\alpha}(\boldsymbol{x})=\Bigl{(}x_{i}^{\alpha_{j}}\Bigr{)}_{1\leq i\leq n,1\leq j\leq l},\quad\text{and}\quad W_{\alpha}(\boldsymbol{x})=\Bigl{(}\chi(\alpha_{j})x_{i}^{\alpha_{j}}\Bigr{)}_{1\leq i\leq n,1\leq j\leq l},$$

$$P_{\lambda }(\mathbf{x})$$

$$Q_{\lambda }(\mathbf{x})$$

$$\sum _{\lambda }{s}_{\lambda }(\mathbf{x})s_{\lambda }(\mathbf{y})=\prod _{i,j}\frac{1}{1-x_i{y}_j}$$

$$\sum _{\lambda }{P}_{\lambda }(\mathbf{x})Q_{\lambda }(\mathbf{y})=\prod _{i,j}\frac{1+x_i{y}_j}{1-x_i{y}_j}.$$

$$\begin{gathered} \sum _K\mathrm{Pf}\left(\begin{array}{cc}A& S([m];K)\\ -{}^{t}S([m];K)& O\end{array}\right)\mathrm{Pf}\left(\begin{array}{cc}B& T([n];K)\\ -{}^{t}T([n];K)& O\end{array}\right) \\ =(-1{)}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}\mathrm{Pf}\left(\begin{array}{cc}A& S{}^{t}T\\ -T{}^{t}S& -B\end{array}\right), \end{gathered}$$

$$\mathrm{Pf}(X)=\sum _{\sigma \in F_{2m}}\mathrm{sgn}(\sigma )\prod _{i=1}^m{x}_{\sigma (2i-1),\sigma (2i)},$$

$$\mathrm{sgn}(\sigma )=(-1{)}^{\mathrm{inv}(i_1,j_1,\dots ,i_m,j_m)},$$

$$\mathrm{Pf}(-X)=(-1{)}^{\left(\genfrac{}{}{0ex}{}{2m}{2}\right)}\mathrm{Pf}X.$$

$$\mathrm{Pf}X=\alpha \mathrm{Pf}{X}^{\prime }+\beta \mathrm{Pf}{X}^{\prime \prime },$$

$$\operatorname{Pf}\bigl{(}{}^{t}\!UXU\bigr{)}=\det(U)\operatorname{Pf}(X).$$

$$\operatorname{Pf}\bigl{(}x_{\sigma(i),\sigma(j)}\bigr{)}_{1\leq i,j\leq n}=\operatorname{sgn}(\sigma)\operatorname{Pf}\bigl{(}x_{i,j}\bigr{)}_{1\leq i,j\leq n}.$$

$$\mathrm{Pf}X=\sum _{i=1}^{k-1}(-1{)}^{k+i-1}{x}_{i,k}\mathrm{Pf}X([n]\setminus \{i,k\})+\sum _{i=k+1}^n(-1{)}^{k+i-1}{x}_{k,i}\mathrm{Pf}X([n]\setminus \{k,i\}).$$

$$\mathrm{Pf}{\left(\frac{\mathrm{Pf}X([n]\cup \{n+i,n+j\})}{\mathrm{Pf}X([n])}\right)}_{1\le i,j\le l}=\frac{\mathrm{Pf}X}{\mathrm{Pf}X([n])}.$$

$$\mathrm{Pf}{\left(\frac{\mathrm{Pf}X(\{i,j\}\cup [l+1,l+n])}{\mathrm{Pf}X([l+1,l+n])}\right)}_{1\le i,j\le l}=\frac{\mathrm{Pf}X}{\mathrm{Pf}X([l+1,l+n])},$$

$$\mathrm{Pf}{\left(\frac{x_j-x_i}{x_j+x_i}\right)}_{1\le i,j\le n}=\prod _{1\le i<j\le n}\frac{x_j-x_i}{x_j+x_i}.$$

$$\mathrm{Pf}\left(\begin{array}{cc}Z& W\\ -{}^{t}W& Z^{\prime }\end{array}\right)=\sum _{I,J}\epsilon (I,J)\mathrm{Pf}Z(I)\mathrm{Pf}{Z}^{\prime }(J)\det W([m]\setminus I;[n]\setminus J),$$

$$\epsilon (I,J)=(-1{)}^{\Sigma (I)+\Sigma (J)+\left(\genfrac{}{}{0ex}{}{m}{2}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)+\left(\genfrac{}{}{0ex}{}{k}{2}\right)},k=m-\#I=n-\#J.$$

$${\pi }_i=\{b\in \pi :\#(b\cap [m])=i\}\text{for i=0,1,2.}$$

$$F_{[m]\bigsqcup [n{]}^{\prime }}\to \underset{(I,J)}{⨆}{F}_I\times S_k\times F_{J^{\prime }},$$

$$\displaystyle\operatorname{inv}\bigl{(}p_{1},p_{2},\dots,p_{2(m-k)},r_{1},s^{\prime}_{\sigma(1)},\dots,r_{k},s^{\prime}_{\sigma(k)},q^{\prime}_{1},q^{\prime}_{2},\dots,q^{\prime}_{2(n-k)}\bigr{)}$$

$$\displaystyle=\operatorname{inv}\bigl{(}p_{1},p_{2},\dots,p_{2(m-k)}\bigr{)}+\operatorname{inv}\bigl{(}r_{1},s^{\prime}_{\sigma(1)},\dots,r_{k},s^{\prime}_{\sigma(k)}\bigr{)}+\operatorname{inv}\bigl{(}q^{\prime}_{1},q^{\prime}_{2},\dots,q^{\prime}_{2(n-k)}\bigr{)}$$

$$+\#\{(i,j)\in I\times ([m]\setminus I):i>j\}+\left(\genfrac{}{}{0ex}{}{k}{2}\right)+\#\{(i^{\prime },j^{\prime })\in ([n{]}^{\prime }\setminus J^{\prime })\times J:i^{\prime }>j^{\prime }\}.$$

$$\operatorname{inv}\bigl{(}r_{1},s^{\prime}_{\sigma(1)},\dots,r_{k},s^{\prime}_{\sigma(k)}\bigr{)}=\operatorname{inv}\bigl{(}\sigma(1),\dots,\sigma(k)\bigr{)}.$$

$$\#\{(i,j)\in I\times ([m]\setminus I):i>j\}=\Sigma (I)-\left(\genfrac{}{}{0ex}{}{m-k+1}{2}\right),$$

$$\#\{(i^{\prime },j^{\prime })\in ([n{]}^{\prime }\setminus J^{\prime })\times J^{\prime }:i^{\prime }>j^{\prime }\}=k(n-k)+(\genfrac{}{}{0ex}{}{n-k+1}{2})-\Sigma (J).$$

$$\left(\genfrac{}{}{0ex}{}{m-k+1}{2}\right)+k(n-k)+\left(\genfrac{}{}{0ex}{}{n-k+1}{2}\right)$$

$$\equiv \left(\genfrac{}{}{0ex}{}{m}{2}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)\mathrm{mod}2.$$

$$\displaystyle\operatorname{inv}\bigl{(}p_{1},p_{2},\dots,p_{2(m-k)},r_{1},s^{\prime}_{\sigma(1)},\dots,r_{k},s^{\prime}_{\sigma(k)},q^{\prime}_{1},q^{\prime}_{2},\dots,q^{\prime}_{2(n-k)}\bigr{)}$$

$$\displaystyle\equiv\operatorname{inv}\bigl{(}p_{1},p_{2},\dots,p_{2(m-k)}\bigr{)}+\operatorname{inv}\bigl{(}\sigma(1),\dots,\sigma(k)\bigr{)}+\operatorname{inv}\bigl{(}q^{\prime}_{1},q^{\prime}_{2},\dots,q^{\prime}_{2(n-k)}\bigr{)}$$

$$+\Sigma (I)+\Sigma (J)+\left(\genfrac{}{}{0ex}{}{m}{2}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)+\left(\genfrac{}{}{0ex}{}{k}{2}\right).$$

$$\mathrm{Pf}\left(\begin{array}{cc}Z& W\\ -{}^{t}W& O_{n,n}\end{array}\right)=\{\begin{array}{ll}\sum _I(-1{)}^{\Sigma (I)+\left(\genfrac{}{}{0ex}{}{m}{2}\right)}\mathrm{Pf}Z(I)\det W([m]\setminus I;[n])& \text{if m>n,}\\ (-1{)}^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)}\det W& \text{if m=n,}\\ 0& \text{if m<n,}\end{array}$$

$$\mathrm{Pf}\left(\begin{array}{cc}Z& O_{m,n}\\ O_{n,m}& Z^{\prime }\end{array}\right)=\{\begin{array}{ll}\mathrm{Pf}Z\cdot \mathrm{Pf}{Z}^{\prime }& \text{if m and n are even,}\\ 0& \text{otherwise}.\end{array}$$

Full text

1 Introduction

The aim of this article is twofold: Firstly we establish Pfaffian analogues of the Cauchy–Binet formula and the Ishikawa–Wakayama minor summation formula [4] for determinants. Secondly we give new transparent proofs to Schur $Q$-function identities by applying general formula for Pfaffians such as these Pfaffian analogues.

Schur $Q$-functions are a family of symmetric functions introduced by Schur [19] in his study on the projective representations of symmetric groups. Schur $Q$-functions play the same role as Schur functions for the linear representation of symmetric groups. Later Hall (unpublished) and Littlewood [12] introduce a family of symmetric functions with parameter $t$, as a common generalization of Schur functions (the $t=0$ case) and Schur $Q$-functions (the $t=-1$ case).

Schur $Q$-functions appear in various situations parallel to Schur functions: the projective representations of symmetric groups [19], the cohomology of Lagrangian or orthogonal Grassmannians [8, 16], the representations of the queer Lie super algebra $\mathfrak{q}(n)$ [20], the BKP hierarchy [23]. Also Schur $Q$-functions are expressed as multivariate generating functions of shifted tableaux.

In this paper we adopt Nimmo’s formula [15, (A13)] as a definition of Schur $P$- and $Q$-functions. This formula is an analogue of the bialternant definition of Schur functions. Let $\boldsymbol{x}=(x_{1},\dots,x_{n})$ be a sequence of $n$ indeterminates. We put

[TABLE]

For a sequence $\alpha=(\alpha_{1},\dots,\alpha_{l})$ of nonnegative integers of length $l$, let $V_{\alpha}(\boldsymbol{x})$ and $W_{\alpha}(\boldsymbol{x})$ be the $n\times l$ matrices given by

[TABLE]

where $\chi(r)=2$ if $r>0$ and $1$ if $r=0$. A strict partition of length $l$ is a strictly decreasing sequence $\lambda=(\lambda_{1},\dots,\lambda_{l})$ of positive integers. We write $l=l(\lambda)$. We define the Schur $P$-function $P_{\lambda}(\boldsymbol{x})$ and the Schur $Q$-function $Q_{\lambda}(\boldsymbol{x})$ corresponding to a strict partition $\lambda$ by putting

[TABLE]

where $\alpha=(\lambda_{1},\dots,\lambda_{l})$ if $n+l$ is even, or $\alpha=(\lambda_{1},\dots,\lambda_{l},0)$ if $n+l$ is odd. Note that $P_{\lambda}(\boldsymbol{x})=Q_{\lambda}(\boldsymbol{x})=0$ if $l>n$.

Many of Schur function identities are easily proved by applying determinant formulas. However some of the known proofs of $Q$-function identities are quite different from the proofs of similar Schur function identities. For example, the Cauchy identity for Schur functions

[TABLE]

can be proved by using the Cauchy–Binet formula for determinants and the evaluation of Cauchy determinant (see [13, I.4, Example 6]). On the other hand, no such direct proof is known for the Cauchy-type identity for Schur $Q$-functions

[TABLE]

See [19, Abschnitt IV], [7, § 4B], [13, III.8] and [2, Chapter 7] for algebraic proofs. One of our motivations is to give an elementary linear algebraic proof to $Q$-function identities.

One of the main results of this paper is the following Pfaffian analogue of the Cauchy–Binet formula, which reduces to the Cauchy–Binet formula for determinants by specializing $A=O$ and $B=O$.

Theorem 1.1**.**

(Theorem 3.2 below) Let $m$ and $n$ be nonnegative integers with the same parity, . Let $A$ and $B$ be $m\times m$ and $n\times n$ skew-symmetric matrices, and let $S$ and $T$ be $m\times l$ and $n\times l$ matrices. Then we have

[TABLE]

where $K$ runs over all subsets of $[l]$ with $\#K\equiv m\equiv n\bmod 2$. (See Section 2 for notations.)

We can give a simple and direct proof to the Cauchy-type identity (1.6) by using this Pfaffian version of the Cauchy–Binet formula as well as the evaluation of Schur Pfaffian. Also we can use a variant to prove the Pragacz–Józefiak–Nimmo identity for skew $Q$-functions [17, 15]. In a forthcoming paper, we take this linear algebraic approach to study generalizations of Schur $P$- and $Q$-functions such as Ivanov’s factorial $P$- and $Q$-functions [5, 6] and the case $t=-1$ of Hall–Littlewood polynomials associated to the classical root systems [14].

This paper is organized as follows. After reviewing basic properties of Pfaffians in Section 2, we give Pfaffian analogues of the Cauchy–Binet formula and the Ishikawa–Wakayama minor-summation formula in Section 3. In Section 4, we apply the Pfaffian analogue of the Sylvester formula to recover Schur’s original definition of $Q$-functions from Nimmo’s formula. In Section 5, we give a proof of the Cauchy-type formula for $Q$-functions by using the Pfaffian analogue of the Cauchy–Binet formula. Section 6 is devoted to a linear algebraic proof of the Pragacz–Józefiak–Nimmo formula for skew $Q$-functions. In Section 7 we use the Pfaffian analogue of the Ishikawa–Wakayama formula to derive a Littlewood-type formula for $Q$-functions.

2 Pfaffians

In this section we review basic properties of Pfaffians and give a Laplace-type expansion formula.

2.1 Basic properties of Pfaffians

Recall the definition and some properties of Pfaffians. (See [3] for some expositions) Let X=\bigl{(}x_{ij}\bigr{)}_{1\leq i,j\leq 2m} be a skew-symmetric matrix of order $2m$. The Pfaffian of $X$, denoted by $\operatorname{Pf}(X)$, is defined by

[TABLE]

where $F_{2m}$ is the set of permutations $\sigma\in S_{2m}$ satisfying $\sigma(1)<\sigma(3)<\dots<\sigma(2m-1)$ and $\sigma(2i-1)<\sigma(2i)$ for $1\leq i\leq m$. Such permutations are in one-to-one correspondence with set-partitions $\pi$ of $\{1,2,\dots,2m\}$ into $m$ disjoint 2-element subsets. If $\sigma\in F_{2m}$ corresponds to a set-partition $\pi=\{\{i_{1},j_{1}\},\dots,\{i_{m},j_{m}\}\}$ with $i_{k}<j_{k}$ for $1\leq k\leq m$, then we have

[TABLE]

where $\operatorname{inv}(\alpha_{1},\dots,\alpha_{2m})$ is the number of pairs $(k,l)$ such that $k<l$ and $\alpha_{k}>\alpha_{l}$. Note that the right hand side is independent of the ordering of blocks of $\pi$. Since $m\equiv\binom{2m}{2}\bmod 2$, it follows from the definition of Pfaffians (2.1) that

[TABLE]

Pfaffians are multilinear in the following sense. Let X=\bigl{(}x_{ij}\bigr{)}_{1\leq i,j\leq n} be a skew-symmetric matrix and fix a row/column index $k$. If the entries of the $k$th row and $k$th column of $X$ are written as $x_{i,j}=\alpha x^{\prime}_{i,j}+\beta x^{\prime\prime}_{i,j}$ for $i=k$ or $j=k$, then

[TABLE]

where $X^{\prime}$ (resp. $X^{\prime\prime}$) is the skew-symmetric matrix obtained from $X$ by replacing the entries $x_{ij}$ for $i=k$ or $j=k$ with $x^{\prime}_{ij}$ (resp. $x^{\prime\prime}_{ij}$).

If $X$ is an $n\times n$ skew-symmetric matrix and $U$ is an $n\times n$ matrix, then we have

[TABLE]

It follows that Pfaffians are alternating, i.e., if $\sigma\in S_{n}$, we have

[TABLE]

Also we see that, if $Y$ is the skew-symmetric matrix obtained from $X$ by adding the $k$th row multiplied by a scalar $\alpha$ to the $l$th row and then adding the $k$th column multiplied by $\alpha$ to the $l$th column, the we have $\operatorname{Pf}Y=\operatorname{Pf}X$.

We use the following notations for submatrices. For a positive integer $n$, we put $[n]=\{1,2,\dots,n\}$. Given a subset $I\subset[n]$, we put $\Sigma(I)=\sum_{i\in I}i$. For an $M\times N$ matrix X=\bigl{(}x_{i,j}\bigr{)}_{1\leq i\leq M,1\leq j\leq N} and subsets $I\subset[M]$ and $J\subset[N]$, we denote by $X(I;J)$ the submatrix of $X$ obtained by picking up rows indexed by $I$ and columns indexed by $J$. If $X$ is a skew-symmetric matrix, then we write $X(I)$ for $X(I;I)$. We use the convention that $\det X(\emptyset;\emptyset)=1$ and $\operatorname{Pf}X(\emptyset)=1$.

For an $n\times n$ skew-symmetric matrix X=\bigl{(}x_{i,j}\bigr{)}_{1\leq i,j\leq n}, we have the following expansion formula along the $k$th row/column:

[TABLE]

Knuth [11] gave the following Pfaffian analogue of the Sylvester identity for determinant.

Proposition 2.1**.**

(Knuth [11, (2.5)]) Let $n$ and $l$ be even integers and let $X$ be an $(n+l)\times(n+l)$ skew-symmetric matrix.

1. (1)

If $\operatorname{Pf}X([n])\neq 0$, then we have

[TABLE] 2. (2)

If $\operatorname{Pf}X([l+1,l+n])\neq 0$, then we have

[TABLE]

where $[l+1,l+n]=\{l+1,l+2,\dots,l+n\}$.

The following evaluation formula of Schur Pfaffian is useful in various places of this paper.

Proposition 2.2**.**

(Schur [19, p. 226], see also [11, Section 4]) Let $n$ be an even integer. For a sequence $\boldsymbol{x}=(x_{1},\dots,x_{n})$ of variables, we have

[TABLE]

2.2 Laplace-type expansion formulas for Pfaffian

The following expansion formula is stated in [1, (12)] without proof.

Proposition 2.3**.**

Let $m$ and $n$ be nonnegative integers such that $m+n$ is even. For an $m\times m$ skew-symmetric matrix Z=\bigl{(}z_{i,j}\bigr{)}_{1\leq i,j\leq m}, an $n\times n$ skew-symmetric matrix Z^{\prime}=\bigl{(}z^{\prime}_{i,j}\bigr{)}_{1\leq i,j\leq n}, and an $m\times n$ matrix W=\bigl{(}w_{i,j}\bigr{)}_{1\leq i\leq m,1\leq j\leq n}, we have

[TABLE]

where the sum is taken over all pairs of even-element subsets $(I,J)$ such that $I\subset[m]$, $J\subset[n]$ and $m-\#I=n-\#J$, and the coefficient $\varepsilon(I,J)$ is given by

[TABLE]

If $m=1$, then the formula (2.8) reduces to the expansion formula (2.4) along the first row/column.

• Proof.

We put $[n]^{\prime}=\{1^{\prime},2^{\prime},\dots,n^{\prime}\}$ and label the rows and columns of $\begin{pmatrix}Z&W\\ -{}^{t}\!W&Z^{\prime}\end{pmatrix}$ by $[m]\sqcup[n]^{\prime}=\{1,2,\dots,m,1^{\prime},2^{\prime},\dots,n^{\prime}\}$ with $1<2<\dots<m<1^{\prime}<2^{\prime}<\dots<n^{\prime}$. For an even-element subset $I$ of $[m]\sqcup[n]^{\prime}$, we denote by $F_{I}$ the set of all set-partitions of $I$ into $2$-element subsets. Given a partition $\pi\in F_{[m]\sqcup[n]^{\prime}}$, we put

[TABLE]

Then there are subsets $I\subset[m]$ and $J^{\prime}\subset[n]^{\prime}$ such that $\pi_{2}\in F_{I}$, $\pi_{0}\in F_{J^{\prime}}$ and $m-\#I=n-\#J^{\prime}$. Moreover, if $[m]\setminus I=\{r_{1},\dots,r_{k}\}$ and $[n]^{\prime}\setminus J^{\prime}=\{s^{\prime}_{1},\dots,s^{\prime}_{k}\}$ with $r_{1}<\dots<r_{k}$, $s_{1}<\dots<s_{k}$, then there exists a unique permutation $\sigma\in S_{k}$ such that \pi_{1}=\bigl{\{}\{r_{1},s^{\prime}_{\sigma(1)}\},\dots,\{r_{k},s^{\prime}_{\sigma^{\prime}(k)}\}\bigr{\}}. The correspondence $\pi\mapsto(\pi_{2},\sigma,\pi_{0})$ gives a bijection

[TABLE]

where $(I,J)$ runs over all pairs of even-element subsets $I\subset[m]$ and $J\subset[n]$ such that $m-\#I=n-\#J$, and $J^{\prime}=\{j^{\prime}:j\in J\}$. Let \pi_{2}=\bigl{\{}\{p_{1},p_{2}\},\dots,\{p_{2(m-k)-1},p_{2(m-k)}\}\bigr{\}} and \pi_{0}=\bigl{\{}\{q^{\prime}_{1},q^{\prime}_{2}\},\dots,\{q^{\prime}_{2(n-k)-1},q^{\prime}_{2(n-k)}\}\bigr{\}} with $p_{2i-1}<p_{2i}$ and $q_{2j-1}<q_{2j}$. Then the inversion number of the permutation associated to $\pi$ is given by

[TABLE]

Since $r_{1}<\dots<r_{k}<s^{\prime}_{1}<\dots<s^{\prime}_{k}$, we have

[TABLE]

Also we have

[TABLE]

Since $m\equiv n\equiv k\bmod 2$ by the assumption, we see that

[TABLE]

Hence we have

[TABLE]

Now (2.8) follows from the definition of Pfaffians (2.1). ∎

By considering the case where $Z^{\prime}$ or $W$ is the zero matrix in Proposition 2.3, we obtain the following corollary. We denote the $p\times q$ zero matrix by $O_{p,q}$ and write simply $O$ for $O_{p,q}$ if there is no confusion on the size.

Corollary 2.4**.**

Suppose that $m+n$ is even.

1. (1)

If $Z$ is an $m\times m$ skew-symmetric matrix and $W$ is an $m\times n$ matrix, then we have

[TABLE]

where $I$ runs over all $(m-n)$-element subsets of $[n]$. 2. (2)

If $Z$ and $Z^{\prime}$ are $m\times m$ and $n\times n$ skew-symmetric matrices respectively, then we have

[TABLE]

• Proof.

(1) If $Z^{\prime}=O$, then we have $\operatorname{Pf}Z^{\prime}(J)=0$ unless $J=\emptyset$.

(2) If $W=O$, then we have $\det W([m]\setminus I;[n]\setminus J)=0$ unless $I=[m]$ and $J=[n]$. ∎

3 Cauchy–Binet type Pfaffian formulas

In this section we give Pfaffian analogues of the Cauchy–Binet formula and the Ishikawa–Wakayama minor-summation formula [4]. These are our main results of this paper.

First we consider the following special case of Proposition 2.3.

Lemma 3.1**.**

Let $m$, $n$ and $l$ be nonnegative integers with $m\equiv n\bmod 2$. We put

[TABLE]

where $E_{l}$ is the $l\times l$ identity matrix. If $Z$ and $Z^{\prime}$ are $(m+l)\times(m+l)$ and $(n+l)\times(n+l)$ skew-symmetric matrices respectively, then we have

[TABLE]

where $K$ runs over all subsets of $[l]$ with $\#K\equiv m\bmod 2$ and $m+K=\{m+k:k\in K\}$, $n+K=\{n+k:k\in K\}$.

• Proof.

We substitute $W=E^{(m+l,n+l)}_{l}$ in Proposition 2.3. Let $I$ and $J$ be even-element subsets of $[m+l]$ and $[n+l]$ respectively such that $m+l-\#I=n+l-\#J$. If $[m]\not\subset I$ or $[n]\not\subset J$, then we have $\det W([m+l]\setminus I;[n+l]\setminus J)=0$. If $[m]\subset I$ and $[n]\subset J$, then we can write $I=[m]\sqcup(m+I^{\prime})$ and $J=[n]\sqcup(n+J^{\prime})$ for some subsets $I^{\prime}$, $J^{\prime}\subset[l]$, and we see that

[TABLE]

Hence $\det W([m+l]\setminus I;[n+l]\setminus J)=0$ unless $I=[m]+(m+K)$ and $J=[n]+(n+K)$ for some subset $K\subset[l]$. In this case,

[TABLE]

and

[TABLE]

This completes the proof. ∎

We use Lemma 3.1 to derive a Pfaffian analogue of the Cauchy–Binet formula.

Theorem 3.2**.**

Let $m$ and $n$ be nonnegative integers with the same parity, . Let $A$ and $B$ be $m\times m$ and $n\times n$ skew-symmetric matrices, and let $S$ and $T$ be $m\times l$ and $n\times l$ matrices. Then we have

[TABLE]

[TABLE]

where $K$ runs over all subsets of $[l]$ with $\#K\equiv m\equiv n\bmod 2$.

Remark 3.3**.**

It follows from (2.9) that both formulas (3.2) and (3.3) reduce to the Cauchy–Binet formula for determinants if we put $A=0$ and $B=0$:

[TABLE]

where $K$ runs over all $m$-element subsets.

• Proof.

Apply Lemma 3.1 to the matrices

[TABLE]

Then we have

[TABLE]

We compute the Pfaffian on the right hand side of (3.1). By using the relation (2.3) with

[TABLE]

and then by using Corollary 2.4, we see that

[TABLE]

Therefore we have

[TABLE]

Since $\#K\equiv n\bmod 2$, we have

[TABLE]

and obtain the desired formula (3.2).

Equation (3.3) is obtained by replacing $B$ with $-B$ in (3.2). In fact, by multiplying the last $k$ rows/columns by $-1$ and then by using (2.2), we have

[TABLE]

Since $\#K\equiv n\bmod 2$, we have

[TABLE]

and obtain (3.3). ∎

Another application of Lemma 3.1 is the following Pfaffian analogue of the Ishikawa–Wakayama minor-summation formula.

Theorem 3.4**.**

Let $m$ be an even integer and $l$ be a positive integer. For an $m\times m$ skew-symmetric matrix $A$, an $l\times l$ skew-symmetric matrix $B$, and an $m\times l$ matrix $S$, we have

[TABLE]

where $K$ runs over all even-element subsets of $[l]$.

Remark 3.5**.**

It follows from (2.9) that (3.5) reduces to the minor-summation formula ([4, Theorem 1]) if $A=O$:

[TABLE]

where $K$ runs over all $m$-element subsets of $[l]$.

• Proof.

We apply Lemma 3.1 (with $n=0$) to the matrices

[TABLE]

Since $\operatorname{Pf}Z^{\prime}(K)=(-1)^{\binom{\#K}{2}}\operatorname{Pf}B(K)$ by (2.2), we have

[TABLE]

By using (2.3) with

[TABLE]

and then by using Corollary 2.4, we obtain

[TABLE]

Hence the proof is completed by using the congruence $\binom{l-\#K}{2}+\binom{\#K}{2}+\binom{l}{2}\equiv 0\bmod 2$. ∎

Remark 3.6**.**

From Lemma 3.1, we can derive the following summation formula for Pfaffians [4, Theorem 3]:

[TABLE]

where $A$ and $B$ are $l\times l$ skew-symmetric matrices, $T$ is an $l\times l$ matrix, and the summation is taken over all pairs of even-element subsets $I$, $J\subset[l]$ such that $\#I=\#J$. In fact, if we consider the case $m=n=0$ of Lemma 3.1, we obtain

[TABLE]

By taking $Z=A$ and $Z^{\prime}=TB{}^{t}\!T$ and using the minor-summation formula (3.6), we obtain (3.7).

4 Schur’s original definition of $Q$-functions

In this section, we recover Schur’s original definition [19] of $Q$-functions from Nimmo’s formula (1.4) by applying the Pfaffian analogue of the Sylvester formula (Proposition 2.1). Macdonald [13, III. 8] proves Part (3) of the following theorem by considering the generating function of Hall–Littlewood functions, And Stembridge’s derivation [21, Theorem 6.1] is based on the combinatorial definition of $Q$-functions and the lattice path method.

Theorem 4.1**.**

(Schur [19])

1. (1)

The generating function of Schur $Q$-functions corresponding to partitions of length $\leq 1$ is given by

[TABLE]

where $Q_{(0)}(\boldsymbol{x})=1$. 2. (2)

The generating function of Schur $Q$-functions corresponding to partitions of length $\leq 2$ is given by

[TABLE]

where $Q_{(0,0)}(\boldsymbol{x})=0$ and

[TABLE]

for positive integers $r$ and $s$. 3. (3)

For a sequence of nonnegative integers $\alpha=(\alpha_{1},\dots,\alpha_{l})$, we put

[TABLE]

Given a strict partition $\lambda$ of length $l$, we have

[TABLE]

where $\lambda^{0}=(\lambda_{1},\dots,\lambda_{l},0)$.

First we show the following stability of Schur $Q$-functions.

Lemma 4.2**.**

For a strict partition $\lambda$, we have

[TABLE]

• Proof.

Let $\boldsymbol{x}=(x_{1},\dots,x_{n})$ and $l=l(\lambda)$. Note that $D(x_{1},\dots,x_{n},0)=(-1)^{n}D(x_{1},\dots,x_{n})$.

If $n+l$ is even, then by definition (1.4) we have

[TABLE]

where $\boldsymbol{1}_{p,q}$ is the all-one matrix of size $p\times q$. By adding the $(n+1)$st column/row to the last column/row and then expanding the resulting Pfaffian along the last column/row, we see that

[TABLE]

If $n+l$ is odd, then we have

[TABLE]

By pulling out the common factor $-1$ from the $(n+1)$st row/column and then moving the $(n+1)$st row/column to the last row/column, we see that

[TABLE]

∎

• Proof of Theorem 4.1.

(1) By the stability (Lemma 4.2), we may assume that $n$ is odd. Then we have

[TABLE]

where $W_{(r)}(\boldsymbol{x})$ is the column vector $\left(\chi(r)x_{i}^{r}\right)_{1\leq i\leq n}$. By using

[TABLE]

we see that

[TABLE]

where $H_{z}(\boldsymbol{x})$ is the column vector with $i$th entry $(1+x_{i}z)/(1-x_{i}z)$. The last Pfaffian is evaluated by using Proposition 2.2 with variables $(x_{1},\dots,x_{n},-1/z)$ and we have

[TABLE]

This complete the proof of (1).

(2) By the stability (Lemma 4.2), we may assume that $n$ is even. Then we have

[TABLE]

and hence obtain

[TABLE]

Applying Proposition 2.2 with variables $(x_{1},\dots,x_{n},-1/z,-1/w)$, we see that

[TABLE]

By splitting the last row/column, we have

[TABLE]

By expanding the last Pfaffian along the last row/column and using (2.7), we have

[TABLE]

Hence we have

[TABLE]

(3) By the stability (Lemma 4.2), we may assume that $n$ is even. We apply the Pfaffian analogue of the Sylvester identity (Proposition 2.1) to the matrix $X$ given by

[TABLE]

Since $\operatorname{Pf}X([n]\sqcup\{n+i,n+j\})/\operatorname{Pf}X([n])=Q_{(\lambda_{i},\lambda_{j})}$ for $i<j$, Schur’s identity (4.3) immediately follows from Proposition 2.1. ∎

Remark 4.3**.**

We can give a direct proof to the Pfaffian identity (4.3) in the case where $n$ is odd, by applying Proposition 2.1 to the matrix given by

[TABLE]

5 Cauchy-type identity for $Q$-functions

In this section, we use the Pfaffian analogue of the Cauchy–Binet formula (Theorem 3.2) to prove the Cauchy-type identity for Schur $Q$-functions, which corresponds to the orthogonality of $Q$-functions. To prove the Cauchy-type identity, Schur [19, Abschnitt IX] (see also [7, § 4B]) used a characterization of $Q_{(n)}$, and Macdonald [13, III.8] appeal to the theory of Hall–Littlewood functions. Also the proof given by Hoffman–Humphreys [2, Chapter 7] is based on the definition of $Q$-functions in terms of vertex operators. Bijective proofs are given by Worley [22, Theorem 6.1.1] and Sagan [18, Corollary 8.3]. Here we give a simple linear algebraic proof.

Theorem 5.1**.**

(Schur [19, p. 231]) For $\boldsymbol{x}=(x_{1},\dots,x_{n})$ and $\boldsymbol{y}=(y_{1},\dots,y_{n})$, we have

[TABLE]

where $\lambda$ runs over all strict partitions.

The following lemma is obvious, so we omit the proof.

Lemma 5.2**.**

Let $n$ be a positive integer and denote by $\mathbb{N}$ the set of nonnegative integers. To a strict partition $\lambda$ we associate the subset $I_{n}(\lambda)\subset\mathbb{N}$ given by

[TABLE]

Then the correspondence $\lambda\mapsto I_{n}(\lambda)$ gives a bijection from the set of all strict partitions to the set of all subsets $I$ of $\mathbb{N}$ with $\#I\equiv n\bmod 2$.

• Proof of Theorem 5.1.

Apply the Pfaffian version of Cauchy–Binet formula (3.3) to the matrices

[TABLE]

It follows from the definition of $P$- and $Q$-functions (1.3) and (1.4) that for a strict partition $\lambda$ we have

[TABLE]

Hence, by using Lemma 5.2 and applying (3.3), we have

[TABLE]

where $\lambda$ runs over all strict partitions and $I$ runs over all subsets of $\mathbb{N}$ with $\#I\equiv n\bmod 2$. Since the $(i,j)$ entry of $S{}^{t}\!T$ is given by

[TABLE]

we can use the evaluation of the Schur Pfaffian (2.7) with variables $(x_{1},\dots,x_{n},-1/y_{1},\dots,\allowbreak-1/y_{n})$ to obtain

[TABLE]

This completes the proof. ∎

6 Pragacz–Józefiak–Nimmo identity for skew $Q$-functions

In this section, we use the Pfaffian analogue of the Cauchy–Binet formula (Theorem 3.2) to prove the Pragacz–Józefiak–Nimmo identity for skew $Q$-functions. Pragacz–Józefiak [17] and Nimmo [15] used differential operators to prove this Pfaffian identity and Stembridge [21, Theorem 6.2] gave a combinatorial proof based on the lattice path method. In the course of our proof, we find a Pfaffian identity which interpolate Nimmo’s identity (1.4) and Schur’s identity (4.3).

Skew $Q$-functions $Q_{\lambda/\mu}(x_{1},\dots,x_{n})$ are uniquely determined by the equation

[TABLE]

where $\lambda$ is a strict partition and the summation is taken over all strict partitions $\mu$.

Theorem 6.1**.**

(Pragacz–Józefiak [17, Theorem 1], Nimmo [15, (2.22)]) For two sequences $\alpha=(\alpha_{1},\dots,\alpha_{l})$ and $\beta=(\beta_{1},\dots,\beta_{m})$ of nonnegative integers, let $M_{\alpha/\beta}(\boldsymbol{x})$ be the $l\times m$ matrix given by

[TABLE]

where $Q_{(k)}(\boldsymbol{x})=0$ for $k<0$. For two strict partitions $\lambda$ and $\mu$, we have

[TABLE]

Note that

[TABLE]

• Proof.

We denote by $\widetilde{Q}_{\lambda/\mu}(\boldsymbol{x})$ the right hand side of (6.1) and prove

[TABLE]

By the stability (Lemma 4.2), we may assume that the length $l=l(\lambda)$ and the number $k$ of variables in $\boldsymbol{y}$ have the same parity.

We apply the Pfaffian analogue of the Cauchy–Binet formula (3.2) to the matrices

[TABLE]

Then, for a strict partition $\mu$, we have

[TABLE]

Hence we have

[TABLE]

And it follows from the definition (1.4) that

[TABLE]

By applying (3.2), we see that

[TABLE]

Also it follows from the generating function (4.1) of $Q_{(r)}$’s that the $(i,j)$ entry of $S{}^{t}\!T$ is given by

[TABLE]

Now we can complete the proof by using the following Theorem. ∎

Theorem 6.2**.**

Let $\boldsymbol{x}=(x_{1},\dots,x_{n})$ and $\boldsymbol{y}=(y_{1},\dots,y_{k})$ be two sequence of variables. For a sequence $\alpha=(\alpha_{1},\dots,\alpha_{l})$ of length $l$, let $N_{\alpha}(\boldsymbol{x}|\boldsymbol{y})$ be the $l\times k$ matrix defined by

[TABLE]

For a strict partition $\lambda$ of length $l$, we have

[TABLE]

where $\lambda^{0}=(\lambda_{1},\dots,\lambda_{l},0)$.

Note that the identity (6.2) reduces to Nimmo’s identity (1.4) if $n=0$ and to Schur’s identity (4.3) if $k=0$.

• Proof.

We denote by $Q^{\prime}_{\lambda}(\boldsymbol{x}|\boldsymbol{y})$ the right hand side of (6.2).

First we show that $Q^{\prime}_{\lambda}(\boldsymbol{x}|\boldsymbol{y})$ is stable with respect to $\boldsymbol{y}$, that is,

[TABLE]

Let $\boldsymbol{y}=(y_{1},\dots,y_{k})$. If $l+k$ is even, then we have by using the stability (Lemma 4.2) of $Q_{\lambda}(\boldsymbol{x})$,

[TABLE]

where $T_{\lambda}(\boldsymbol{x})$ is the column vector \bigl{(}Q_{\lambda_{i}}(\boldsymbol{x})\bigr{)}_{1\leq i\leq l}. By adding the $(l+1)$st row/column multiplied by $-1$ to the last row/column and then expanding the resulting Pfaffian along the last row/column, we see that

[TABLE]

If $l+k$ is odd, then by moving the last row/column to the $(l+1)$st row/column we have

[TABLE]

Next we use the Sylvester formula for Pfaffians (Proposition 2.1) to prove

[TABLE]

where $\lambda^{0}=(\lambda_{1},\dots,\lambda_{l(\lambda)},0)$ and the matrix $S^{\prime}_{\alpha}(\boldsymbol{x}|\boldsymbol{y})$ is defined by

[TABLE]

By the stability (6.3), we may assume $k$ is even. In this case the identity (6.4) can be obtained by applying (2.6) to the matrix

[TABLE]

By comparing two Pfaffian identities (4.3) and (6.4), the proof of Theorem 6.2 is reduced to showing

[TABLE]

We prove these equality by considering the generating functions. If we put

[TABLE]

then by virtue of (4.1) and (4.2) the identities (6.5) and (6.6) follow from

[TABLE]

and

[TABLE]

respectively.

By the stability (6.3) we may assume $k$ is odd for the proof of (6.7). If $k$ is odd, then

[TABLE]

Since $\sum_{r\geq 0}Q(\boldsymbol{x},y_{j})z^{j}=F_{z}(\boldsymbol{x})(1+y_{j}z)/(1-y_{j}z)$ by (4.1), we have

[TABLE]

where $H_{z}(\boldsymbol{y})$ is the column vector $\left((1+y_{i}z)/(1-y_{i}z)\right)_{1\leq i\leq k}$. By applying Proposition 2.2 with variables $(-1/z,y_{1},\dots,y_{n})$, we have

[TABLE]

and obtain (6.7).

By the stability (6.3) we may assume $k$ is even for the proof of (6.8). If $k$ is even, then

[TABLE]

for $r$, $s\geq 0$. Hence it follows from (4.1) and (4.2) that

[TABLE]

where

[TABLE]

By splitting the first row/column and then pulling out the common factor $F_{z}(\boldsymbol{x})$ and $F_{w}(\boldsymbol{x})$ from the $1$st and $2$nd rows/columns, we see that

[TABLE]

The first Pfaffian is evaluated by using the Schur Pfaffian with $(-1/z,-1/w,y_{1},\dots,y_{n})$ and we see that

[TABLE]

By expanding the second Pfaffian along the first column/row, we see that it equals to

[TABLE]

Therefore we obtain

[TABLE]

This completes the proof of Theorem 6.2 and hence Theorem 6.1. ∎

7 Littlewood-type identity for $Q$-functions

In this section, we prove the following Littlewood-type identity for $Q$-functions. This identity is a special case ($t=\sqrt{-1}$) of [10, (1.21)] for Hall–Littlewood functions, which is essentially proved in [9] by using the representation theory of finite Chevalley groups.

Theorem 7.1**.**

(Kawanaka [10]) For $\boldsymbol{x}=(x_{1},\dots,x_{n})$, we have

[TABLE]

where $\lambda$ runs over all strict partitions.

Remark 7.2**.**

The right hand side of (7.1) is one of the simplest example of products involving the factor $\prod_{1\leq i<j\leq n}(1+x_{i}x_{j})/(1-x_{i}x_{j})$ that is a(n infinite) linear combination of $P$- or $Q$-functions. Recall that a symmetric polynomial $f(\boldsymbol{x})$ is a linear combination of Schur $Q$-functions if and only if $f(t,-t,x_{3},\dots,x_{n})$ is independent of $t$. (See [13, III (8.5)] for example.) Consider a symmetric power series of the form

[TABLE]

where $\{\alpha_{1},\dots,\alpha_{r}\}\cap\{\beta_{1},\dots,\beta_{s}\}=\emptyset$. Then

[TABLE]

is independent of $t$ if and only if

[TABLE]

as multisets. Thus $f_{n}(\boldsymbol{x})$ is an infinite linear combination of $P$-functions if and only if $r=s$ and $\alpha_{1}=\pm\sqrt{-1}$, $\beta_{1}=\pm 1$ and $\alpha_{k}=-\beta_{k}$ for $2\leq k\leq r$ up to permutation of $\alpha_{1},\dots,\alpha_{r}$ and $\beta_{1},\dots,\beta_{r}$.

• Proof of Theorem 7.1.

By the stability (Lemma 4.2), we may assume $n$ is even. We apply the Pfaffian version of the minor-summation formula (Theorem 3.4) to the matrices

[TABLE]

and the skew-symmetric matrix $B$ whose $(i,j)$ entry, $0\leq i<j$, is given by

[TABLE]

where $\alpha=1+\sqrt{-1}$.

By using (2.2), (2.4) and the induction on $\#I$, we see that the subpfaffian $\operatorname{Pf}B(I)$ of $B$ corresponding to a even-element subset $I\subset\mathbb{N}$ is given by

[TABLE]

Since $n$ is even, strict partitions $\lambda$ are in bijection with even-element subsets $I_{n}(\lambda)$ of $\mathbb{N}$ by Lemma 5.2 and

[TABLE]

Also it follows from Nimmo’s identity (1.3) that

[TABLE]

Therefore we have

[TABLE]

where $I$ runs over all even-element subsets of $\mathbb{N}$ and $\lambda$ runs over all strict partitions.

By direct computations, we see that the $(i,j)$-entry of $SB{}^{t}\!S$ is equal to

[TABLE]

and the $(i,j)$ entry of $A-SB{}^{t}\!S$ is equal to

[TABLE]

Hence, by using Proposition 2.2 with variables $(x_{1}-1/x_{1},\dots,x_{n}-1/x_{n})$, we have

[TABLE]

This completes the proof. ∎

By considering the real and imaginary parts of Theorem 7.1, we obtain

Corollary 7.3**.**

If we put

[TABLE]

then we have

[TABLE]

where $e_{k}=e_{k}(\boldsymbol{x})$ is the $k$th elementary symmetric polynomial.

• Proof.

Since we have

[TABLE]

we obtain this corollary from Theorem 7.1. ∎