# Free fermions and Schur expansions of multi-Schur functions

Shinsuke Iwao  Department of Mathematics, Tokai University, 4-1-1, Kitakaname,
Hiratsuka, Kanagawa 259-1292, Japan<EMAIL_ADDRESS>
###### Abstract.

Multi-Schur functions are symmetric functions that generalize the
supersymmetric Schur functions, the flagged Schur functions, and the refined
dual Grothendieck functions, which have been intensively studied by Lascoux.
In this paper, we give a new free-fermionic presentation of them. The multi-
Schur functions are indexed by a partition and two “tuples of tuples” of
indeterminates. We construct a family of linear bases of the fermionic Fock
space that are indexed by such data and prove that they correspond to the
multi-Schur functions through the boson-fermion correspondence. By focusing on
some special bases, which we call refined bases, we give a straightforward
method of expanding a multi-Schur function in the refined dual Grothendieck
polynomials. We also present a sufficient condition for a multi-Schur function
to have its Hall-dual function in the completed ring of symmetric functions.

Keywords. multi-Schur functions, free fermions, boson-fermion correspondence,

MSC Classification. 05E05, 13M10

##  1\. Introduction

In the series of papers [3, 4] on classical integrable systems, Date-Jimbo-
Miwa showed that the standard basis of the free-fermion Fock space naturally
corresponds to the Schur functions through the boson-fermion correspondence.
This result provides a useful tool for studying algebraic properties of
symmetric polynomials such as the Jacobi-Trudi formula and the Cauchy identity
(See [21, §9]). The fermionic description has been generalized by Lam [13] to
a wider class of symmetric functions including Hall-Littlewood polynomials and
Macdonald polynomials.

In [7, 8, 9], the author of this paper gave new fermionic presentations of the
(dual) stable Grothendieck polynomials [15, 16] and the $K$-theoretic
$Q$-functions [6]. A key method used there is to construct linear bases of the
Fock space that correspond with desired symmetric functions.

This paper aims to extend the previous results to the multi-Schur functions,
which are intensively studied in Lascoux’s textbook [14]. As proved by Motegi-
Scrimshaw [24], Amanov-Yeliussizov [2], and Kim [11, 12], the multi-Schur
functions generalize various important symmetric functions such as the refined
dual Grothendieck polynomials [17]. We show that, as with the stable
Grothendieck polynomials, the multi-Schur functions have a useful fermionic
description. Many algebraic identities such as the Jacobi-Trudi formula are
derived from this description. This result is a fermionic counterpart to the
combinatorial/algebraic approaches concerning symmetric Grothendieck
polynomials [19, 20, 22, 23, 25] and their variants [5, 26, 27, 28].

The fermionic presentation is useful in the calculation of the Hall inner
product. Let $\Lambda(X)$ be the ring of symmetric functions in infinitely
many variables $X_{1},X_{2},\dots$ (§3.1). As noted by various authors (see,
for example, [8, §3]), we often need a certain extended ring
$\widehat{\Lambda}(X)$ of $\Lambda(X)$ to deal with dual functions. We show
that many (but not all) multi-Schur functions have their dual in the extended
ring $\widehat{\Lambda}(X)$. A sufficient condition for a multi-Schur function
to have the dual is given (assumption ($\ast$ ‣ 4.7) in Lemma 4.7).

Each symmetric function in this paper is parameterized by a pair of tuples of
tuples ${\bf x}/{\bf y}$ with

${\bf x}=(x^{(1)},x^{(2)},\dots),\qquad{\bf y}=(y^{(1)},y^{(2)},\dots),$

where $x^{(i)}=(x^{(i)}_{1},x^{(i)}_{2},\dots,x^{(i)}_{M_{i}})$ and
$y^{(i)}=(y^{(i)}_{1},y^{(i)}_{2},\dots,y^{(i)}_{N_{i}})$ are tuples of
finitely many indeterminates (might be empty). The symbol
$\mbox{\boldmath{$\emptyset$}}=(\emptyset,\emptyset,\dots)$ will denote the
tuple of empty tuples.

For $t=(t_{1},t_{2},\dots)$, we let $t[i]=(t_{1},\dots,t_{i-1})$ be the tuple
consisting of the first $i-1$ letters of $t$. In the sequel, we often focus on
the tuple of tuples $[{\bf t}]$ that is written as

(1) $[{\bf t}]:=(t[1],t[2],\dots).$

We will call a tuple of the form (1) a refined tuple. We prove that every
refined tuple possesses the “orthonormality” property (Theorem 3.2), which is
a key tool for calculating (generalized) Schur expansions.

In this paper, we present two families of symmetric functions:
$s_{\lambda}(X)_{{\bf x}/{\bf y}}$ and $s_{\lambda}(X)^{{\bf x}}$. The former
is the multi-Schur function in $\Lambda(X)$ and the latter is some new
symmetric function in $\widehat{\Lambda}(X)$. The $s_{\lambda}(X)_{{\bf
x}/{\bf y}}$ is defined for any ${\bf x}$ and ${\bf y}$ while
$s_{\lambda}(X)^{{\bf x}}$ is defined only if ${\bf x}$ satisfies a certain
finiteness condition (see ($\ast$ ‣ 4.7) in Lemma 4.7). We will see that if
${\bf x}=[{\bf t}]$ and ${\bf y}=\mbox{\boldmath{$\emptyset$}}$ for some
refined tuple $[{\bf t}]$, $s_{\lambda}(X)_{[{\bf t}]}$ and
$s_{\lambda}(X)^{[{\bf t}]}$ are Hall dual with each other (Proposition 4.9).

The following is a list of symmetric functions in our scope.

x and y |  $s_{\lambda}(X)_{{\bf x}/{\bf y}}$ in $\Lambda(X)$ |  $s_{\lambda}(X)^{{\bf x}}$ in $\widehat{\Lambda}(X)$
---|---|---
General ${\bf x}$ and ${\bf y}$ | Multi-Schur function | —
|  ${\bf x}=(x^{(1)},x^{(2)},\dots)$, ${\bf y}=\mbox{\boldmath{$\emptyset$}}$,
---
$x^{(i)}=(x_{1},\dots,x_{f_{i}})$,
where $f=(f_{1}\leq f_{2}\leq\dots)$
is a flagging
|  $s_{\lambda}(0)_{{\bf x}}$ is
---
the flagged
Schur function
in $x_{1},x_{2},\dots$
—
|  ${\bf x}$ satisfying
---
the assumption ($\ast$ ‣ 4.7)
in Lemma 4.7,
${\bf y}=\mbox{\boldmath{$\emptyset$}}$.
| A generalization
---
of a refined dual
Grothendieck
function
| A generalization
---
of a refined
Grothendieck
function
|  ${\bf x}=[{\bf t}]$, ${\bf y}=\mbox{\boldmath{$\emptyset$}}$
---
| Refined dual
---
Grothendieck
function
| Refined
---
Grothendieck
function
|  ${\bf x}=[{\bf t}]$, ${\bf y}=\mbox{\boldmath{$\emptyset$}}$
---
with $t=(-\beta,-\beta,\dots)$
| Dual stable
---
Grothendieck
polynomial
| Stable
---
Grothendieck
polynomial

The paper is organized as follows: In Section 2, we give a brief review of
free-fermions and the boson-fermion correspondence. In Section 3, we introduce
the definition of the refined bases and prove their orthonormality, which is
the main theorem of this paper. Section 4 contains free-fermionic
presentations of dual functions. We also prove the Schur expansion of dual
functions. In Section 5, we extend our result to the skew dual multi-Schur
functions.

##  2\. Free-fermionic presentations of multi-Schur functions

###  2.1. Wick’s theorem

In this section, we review necessary definitions and facts about the boson-
fermion correspondence according to the standard textbooks [10, 21].
Throughout the paper, we write $[A,B]=AB-BA$ and $[A,B]_{+}=AB+BA$.

Assume that $k$ is a field of characteristic $0$. Let $\mathcal{A}$ be the
$k$-algebra of free fermions $\psi_{n}$, $\psi_{n}^{\ast}$
($n\in\mathord{\mathbb{Z}}$) with

(2)
$[\psi_{m},\psi_{n}]_{+}=[\psi^{\ast}_{m},\psi^{\ast}_{n}]_{+}=0,\qquad[\psi_{m},\psi^{\ast}_{n}]_{+}=\delta_{m,n}.$

Let $|0\rangle$, $\langle 0|$ denote the vacuum vectors:

$\psi_{m}|0\rangle=\psi^{\ast}_{n}|0\rangle=0,\quad\langle 0|\psi_{n}=\langle
0|\psi^{\ast}_{m}=0,\qquad m<0,\ n\geq 0.$

The Fock space (over $k$) is the $k$-space $\mathcal{F}$ generated by the
vectors

(3)
$\psi_{n_{1}}\psi_{n_{2}}\cdots\psi_{n_{r}}\psi^{\ast}_{m_{1}}\psi^{\ast}_{m_{2}}\cdots\psi^{\ast}_{m_{s}}|0\rangle,\
(r,s\geq 0,\ n_{1}>\dots>n_{r}\geq 0>m_{s}>\dots>m_{1}).$

We also consider the $k$-space $\mathcal{F}^{\ast}$ generated by the vectors

(4) $\langle
0|\psi_{m_{s}}\cdots\psi_{m_{2}}\psi_{m_{1}}\psi^{\ast}_{n_{r}}\cdots\psi^{\ast}_{n_{2}}\psi^{\ast}_{n_{1}},\
(r,s\geq 0,\ n_{1}>\dots>n_{r}\geq 0>m_{s}>\dots>m_{1}).$

The vectors (3) and (4) form bases of $\mathcal{F}$ and $\mathcal{F}^{\ast}$
respectively (see, for example, [10, §4 and §5.2]). The commutation relation
(2) determines the left $\mathcal{A}$-module structure of $\mathcal{F}$ and
the right $\mathcal{A}$-module structure of $\mathcal{F}^{\ast}$.

There exists an anti-algebra involution on $\mathcal{A}$ defined by

(5)
${}^{\ast}:\mathcal{A}\to\mathcal{A};\quad\psi_{n}\leftrightarrow\psi_{n}^{\ast},$

with $(ab)^{\ast}=b^{\ast}a^{\ast}$ and $(a^{\ast})^{\ast}=a$. The transpose
is the $k$-linear involution $\mathcal{F}\leftrightarrow{\mathcal{F}}^{\ast}$
given by $X|0\rangle\leftrightarrow\langle 0|{X}^{\ast}$.

Let

${\mathcal{F}}^{\ast}\otimes_{k}\mathcal{F}\to k,\quad\langle
w|\otimes|v\rangle\mapsto\langle{w}|v\rangle$

be the vacuum expectation value [21, §4.5], that is, a unique $k$-bilinear
form with (i) $\langle 0|0\rangle=1$, (ii) $(\langle
w|\psi_{n})|v\rangle=\langle w|(\psi_{n}|v\rangle)$, and (iii) $(\langle
w|\psi_{n}^{\ast})|v\rangle=\langle w|(\psi_{n}^{\ast}|v\rangle)$. From these
properties, we may write $\langle w|X|v\rangle=(\langle{w}|X)|v\rangle=\langle
w|(|X|v\rangle)$ for any $X\in\mathcal{A}$. We often use the abbreviation
$\langle X\rangle=\langle 0|X|0\rangle$.

######  Theorem 2.1 (Wick’s theorem (see [1, §2], [21, Exercise 4.2]) ).

Let $\\{m_{1},\dots,m_{r}\\}$ and $\\{n_{1},\dots,n_{r}\\}$ be sets of
integers. Then we have

$\langle\psi_{m_{1}}\cdots\psi_{m_{r}}\psi^{\ast}_{n_{r}}\cdots\psi^{\ast}_{n_{1}}\rangle=\det(\langle\psi_{m_{i}}\psi^{\ast}_{n_{j}}\rangle)_{1\leq
i,j\leq r}.$

For an integer $m$, we define the shifted vacuum vectors $|m\rangle$, $\langle
m|$ as

$|m\rangle=\begin{cases}\psi_{m-1}\psi_{m-2}\cdots\psi_{0}|0\rangle,&m\geq
0,\\\
\psi^{\ast}_{m}\cdots\psi^{\ast}_{-2}\psi^{\ast}_{-1}|0\rangle,&m<0\end{cases}$

and $\langle m|=(|m\rangle)^{\ast}$. Then, from Wick’s theorem, we have:

######  Corollary 2.2.

Suppose that $m_{1}>m_{2}>\dots>m_{r}\geq-r$ and
$n_{1}>n_{2}>\dots>n_{s}\geq-r$. Then we have

$\langle-r|\psi^{\ast}_{m_{r}}\cdots\psi^{\ast}_{m_{1}}\psi_{n_{1}}\cdots\psi_{n_{r}}|-r\rangle=\begin{cases}1,&\mbox{if
}m_{i}=n_{i}\mbox{ for all }i,\\\ 0,&\mbox{otherwise}.\end{cases}$

###  2.2. Supersymmetric Schur functions and the boson-fermion correspondence

We denote by $:\bullet:$ the normal ordering (see [1, §2], [21, §5.2]) of
free-fermions. For $m\in\mathord{\mathbb{Z}}$, let
$a_{m}=\sum_{i\in\mathord{\mathbb{Z}}}:\psi_{i}\psi^{\ast}_{i+m}:$ be the
Heisenberg operator acting on $\mathcal{F}$. The $a_{m}$ satisfies the
commutation relations [21, §5.3]:

(6)
$[a_{m},a_{n}]=m\delta_{m+n,0},\qquad[a_{m},\psi_{n}]=\psi_{n-m},\qquad[a_{m},\psi^{\ast}_{n}]=-\psi^{\ast}_{n+m}.$

Let $x=(x_{1},x_{2},\dots,x_{m})$ and $y=(y_{1},y_{2},\dots,y_{n})$ be tuples
of finitely many indeterminates (possibly empty). We define the complete
supersymmetric functions $h_{i}(x/y)$ by the generating function

$\sum_{i=0}^{\infty}h_{i}(x/y)z^{i}=\prod_{j=1}^{m}\frac{1}{1-x_{j}z}\prod_{l=1}^{n}(1-y_{l}z).$

When $y_{i}=0$ for all $i$, the $h_{i}(x/y)$ reduces to the ordinal complete
symmetric polynomial $h_{i}(x)$. Since any symmetric polynomial is a
polynomial in $h_{1}(x),h_{2}(x),\dots$, we define the corresponding
supersymmetric function by replacing $h_{i}(x)$ with $h_{i}(x/y)$. For
example, the $i$-th supersymmetric power sum $p_{i}(x/y)$ is expressed as
$p_{i}(x/y)=x_{1}^{i}+x_{2}^{i}+\cdots+x_{m}^{i}-y_{1}^{i}-y_{2}^{i}-\dots-
y_{n}^{i}$.

Let $s_{\lambda}(x/y)$ be the supersymmetric Schur function [18, §I-5. Example
23]. Then they satisfy

(7) $\displaystyle
s_{\lambda}(x/y)=\sum_{\mu}(-1)^{\left|{\lambda-\mu}\right|}s_{\mu}(x)s_{(\lambda/\mu)^{\prime}}(y),\qquad\left|{\lambda}\right|=\sum_{i}\lambda_{i},$
(8) $\displaystyle
s_{\lambda}(x/y)=(-1)^{\left|{\lambda}\right|}s_{\lambda^{\prime}}(y/x),$

where $\lambda^{\prime}$ is the transpose of $\lambda$. The following
equations are useful:

(9) $\displaystyle
h_{i}(x/y)=h_{i}(x)e_{0}(y)-h_{i-1}(x)e_{1}(y)+\cdots+(-1)^{i}h_{0}(x)e_{i}(y),$
(10) $\displaystyle
h_{i}(\emptyset/y)=(-1)^{i}e_{i}(y)=e_{i}(-y),\qquad\mbox{where}\quad-y=(-y_{1},\dots,-y_{n}),$
(11) $\displaystyle s_{\lambda}((x\cup p)/(y\cup
q))=\sum_{\mu\subset\lambda}s_{\mu}(x/y)s_{\lambda/\mu}(p/q),$ (12)
$\displaystyle h_{n}\left((x\cup p)/(y\cup
q)\right)=\sum_{i+j=n}h_{i}(x/y)h_{j}(p/q),$

where $x\cup p$ be the tuple consisted of all elements of $x$ and $p$.

Let us consider the formal sums $H(x)$ and $H(x/y)$ defined by

$H(x)=\sum_{n>0}\frac{p_{n}(x)}{n}a_{n},\qquad
H(x/y)=\sum_{n>0}\frac{p_{n}(x/y)}{n}a_{n}=H(x)-H(y).$

If the base field $k$ contains all rational functions in
$x_{1},\dots,x_{m},y_{1},\dots,y_{n}$, the $H(x/y)$ defines a $k$-linear map
from $\mathcal{F}$ to itself. From (6), we have the commutation relation

(13) $e^{H(x/y)}\psi_{n}e^{-H(x/y)}=\sum_{i=0}^{\infty}h_{i}(x/y)\psi_{n-i}.$

Let $\mathcal{F}^{0}_{(r)}$ and $\mathcal{F}^{0}$ be the subspace of
$\mathcal{F}$ defined as

$\displaystyle\mathcal{F}^{0}_{(r)}:=\big{\\{}\psi_{n_{1}}\dots\psi_{n_{r}}|-r\rangle\in\mathcal{F}\
;\ n_{1},n_{2},\dots,n_{r}\in\mathord{\mathbb{Z}}\big{\\}},$
$\displaystyle\mathcal{F}^{0}:=\bigcup_{r=0}^{\infty}\mathcal{F}^{0}_{(r)}.$

Since $\psi_{-r}|-r\rangle=|-r+1\rangle$, there exists the natural inclusion

(14)
$\mathcal{F}^{0}_{(0)}\subset\mathcal{F}^{0}_{(1)}\subset\cdots\subset\mathcal{F}^{0}_{(r)}\subset\cdots\subset\mathcal{F}^{0}.$

The boson-fermion correspondence is the homomorphism

(15) $|v\rangle\mapsto\langle 0|e^{H(x)}|v\rangle,\quad x=(x_{1},x_{2},\dots)$

from $\mathcal{F}^{0}$ to the ring of symmetric functions $\Lambda(x)$ in
$x_{1},x_{2},\dots$. This map is in fact isomorphic [21, Lemma 9.5], [10,
Theorem 6.1].

For a partition $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$, that
is, a weakly decreasing sequence of nonnegative integers, we let

(16)
$|\lambda\rangle:=|\lambda_{1},\dots,\lambda_{r}\rangle=\psi_{\lambda_{1}-1}\psi_{\lambda_{2}-2}\cdots\psi_{\lambda_{r}-r}|-r\rangle\in\mathcal{F}^{0}_{(r)}.$

The image of $|\lambda\rangle$ under (15) coincides with the Schur function
$s_{\lambda}(x)$. This implies the fact that the composition

$\mathcal{F}^{0}_{(r)}\hookrightarrow\mathcal{F}^{0}\to\Lambda(x)\stackrel{{\scriptstyle
x_{r+1}=x_{r+2}=\cdots=0}}{{\to}}\Lambda(x_{1},\dots,x_{r})$

is isomorphic, where $\Lambda(x_{1},\dots,x_{r})$ is the ring of symmetric
polynomials in $x_{1},\dots,x_{r}$.

By replacing $h_{i}(x)$ with $h_{i}(x/y)$, we obtain the following theorem:

######  Theorem 2.3.

We have $s_{\lambda}(x/y)=\langle 0|e^{H(x/y)}|\lambda\rangle$, where
$s_{\lambda}(x/y)$ is the supersymmetric Schur function labeled by $\lambda$.

We always identify the partition $(\lambda_{1},\dots,\lambda_{r},0)$ with
$(\lambda_{1},\dots,\lambda_{r})$. This is compatible with the natural
inclusion (14) since
$|\lambda_{1},\dots,\lambda_{r-1},0\rangle=|\lambda_{1},\dots,\lambda_{r-1}\rangle$.
We let $\ell(\lambda)$ be the length of $\lambda$, that is, the number of non-
zero entries of $\lambda$.

###  2.3. Multi-Schur functions

Let $\psi(z)=\sum_{n\in\mathord{\mathbb{Z}}}\psi_{n}z^{n}$ and
$\psi^{\ast}(z)=\sum_{n\in\mathord{\mathbb{Z}}}\psi^{\ast}_{n}z^{n}$ be the
generating functions of $\psi_{n}$ and $\psi^{\ast}_{n}$. They satisfy the
commutation relations $[a_{n},\psi(z)]=z^{n}\psi(z)$ and
$[a_{n},\psi^{\ast}(z)]=-z^{-n}\psi^{\ast}(z)$, which are given from (6). By
using (13), we have

(17) $\displaystyle e^{H(x/y)}\psi(z)e^{-H(x/y)}$
$\displaystyle=\left(\sum_{i=0}^{\infty}h_{i}(x/y)z^{i}\right)\cdot\psi(z)$
$\displaystyle=\prod_{i=1}^{m}\frac{1}{1-x_{i}z}\prod_{j=1}^{n}(1-y_{j}z)\cdot\psi(z)$

and

(18) $\displaystyle e^{H(x/y)}\psi^{\ast}(z)e^{-H(x/y)}$
$\displaystyle=\left(\sum_{i=0}^{\infty}h_{i}(y/x)z^{-i}\right)\cdot\psi^{\ast}(z)$
$\displaystyle=\prod_{i=1}^{m}(1-x_{i}z^{-1})\prod_{j=1}^{n}\frac{1}{1-y_{j}z^{-1}}\cdot\psi^{\ast}(z).$

Let ${\bf x}=(x^{(1)},x^{(2)},\dots)$ and ${\bf y}=(y^{(1)},y^{(2)},\dots)$ be
tuples of tuples where

$x^{(i)}=(x^{(i)}_{1},x^{(i)}_{2},\dots,x^{(i)}_{M_{i}}),\quad
y^{(i)}=(y^{(i)}_{1},y^{(i)}_{2},\dots,y^{(i)}_{N_{i}})$

are tuples of finitely many indeterminates (see Section 1). For
$\ell(\lambda)\leq r$, we define the polynomial $S_{\lambda}({\bf x}/{\bf
y})=S_{\lambda}(x^{(1)}/y^{(1)},x^{(2)}/y^{(2)},\dots,x^{(r)}/y^{(r)})$ by

(19) $S_{\lambda}({\bf x}/{\bf y}):=\\\ \langle
0|\left(e^{H(x^{(1)}/y^{(1)})}\psi_{\lambda_{1}-1}e^{-H(x^{(1)}/y^{(1)})}\right)\left(e^{H(x^{(2)}/y^{(2)})}\psi_{\lambda_{2}-2}e^{-H(x^{(2)}/y^{(2)})}\right)\\\
\dots\left(e^{H(x^{(r)}/y^{(r)})}\psi_{\lambda_{r}-r}e^{-H(x^{(r)}/y^{(r)})}\right)|-r\rangle.$

######  Proposition 2.4.

$S_{\lambda}({\bf x}/{\bf y})$ is equal to the determinant

(20) $\det\left(h_{\lambda_{i}-i+j}(x^{(i)}/y^{(i)})\right)_{1\leq i,j\leq
r}.$

###### Proof.

By Wick’s theorem 2.1, we rewrite the vacuum expectation value (19) as

$\displaystyle\det\left(\langle
0|e^{H(x^{(i)}/y^{(i)})}\psi_{\lambda_{i}-i}e^{-H(x^{(i)}/y^{(i)})}\psi_{-j}^{\ast}|0\rangle\right)_{1\leq
i,j\leq r}$
$\displaystyle=\det\left(\sum_{n=0}^{\infty}h_{n}(x^{(i)}/y^{(i)})\langle
0|\psi_{\lambda_{i}-i-n}\psi_{-j}^{\ast}|0\rangle\right)_{1\leq i,j\leq r}$
$\displaystyle=\det\left(h_{\lambda_{i}-i+j}(x^{(i)}/y^{(i)})\right)_{1\leq
i,j\leq r}.$

∎

The determinant (20) is nothing but the Jacobi-Trudi formula for the multi-
Schur function [14, §1.4] (see [24, §2.3] also)111In the terminology of
Lascoux’s textbook [14], the multi-Schur function $S_{\lambda}({\bf x}/{\bf
y})$ is expressed as
$S_{I}(\mathbb{A}_{1}-\mathbb{B}_{1},\mathbb{A}_{2}-\mathbb{B}_{2},\dots,\mathbb{A}_{r}-\mathbb{B}_{r})$
with $I=(\lambda_{r},\lambda_{r-1},\dots,\lambda_{1})$,
$\mathbb{A}_{i}=x^{(r-i+1)}$, and $\mathbb{B}_{i}=y^{(r-i+1)}$..

######  Example 2.5 (Supersymmetric Schur functions).

When

${\bf x}=(x,\emptyset,\emptyset,\dots),\quad{\bf
y}=(y,\emptyset,\emptyset,\dots),\quad\mbox{with}\quad x=(x_{1},\dots,x_{m}),\
y=(y_{1},\dots,y_{n}),$

the multi-Schur function $S_{\lambda}({\bf x}/{\bf y})$ reduces to the
supersymmetric Schur function $s_{\lambda}(x/y)$.

######  Example 2.6 (Flagged Schur functions).

Let $f=(f_{1}\leq f_{2}\leq\dots)$ be a flagging, that is, a weakly increasing
string of positive integers. When

${\bf x}=(x^{(1)},x^{(2)},\dots),\quad{\bf
y}=\mbox{\boldmath{$\emptyset$}},\quad\mbox{with}\quad
x^{(i)}=(x_{1},\dots,x_{f_{i}}),$

the multi-Schur function $S_{\lambda}({\bf x}/{\bf y})$ reduces to the flagged
Schur function $s_{\lambda}^{f}(x)$.

######  Example 2.7 (Refined dual Grothendieck functions).

Motegi-Scrimshaw [24, §3] proved that when

${\bf x}=(x^{(1)},x^{(2)},\dots),\quad{\bf
y}=\mbox{\boldmath{$\emptyset$}},\quad\mbox{with}\quad
x^{(i)}=(x_{1},\dots,x_{n},t_{1},\dots,t_{i}),$

the multi-Schur function $S_{\lambda}({\bf x}/{\bf y})$ reduces to the refined
dual Grothendieck polynomial $g_{\lambda}(x;t)$. It moreover reduces to the
ordinal dual Grothendieck polynomial when $t_{1}=\dots=t_{r}=-\beta$.

Since

(21) $e^{H(x/y)}|-r\rangle=|-r\rangle,$

the expression (19) is simply rewritten as

(22) $\langle
0|e^{H(u^{(1)}/v^{(1)})}\psi_{\lambda_{1}-1}e^{H(u^{(2)}/v^{(2)})}\psi_{\lambda_{2}-2}\cdots
e^{H(u^{(r)}/v^{(r)})}\psi_{\lambda_{r}-r}|-r\rangle,$

where $u^{(i)}=x^{(i)}\cup y^{(i-1)}$ and $v^{(i)}=x^{(i-1)}\cup y^{(i)}$.

######  Remark 2.8.

When $x^{(i)}=(x_{1},\dots,x_{n},\overbrace{-\beta,\dots,-\beta}^{i-1})$ and
${\bf y}=\mbox{\boldmath{$\emptyset$}}$, (22) reduces to

(23) $\langle
0|e^{H(x_{1},\dots,x_{n})}\psi_{\lambda_{1}-1}e^{H(-\beta)}\psi_{\lambda_{2}-2}\cdots
e^{H(-\beta)}\psi_{\lambda_{r}-r}|-r\rangle,$

which is exactly the same as the free fermionic presentation of the dual
Grothendieck polynomial $g^{\beta}_{\lambda}(x_{1},\dots,x_{n})$ in the
author’s previous paper [8, §4.1].

##  3\. Generalized Schur expansion

In the sequel, we fix the following four tuples of tuples

${\bf x}=(x^{(1)},x^{(2)},\dots),\ {\bf y}=(y^{(1)},y^{(2)},\dots),\ {\bf
p}=(p^{(1)},p^{(2)},\dots),\ {\bf q}=(q^{(1)},q^{(2)},\dots).$

Suppose that all of the indeterminates here are contained in the base field
$k$.

###  3.1. The symmetric function $s_{\lambda}(X)_{{\bf x}/{\bf y}}$

Let $\Lambda(X)$ be the $k$-algebra of symmetric functions in
$X_{1},X_{2},\dots$. There exists a nondegenerate $k$-bilinear form

$\Lambda(X)\otimes_{k}\Lambda(X)\to k;\quad f\otimes g\mapsto\langle
f,g\rangle\qquad\mbox{with}\quad\langle
s_{\lambda},s_{\mu}\rangle=\delta_{\lambda,\mu},$

that is called the Hall inner product.

######  Lemma 3.1.

Let $\\{|\lambda\rangle_{1}\\}_{\lambda}$ and
$\\{|\lambda\rangle_{2}\\}_{\lambda}$ be two $k$-basis of $\mathcal{F}^{0}$
labeled by all partitions $\lambda$ (possibly $\lambda=\emptyset$). Then the
following three conditions are equivalent:

  1. (i)

If $f_{\lambda}(X)=\langle 0|e^{H(X)}|\lambda\rangle_{1}$ and
$g_{\mu}(X)=\langle 0|e^{H(X)}|\mu\rangle_{2}$, then $\langle
f_{\lambda},g_{\mu}\rangle=\delta_{\lambda,\mu}$.

  2. (ii)

$\sum_{\lambda}f_{\lambda}(X)g_{\lambda}(Y)=\prod_{i,j}(1-X_{i}Y_{j})^{-1}$.

  3. (iii)

If $(|\mu\rangle_{2})^{\ast}={}_{2}\langle\mu|$, then
${}_{2}\langle\mu|\lambda\rangle_{1}=\delta_{\lambda,\mu}$.

###### Proof.

See [18, I.§4] for proofs. ∎

For a partition $\lambda$ and an integer $r\geq\ell(\lambda)$, let

(24) $|\lambda\rangle_{{\bf x}/{\bf
y}}:=\left(e^{H(x^{(1)}/y^{(1)})}\psi_{\lambda_{1}-1}e^{-H(x^{(1)}/y^{(1)})}\right)\left(e^{H(x^{(2)}/y^{(2)})}\psi_{\lambda_{2}-2}e^{-H(x^{(2)}/y^{(2)})}\right)\\\
\cdots\left(e^{H(x^{(r)}/y^{(r)})}\psi_{\lambda_{r}-r}e^{-H(x^{(r)}/y^{(r)})}\right)|-r\rangle.$

The definition (24) does not depend on the choice of $r$ because

$e^{H(x/y)}\psi_{-s}e^{-H(x/y)}|-s\rangle=\psi_{-s}|-s\rangle=|-s-1\rangle.$

Since the vector $|\lambda\rangle_{{\bf x}/{\bf y}}$ is expanded as

$|\lambda\rangle_{{\bf x}/{\bf
y}}=|\lambda\rangle+\sum_{\left|{\mu}\right|<\left|{\lambda}\right|}\alpha_{\mu}|\mu\rangle,\qquad\alpha_{\mu}\in
k,$

the set $\\{|\lambda\rangle_{{\bf x}/{\bf y}}\\}_{\lambda}$ forms a $k$-basis
of $\mathcal{F}^{0}$.

Define the new symmetric function $s_{\lambda}(X)_{{\bf x}/{\bf
y}}\in\Lambda(X)$ by

(25) $s_{\lambda}(X)_{{\bf x}/{\bf y}}:=\langle
0|e^{H(X)}|\lambda\rangle_{{\bf x}/{\bf y}}.$

When $X_{i}=0$ for all $i$, $s_{\lambda}(X)_{{\bf x}/{\bf y}}$ reduces to the
multi-Schur function $S_{\lambda}({\bf x}/{\bf y})$.

###  3.2. Refined bases and orthonormality

What we are interested in is the transition matrix between the two bases
$\\{|\lambda\rangle_{{\bf x}/{\bf y}}\\}$ and $\\{|\mu\rangle_{{\bf p}/{\bf
q}}\\}$ of $\mathcal{F}^{0}$. Unfortunately, we have no explicit formula for
general bases at this stage. Instead, we present a certain subclass of bases
whose transition matrices are directly calculated from our fermionic
presentation.

Set $t=(t_{1},t_{2},\dots)$. We consider the refined tuple $[{\bf
t}]=(t[1],t[2],\dots)$, which we have defined in (1). When ${\bf x}=[{\bf t}]$
and ${\bf y}=\mbox{\boldmath{$\emptyset$}}$, the vector $|\lambda\rangle_{{\bf
x}/{\bf y}}$ (24) reduces to

(26) $\begin{split}|\lambda\rangle_{[{\bf
t}]}=\psi_{\lambda_{1}-1}e^{H(t_{1})}\psi_{\lambda_{2}-2}e^{H(t_{2})}\cdots
e^{H(t_{r-1})}\psi_{\lambda_{r}-r}e^{H(t_{r})}|-r\rangle.\end{split}$

To discuss the “orthonormality” property, we also define the transposed vector

(27) ${}_{[{\bf
t}]}\langle\lambda|:=\langle-r|e^{-H(t_{r})}\psi^{\ast}_{\lambda_{r}-r}e^{-H(t_{r-1})}\cdots
e^{-H(t_{2})}\psi^{\ast}_{\lambda_{2}-2}e^{-H(t_{1})}\psi^{\ast}_{\lambda_{1}-1}$

for sufficiently large $r$. The definition (27) may seem to depend on the
choice of $r$. However, we will prove later that it is in fact independent of
the choice of $r$ when $r$ is sufficiently large. See Remark 4.8.

The following is the main theorem of this paper:

######  Theorem 3.2 (Orthonormality of refined basis).

We have

${}_{[{\bf t}]}\langle\mu|\lambda\rangle_{[{\bf t}]}=\delta_{\lambda,\mu}.$

###### Proof.

The proof is essentially the same as the proof of [8, Proposition 4.1]. We
here note the equations

(28) $\displaystyle
e^{-H(t_{i})}\psi_{\lambda_{i}-i}e^{H(t_{i})}=\psi_{\lambda_{i}-i}-t_{i}\psi_{\lambda_{i}-i-1},$
(29) $\displaystyle
e^{-H(t_{i})}\psi^{\ast}_{\lambda_{i}-i}e^{H(t_{i})}=\psi^{\ast}_{\lambda_{i}-i}+t_{i}\psi^{\ast}_{\lambda_{i}-i+1}+t_{i}^{2}\psi^{\ast}_{\lambda_{i}-i+2}+\cdots,$

which are obtained from (17–18).

Let $0\leq s\leq r$. We temporally use the notation

$\begin{split}|\lambda,s\rangle_{[{\bf
t}]}=\psi_{\lambda_{1}-1}e^{H(t_{1})}\psi_{\lambda_{2}-2}e^{H(t_{2})}\cdots\psi_{\lambda_{s}-s}e^{H(t_{s})}|-r\rangle\end{split}$

and prove the theorem by induction on $s$. The theorem follows from the three
facts:

  1. (I).

If $N\geq\lambda_{1}$, then $\psi^{\ast}_{N}|\lambda,s\rangle_{[{\bf t}]}=0$.

  2. (II).

If $N\geq\mu_{1}$, then ${}_{[{\bf t}]}\langle\mu,s|\psi_{N}=0$.

  3. (III).

If $\lambda_{1}=\mu_{1}$,
$\lambda^{\prime}=(\lambda_{2}-1,\dots,\lambda_{r}-1)$,
$\mu^{\prime}=(\mu_{2}-1,\dots,\mu_{r}-1)$, and ${\bf
t}^{\prime}=(t_{2},t_{3},\dots)$. Then

${}_{[{\bf t}]}\langle\mu,s|\lambda,s\rangle_{[{\bf t}]}={}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}.$

(I) follows from the fact that $|\lambda,s\rangle_{[{\bf t}]}$ is a linear
combination of vectors of the form
$\psi_{n_{1}}\psi_{n_{2}}\dots\psi_{n_{s}}|-r\rangle$ with $N>n_{i}$ for each
$i$. We prove (II) by induction on $s$. If $s=0$, (II) is obvious. For general
$s>0$, we have

$\displaystyle{}_{[{\bf t}]}\langle\lambda,s|\psi_{N}$
$\displaystyle={}_{[{\bf
t}]}\langle\lambda^{\prime},s-1|e^{-H(t_{1})}\psi^{\ast}_{\mu_{1}-1}\psi_{N}$
$\displaystyle=-{}_{[{\bf
t}]}\langle\lambda^{\prime},s-1|e^{-H(t_{1})}\psi_{N}\psi^{\ast}_{\mu_{1}-1}$
$\displaystyle=-{}_{[{\bf
t}]}\langle\lambda^{\prime},s-1|(\psi_{N}-t_{1}\psi_{N-1})e^{-H(t_{1})}\psi^{\ast}_{\mu_{1}-1}.$

By induction hypothesis, the last expression should be zero, which concludes
(II). (III) is proved as

$\displaystyle{}_{[{\bf t}]}\langle\mu,s|\lambda,s\rangle_{[{\bf t}]}$
$\displaystyle={}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|e^{-H(t_{1})}\psi^{\ast}_{\mu_{1}-1}\psi_{\lambda_{1}-1}e^{H(t_{1})}|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}$ $\displaystyle={}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|e^{-H(t_{1})}(1-\psi_{\lambda_{1}-1}\psi^{\ast}_{\mu_{1}-1})e^{H(t_{1})}|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}$ $\displaystyle={}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}-{}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|e^{-H(t_{1})}\psi_{\lambda_{1}-1}\psi^{\ast}_{\mu_{1}-1}e^{H(t_{1})}|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}$ $\displaystyle={}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}$ $\displaystyle\hskip
30.00005pt-\sum_{m=0}^{\infty}t_{1}^{m}\cdot{}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|e^{-H(t_{1})}\psi_{\lambda_{1}-1}e^{H(t_{1})}\psi^{\ast}_{\mu_{1}-1+m}|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}$
$\displaystyle\stackrel{{\scriptstyle\mathrm{(I)}}}{{=}}{}_{[{\bf
t}^{\prime}]}\langle\mu^{\prime},s-1|\lambda^{\prime},s-1\rangle_{[{\bf
t}^{\prime}]}.$

By using (I–III) repeatedly, we conclude the proof. ∎

Theorem 3.2 provides a straightforward method for giving the linear expansion
$|\lambda\rangle_{{\bf x}/{\bf
y}}=\sum_{\mu}C_{\lambda}^{\mu}|\mu\rangle_{[{\bf t}]}$. By using the
orthonormality, we obtain the simple formula

(30) $C_{\lambda}^{\mu}={}_{[{\bf t}]}\langle\mu|\lambda\rangle_{{\bf x}/{\bf
y}},$

which implies the generalized Schur expansion

(31) $s_{\lambda}(X)_{{\bf x}/{\bf y}}=\sum_{\mu}{}_{[{\bf
t}]}\langle\mu|\lambda\rangle_{{\bf x}/{\bf y}}\cdot s_{\mu}(X)_{[{\bf t}]}.$

######  Example 3.3 (Schur expansion of $s_{\lambda}(X)_{{\bf x}/{\bf y}}$).

If $t=(0,0,0,\dots)$, then $|\lambda\rangle_{[{\bf t}]}=|\lambda\rangle$.
Assuming $r\geq\max[\ell(\lambda),\ell(\mu)]$, we obtain the determinantial
formula

$\displaystyle C_{\lambda}^{\mu}=\langle{\mu}|\lambda\rangle_{{\bf x}/{\bf
y}}$
$\displaystyle=\det\left(\sum_{l=0}^{\infty}h_{l}(x^{(i)}/y^{(i)})\cdot\langle-r|\psi^{\ast}_{\mu_{j}-j}\psi_{\lambda_{i}-i-l}|-r\rangle\right)_{1\leq
i,j\leq r}$
$\displaystyle=\det\left(h_{\lambda_{i}-\mu_{j}-i+j}(x^{(i)}/y^{(i)})\right)_{1\leq
i,j\leq r}.$

Since $\lambda_{i}=0$ for $i>\ell(\lambda)$ and $h_{n}(x^{(i)}/y^{(i)})=0$ for
$n<0$, it follows that
$\mu\not\subset\lambda\Rightarrow\langle{\mu}|\lambda\rangle_{{\bf x}/{\bf
y}}=0$. This fact gives the Schur expansion

(32) $s_{\lambda}(X)_{{\bf x}/{\bf
y}}=\sum_{\mu\subset\lambda}\det\left(h_{\lambda_{i}-\mu_{j}-i+j}(x^{(i)}/y^{(i)})\right)_{1\leq
i,j\leq\ell(\lambda)}s_{\mu}(X).$

######  Example 3.4 (Schur expansion of refined dual Grothendieck functions).

Let $r\geq\ell(\lambda)$. By substituting ${\bf x}=[{\bf t}]$ and ${\bf
y}=\mbox{\boldmath{$\emptyset$}}$ to (32), we have the Schur expansion

$g_{\lambda}(X;t_{1},\dots,t_{r-1})=\sum_{\mu\subset\lambda}\det\left(h_{\lambda_{i}-\mu_{j}-i+j}(t_{1},\dots,t_{i-1})\right)_{1\leq
i,j\leq\ell(\lambda)}s_{\mu}(X).$

######  Remark 3.5.

The function

$S_{\lambda/\mu}({\bf x}/{\bf
y}):=\det\left(h_{\lambda_{i}-\mu_{j}-i+j}(x^{(i)}/y^{(i)})\right)_{i,j}$

is called the skew multi-Schur function [14, §1.4].

######  Example 3.6 (Expansion of $s_{\lambda}(X)_{{\bf x}/{\bf y}}$ in
refined dual Grothendieck functions).

Let $t=(t_{1},t_{2},\dots)$ and $r\geq\max[\ell(\lambda),\ell(\mu)]$. Then we
have

$\displaystyle{}_{[{\bf t}]}\langle{\mu}|\lambda\rangle_{{\bf x}/{\bf y}}$
$\displaystyle=\det\left(\sum_{m=0}^{\infty}\sum_{l=0}^{\infty}h_{m}(t_{j},t_{j+1},\dots,t_{r})\cdot
h_{l}(x^{(i)}/y^{(i)}\cup(t_{1},\dots,t_{r}))\cdot\right.$
$\displaystyle\hskip
180.00027pt\left.\langle-r|\psi^{\ast}_{\mu_{j}-j+m}\psi_{\lambda_{i}-i-l}|-r\rangle\right)_{1\leq
i,j\leq r}$
$\displaystyle=\det\left(\sum_{\begin{subarray}{c}l+m=\lambda_{i}-\mu_{j}-i+j\\\
0\leq l\leq\lambda_{i}-i+r\end{subarray}}h_{m}(t_{j},t_{j+1},\dots,t_{r})\cdot
h_{l}(x^{(i)}/(y^{(i)}\cup(t_{1},\dots,t_{r})))\right)_{1\leq i,j\leq r}$
$\displaystyle=\det\left(h_{\lambda_{i}-\mu_{j}-i+j}\left(x^{(i)}/(y^{(i)}\cup(t_{1},\dots,t_{j-1}))\right)\right)_{1\leq
i,j\leq r},$

which implies $\mu\not\subset\lambda\Rightarrow{}_{[{\bf
t}]}\langle{\mu}|\lambda\rangle_{{\bf x}/{\bf y}}=0$. Finally, we have the
generalized Schur expansion

$\displaystyle s_{\lambda}(X)_{{\bf x}/{\bf y}}=$
$\displaystyle\sum_{\mu\subset\lambda}\det\left(h_{\lambda_{i}-\mu_{j}-i+j}\left(x^{(i)}/(y^{(i)}\cup(t_{1},\dots,t_{j-1}))\right)\right)_{1\leq
i,j\leq\ell(\lambda)}\cdot g_{\mu}(X;t_{1},\dots,t_{r-1}).$

##  4\. Dual symmetric functions

This section aims to study the dual functions of multi-Schur functions.
Unfortunately, since $\Lambda(X)$ is infinite-dimensional, not every linear
basis has its dual in $\Lambda(X)$. Thus we often need to use the extended
ring $\widehat{\Lambda}(X)$ (see [7], for example) to obtain desired dual
functions. As the ring $\widehat{\Lambda}(X)$ has a linear topology and is
complete, an element of $\widehat{\Lambda}(X)$ can be specified by giving a
convergent sequence of symmetric functions.

###  4.1. $r$-truncated dual function $s^{r}_{\lambda}(X)^{\bf x}$

We will use the “dual operator”

$H^{\ast}(x/y):=\sum_{n>0}\frac{p_{n}(x/y)}{n}a_{-n}$

to define dual functions. However, since the sum

$e^{-H^{\ast}(x/y)}\psi_{n}e^{H^{\ast}(x/y)}=\psi_{n}-e_{1}(x/y)\psi_{n+1}+e_{2}(x/y)\psi_{n+2}+\cdots$

is finite only if $y=\emptyset$, we should assume $y$ to be empty to have a
well-defined linear map from $\mathcal{F}$ to itself.

For $r\geq\ell(\lambda)$, set

(33) $\begin{split}|\lambda\rangle_{r}^{{\bf
x}}:=\left(e^{-H^{\ast}(x^{(1)})}\psi_{\lambda_{1}-1}e^{H^{\ast}(x^{(1)})}\right)\left(e^{-H^{\ast}(x^{(2)})}\psi_{\lambda_{2}-2}e^{H^{\ast}(x^{(2)})}\right)\\\
\cdots\left(e^{-H^{\ast}(x^{(r)})}\psi_{\lambda_{r}-r}e^{H^{\ast}(x^{(r)})}\right)|-r\rangle.\end{split}$

Since all the expressions on the right hand side of (33) are finite sums, the
$|\lambda\rangle_{r}^{{\bf x}}$ is contained in the Fock space $\mathcal{F}$.
Let us define the new symmetric function

$s^{r}_{\lambda}(X)^{{\bf x}}:=\langle 0|e^{H(X)}|\lambda\rangle_{r}^{{\bf
x}}.$

We call $s^{r}_{\lambda}(X)^{{\bf x}}$ the $r$-truncated dual function (see
[7, Section 3]).

######  Example 4.1 (Truncated Grothendieck function).

Let $t=(-\beta,-\beta,\dots)$. Then $s^{r}_{\lambda}(X)^{[{\bf t}]}$ coincides
with the $r$-truncated Grothendieck function $G^{r}_{\lambda}(X)$ introduced
in [8, §3.1]. Even if $r$ tends to be sufficiently large, the sequence of
$r$-truncated Grothendieck functions does not stable: $G^{r}_{\lambda}(X)\neq
G^{r+1}_{\lambda}(X)\neq G^{r+2}_{\lambda}(X)\neq\cdots$. Instead, the
sequence converges to some element in the topological ring
$\widehat{\Lambda}(X)$. The limit
$G_{\lambda}(X):=\lim\limits_{r\to\infty}G_{\lambda}^{r}(X)$ is called the
stable Grothendieck polynomial.

######  Example 4.2 (Schur expansion of $s^{r}_{\lambda}(X)^{{\bf x}}$).

For ${\bf x}=(x^{(1)},x^{(2)},\dots)$, let

$\overline{x}^{(i)}=\begin{cases}x^{(i)},&1\leq i\leq r,\\\
\emptyset,&i>r\end{cases}$

be a “$r$-truncated” tuple. For any $R\geq\max[\ell(\mu),r]$, we have

$\displaystyle\langle\mu|\lambda\rangle_{r}^{{\bf x}}$
$\displaystyle=\det\left(\sum_{m=0}^{\infty}(-1)^{m}e_{m}(\overline{x}^{(i)})\langle-R|\psi^{\ast}_{\mu_{j}-j}\psi_{\lambda_{i}-i+m}|-R\rangle\right)_{1\leq
i,j\leq R}$
$\displaystyle=\det\left(e_{-\lambda_{i}+\mu_{j}+i-j}(-\overline{x}^{(i)})\right)_{1\leq
i,j\leq R}.$

Because $e_{n}(\emptyset)=\delta_{n,0}$ and $\lambda_{i}=0$ $(i>r)$, the
determinant $\langle\mu|\lambda\rangle_{r}^{{\bf x}}$ must vanish if
$\ell(\mu)>r$. Hence we have the Schur expansion

$s^{r}_{\lambda}(X)^{{\bf x}}=\sum_{\begin{subarray}{c}\mu\supset\lambda\\\
\ell(\mu)\leq
r\end{subarray}}\det\left(e_{-\lambda_{i}+\mu_{j}+i-j}(-x^{(i)})\right)_{1\leq
i,j\leq r}s_{\mu}(X).$

######  Example 4.3 (Schur expansion of $s^{r}_{\lambda}(X)^{[{\bf t}]}$).

Assume $t=(t_{1},t_{2},\dots)$ and $r\geq\ell(\lambda)$. We call
$G^{r}_{\lambda}(X;t_{1},\dots,t_{r-1}):=s^{r}_{\lambda}(X)^{[{\bf t}]}$ the
$r$-truncated refined Grothendieck function. From Example 4.2, we have the
Schur expansion

$G^{r}_{\lambda}(X;t_{1},\dots,t_{r-1})=\sum_{\begin{subarray}{c}\mu\supset\lambda\\\
\ell(\mu)\leq
r\end{subarray}}\det\left(e_{-\lambda_{i}+\mu_{j}+i-j}(-t_{1},\dots,-t_{i-1})\right)_{1\leq
i,j\leq r}s_{\mu}(X).$

###  4.2. $r$-th stable dual function $s^{[r]}_{\lambda}(X)^{\bf x}$

Seeing Example 4.1, one might expect that the sequence

(34) $s^{1}_{\lambda}(X)^{{\bf x}},s^{2}_{\lambda}(X)^{{\bf x}},\dots$

converges in $\widehat{\Lambda}(X)$, but this is not the case for general
${\bf x}$. Here we do not go into details of the topology of
$\widehat{\Lambda}(X)$ but just recall the following fact:

######  Lemma 4.4.

Let $f_{1}(X),f_{2}(X),\dots\in\Lambda(X)$ be a sequence of symmetric
functions. Then the sequence converges in $\widehat{\Lambda}(X)$ if for any
$n>0$ there exists some $M>0$ such that

$i,j>M\quad\Rightarrow\quad
f_{i}(X_{1},\dots,X_{n},0,0,\dots)=f_{j}(X_{1},\dots,X_{n},0,0,\dots).$

###### Proof.

See [7, §3.1]. ∎

For $r\geq\ell(\lambda)$, we define the vector

(35) $\begin{split}|\lambda\rangle^{{\bf
x}}_{[r]}:=\left(e^{-H^{\ast}(x^{(1)})}\psi_{\lambda_{1}-1}e^{H^{\ast}(x^{(1)})}\right)\left(e^{-H^{\ast}(x^{(2)})}\psi_{\lambda_{2}-2}e^{H^{\ast}(x^{(2)})}\right)\\\
\cdots\left(e^{-H^{\ast}(x^{(r)})}\psi_{\lambda_{r}-r}e^{H^{\ast}(x^{(r)})}\right)e^{-H^{\ast}(x^{(r+1)})}|-r\rangle,\end{split}$

which “approximates” $|\lambda\rangle^{\bf x}_{r}$. Note that
$|\lambda\rangle^{\bf x}_{r}$ and $|\lambda\rangle^{\bf x}_{[r]}$ are indeed
different since $e^{-H^{\ast}(x^{(r+1)})}|-r\rangle\neq|-r\rangle$ in general.
Furthermore, $|\lambda\rangle^{{\bf x}}_{[r]}$ is not always an element of
$\mathcal{F}$ because $e^{-H^{\ast}(x^{(r+1)})}|-r\rangle$ may not be a finite
sum of vectors (3).

Let us introduce the “formal” symmetric function

$s^{[r]}_{\lambda}(X)^{\bf x}:=\langle 0|e^{H(X)}|\lambda\rangle_{[r]}^{\bf
x}.$

In the sequel, we prove that $s^{[r]}_{\lambda}(X)^{\bf x}$ is an element of
$\widehat{\Lambda}(X)$.

######  Lemma 4.5 ([8, Lemma 3.3], [7, Lemma 3.7]).

Let $x=(x_{1},\dots,x_{p})$ be a tuple of $p$ indeterminates. Suppose $r>0$.
Then there exists a certain sequence $Y_{1},Y_{2},\dots\in\mathcal{A}$ that
satisfies the following properties:

  1. (1)

The vector $e^{-H^{\ast}(x)}|-r\rangle$ is formally expanded as

$e^{-H^{\ast}(x)}|-r\rangle=|-r\rangle+\sum_{i=1}^{\infty}Y_{i}|-r-i\rangle.$

  2. (2)

For any $m_{1},\dots,m_{r}\in\mathord{\mathbb{Z}}$, the vector
$\psi_{m_{1}}\dots\psi_{m_{r}}Y_{i}|-r-i\rangle$ is contained in
$\mathcal{F}^{0}_{(r+i+1)}\setminus\mathcal{F}^{0}_{(r+i)}$.

  3. (3)

The element $\psi_{-r+p-1}\cdots\psi_{-r+1}\psi_{-r}Y_{i}$ vanishes.

###### Proof.

The proof is a variant of [8, Lemma 3.3]. Here we present a sketch of the
proof. Let $Z_{r}:=\sum_{l=1}^{p}(-1)^{l}e_{l}(x_{1},\dots,x_{p})\psi_{-r+l}$.
Then we have $e^{-H^{\ast}(x)}\psi_{-n}=(\psi_{-n}+Z_{n})e^{-H^{\ast}(x)}$. By
using this, we can prove that

$Y_{i}:=(\psi_{-r-1}+Z_{r+1})(\psi_{-r-2}+Z_{r+2})\cdots(\psi_{-r-i+1}+Z_{r+i-1})Z_{r+i}$

satisfies the desired properties (1–3). For details, see [8, §3]. ∎

By using Lemma 4.5 (1), we have the infinite series

(36) $\langle 0|e^{H(X)}|\lambda\rangle^{{\bf x}}_{[r]}=\langle
0|e^{H(X)}|\lambda\rangle^{{\bf x}}_{r}+\sum_{i=1}^{\infty}\langle
0|e^{H(X)}EY_{i}|-r-i\rangle,$

where

$E=\left(e^{-H^{\ast}(x^{(1)})}\psi_{\lambda_{1}-1}e^{H^{\ast}(x^{(1)})}\right)\cdots\left(e^{-H^{\ast}(x^{(r)})}\psi_{\lambda_{r}-r}e^{H^{\ast}(x^{(r)})}\right).$

Applying Lemma 4.5 (2) to (36), we obtain the formal series

(37) $s_{\lambda}^{[r]}(X)^{{\bf x}}=s_{\lambda}^{r}(X)^{{\bf
x}}+\sum_{i=1}^{\infty}o_{i}(X),\quad\mbox{where}\quad
o_{i}(X_{1},\dots,X_{r+i},0,0,\dots)=0.$

From Lemma 4.4, we find that the right hand side of (37) converges in
$\widehat{\Lambda}(X)$. Moreover, we have

(38) $s_{\lambda}^{[r]}(X_{1},\dots,X_{r},0,\dots)^{\bf
x}=s_{\lambda}^{r}(X_{1},\dots,X_{r},0,\dots)^{\bf x}.$

We call $s_{\lambda}^{[r]}(X)^{\bf x}$ the $r$-th stable dual function.

###  4.3. Stable dual function $s_{\lambda}(X)^{\bf x}$

Next, we consider letting $r$ tend to infinity.

######  Lemma 4.6 (See [8, Lemma 3.5]).

Let $t=(t_{1},t_{2},\dots,t_{p})$ be a tuple. For $n\in\mathord{\mathbb{Z}}$
and $p>0$, define

$I_{p}=\psi_{n}e^{-H^{\ast}(t_{1})}\psi_{n-1}e^{-H^{\ast}(t_{2})}\cdots\psi_{n-p+1}e^{-H^{\ast}(t_{p})}.$

Then we have

  1. (1)

$I_{p}=\psi_{n}\psi_{n-1}\cdots\psi_{n-p+1}e^{-H^{\ast}(t_{1},\dots,t_{p})}$,

  2. (2)

$I_{p}|n-p+1\rangle=|n+1\rangle$.

###### Proof.

We prove (1) by induction on $p>0$. For $p=1$, the equation is obvious. For
general $p>1$, we have

$\displaystyle I_{p+1}$ $\displaystyle=I_{p}\psi_{n-p}e^{-H^{\ast}(t_{p+1})}$
$\displaystyle=\psi_{n}\psi_{n-1}\cdots\psi_{n-p-1}e^{-H^{\ast}(t_{1},\dots,t_{p})}\psi_{n-p}e^{-H^{\ast}(t_{p+1})}\qquad(\mbox{induction
hypothesis})$
$\displaystyle=\psi_{n}\psi_{n-1}\cdots\psi_{n-p-1}\left\\{\sum_{l=0}^{p}(-1)^{l}e_{l}(t)\psi_{n-p+l}\right\\}e^{-H^{\ast}(t_{1},\dots,t_{p})}e^{-H^{\ast}(t_{p+1})}$
$\displaystyle=\psi_{n}\psi_{n-1}\dots\psi_{n-p}e^{-H^{\ast}(t_{1},\dots,t_{p},t_{p+1})},\qquad\
(\because\psi_{m}^{2}=0)$

which implies (1). (2) is proved by

$\displaystyle I_{p}|n-p+1\rangle$
$\displaystyle\stackrel{{\scriptstyle\mathrm{(1)}}}{{=}}\psi_{n}\psi_{n-1}\cdots\psi_{n-p+1}e^{-H^{\ast}(t_{1},\dots,t_{p})}|n-p+1\rangle$
$\displaystyle=\psi_{n}\psi_{n-1}\cdots\psi_{n-p+1}|n-p+1\rangle\qquad(\mbox{Lemma
\ref{lemma:technical_lemma_1}}.\,(1),(3))$ $\displaystyle=|n+1\rangle.$

∎

######  Lemma 4.7.

Let ${\bf x}=(x^{(1)},x^{(2)},\dots)$ be a tuple of tuples. Assume that there
exist some $R>0$ and $t_{1},t_{2},\dots\in k$ such that

($\ast$) $x^{(R+p)}\setminus x^{(R)}=(t_{1},\dots,t_{p})\qquad\mbox{for all
}\quad p>0$

Then we have $|\lambda\rangle^{{\bf x}}_{[r]}=|\lambda\rangle^{{\bf
x}}_{[r+1]}=|\lambda\rangle^{{\bf x}}_{[r+2]}=\cdots$ for sufficiently large
$r$.

###### Proof.

Let $n=-\max[R,\ell(\lambda)]-1$. Then if $r>-n$, we have

$\begin{split}|\lambda\rangle^{{\bf
x}}_{[r]}:=\left(e^{-H^{\ast}(x^{(1)})}\psi_{\lambda_{1}-1}e^{H^{\ast}(x^{(1)})}\right)\left(e^{-H^{\ast}(x^{(2)})}\psi_{\lambda_{2}-2}e^{H^{\ast}(x^{(2)})}\right)\\\
\cdots\left(e^{-H^{\ast}(x^{(n-1)})}\psi_{\lambda_{-n-1}+n+1}e^{H^{\ast}(x^{(n-1)})}\right)I_{p}|-r\rangle,\end{split}$

where $p=r+n+1$. From Lemma 4.6 (2), it follows that $|\lambda\rangle^{{\bf
x}}_{[r]}=|\lambda\rangle^{{\bf x}}_{[r+1]}=|\lambda\rangle^{{\bf
x}}_{[r+2]}=\cdots$. ∎

For ${\bf x}$ satisfying the assumption ($\ast$ ‣ 4.7) of Lemma 4.7, we write
$|\lambda\rangle^{{\bf x}}:=|\lambda\rangle^{{\bf x}}_{[r]}$ for sufficiently
large $r$. We define the dual stable function $s_{\lambda}(X)^{\bf x}$ as

$s_{\lambda}(X)^{{\bf x}}:=\langle 0|e^{H(X)}|\lambda\rangle^{{\bf x}}.$

From (37), we have

$s_{\lambda}(X)^{\bf x}=\lim_{r\to\infty}s_{\lambda}^{r}(X)^{\bf
x}=\lim_{r\to\infty}s_{\lambda}^{[r]}(X)^{\bf x}.$

Because any refined tuple $[{\bf t}]$ satisfies the assumption ($\ast$ ‣ 4.7),
there always exists the stable dual function $s_{\lambda}(X)^{[{\bf t}]}$
associated with $[{\bf t}]$.

######  Remark 4.8.

The vector ${}_{[{\bf t}]}\langle\lambda|$ in (27) is just the transpose of
$|\lambda\rangle^{[{\bf t}]}_{[r]}$. This implies the fact that (27) is
independent of the choice on sufficiently large $r$.

###  4.4. Duality between $s_{\lambda}(X)_{[{\bf t}]}$ and
$s_{\lambda}(X)^{[{\bf t}]}$

The Hall inner product can be extended continuously to the bilinear map
$\widehat{\Lambda}(X)\otimes_{k}\Lambda(X)\to k$. Lemma 3.1 is also available
for the extended bilinear form.

######  Proposition 4.9.

We have $\langle s_{\lambda}(X)^{[{\bf t}]},s_{\mu}(X)_{[{\bf
t}]}\rangle=\delta_{\lambda,\mu}$.

###### Proof.

It is a direct consequence of Lemma 3.1 and Theorem 3.2. ∎

Let ${}^{[{\bf t}]}\langle\lambda|=(|\lambda\rangle_{[{\bf t}]})^{\ast}$ be
the transposed vector. Since ${}^{[{\bf t}]}\langle\mu|\lambda\rangle^{[{\bf
t}]}={}_{[{\bf t}]}\langle\lambda|\mu\rangle_{[{\bf
t}]}=\delta_{\lambda,\mu}$, we have the Schur expansion

$s_{\lambda}(X)^{{\bf x}}=\sum_{\mu}{}^{[{\bf
t}]}\langle\mu|\lambda\rangle^{{\bf x}}\cdot s_{\mu}(X)^{[{\bf t}]}.$

######  Example 4.10 (Expansion of $s^{r}_{\lambda}(X)^{{\bf x}}$ in stable
refined Grothendieck polynomials).

Let ${\bf t}=(t_{1},t_{2},\dots)$, $r\geq\ell(\lambda)$, and
$R\geq\max[r,\ell(\mu)]$. Then the expansion of $|\lambda\rangle_{r}^{{\bf
x}}$ in $\\{|\mu\rangle^{[{\bf t}]}\\}$ can be calculated as follows:

$\displaystyle{}^{[{\bf t}]}\langle\mu|\lambda\rangle_{r}^{{\bf x}}$
$\displaystyle=\det\left(\sum_{m=0}^{\infty}\sum_{l=0}^{\infty}h_{l}(t_{1},\dots,t_{j-1})e_{m}(-\overline{x}^{(i)})\langle-R|\psi^{\ast}_{\mu_{j}-j-l}\psi_{\lambda_{i}-i+m}|-R\rangle\right)_{1\leq
i,j\leq R}$
$\displaystyle=\det\left(\sum_{\begin{subarray}{c}m+l=-\lambda_{i}+\mu_{j}+i-j\\\
0\leq
l\leq\mu_{j}-j+R\end{subarray}}h_{l}(t_{1},\dots,t_{j-1})e_{m}(-\overline{x}^{(i)})\right)_{1\leq
i,j\leq R}$
$\displaystyle=\det\left(h_{-\lambda_{i}+\mu_{j}+i-j}((t_{1},\dots,t_{j-1})/\overline{x}^{(i)})\right)_{1\leq
i,j\leq R}.$

Hence, we obtain the expansion

$\displaystyle s^{r}_{\lambda}(X)^{{\bf x}}=$
$\displaystyle\sum_{\mu\supset\lambda}\det\left(h_{-\lambda_{i}+\mu_{j}+i-j}((t_{1},\dots,t_{j-1})/\overline{x}^{(i)})\right)_{1\leq
i,j\leq\max[r,\ell(\mu)]}G_{\mu}(X;t_{1},\dots,t_{\ell(\mu)-1}),$

where $\overline{x}^{(i)}$ is the $r$-truncated tuple of $x^{(i)}$.

######  Example 4.11 (Expansion of $s_{\lambda}(X)^{{\bf x}}$ in stable
refined Grothendieck polynomials).

Let ${\bf x}$ be a tuple of tuples satisfying the assumption ($\ast$ ‣ 4.7) of
Lemma 4.7. Let $R$ be the positive integer in ($\ast$ ‣ 4.7). Then, for
$S\geq\max[R,\ell(\mu)]$, we have

$\displaystyle{}^{[{\bf t}]}\langle\mu|\lambda\rangle^{{\bf x}}$
$\displaystyle=\det\left(\sum_{m=0}^{\infty}\sum_{l=0}^{\infty}h_{l}((t_{1},\dots,t_{j-1})/x^{(S+1)})e_{m}(-x^{(i)}/-x^{(S+1)})\right.$
$\displaystyle\hskip
160.00024pt\left.\langle-S|\psi^{\ast}_{\mu_{j}-j-l}\psi_{\lambda_{i}-i+m}|-S\rangle\right)_{1\leq
i,j\leq S}$
$\displaystyle=\det\left(\sum_{\begin{subarray}{c}m+l=-\lambda_{i}+\mu_{j}+i-j\\\
0\leq
l\leq\mu_{j}-j+S\end{subarray}}h_{l}((t_{1},\dots,t_{j-1})/x^{(S+1)})e_{m}(-x^{(i)}/-x^{(S+1)})\right)_{1\leq
i,j\leq S}$
$\displaystyle=\det\left(h_{-\lambda_{i}+\mu_{j}+i-j}((t_{1},\dots,t_{j-1})/x^{(i)})\right)_{1\leq
i,j\leq S}.$

Hence, we have

$\displaystyle s_{\lambda}(X)^{{\bf x}}=$
$\displaystyle\sum_{\mu\supset\lambda}\det\left(h_{-\lambda_{i}+\mu_{j}+i-j}((t_{1},\dots,t_{j-1})/x^{(i)})\right)_{1\leq
i,j\leq\max[R,\ell(\mu)]}G_{\mu}(X;t_{1},\dots,t_{\ell(\mu)-1}).$

######  Example 4.12 (stable Grothendieck polynomials).

Let $t=(-\beta,-\beta,\dots)$. Then $s_{\lambda}(X)^{[{\bf t}]}$ coincides
with the stable Grothendieck polynomial $G_{\lambda}(X)$. From Example 4.11,
we have the Schur expansion

$\displaystyle G_{\lambda}(X)$
$\displaystyle=\sum_{\mu\supset\lambda}\det\left(h_{-\lambda_{i}+\mu_{j}+i-j}(\emptyset/(\overbrace{-\beta,\dots,-\beta}^{i-1}))\right)_{1\leq
i,j\leq\ell(\mu)}s_{\mu}(X)$
$\displaystyle=\sum_{\mu\supset\lambda}\det\left(h_{-\lambda_{i}+\mu_{j}+i-j}(\emptyset/(\overbrace{-\beta,\dots,-\beta}^{i-1}))\right)_{1\leq
i,j\leq\ell(\mu)}s_{\mu}(X)$
$\displaystyle=\sum_{\mu\supset\lambda}\det\left(\binom{i-1}{-\lambda_{i}+\mu_{j}+i-j}\beta^{-\lambda_{i}+\mu_{j}+i-j}\right)_{1\leq
i,j\leq\ell(\mu)}s_{\mu}(X).$

##  5\. Multi-Schur functions for skew shapes

In this section, we introduce a multi-Schur function corresponding to a skew
shape. By the fermionic presentation, we can easily prove the branching
formula, which is a desirable property for skew symmetric functions.

Let ${\bf p}$ be a tuple of tuples satisfying the assumption ($\ast$ ‣ 4.7) in
Lemma 4.7.

######  Definition 5.1.

We define the symmetric function $s_{\lambda/\mu}(X)_{({\bf x}/{\bf y})/{\bf
p}}$ by

$s_{\lambda/\mu}(X)_{({\bf x}/{\bf y})/{\bf p}}={}_{{\bf
p}}\langle\mu|e^{H(X)}|\lambda\rangle_{{\bf x}/{\bf y}}.$

Let $X=(X_{1},X_{2},\dots)$ and $Y=(Y_{1},Y_{2},\dots)$ be two strings of
indeterminates. For a symmetric function $f(X)=f(X_{1},X_{2},\dots)$, we write

$f(X,Y):=f(X_{1},X_{2},\dots,Y_{1},Y_{2},\dots).$

######  Theorem 5.2.

Let $t=(t_{1},t_{2},\dots)$. If ${\bf p}=[{\bf t}]$, we have

(39) $s_{\lambda}(X,Y)_{{\bf x}/{\bf y}}=\sum_{\mu}s_{\lambda/\mu}(X)_{({\bf
x}/{\bf y})/[{\bf t}]}\cdot s_{\mu}(Y)_{[{\bf t}]}$

###### Proof.

Since any $|v\rangle\in\mathcal{F}^{0}$ is decomposed as
$|v\rangle=\sum_{\mu}|\mu\rangle_{[{\bf t}]}\cdot{}_{[{\bf
t}]}\langle\mu|v\rangle$, we have $s_{\lambda}(X,Y)_{{\bf x}/{\bf y}}=\langle
0|e^{H(Y)}e^{H(X)}|\lambda\rangle_{{\bf x}/{\bf y}}=\sum_{\mu}\langle
0|e^{H(Y)}|\mu\rangle_{[{\bf t}]}\cdot{}_{[{\bf
t}]}\langle\mu|e^{H(X)}|\lambda\rangle_{{\bf x}/{\bf y}}$, which implies the
desired formula. ∎

If ${\bf x}={\bf p}=[{\bf t}]$ and ${\bf y}=\mbox{\boldmath{$\emptyset$}}$, we
write

(40) $s_{\lambda/\mu}(X)_{[{\bf t}]}:=s_{\lambda/\mu}(X)_{[{\bf t}]/[{\bf
t}]}.$

Because ${}_{[{\bf t}]}\langle 0|=\langle 0|$ and

$s_{\lambda/\emptyset}(X)_{[{\bf t}]/[{\bf t}]}={}_{[{\bf t}]}\langle
0|e^{H(X)}|\lambda\rangle_{[{\bf t}]}=\langle 0|e^{H(X)}|\lambda\rangle_{[{\bf
t}]}=s_{\lambda}(X)_{[{\bf t}]},$

the notation (40) does not cause any confusion.

######  Corollary 5.3 (branching formula).

We have

$s_{\lambda}(X,Y)_{[{\bf t}]}=\sum_{\mu}s_{\lambda/\mu}(X)_{[{\bf t}]}\cdot
s_{\mu}(Y)_{[{\bf t}]}.$

From Corollary 5.3, we would say that “$s_{\lambda/\mu}(X)_{[{\bf t}]}$ is a
skew version of the multi-Schur function”. However, we should note that there
exists another notion of “skew multi-Schur functions” derived from the Jacobi-
Trudi formula. See Remark 5.4.

######  Remark 5.4.

Now assume $X=0$ and ${\bf p}=\mbox{\boldmath{$\emptyset$}}$. In this case, we
have

$s_{\lambda/\mu}(0,0,\dots)_{({\bf x}/{\bf
y})/\mbox{\boldmath{$\emptyset$}}}=\langle\mu|e^{H(0,0,\dots)}|\lambda\rangle_{{\bf
x}/{\bf y}}=\langle\mu|\lambda\rangle_{{\bf x}/{\bf y}}=s_{\lambda/\mu}({\bf
x}/{\bf y}),$

where $s_{\lambda/\mu}({\bf x}/{\bf y})$ is the “skew multi-Schur function” in
Remark 3.5. Obviously, this fact does not fit with our definition of skew
multi-Schur function (40).

######  Proposition 5.5 (Determinantial formula for $s_{\lambda/\mu}(X)_{({\bf
x}/{\bf y})/{\bf p}}$).

Assume $r\geq\ell(\lambda)$. Then we have

$s_{\lambda/\mu}(X)_{({\bf x}/{\bf y})/{\bf
p}}=\det\left(\sum_{m+n=\lambda_{i}-\mu_{j}-i+j}h_{m}\left(x^{(i)}/(y^{(i)}\cup
p^{(j)})\right)h_{n}(X)\right)_{1\leq i,j\leq r}.$

###### Proof.

Assume $r\geq\ell(\lambda)$. By Wick’s theorem 2.1, we have

$\displaystyle s_{\lambda/\mu}(X)_{({\bf x}/{\bf y})/{\bf p}}$
$\displaystyle=\det\left(\langle-r|e^{H(p^{(j)})}\psi^{\ast}_{\mu_{j}-j}e^{-H(p^{(j)})}e^{H(X)}e^{H(x^{(i)}/y^{(i)})}\psi_{\lambda_{i}-i}e^{-H(x^{(i)}/y^{(i)})}|-r\rangle\right)$
$\displaystyle=\det\left(\sum_{m=0}^{\infty}\sum_{l=0}^{\infty}\sum_{n=0}^{\infty}h_{m}(\emptyset/p^{(j)})h_{l}(x^{(i)}/y^{(i)})h_{n}(X)\langle-r|\psi^{\ast}_{\mu_{j}-j+m}\psi_{\lambda_{i}-i-l-n}|-r\rangle\right)$
$\displaystyle=\det\left(\sum_{\begin{subarray}{c}m+l+n=\lambda_{i}-\mu_{j}-i+j\\\
l+n\leq\lambda_{i}-i+r\end{subarray}}h_{m}(\emptyset/p^{(j)})h_{l}(x^{(i)}/y^{(i)})h_{n}(X)\right)$
$\displaystyle=\det\left(\sum_{m+n=\lambda_{i}-\mu_{j}-i+j}h_{m}\left(x^{(i)}/(y^{(i)}\cup
p^{(j)})\right)h_{n}(X)\right).$

∎

######  Example 5.6 (Jacobi-Trudi formula for
$g_{\lambda/\mu}(X;t_{1},\dots,t_{r-1})$).

Let ${\bf x}={\bf p}=[{\bf t}]$ and ${\bf y}=\mbox{\boldmath{$\emptyset$}}$.
Assume $r\geq\ell(\lambda)$. Then Proposition 5.5 implies

$\displaystyle s_{\lambda/\mu}(X)_{[{\bf
t}]}=\det\left(\sum_{m=0}^{\infty}h_{m}\left((t_{1},\dots,t_{i-1})/(t_{1},\dots,t_{j-1})\right)h_{\lambda_{i}-\mu_{j}-i+j-m}(X)\right)_{1\leq
i,j\leq r},$

where

$h_{m}\left((t_{1},\dots,t_{i-1})/(t_{1},\dots,t_{j-1})\right)=\begin{cases}h_{m}(t_{i},t_{i+1},\dots,t_{j-1}),&i<j,\\\
\delta_{m,0},&i=j,\\\ e_{m}(-t_{j},-t_{j+1},\dots,-t_{i-1}),&i>j.\end{cases}$

This determinantal formula coincides with the Jacobi-Trudi formula of the
refined dual skew Grothendieck function given in [24, Theorem 4.15]. Thus we
have

(41) $s_{\lambda/\mu}(X)_{[{\bf t}]}=g_{\lambda/\mu}(X;t_{1},\dots,t_{r-1}).$

######  Example 5.7 (Jacobi-Trudi formula for $g_{\lambda/\mu}(X)$).

Put $t=(-\beta,-\beta,\dots)$. Since
$h_{m}(\overbrace{-\beta,\dots,-\beta}^{N})=\binom{m+N-1}{m}(-\beta)^{m}=\binom{-N}{m}\beta^{m}$
and $e_{m}(\overbrace{\beta,\dots,\beta}^{N})=\binom{N}{m}\beta^{m}$, we have
the Jacobi-Trudi formula

$\displaystyle
g_{\lambda/\mu}(X)=\det\left(\sum_{m=0}^{\infty}\binom{j-i}{m}\beta^{m}h_{\lambda_{i}-\mu_{j}-i+j-m}(X)\right).$

## Acknowledgments

The author is very grateful to Hiroshi Naruse for letting me know the
inspiring preprint [24]. This work is partially supported by JSPS Kakenhi
Grant Number 19K03605.

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