Elementary proof and application of the generating function for generalized Hall-Littlewood functions

TL;DR

This paper introduces a generalized form of Hall-Littlewood symmetric functions via formal group laws, provides an elementary proof of their generating function, and explores some applications of this formula.

Contribution

It presents a new generalization of Hall-Littlewood functions and offers an elementary proof of their generating function, expanding their theoretical framework.

Findings

Elementary proof of the generating function formula

New generalization of Hall-Littlewood functions

Applications of the generating function formula

Abstract

In this note we define a generalization of Hall-Littlewood symmetric functions using formal group law and give an elementary proof of the generating function formula for the generalized Hall-Littlewood symmetric functions. We also give some applications of this formula.

Equations

$$F(x,y)=x+y+\sum _{i,j\ge 1}{a}_{i,j}{x}^i{y}^j,$$

$$P_F(z):=1+\sum _{i=1}^{\infty }{a}_{1,i}{z}^i.$$

$$F_1(x,y):={\partial }_{x}F(x,y)=1+\sum _{i,j\ge 1}i{a}_{i,j}{x}^{i-1}{y}^j.$$

$$F(x,F(y,z))=F(F(x,y),z).$$

$$F_1(x,F(y,z))=F_1(F(x,y),z)F_1(x,y).$$

$$1=F_1(y,\bar{y})F_1(0,y).$$

$$P_F(y,y)=\underset{x\to y}{\lim }\frac{x-y}{F(x,\bar{y})}=\frac{1}{F_1(y,\bar{y})}=F_1(0,y)=P_F(y),$$

$${\mathcal{Q}}_{{\lambda }_1,\dots ,{\lambda }_r}^F(x_1,\dots ,x_n;T,H):=\sum _{w\in S_n/(S_1^r\times S_{n-r})}w\left(x_1^{[{\lambda }_1]}\cdots x_r^{[{\lambda }_r]}\prod _{1\le i\le r}\prod _{i<j\le n}\frac{T(x_i,x_j)}{F(x_i,{\bar{x}}_j)}\right),$$

$$[z^{-k}]\left(\frac{H(z)}{P_F(z)}\prod _{i=1}^n\frac{T(z,x_i)}{F(z,{\bar{x}}_i)}\right)=\sum _{i=1}^n{x}_i^{k-1}H(x_i)T(x_i,x_i)\prod _{j\ne i}\frac{T(x_i,x_j)}{F(x_i,{\bar{x}}_j)}.$$

$$\frac{H(z)}{P_F(z)}\prod _{i=1}^n\frac{T(z,x_i)}{F(z,{\bar{x}}_i)}=\frac{H(z)}{P_F(z)}\prod _{i=1}^n\frac{P_F(z,x_i)T(z,x_i)}{z-x_i}.$$

$$R_F(z):=z^{\ell }\frac{H(z)}{P_F(z)}\prod _{i=1}^n\left(P_F(z,x_i)T(z,x_i)\right)-\sum _{i=1}^n{x}_i^{\ell }H(x_i)T(x_i,x_i)\prod _{j\ne i}\frac{T(x_i,x_j)(z-x_j)}{F(x_i,{\bar{x}}_j)}.$$

$$R_F(z)=B(z)\prod _{i=1}^n(z-x_i),$$

$$z^{\ell }\frac{H(z)}{P_F(z)}\prod _{i=1}^n\frac{P_F(z,x_i)T(z,x_i)}{z-x_i}=B(z)+\sum _{i=1}^n\frac{x_i^{\ell }H(x_i)T(x_i,x_i)}{z-x_i}\prod _{j\ne i}\frac{T(x_i,x_j)}{F(x_i,{\bar{x}}_j)}.$$

$$\frac{1}{P_F(z)}\prod _{i=1}^n\frac{F(z,[t]{\bar{x}}_i)}{F(z,{\bar{x}}_i)}=N_F(z;x_1,\dots ,x_n;t)+\sum _{i=1}^n\frac{F(x_i,[t]{\bar{x}}_i)}{z-x_i}\prod _{j\ne i}\frac{F(x_i,[t]{\bar{x}}_j)}{F(x_i,{\bar{x}}_j)},$$

$$N_{F_{CK}}(z;x_1,\dots ,x_n;t)=\frac{1}{1+\beta z},$$

$$N_{F_{Ell}}(z;x_1;t)=\frac{1}{1+\beta z+\gamma z^2}+\frac{\gamma w_1(x_1+w_1+\beta x_1{w}_1)}{(1-\gamma w_1{x}_1)(1-\gamma w_{1}z)},$$

$$\frac{z}{F(z,[t]\bar{z})}\frac{1}{P_F(z)}\prod _{i=1}^n\frac{F(z,[t]{\bar{x}}_i)}{F(z,{\bar{x}}_i)}=M_F(z;x_1,\dots ,x_n;t)+\sum _{i=1}^n\frac{x_i}{z-x_i}\prod _{j\ne i}\frac{F(x_i,[t]{\bar{x}}_j)}{F(x_i,{\bar{x}}_j)},$$

$$M_{F_{CK}}(z;x_1,\dots ,x_n;t=-1)=\frac{1}{1+\beta z}-\frac{{\prod }_{i=1}^n(1+\beta x_i)}{2+\beta z}$$

$$A(z_1,\dots ,z_r):=\prod _{i=1}^{r}A(z_i)\prod _{1\le i<j\le r}\frac{F(z_j,{\bar{z}}_i)}{T(z_j,z_i)}.$$

$$[z_1^{-{\lambda }_1}\cdots z_r^{-{\lambda }_r}]A(z_1,\dots ,z_r)={\mathcal{Q}}_{\lambda }^F(x_1,\dots ,x_n;T,H).$$

$$A(z_1,\dots ,z_r)=A(z_2,\dots ,z_r)\times A(z_1)\prod _{j=2}^r\frac{F(z_j,{\bar{z}}_1)}{T(z_j,z_1)}.$$

$$\prod _{j=2}^r\frac{F(z_j,{\bar{z}}_1)}{T(z_j,z_1)}=\sum _{m\ge 0}{f}_m(z_2,\dots ,z_r)z_1^m.$$

$$[z_1^{-{\lambda }_1}]A(z_1,\dots ,z_r)=A(z_2,\dots ,z_r)\times \left(\sum _{m\ge 0}{f}_m(z_2,\dots ,z_r){\mathcal{Q}}_{{\lambda }_1+m}(x_1,\dots ,x_n)\right).$$

$${\mathcal{Q}}_{{\lambda }_1+m}(x_1,\dots ,x_n)=\sum _{i=1}^n{x}_i^{{\lambda }_1+m}H(x_i)\frac{T(x_i,x_i)}{x_i}\prod _{1\le j\le n,j\ne i}\frac{T(x_i,x_j)}{F(x_i,{\bar{x}}_j)}.$$

$$\sum _{i=1}^n\left({\mathcal{Q}}_{({\lambda }_2,\dots ,{\lambda }_r)}^{(i)}(x_1,\dots ,x_n)\right)x_i^{{\lambda }_1}H(x_i)\frac{T(x_i,x_i)}{x_i}\prod _{1\le j\le n,j\ne i}\frac{T(x_i,x_j)}{F(x_i,{\bar{x}}_j)},$$

$$=\sum _{w\in S_n/(S_1^r\times S_{n-r})}w\left((x_1|b{)}^{[{\lambda }_1]}\cdots (x_r|b{)}^{[{\lambda }_r]}\prod _{1\le i\le r}\prod _{i<j\le n}\frac{T(x_i,x_j)}{F(x_i,{\bar{x}}_j)}\right),$$

$$\begin{array}[]{lll}\mathcal{Q}_{\lambda_{1},\ldots,\lambda_{r}}(x_{1},\ldots,x_{n})&=&\det\left([z_{i}^{-(\lambda_{i}+j-i)}]\displaystyle\frac{1}{(1+\beta z_{i})^{r-i}}A(z_{i})\right)_{r\times r}\\[8.5359pt] &=&\det\left([z_{i}^{-(\lambda_{i}+j-i)}]\displaystyle\frac{1}{(1+\beta z_{i})^{j-i}}A(z_{i})\right)_{r\times r},\end{array}$$

$${\mathcal{Q}}_{{\lambda }_1,\dots ,{\lambda }_r}(x_1,\dots ,x_n)=\mathrm{Pf}{\left([z_i^{-{\lambda }_i}{z}_j^{-{\lambda }_j}]\frac{1}{(1+\beta z_i{)}^{r-i-1}}\frac{1}{(1+\beta z_i{)}^{r-j}}A(z_i)A(z_j)\frac{z_j\ominus z_i}{z_j\oplus z_i}\right)}_{r\times r}.$$

$$\frac{z_j\ominus z_i}{z_j}=\frac{1-z_i/z_j}{1+\beta z_i}=1-\frac{{\bar{z}}_i}{{\bar{z}}_j},$$

$$N_F(z;x_1,\dots ,x_n;t)=\frac{1}{1+\beta z}.$$

Full text

1. Introduction

Let $r$ and $n$ $(1\leq r\leq n)$ be positive integers. Given a sequence $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ of positive integers of length $r$, a formal power series $T(x,y)$, and a formal Laurent series $H(z)$, we define the generalized Hall–Littlewood symmetric function $\mathcal{Q}^{F}_{\lambda_{1},\ldots,\lambda_{r}}(x_{1},\ldots,x_{n};T,H)$ using a formal group law $F(x,y)$. In the case of the additive group law $F(x,y)=x+y$ with $T(x,y)=x-ty$ and $H(z)=1$ this becomes the usual Hall–Littlewood $Q$-symmetric function defined in [Mac](cf. Definition 2 below). Nakagawa and the author have previously defined the universal Hall–Littlewood function $H^{L}_{\lambda}(x_{1},\ldots,x_{n};t)$ for a partition $\lambda$ [NN]. When $\lambda$ is a strict partition, definition of [NN] provides a special case of a generalized Hall–Littlewood function, but in general functions are different.

We do not discuss in this paper related geometries here, but we formulate the generating function and the proof of its properties in almost the same way as in [Mac]. After this elementary proof, we found another simple proof by the means of the push-forward formula [NN2]. However, the proof presented in this paper only uses elementary algebraic manipulation and does not use any topological arguments, such as Quillen’s push-forward formula. Actually, Quillen’s formula ([Qui, Theorem 1]) can be derived from Lemma 4 below, applied to $\ell=0,k=1$ and $T(x,y)=1$, as well as Bressler-Evens formula [NN, Theorem 2.5, Corollary 2.6].

2.

Generating function formulas and their proofs

2.1. Formal group law

A formal group law $F(x,y)$ is a formal power series in two variables $x$ and $y$,

[TABLE]

such that $a_{i,j}$’s satisfy the following two properties:

• (1)

commutativity $F(x,y)=F(y,x)$, and

• (2)

associativity $F(x,F(y,z))=F(F(x,y),z)$

(cf. [LM]) .

For a given formal group law $F$, we define an associated formal power series $P_{F}(z)$ as follows:

[TABLE]

In addition, set $P_{F}(x,y):=\displaystyle\frac{x-y}{F(x,\bar{y})}$, where $\bar{y}$ is the inverse of $y$, i.e., it satisfies $F(y,\bar{y})=0$.

Lemma 1**.**

$P_{F}(x,y)$* is invertible and $P_{F}(x,x)=P_{F}(x)$.*

Proof. We use the following notation for the derivative with respect to the first variable of $F$:

[TABLE]

Then, by definition, $P_{F}(x)=F_{1}(0,x)$. We can use associativity of $F$

[TABLE]

Differentiating both sides with respect to variable $x$, we get

[TABLE]

By substituting $x=0$ and $z=\bar{y}$, we have

[TABLE]

Finally, using this equation, we can deduce

[TABLE]

which is what we wanted to prove.

QED

For the rest of this paper, we assume that the formal group law $F(x,y)$ is written as $F(x,y)=\exp_{F}(\log_{F}(x)+\log_{F}(y))$, using appropriate formal power series $\exp_{F}$ and $\log_{F}$. Then we can define formal multiplication by a variable $t$ as $[t]_{F}x:=\exp_{F}(t\log_{F}(x))$. The inverse of $x$ can be written as $\bar{x}=[-1]_{F}x$. We sometimes drop the $F$ notation and just write $[t]x$.

2.2. Generalized Hall–Littlewood function

We now generalize the classical Hall–Littlewood function using the formal group law $F$. Actually, we further generalize Hall–Littlewood function using a formal power series $T(x,y)$ and a formal Laurent series $H(z)$. For a formal Laurent series $L(z)$, we use notation $[z^{i}]L(z)$ for the coefficient of $z^{i}$ in $L(z)$. We also allow the case of $L(z)=\sum_{i\in\mathbb{Z}}c_{i}z^{i}$ with coefficients to be non-zero for all $i\in\mathbb{Z}$. For example, $\frac{1}{1+\beta z}+\frac{x}{z-x}=\cdots+(-\beta)^{2}z^{2}+(-\beta)z+1+(x/z)+(x/z)^{2}+\cdots$. Given $T(x,y)$ and $H(z)$, we define the generalized Hall–Littlewood function $\mathcal{Q}^{F}_{\lambda_{1},\ldots,\lambda_{r}}(x_{1},\ldots,x_{n};T,H)$ for a positive integer sequence $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ as follows. Let $S_{n}$ be symmetric group acting on the variables $x_{1},\ldots,x_{n}$.

Definition 2**.**

The generalized Hall–Littlewood function associated with $F,T,$ and $H$ is

[TABLE]

where $x^{[k]}:=H(x)T(x,{x})x^{k-1}$ for any positive integer $k$.

In the following, we sometimes drop the $F,T,H$ notations and simply write this function as $\mathcal{Q}_{\lambda}(x_{1},\ldots,x_{n})$.

Definition 3**.**

We focus on the following two cases:

1. (1)

$T(x,y)=F(x,[t]\bar{y})$, $H(z)=1$ and define

$HQ_{\lambda}(x;t):=$**

$\displaystyle\sum_{w\in S_{n}/(S_{1}^{r}\times S_{n-r})}w\left(F(x_{1},[t]\bar{x}_{1})x_{1}^{\lambda_{1}-1}\cdots F(x_{r},[t]\bar{x}_{r})x_{r}^{\lambda_{r}-1}\displaystyle\prod_{1\leq i\leq r}\prod_{i<j\leq n}\frac{F(x_{i},[t]\bar{x}_{j})}{F(x_{i},\bar{x}_{j})}\right);$* * 2. (2)

$T(x,y)=F(x,[t]\bar{y})$, $H(z)=\frac{z}{T(z,z)}$ and define

$HP_{\lambda}(x;t):=\displaystyle\sum_{w\in S_{n}/(S_{1}^{r}\times S_{n-r})}w\left(x_{1}^{\lambda_{1}}\cdots x_{r}^{\lambda_{r}}\displaystyle\prod_{1\leq i\leq r}\prod_{i<j\leq n}\frac{F(x_{i},[t]\bar{x}_{j})}{F(x_{i},\bar{x}_{j})}\right).$**

When $F(x,y)=x+y$, $HQ_{\lambda}(x;t)$ becomes the Hall–Littlewood $Q$-function and for a strict partiton $\lambda$, $HP_{\lambda}(x;t)$ becomes the Hall–Littlewood $P$-function defined in [Mac]. For the connective $K$-theory case $F=F_{CK}(x,y)=x+y+\beta xy$ with $t=-1$, they are the $K$-theoretic Schur $Q$- and $P$-functions $GQ_{\lambda}(x)$ and $GP_{\lambda}(x)$ respectively defined in [IN].

Lemma 4**.**

Let $T(x,y)$ be a formal power series in $x$ and $y$, and let $H(z)$ be a formal Laurent series in $z$ with a pole at $z=0$ of order $\ell\geq 0$, i.e., $H(z)=\sum_{i\geq-\ell}c_{i}z^{i}$ with $c_{-\ell}\neq 0$. Then, for $k>\ell$, we have

[TABLE]

Proof. The expression under consideration can be written as

[TABLE]

We define

[TABLE]

This is a formal power series in variable $z$. When specializing $z=x_{k}$ ($k=1,2,\ldots,n$), it vanishes. Therefore, we can write

[TABLE]

where $B(z)$ is a formal power series in $z$. By dividing this by $\displaystyle\prod_{i=1}^{n}(z-x_{i})$, we get

[TABLE]

By taking the coefficients of $z^{\ell-k}$ in both sides, we obtain the claim of the lemma. QED

Corollary 5**.**

Setting $T(x,y)=F(x,[t]\bar{y})$ and $H(z)=1$, we get the equality

[TABLE]

where $N_{F}(z;x_{1},\ldots,x_{n};t)$ is a formal power series in variable $z$.

Remark 6**.**

In the connective $K$-theory case $F_{CK}(x,y)=x+y+\beta xy$ we can prove that

[TABLE]

(cf. Proposition 12).

In the elliptic cohomology case $F_{Ell}(x,y)=\frac{x+y+\beta xy}{1-\gamma xy}$ we have

[TABLE]

where $w_{1}=[t]\bar{x}_{1}$.

Corollary 7**.**

Setting $T(x,y)=F(x,[t]\bar{y})$ and $H(z)=\frac{z}{T(z,z)}$, we obtain the equality

[TABLE]

where $M_{F}(z;x_{1},\ldots,x_{n};t)$ is a formal power series in variable $z$.

Remark 8**.**

In the connective K-theory case $F_{CK}(x,y)=x+y+\beta xy$ we can prove that

[TABLE]

(cf. Proposition 13).

Theorem 9**.**

Assume that $T(x,y)$ is a formal power series and $T(x,0)=x$. Assume further that $H(z)$ is a formal Laurent series in $z$ with a pole at zero of order $\ell\geq 0$.

Let $A(z):=\displaystyle\frac{H(z)}{P_{F}(z)}\prod_{i=1}^{n}\frac{T(z,{x}_{i})}{F(z,\bar{x}_{i})}$. Set

[TABLE]

Then for a sequence of positive integers $\lambda_{1},\ldots,\lambda_{r}>\ell$ , $(r\leq n)$, we have

[TABLE]

Proof. (See [Mac] p.211 proof of (2.15)). We will show that for a sequence of positive integers $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ the coefficient of $z^{-\lambda}:=z_{1}^{-\lambda_{1}}\cdots z_{r}^{-\lambda_{r}}$ in $A(z_{1},\ldots,z_{r})$ is $\mathcal{Q}_{\lambda}(x_{1},\ldots,x_{n})$. For the case $r=1$ it follows by Lemma 4. Assume $r\geq 2$. By definition,

[TABLE]

We can expand the product in the formula above as a formal power series in $z_{1}$ as follows.

[TABLE]

On the other hand, by Lemma 4 the coefficient of $z_{1}^{-\lambda_{1}-m}$ in $A(z_{1})$ is $\mathcal{Q}_{\lambda_{1}+m}(x_{1},\ldots,x_{n})$. Therefore, the cofefficient of $z_{1}^{-\lambda_{1}}$ in $A(z_{1},\ldots,z_{r})$ is

[TABLE]

However, by definition,

[TABLE]

Therefore, we can substitute this into the previous equation to obtain

\begin{array}[]{ccl}&&[z_{1}^{-\lambda_{1}}]A(z_{1},\ldots,z_{r})\\ &=&A(z_{2},\ldots,z_{r})\times\left(\displaystyle\sum_{m\geq 0}f_{m}(z_{2},\ldots,z_{r})\sum_{i=1}^{n}x_{i}^{\lambda_{1}+m}H(x_{i})\frac{T(x_{i},{x}_{i})}{x_{i}}\prod_{1\leq j\leq n,j\neq i}\frac{T(x_{i},{x}_{j})}{F(x_{i},\bar{x}_{j})}\right)\\[22.76228pt] &=&A(z_{2},\ldots,z_{r})\times\left(\displaystyle\sum_{i=1}^{n}\left(\sum_{m\geq 0}x_{i}^{m}f_{m}(z_{2},\ldots,z_{r})\right)x_{i}^{\lambda_{1}}H(x_{i})\frac{T(x_{i},{x}_{i})}{x_{i}}\prod_{1\leq j\leq n,j\neq i}\frac{T(x_{i},{x}_{j})}{F(x_{i},\bar{x}_{j})}\right)\\[22.76228pt] &=&\displaystyle\sum_{i=1}^{n}\left(A(z_{2},\ldots,z_{r})\prod_{j=2}^{r}\frac{F(z_{j},\bar{x}_{i})}{T(z_{j},{x}_{i})}\right)x_{i}^{\lambda_{1}}H(x_{i})\frac{T(x_{i},{x}_{i})}{x_{i}}\prod_{1\leq j\leq n,j\neq i}\frac{T(x_{i},{x}_{j})}{F(x_{i},\bar{x}_{j})}\\[22.76228pt] &=&\displaystyle\sum_{i=1}^{n}\left(A^{(i)}(z_{2},\ldots,z_{r})\right)x_{i}^{\lambda_{1}}H(x_{i})\frac{T(x_{i},{x}_{i})}{x_{i}}\prod_{1\leq j\leq n,j\neq i}\frac{T(x_{i},{x}_{j})}{F(x_{i},\bar{x}_{j})},\\ \end{array} where we denoted by $A^{(i)}(z_{2},\ldots,z_{r})$ the result of setting $x_{i}=0$ in $A(z_{2},\ldots,z_{r})$. Here we used the property $T(x,0)=x$. By the induction hypothesis, the coefficient of $z_{2}^{-\lambda_{2}}\cdots z_{r}^{-\lambda_{r}}$ in $A(z_{2},\ldots,z_{r})$ is $\mathcal{Q}_{(\lambda_{2},\ldots,\lambda_{r})}(x_{1},\ldots,x_{n})$; therefore, the coefficient of $z_{1}^{-\lambda_{1}}z_{2}^{-\lambda_{2}}\cdots z_{r}^{-\lambda_{r}}$ in $A(z_{1},\ldots,z_{r})$ is

[TABLE]

which can easily be seen to equal $\mathcal{Q}_{(\lambda_{1},\ldots,\lambda_{r})}(x_{1},\ldots,x_{n}).$ QED

Remark 10**.**

We can modify the above proof so that each $A(z)$ has a different $H(z)$. This gives us a formula for the factorial version and its determinant-Pfaffian formula. For example, for a given sequence $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ of positive integers, we have

$[z^{-\lambda}]\left(A_{1}^{(\lambda_{1}-1)}(z_{1})\cdots A_{r}^{(\lambda_{r}-1)}(z_{r})\displaystyle\prod_{1\leq i<j\leq r}\frac{F(z_{j},\bar{z}_{i})}{T(z_{j},{z}_{i})}\right)$**

[TABLE]

where $A_{i}^{(k)}(z):=\displaystyle\frac{H_{i}^{(k)}(z)}{P_{F}(z)}\prod_{i=1}^{n}\frac{T(z,{x}_{i})}{F(z,\bar{x}_{i})}$ , if we set $H_{i}^{(k)}(z)=(z|b)^{k}/z^{k}$, $(z|b)^{k}=\displaystyle\prod_{i=1}^{k}F(z,b_{i})$ as the factorial power and $(x|b)^{[k]}:=T(x,{x})(x|b)^{k-1}$ (cf. similar formulas in [NN2]).

We set $x\oplus y:=F_{CK}(x,{y})=x+y+\beta xy$ and $x\ominus y:=F_{CK}(x,\bar{y})=\frac{x-y}{1+\beta y}$. A geometric proof of Corollary 11 below is given in [HIMN].

Corollary 11**.**

(Determinant-Pfaffian formula) Consider the connective $K$-theory case $F=F_{CK}(x,y)$ and assume $\lambda$ is a partition of length $r$, $T(x,y)=F(x,[t]\bar{y})$, and $H(z)=1$ or $H(z)=\frac{z}{T(z,z)}$. Then for $t=0$, we have the determinantal formulas

[TABLE]

and for $t=-1$, if we assume $r$ is even, we have the Pfaffian formula

[TABLE]

Proof. For the determinantal formulas, using the identity

[TABLE]

the assertion follows from Theorem 9. For the Pfaffian formula, by Theorem 9, we have

$\mathcal{Q}_{\lambda}(x)=[z^{-\lambda}]A(z_{1})\cdots A(z_{r})\displaystyle\prod_{1\leq i<j\leq r}\frac{z_{j}\ominus z_{i}}{z_{j}\oplus z_{i}}\\ =[z^{-\lambda}]A(z_{1})\cdots A(z_{r})\displaystyle\prod_{i=1}^{r-1}\frac{1}{(1+\beta z_{i})^{r-i}}\prod_{1\leq i<j\leq r}\frac{z_{j}-z_{i}}{z_{j}\oplus z_{i}}\\ =[z^{-\lambda}]A(z_{1})\cdots A(z_{r})\displaystyle\prod_{i=1}^{r-1}\frac{1}{(1+\beta z_{i})^{r-i}}{\rm Pf}\left(\frac{z_{j}-z_{i}}{z_{j}\oplus z_{i}}\right)\\ =[z^{-\lambda}]\displaystyle{\rm Pf}\left(A(z_{i})A(z_{j})\frac{1}{(1+\beta z_{i})^{r-i}}\frac{1}{(1+\beta z_{j})^{r-j}}\frac{z_{j}-z_{i}}{z_{j}\oplus z_{i}}\right)\\ =\displaystyle{\rm Pf}\left([z_{i}^{-\lambda_{i}}z_{j}^{-\lambda_{j}}]A(z_{i})A(z_{j})\frac{1}{(1+\beta z_{i})^{r-i-1}}\frac{1}{(1+\beta z_{j})^{r-j}}\frac{z_{j}\ominus z_{i}}{z_{j}\oplus z_{i}}\right).$

QED

2.3. Non-negative power part

Proposition 12**.**

For $F=F_{CK}(x,y)=x+y+\beta xy,$ we have

[TABLE]

Proof. We will prove

[TABLE]

Then we can substitute $w_{i}=[t]x_{i}$ ($i=1,\ldots,n$) to get the assertion of the proposition. Multiplying by $(1+\beta z)\prod_{i=1}^{n}(z\ominus x_{i})$, we must prove

[TABLE]

But both sides are polynomial in $z$ of degree $n$, and the equation has $n$ solutions $z=x_{i}$ ($i=1,\ldots,n$). So we need to prove that the top degree terms have the same coefficient. It gives the equality

[TABLE]

QED

Proposition 13**.**

$t=-1$* and $F=F_{CK}$ is the connective $K$-theory case, and we have*

[TABLE]

Note that $1+\beta GP_{1}(x_{1},\ldots,x_{n})=\displaystyle\prod_{i=1}^{n}(1+\beta x_{i})$.

Proof. We need to show

[TABLE]

For this we multiply by $(1+\beta z)(2+\beta z)\displaystyle\prod_{i=1}^{n}(z\ominus x_{i})$ both sides.

Therefore, we need to show

$\displaystyle\prod_{i=1}^{n}{(z\oplus x_{i})}=\left((2+\beta z)-(1+\beta z)\prod_{i=1}^{n}(1+\beta x_{i})\right)\prod_{i=1}^{n}{(z\ominus x_{i})}$

$\hskip 85.35826pt\displaystyle+(1+\beta z)(2+\beta z)\sum_{i=1}^{n}\frac{z_{i}}{1+\beta x_{i}}\prod_{j\neq i}\left(\frac{x_{i}\oplus x_{j}}{x_{i}\ominus x_{j}}(z\ominus x_{j})\right).$

This holds for $z=x_{1},\ldots,x_{n}$. So we need to check that the coefficient of $z^{n+1}$ on the right hand side is zero, and that the coefficients of $z^{n}$ are the same for both sides. The coefficient of $z^{n+1}$ on the right hand side is

[TABLE]

The coefficient of $z^{n}$ on the right hand side is

$\displaystyle\frac{3\beta{GP_{1}}+\beta^{2}(GP_{2}-(x_{1}+\cdots+x_{n})GP_{1})}{1+\beta GP_{1}}+\frac{2-(1+\beta GP_{1})+(x_{1}+\cdots+x_{n})\beta^{2}GP_{1}}{1+\beta GP_{1}}\\ =1+\beta GP_{1}.$

As the left hand side is $\displaystyle\prod_{i=1}^{n}(z(1+\beta x_{i})+x_{i})$, the coefficient of $z^{n}$ is

[TABLE]

so we get the result.

QED

3. Application

In this section we explain how the generating function can be used to give a formula such as the Pieri rule.

3.1. Two row in terms of one rows

We explain the case of $K$-theory and $t=-1$, i.e. we express $GQ_{k,\ell}$ in terms of $GQ_{p}\times GQ_{q}$.

Lemma 14**.**

We have the identity

[TABLE]

where $g_{i,j}:=(-1)^{i}\beta^{i-j}\left(2\binom{i}{j}+\binom{i}{j+1}-\delta_{j,0}\right)$ and $\delta_{i,j}$ is the Kronecker delta.

Proof. The assertion follows from the following equality:

[TABLE]

QED

Proposition 15**.**

For $k>\ell>0$ we have

[TABLE]

where

[TABLE]

Proof. We will compare the coefficients of $z_{1}^{-k}z_{2}^{-\ell}$ on both side of

[TABLE]

We expand $A(z)=\sum_{m\in\mathbb{Z}}{\bf GQ}_{m}z^{m}$. Then if $m>0$, ${\bf GQ}_{m}=GQ_{m}$, and if $m\leq 0$, then ${\bf GQ}_{m}=(-\beta)^{-m}$ by Proposition 12. By Lemma 14, it follows that

[TABLE]

Then the following binomial equality enables us to achieve the desiered result:

[TABLE]

Note that this equation is derived from the comparison of coefficients of $x^{j}$ in

[TABLE]

Proposition 16**.**

For $k>\ell>0$, we have

[TABLE]

where if $\ell\geq 2$

[TABLE]

and if $\ell=1$

[TABLE]

Proof. By Proposition 13, we have

[TABLE]

So, we must consider the case where $j=\ell-1,\ell$. If $\ell\geq 2$,

$d_{i,\ell-1}=\displaystyle g_{i,\ell-1}+\sum_{s=0}^{i-\ell}g_{i,\ell+s}\left(\frac{-\beta}{2}\right)^{s+1}=(-1)^{i}\beta^{i-(\ell-1)}2\binom{i}{\ell-1},$

and

$d_{i,\ell}=\displaystyle\sum_{s=0}^{i-\ell}g_{i,\ell+s}\left(1-\frac{1}{2^{s+1}}\right)(-\beta)^{s}=(-1)^{i}\beta^{i-\ell}\binom{i-1}{\ell-1}.$

If $\ell=1$, we find that

\begin{array}[]{ccl}d_{i,0}&=&g_{i,0}+\displaystyle\sum_{s=0}^{i-1}\left(\frac{-\beta}{2}\right)^{s+1}g_{i,s+1}\\ &=&(-\beta)^{i}\left((i+1)-\frac{1}{2}(2\binom{i}{1}+\binom{i}{2})+\frac{1}{2^{2}}(2\binom{i}{2}+\binom{i}{3})-\cdots\right)\\ &=&(-\beta)^{i},\\ \text{ and}\\ d_{i,1}&=&\displaystyle\sum_{s=0}^{i-1}(-\beta)^{s}\left(1-\frac{1}{2^{s+1}}\right)g_{i,s+1}\\ &=&(-1)^{i}\beta^{i-1}\left((1-\frac{1}{2})(2\binom{i}{1}+\binom{i}{2})-(1-\frac{1}{2^{2}})(2\binom{i}{2}+\binom{i}{3})+\cdots\right)\\ &=&(-1)^{i}\beta^{i-1}\left(2\binom{i-1}{0}+\binom{i-1}{1}-\frac{1}{2}(2\binom{i}{1}+\binom{i}{2})+\frac{1}{2^{2}}(2\binom{i}{2}+\binom{i}{3})-\cdots\right)\\ &=&(-1)^{i}\beta^{i-1}.\\ \end{array}

QED

3.2. Pieri rule for

multiplying one row by another

For $F=F_{CK}(x,y)=x+y+\beta xy$, using that

[TABLE]

and Lemma 17, we can obtain the following Pieri rule below (Proposition 18).

Lemma 17**.**

We have the following equality:

[TABLE]

Proof. The assertion can be shown in a straightforward way as follows.

[TABLE]

$=(1+\beta z_{1})^{2}+(2+\beta z_{1})(1+\beta z_{1})((z_{1}/z_{2})+(z_{1}/z_{2})^{2}+(z_{1}/z_{2})^{3}+\cdots)$

QED

Proposition 18**.**

*For $k\geq l>0$, we have the following:

\begin{array}[]{lll}(1)\;GQ_{k}\times GQ_{l}&=&(GQ_{k,l}+2\beta GQ_{k+1,l}+\beta^{2}GQ_{k+2,l})\\ &&+\displaystyle\sum_{i=1}^{l-1}(2GQ_{k+i,l-i}+3\beta GQ_{k+1+i,l-i}+\beta^{2}GQ_{k+2+i,l-i})\\ &&+(2GQ_{k+l,0}+\beta GQ_{k+l+1,0}),\end{array}**

$(2)\;GP_{k}\times GP_{1}=GP_{k,1}+\beta GP_{k+1,1}+GP_{k+1}.$**

For $k\geq l\geq 2$,

\begin{array}[]{lll}GP_{k}\times GP_{l}&=&(GP_{k,l}+2\beta GP_{k+1,l}+\beta^{2}GP_{k+2,l})\\ &&+\displaystyle\sum_{i=1}^{l-2}(2GP_{k+i,l-i}+3\beta GP_{k+1+i,l-i}+\beta^{2}GP_{k+2+i,l-i})\\ &&+(2GP_{k+l,1}+2\beta GP_{k+l+1,1})\\ &&+GP_{k+l}.\end{array}**

Proof. These follow directly from Lemma 17. QED

Acknowledgements

The author is gratefull to Natasha Rozhkovskaya for checking and correcting English expressions to make this document readable. This work was partially supported by the Grant-in-Aid for Scientific Research (B) 16H03921, Japan Society for the Promotion of Science.