Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $\beta$ -Grothendieck polynomials

Jirui Guo School of Mathematical Sciences,
Institute for Advanced Study,
Key Laboratory of Intelligent Computing and Applications,
Tongji University, Shanghai 200092, China

(jrkwok@tongji.edu.cn)

Abstract

A GL $(n)$ quantum integrable system generalizing the asymmetric five vertex spin chain is shown to encode the ring relations of the equivariant quantum cohomology and equivariant quantum K-theory ring of flag varieties. We also show that the Bethe ansatz states of this system generate the double $\beta$ -Grothendieck polynomials.

1 Introduction

The mathematical structure and exact solutions of quantum integrable systems have proved to be useful tools in providing information about physical phenomena, as well as revealing certain algebraic results. For example, in [2], it was shown that the underlying Frobenius algebra of the asymmetric six vertex model, which depends on a parameter $\beta$ , is exactly the equivariant quantum cohomology of the Grassmannians when $\beta=0$ , and the equivariant quantum K-theory ring of the Grassmannians when $\beta=-1$ . The Bethe/Gauge correspondence of this model was later studied in [3].

This work aims at generalizing the idea of [2], establishing a relationship between a GL $(n)$ integrable model and the equivariant quantum cohomology/quantum K-theory ring of general flag varieties. We will also show that the Bethe ansatz states of this model generate the double $\beta$ -Grothendieck polynomials. When $\beta=0$ these polynomials reduce to the double Schubert polynomials representing Schubert classes in the equivariant cohomology ring, and when $\beta=-1$ they reduce to the double Grothendieck polynomials representing Schubert classes in the equivariant K-theory ring.

The relationships between the quantum cohomology/K-theory of flag varieties and Toda lattices were discussed in [4] and [5]. Moreover, it was shown that quantum K-theory ring of the cotangent bundle of flag varieties can be realized by the XXZ model, and in certain limit it reduces to the quantum K-theory ring of flag varieties [6]. Double $\beta$ -Grothendieck polynomials can also be realized by the partition functions of certain 2d lattice models [7, 8, 9].

Our main results are as follows:

•

The Bethe ansatz equations of the GL $(n)$ quantum integrable system defined by the R-matrix (2.3)

R^{(n)}(x,y)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&0&I_{n-1}&0\\ 0&(1+\beta(x\ominus y))I_{n-1}&(x\ominus y)I_{n-1}&0\\ 0&0&0&R^{(n-1)}(x,y)\end{array}\right),

where

R^{(2)}(x,y)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&0&1&0\\ 0&1+\beta(x\ominus y)&x\ominus y&0\\ 0&0&0&1\end{array}\right),

are derived (Theorem 3.1).

•

It is shown that the Bathe ansatz equations give rise to the Whitney type ring relations of the equivariant quantum cohomology ring of the partial flag variety $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ when $\beta=0$ in terms of the Chern classes (Eq.(3.22)):

c(\mathcal{S}_{i})\cdot c(\mathcal{S}_{i+1}/\mathcal{S}_{i})=c(\mathcal{S}_{i+1})+(-1)^{k_{n-i-1}-k_{n-i}}q_{n-i}c(\mathcal{S}_{i-1}),

and give rise to the ring relations of the equivariant quantum K-theory ring of $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ when $\beta=-1$ in terms of the $\lambda_{y}$ classes (Eq.(3.27)):

\lambda_{y}(\mathcal{S}_{i})*\lambda_{y}(\mathcal{S}_{i+1}/\mathcal{S}_{i})=\lambda_{y}(\mathcal{S}_{i+1})+\frac{q_{n-i}}{1-q_{n-i}}y^{k_{n-i-1}-k_{n-i}}\left(\lambda_{y}(\mathcal{S}_{i-1})-\lambda_{y}(\mathcal{S}_{i})\right)*\det(\mathcal{S}_{i+1}/\mathcal{S}_{i}),

where $\mathcal{S}_{i}$ is the $i$ th tautological bundle of the flag variety.

•

Given the matrix elements $B_{i}(u)$ of the monodromy matrix as in Eq.(2.22), and the pseudo vacuum state $\ket{\Omega^{(0)}}$ as in (3.2), the expansion coefficients in the natural basis of the quantum state $B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}$ are exactly the double $\beta$ -Grothendieck polynomials $\mathcal{G}^{(\beta)}_{\pi}$ (Theorem 4.6):

B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}=\sum_{\pi\in S_{N}}\mathcal{G}^{(\beta)}_{\pi}(\sigma_{1},\cdots,\sigma_{N-1};\ominus t_{1},\cdots,\ominus t_{N-1})\ket{\omega_{N}\pi^{-1}},

where $\omega_{N}$ is the permutation with maximal length in the permutation group $S_{N}$ .

•

When the elementary symmetric polynomial $e_{l}(\sigma^{(i)}_{1},\cdots,\sigma^{(i)}_{N-i})$ is identified with $e_{l}(x_{1},\cdots,x_{N-i})$ for all $i=1,\cdots,N-1$ and $l=1,\cdots,N-i$ , the Bethe ansatz state (3.4) (in the full flag case) has the expansion (Theorem 4.11):

	$\displaystyle\ket{\psi^{(0)}}$	$\displaystyle=\sum_{i_{1},\cdots,i_{k_{1}}}\psi_{i_{1}\cdots i_{k_{1}}}^{(1)}B_{i_{1}}(\sigma_{1}^{(1)})\cdots B_{i_{k_{1}}}(\sigma_{k_{1}}^{(1)})\ket{\Omega^{(0)}}$
		$\displaystyle=\sum_{w\in S_{N}}\mathcal{G}^{(\beta)}_{w}(x_{1},\cdots,x_{N-1};\ominus t_{1},\cdots,\ominus t_{N-1})\ket{\omega_{N}w^{-1}},$

where $x_{i}$ can be interpreted as the first Chern class of the line bundle $\mathcal{S}_{i}/\mathcal{S}_{i-1}$ when $\beta=0$ , and $1-x_{i}$ can be interpreted as the K-theory class of $\mathcal{S}_{i}/\mathcal{S}_{i-1}$ when $\beta=-1$ (This means $\sigma^{(N-i)}_{a}$ ( $\beta=0$ ) or $1-\sigma^{(N-i)}_{a}$ ( $\beta=-1$ ), $a=1,\cdots,i$ , should be interpreted as the Chern roots of $\mathcal{S}_{i}$ ).

This paper is organized as follows. In Sec. 2, the basic ingredients of the GL $(n)$ quantum integrable system generalizing the asymmetric five vertex spin chain are introduced. In Sec. 3, the Bethe ansatz equations are derived, which are shown to give rise to the ring relations of the equivariant quantum cohomology/K-theory ring of the flag varieties. Sec. 4 consists of a series of propositions leading to our main results on the relationship between the quantum integrable model under study and the double $\beta$ -Grothendieck polynomials. In the appendix, we present some background of the quantum cohomology/K-theory of flag varieties and the double $\beta$ -Grothendieck polynomials.

2 The GL $(n)$ asymmetric five vertex model

In this section, we present the basic ingredients of the GL $(n)$ asymmetric five vertex model. Let $V=\mathbb{C}^{n}$ , we define the R-matrix $R^{(n)}(x,y)\in\mathrm{End}(V\otimes V)$ as follows:
For $n=2$ , the R-matrix is defined as

R^{(2)}(x,y)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&0&1&0\\ 0&1+\beta(x\ominus y)&x\ominus y&0\\ 0&0&0&1\end{array}\right),

(2.1)

where $\beta$ is a formal variable, and

x\ominus y\equiv\frac{x-y}{1+\beta y}.

(2.2)

For $n>2$ , under the decomposition $V\cong\mathbb{C}\oplus\mathbb{C}^{n-1}$ , $R^{(n)}(x,y)$ is defined as follows:

R^{(n)}(x,y)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&0&I_{n-1}&0\\ 0&(1+\beta(x\ominus y))I_{n-1}&(x\ominus y)I_{n-1}&0\\ 0&0&0&R^{(n-1)}(x,y)\end{array}\right),

(2.3)

where $I_{r}$ stands for the identity matrix of dimension $r$ . The entries of the R-matrix can be written as

R^{(n)}_{ij,kl}(x,y)=(1+\delta_{k<l}\beta(x\ominus y))\delta_{il}\delta_{jk}+\delta_{k>l}(x\ominus y)\delta_{ik}\delta_{jl},

(2.4)

where $i,j$ are row indices, $k,l$ are colomn indices, $\{i,k\}$ and $\{j,l\}$ are indices for the first and second factor of $V\otimes V$ respectively. We also define

\delta_{i<j}=\left\{\begin{array}[]{ll}1,&~{}\mathrm{if}~{}i<j,\\ 0,&~{}\mathrm{if}~{}i\geq j,\end{array}\right.\quad\delta_{i>j}=\left\{\begin{array}[]{ll}1,&~{}\mathrm{if}~{}i>j,\\ 0,&~{}\mathrm{if}~{}i\leq j.\end{array}\right.

The R-matrix (2.3) appeared in [9] as the $q\rightarrow 0$ limit of a Drinfeld twist of the R-matrix for an evaluation module of $U_{q}(\widehat{\mathrm{sl}}_{n+1})$ , but here we show that $R^{(n)}(x,y)$ satisfies the Yang-Baxter equation by a direct computation:

Theorem 2.1.

The R-matrix defined by Eq.(2.3) satisfies the Yang-Baxter equation

R^{(n)}_{12}(x,y)R^{(n)}_{13}(x,z)R^{(n)}_{23}(y,z)=R^{(n)}_{23}(y,z)R^{(n)}_{13}(x,z)R^{(n)}_{12}(x,y)

(2.5)

for all $n>1$ , where $R^{(n)}_{ab}$ acts on the $a$ th and $b$ th factor of $V\otimes V\otimes V$ .

Proof.

A direct computation shows that Eq.(2.5) holds for $n=2$ . Now suppose $R^{(n-1)}$ satisfies the Yang-Baxter equation for some $n>2$ . Let $i_{ab}$ denote the identity map from the $a$ th factor to the $b$ th factor of $V\otimes V\otimes V$ . Plugging (2.3) into Eq.(2.5), we find that Eq.(2.5) decomposes into the following identities:

	$\displaystyle i_{21}i_{32}=i_{31},\quad i_{13}=i_{23}i_{12},$		(2.6)
	$\displaystyle i_{12}i_{31}i_{23}=i_{32}i_{13}i_{21},\quad i_{21}i_{13}i_{32}=i_{23}i_{31}i_{12},$		(2.7)
	$\displaystyle(x-y)i_{31}i_{23}+(y-z)i_{21}=i_{21}(x-z),\quad(x-z)i_{12}=(x-y)i_{32}i_{13}+(y-z)i_{12},$		(2.8)
	$\displaystyle R^{(n-1)}_{12}(x,y)i_{31}R^{(n-1)}_{23}(y,z)=i_{32}R^{(n-1)}_{13}(x,z)i_{21},$		(2.9)
	$\displaystyle i_{12}R^{(n-1)}_{13}(x,z)i_{23}=R_{23}^{(n-1)}(y,z)i_{13}R_{12}^{(n-1)}(x,y),$		(2.10)
	$\displaystyle(x-z)(1+\beta y)R_{12}^{(n-1)}(x,y)i_{32}=(y-z)(1+\beta x)i_{31}i_{12}+(x-y)(1+\beta z)i_{32}R_{13}^{(n-1)}(x,z),$		(2.11)
	$\displaystyle(y-z)(1+\beta x)i_{21}i_{13}+(x-y)(1+\beta z)R_{13}^{(n-1)}(x,z)i_{23}=(x-z)(1+\beta y)i_{23}R_{12}^{(n-1)}(x,y),$		(2.12)
	$\displaystyle R^{(n-1)}_{12}(x,y)R^{(n-1)}_{13}(x,z)R^{(n-1)}_{23}(y,z)=R^{(n-1)}_{23}(y,z)R^{(n-1)}_{13}(x,z)R^{(n-1)}_{12}(x,y).$		(2.13)

Eq.(2.6)-(2.8) are direct consequences of the definition of $i_{ab}$ . Eq.(2.9) and (2.10) can be easily derived from the identities

\sum_{k,l}R^{(m)}_{ij,lk}(x,y)R^{(m)}_{kl,st}(y,z)=R^{(m)}_{ij,st}(x,z)

(2.14)

and

\sum_{k,l}R^{(m)}_{ij,lk}(y,z)R^{(m)}_{kl,st}(x,y)=R^{(m)}_{ij,st}(x,z)

(2.15)

respectively, while Eq.(2.11) and (2.12) can be easily derived from

(y-z)(1+\beta x)R^{(m)}(x,x)+(x-y)(1+\beta z)R^{(m)}(x,z)=(x-z)(1+\beta y)R^{(m)}(x,y),

(2.16)

where $m\geq 2$ . Eq.(2.14)-(2.16) can all be proved by simple induction on $m$ . Finally, Eq.(2.13) is the induction hypothesis. Therefore, by induction, the Yang-Baxter equation holds for all $n\geq 2$ . ∎

To construct an integrable system, we take $N$ copies of $V\cong\mathbb{C}^{n}$ and label them by $V_{i},i=1,\cdots,N$ . Let $V_{a}\cong V$ be an auxiliary space. The monodromy matrix is defined by

T^{(n)}_{a}(u)=R_{aN}(u,t_{N})R_{a,N-1}(u,t_{N-1})\cdots R_{a2}(u,t_{2})R_{a1}(u,t_{1}),

(2.17)

where $R_{ai}$ acts on $V_{a}\otimes V_{i}$ . The Yang-Baxter equation (2.5) leads to the following RTT relation:

R^{(n)}_{ab}(x,y)T^{(n)}_{a}(x)T^{(n)}_{b}(y)=T^{(n)}_{b}(y)T^{(n)}_{a}(x)R^{(n)}_{ab}(x,y).

(2.18)

We define the transfer matrix to be

t^{(n)}(u)=\mathrm{Tr}_{V_{a}}(M_{n}T^{(n)}_{a}(u)),

(2.19)

where $M_{n}=\mathrm{diag}(b_{0},b_{1},\cdots,b_{n-1})$ acts on $V_{a}$ , imposing a twisted periodic boundary condition. In the following, we also need

M_{l}:=\mathrm{diag}(b_{n-l},\cdots,b_{n-1}),\quad l=1,\cdots,n.

(2.20)

Eq.(2.18) implies that¹¹1Notice that for any diagonal matrix $M$ acting on the auxiliary space, we have $R^{(n)}_{ab}(x,y)M_{a}M_{b}=M_{b}M_{a}R^{(n)}_{ab}(x,y)$ .

[t(u),t(v)]=0.

(2.21)

Under the decomposition $V_{a}=\mathbb{C}\oplus\mathbb{C}^{n-1}$ , the monodromy matrix can be written in the following form:

T^{(n)}_{a}(u)=\left(\begin{array}[]{cccc}A(u)&B_{1}(u)&\cdots&B_{n-1}(u)\\ C_{1}(u)&D_{11}(u)&\cdots&D_{1,n-1}(u)\\ \vdots&\vdots&\ddots&\vdots\\ C_{n-1}(u)&D_{n-1,1}(u)&\cdots&D_{n-1,n-1}(u)\end{array}\right)

(2.22)

with $A(u),B_{i}(u),C_{i}(u),D_{ij}(u)$ being operators acting on the Hilbert space $\mathcal{H}=V_{1}\otimes V_{2}\otimes\cdots\otimes V_{N}$ . The RTT relation (2.18) yields the following commutation ralations:

$\displaystyle A(x)B_{i}(y)$	$\displaystyle=\frac{1}{y\ominus x}\left[B_{i}(y)A(x)-B_{i}(x)A(y)\right],$	(2.23)
$\displaystyle D_{ij}(x)B_{k}(y)$	$\displaystyle=\sum_{s,l}\left[\frac{1}{x\ominus y}B_{s}(y)D_{il}(x)R^{(n-1)}_{ls,jk}(x,y)\right]+\frac{1}{y\ominus x}B_{j}(x)D_{ik}(y),$	(2.24)
$\displaystyle B_{i}(x)B_{j}(y)$	$\displaystyle=\sum_{r,s}B_{r}(y)B_{s}(x)R^{(n-1)}_{sr,ij}(x,y).$	(2.25)

3 Bethe ansatz equations and the equivariant quantum
cohomology/K-theory ring of flag varieties

In this section, we derive the Bethe ansatz equations of the GL $(n)$ five vertex model introduced in the last section following the method proposed in [10], and discuss their relationship with the equivariant quantum cohomology and quantum K-theory ring of flag varieties.

Let $V=\mathbb{C}^{n}$ be spanned by $v_{0},v_{1},\cdots,v_{n-1}$ , where $v_{i}:=\ket{i}$ is the $(i+1)$ th unit vector of $\mathbb{C}^{n}$ . Define

\ket{n_{1},n_{2},\cdots,n_{N}}:=v_{n_{1}}\otimes v_{n_{2}}\otimes\cdots\otimes v_{n_{N}},

then a basis of the Hilbert space $\mathcal{H}=V_{1}\otimes V_{2}\otimes\cdots V_{N}$ can be chosen to be

\{\ket{n_{1},n_{2},\cdots,n_{N}}|~{}0\leq n_{i}\leq n-1~{}\}.

(3.1)

We call the basis (3.1) the natural basis of the Hilbert space $\mathcal{H}$ . The pseudo vacuum is taken to be

\ket{\Omega^{(0)}}=\ket{0,0,\cdots,0}\in V_{1}\otimes V_{2}\otimes\cdots\otimes V_{N}.

(3.2)

From Eq.(2.3) and (2.22), it is easy to compute

A(u)\ket{\Omega^{(0)}}=\ket{\Omega^{(0)}},\quad D_{ij}(u)\ket{\Omega^{(0)}}=\prod_{k=1}^{N}(u\ominus t_{k})\delta_{ij}\ket{\Omega^{(0)}}.

(3.3)

The Bethe ansatz state is taken to be of the form

\ket{\psi^{(0)}}=\sum_{i_{1},\cdots,i_{k_{1}}}\psi_{i_{1}\cdots i_{k_{1}}}^{(1)}B_{i_{1}}(\sigma_{1}^{(1)})\cdots B_{i_{k_{1}}}(\sigma_{k_{1}}^{(1)})\ket{\Omega^{(0)}},

(3.4)

where $\sigma_{\alpha_{1}}^{(1)}$ , $\alpha_{1}=1,\cdots,k_{1}$ , are parameters to be determined, and the expansion coefficients $\psi_{i_{1}\cdots i_{k_{1}}}^{(1)}$ depend on $n-2$ sets of parameters

\sigma^{(2)}_{\alpha_{2}},\cdots,\sigma^{(n-1)}_{\alpha_{n-1}}

with $\alpha_{s}=1,2,\cdots,k_{s}$ , where $\sigma^{(s)}_{a}\neq\sigma^{(s)}_{b}$ for $a\neq b$ and $1\leq s\leq n-1$ . Now we can prove the following

Theorem 3.1.

If the parameters $\sigma^{(1)}_{\alpha_{1}},\cdots,\sigma^{(n-1)}_{\alpha_{n-1}}$ satisfy the following Bethe ansatz equations:

	$\displaystyle\prod_{a=1}^{k_{1}}(1+\beta(\sigma^{(1)}_{a}\ominus\sigma^{(1)}_{\alpha_{1}}))\prod_{i=1}^{N}(\sigma^{(1)}_{\alpha_{1}}\ominus t_{i})=(-1)^{k_{1}-1}q_{1}\prod_{b=1}^{k_{2}}(\sigma^{(2)}_{b}\ominus\sigma^{(1)}_{\alpha_{1}}),$		(3.5)
	$\displaystyle\prod_{a=1}^{k_{m}}(1+\beta(\sigma^{(m)}_{a}\ominus\sigma^{(m)}_{\alpha_{m}}))\prod_{i=1}^{k_{m-1}}(\sigma^{(m)}_{\alpha_{m}}\ominus\sigma^{(m-1)}_{i})=(-1)^{k_{m}-1}q_{m}\prod_{b=1}^{k_{m+1}}(\sigma^{(m+1)}_{b}\ominus\sigma^{(m)}_{\alpha_{m}}),$		(3.6)
	$\displaystyle\prod_{a=1}^{k_{n-1}}(1+\beta(\sigma^{(n-1)}_{a}\ominus\sigma^{(n-1)}_{\alpha_{n-1}}))\prod_{i=1}^{k_{n-2}}(\sigma^{(n-1)}_{\alpha_{n-1}}\ominus\sigma^{(n-2)}_{i})=(-1)^{k_{n-1}-1}q_{n-1},$		(3.7)

where $m=2,\cdots,n-2,\quad\alpha_{s}=1,\cdots,k_{s},\quad q_{s}:=b_{s-1}/b_{s},~{}s=1,\cdots,n-1$ , then the state (3.4) is a common eigenstate of the transfer matrices $t(u)$ defined by (2.19).

Proof.

From the commutation relations (2.23) and (2.24), one can compute

		$\displaystyle A(u)\ket{\psi^{(0)}}=\prod_{a=1}^{k_{1}}(\sigma_{a}^{(1)}\ominus u)^{-1}\ket{\psi^{(0)}}+\ket{\phi_{A}},$		(3.8)
		$\displaystyle D_{ij}(u)\ket{\psi^{(0)}}=\frac{\prod_{l=1}^{N}(u\ominus t_{l})}{\prod_{a=1}^{k_{1}}(u\ominus\sigma^{(1)}_{a})}$
		$\displaystyle\times\sum_{i_{1},\cdots,i_{k_{1}}}\sum_{j_{1},\cdots,j_{k_{1}}}\left\{[T_{ij}^{(n-1)}(u;\sigma^{(1)})]_{i_{1}\cdots i_{k_{1}},j_{1}\cdots j_{k_{1}}}\psi_{j_{1}\cdots j_{k_{1}}}^{(1)}B_{i_{1}}(\sigma_{1}^{(1)})\cdots B_{i_{k_{1}}}(\sigma_{k_{1}}^{(1)})\right\}\ket{\Omega^{(0)}}+\ket{\phi_{D}},$

where $\ket{\phi_{A}}$ and $\ket{\phi_{D}}$ contain all the unwanted terms that are not in the subspace spanned by states of the form $B_{i_{1}}(\sigma_{1}^{(1)})\cdots B_{i_{k_{1}}}(\sigma_{k_{1}}^{(1)})\ket{\Omega^{(0)}}$ , and

		$\displaystyle[T_{ij}^{(n-1)}(u;\sigma^{(1)})]_{i_{1}\cdots i_{k_{1}},j_{1}\cdots j_{k_{1}}}$		(3.9)
		$\displaystyle=\sum_{\mu_{1}\cdots\mu_{k_{1}-1}}R^{(n-1)}_{ii_{k_{1}},\mu_{k_{1}-1}j_{k_{1}}}(u,\sigma^{(1)}_{k_{1}})\cdots R^{(n-1)}_{\mu_{2}i_{2},\mu_{1}j_{2}}(u,\sigma^{(1)}_{2})R^{(n-1)}_{\mu_{1}i_{1},jj_{1}}(u,\sigma^{(1)}_{1}).$		(3.9)

Eq.(3.9) suggests that $T^{(n-1)}$ can be viewed as the monodromy matrix of the GL $(n-1)$ asymmetric five vertex model with $k_{1}$ sites and R-matrix $R^{(n-1)}$ , where $\sigma^{(1)}_{a},a=1,\cdots,k_{1}$ are the equivariant parameters. Correspondingly, the transfer matrix can be defined as

t^{(n-1)}(u;\sigma^{(1)})=\mathrm{Tr}(M_{n-1}T^{(n-1)}(u;\sigma^{(1)})),

where $M_{l}$ is defined by (2.20). Let

\psi_{i_{1}\cdots i_{k_{1}}}^{(1)}=\braket{i_{1}i_{2}\cdots i_{k_{1}}}{\psi^{(1)}},\quad i_{s}=1,\cdots,n-1.

If $\ket{\psi^{(1)}}$ is an eigenstate of $t^{(n-1)}(u;\sigma^{(1)})$ with eigenvalue $\Lambda^{(1)}(u)$ , i.e.

t^{(n-1)}(u;\sigma^{(1)})\ket{\psi^{(1)}}=\Lambda^{(1)}(u)\ket{\psi^{(1)}},

(3.10)

then, as a consequence of Eq.(3.8)

\mathrm{Tr}(M_{n-1}D(u))\ket{\psi^{(0)}}=\frac{\prod_{i=1}^{N}(u\ominus t_{i})}{\prod_{a=1}^{k_{1}}(u\ominus\sigma^{(1)}_{a})}\Lambda^{(1)}(u)\ket{\psi^{(0)}}+\ket{\tilde{\phi}_{D}},

where $\ket{\tilde{\phi}_{D}}$ is the sum of the unwanted terms. Then

t^{(n)}(u)\ket{\psi^{(0)}}=\left[b_{0}A(u)+\mathrm{Tr}(M_{n-1}D(u))\right]\ket{\psi^{(0)}}=\Lambda^{(0)}(u)\ket{\psi^{(0)}}+\ket{\phi^{(n)}},

where

\Lambda^{(0)}(u)=b_{0}\prod_{a=1}^{k_{1}}(\sigma_{a}^{(1)}\ominus u)^{-1}+\frac{\prod_{i=1}^{N}(u\ominus t_{i})}{\prod_{a=1}^{k_{1}}(u\ominus\sigma^{(1)}_{a})}\Lambda^{(1)}(u),

(3.11)

and $\ket{\phi^{(n)}}$ is the sum of the unwanted terms. For $\ket{\psi^{(0)}}$ to be an eigenstate of $t^{(n)}(u)$ for all $u$ , the unwanted terms must vanish. As in the usual algebraic Bethe ansatz method, vanishing of the unwanted terms $\ket{\phi^{(n)}}$ amounts to $\Lambda^{(0)}(u)$ being holomorphic, i.e.

\mathrm{Res}(\Lambda^{(0)}(u),\sigma^{(1)}_{a})=0

(3.12)

for all $a=1,\cdots,k_{1}$ . Eq.(3.12) gives us the equations for $\sigma_{a}^{(1)}$ :

\prod_{\alpha=1}^{k_{1}}(1+\beta\sigma_{\alpha}^{(1)})\prod_{i=1}^{N}(\sigma_{a}^{(1)}\ominus t_{i})\Lambda^{(1)}(\sigma_{a}^{(1)})=(-1)^{k_{1}-1}b_{0}(1+\beta\sigma_{a}^{(1)})^{k_{1}},\quad a=1,\cdots,k_{1}.

(3.13)

Thus we have reduced the eigenvalue problem for the GL $(n)$ system to the eigenvalue problem (3.10) for a GL $(n-1)$ system. Clearly, we can continue to reduce the rank in exactly the same way. At each step, a set of parameters $\sigma^{(m)}_{\alpha_{m}},\alpha_{m}=1,\cdots,k_{m}$ , has to be introduced to construct the Bethe ansatz state

\ket{\psi^{(m-1)}}=\sum_{i_{1},\cdots,i_{k_{m}}}\psi^{(m)}_{i_{1},\cdots,i_{k_{m}}}B^{(m-1)}_{i_{1}}(\sigma^{(m)}_{1})\cdots B^{(m-1)}_{i_{k_{m}}}(\sigma^{(m)}_{k_{m}})\ket{\Omega^{(m-1)}},

(3.14)

where

\psi^{(m)}_{i_{1},\cdots,i_{k_{m}}}=\braket{i_{1},\cdots,i_{k_{m}}}{\psi^{(m)}},

(3.15)

and $B^{(m-1)}_{i}$ ’s are the $B_{i}$ operators defined as matrix elements of the monodromy matrix in the corresponding step as in (2.22) and $\sigma^{(m-1)}_{a},a=1,\cdots,k_{m-1}$ , serve as the equivariant parameters. As in the $m=1$ case, for (3.14) to be an eigenstate, the following equations have to be satisfied:

\prod_{b=1}^{k_{m}}(1+\beta\sigma_{b}^{(m)})\prod_{a=1}^{k_{m-1}}(\sigma_{\alpha_{m}}^{(m)}\ominus\sigma_{a}^{(m-1)})\Lambda^{(m)}(\sigma_{\alpha_{m}}^{(m)})=(-1)^{k_{m}-1}b_{m-1}(1+\beta\sigma_{\alpha_{m}}^{(m)})^{k_{m}},\quad\alpha_{m}=1,\cdots,k_{m},

(3.16)

accordingly, the eigenvalues satisfy

\Lambda^{(m-1)}(u)=b_{m-1}\prod_{\alpha=1}^{k_{m}}(\sigma_{\alpha}^{(m)}\ominus u)^{-1}+\frac{\prod_{a=1}^{k_{m-1}}(u\ominus\sigma_{a}^{(m-1)})}{\prod_{b=1}^{k_{m}}(u\ominus\sigma^{(m)}_{b})}\Lambda^{(m)}(u)

(3.17)

for $m=2,\cdots,n-2$ .

This procedure continues until we get a GL $(2)$ system with R-matrix given by Eq.(2.1). This GL $(2)$ system can be solved exactly by the algebraic Bethe ansatz method as for other spin chain models, leading to the Bethe ansatz equations Eq.(3.7) and

\Lambda^{(n-2)}(u)=b_{n-2}\prod_{\alpha=1}^{k_{n-1}}(\sigma_{\alpha}^{(n-1)}\ominus u)^{-1}+b_{n-1}\frac{\prod_{a=1}^{k_{n-2}}(u\ominus\sigma_{a}^{(n-2)})}{\prod_{b=1}^{k_{n-1}}(u\ominus\sigma^{(n-1)}_{b})}.

(3.18)

Eq.(3.18) then allows us to solve all the eigenvalues $\Lambda^{(m)},m=0,1,\cdots,n-3$ , from Eq.(3.17) and (3.11) recursively. Plugging these eigenvalues into Eq.(3.16) and (3.13), we get the equations Eq.(3.6) and (3.5). ∎

Now let us restrict to the cases in which $k_{n-1}<\cdots<k_{2}<k_{1}<N$ and compare Eq.(3.5)-(3.7) with the ring relations of the equivariant quantum cohomology and equivariant quantum K-theory ring of flag varieties. The reader may refer to the appendix and references therein for a review of the quantum cohomology and quantum K-theory ring of flag varieties.

When $\beta=0$ , we can identify $\sigma_{a}^{(n-i)}$ with the Chern roots of $\mathcal{S}_{i}$ , the $i$ th tautological bundle of the flag variety $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ , i.e. the $j$ th elementary symmetric polynomial $e_{j}(\sigma_{1}^{(n-i)},\cdots,\sigma_{k_{n-i}}^{(n-i)})$ is identified with the $j$ th Chern class of $\mathcal{S}_{i}$ :

e_{j}(\sigma_{1}^{(n-i)},\cdots,\sigma_{k_{n-i}}^{(n-i)})=c_{j}(\mathcal{S}_{i}),\quad j=1,\cdots,k_{n-i}.

Under this identification, it was checked in [11, 12] that Eq.(3.5)-(3.7) specialized to $\beta=0$ are exactly the ring relations of the equivariant quantum cohomology of $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ with the equivariant parameters given by $(t_{1},\cdots,t_{N})$ .

When $\beta=-1$ , we can identify $1-\sigma_{a}^{(n-i)}$ with the Chern roots of $\mathcal{S}_{i}$ , i.e. the $j$ th elementary symmetric polynomial $e_{j}(1-\sigma_{1}^{(i)},\cdots,1-\sigma_{k_{n-i}}^{(i)})$ is identified with $\wedge^{j}\mathcal{S}_{i}$ in the Grothendieck group of the flag variety $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ :

e_{j}(1-\sigma_{1}^{(n-i)},\cdots,1-\sigma_{k_{n-i}}^{(n-i)})=\wedge^{j}\mathcal{S}_{i},\quad j=1,\cdots,k_{n-i}.

It was argued in [12] that Eq.(3.5)-(3.7) specialized to $\beta=-1$ are the ring relations of the (predicted) equivariant quantum K-theory ring of $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ with the equivariant parameters given by $(1-t_{1},\cdots,1-t_{N})$ .

Indeed, as in [12], one can use Vieta’s formula²²2Vieta’s formula: If $r_{1},r_{2},\cdots,r_{n}$ are roots of the polynomial $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$ , then $e_{l}(r_{1},\cdots,r_{n})=(-1)^{l}a_{n-l}/a_{n}$ . to convert Eq.(3.5)-(3.7) into the ring relations of the equivariant quantum cohomology ring ( $\beta=0$ ) or the equivariant K-theory ring ( $\beta=-1$ ) of the flag variety $\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N)$ .

When $\beta=0$ , Eq.(3.5)-(3.7) reduces to

\prod_{\alpha=1}^{k_{m-1}}(\sigma^{(m)}_{a}-\sigma^{(m-1)}_{\alpha})=(-1)^{k_{m}-1}q_{m}\prod_{\gamma=1}^{k_{m+1}}(\sigma^{(m+1)}_{\gamma}-\sigma^{(m)}_{a})

(3.19)

for $a=1,\cdots,k_{m},m=1,2,\cdots,n-1$ , where we have defined $\sigma^{(0)}_{i}=t_{i},k_{0}=N,k_{n}=0$ . Eq.(3.19) tells us that $\sigma^{(m)}_{a},a=1,\cdots,k_{m}$ , are roots of the polynomial

\sum_{i=0}^{k_{m-1}}(-1)^{i}x^{k_{m-1}-i}e_{i}(\sigma^{(m-1)})+(-1)^{k_{m}-k_{m+1}}q_{m}\sum_{j=0}^{k_{m+1}}(-1)^{j}x^{k_{m+1}-j}e_{j}(\sigma^{(m+1)}),

(3.20)

where $e_{i}(\sigma^{(s)})$ is the $i$ th elementary symmetric polynomial in the variables $\sigma^{(s)}_{a}$ , $a=1,\cdots,k_{s}$ . Notice that the coefficient of $x^{k_{m-1}-l}$ in (3.20) is³³3We set $e_{l}(x_{1},\cdots,x_{s})=0$ for $l<0$ or $l>s$ .

(-1)^{l}e_{l}(\sigma^{(m-1)})+(-1)^{k_{m-1}-k_{m}+l}q_{m}e_{k_{m+1}-k_{m-1}+l}(\sigma^{(m+1)}).

Assume the other $k_{m-1}-k_{m}$ roots of (3.20) are $\eta^{(m)}_{b},b=1,\cdots,k_{m-1}-k_{m}$ , then Vieta’s formula yields

\sum_{i=0}^{l}e_{i}(\sigma^{(m)})e_{l-i}(\eta^{(m)})=e_{l}(\sigma^{(m-1)})+(-1)^{k_{m-1}-k_{m}}q_{m}e_{k_{m+1}-k_{m-1}+l}(\sigma^{(m+1)}).

(3.21)

If we interpret $\sigma^{(m)}_{a},a=1,\cdots,k_{m},$ as Chern roots of $\mathcal{S}_{n-m}$ and $\eta^{(m)}_{b},b=1,\cdots,k_{m-1}-k_{m},$ as Chern roots of $\mathcal{S}_{n-m+1}/\mathcal{S}_{n-m}$ , i.e. $e_{i}(\sigma^{(m)})=c_{i}(\mathcal{S}_{n-m})$ , $e_{i}(\eta^{(m)})=c_{i}(\mathcal{S}_{n-m+1}/\mathcal{S}_{n-m})$ , then Eq.(3.21) becomes⁴⁴4 $\mathcal{S}_{0}$ is the trivial vector bundle of rank zero, so $c({\mathcal{S}_{0}})=1$ . $\mathcal{S}_{n}$ is the trivial vector bundle of rank $N$ , but since it carries the $T$ -action, we have $c(\mathcal{S}_{n})=\sum_{i=0}^{N}e_{i}(t_{1},\cdots,t_{N})$ .

c(\mathcal{S}_{i})\cdot c(\mathcal{S}_{i+1}/\mathcal{S}_{i})=c(\mathcal{S}_{i+1})+(-1)^{k_{n-i-1}-k_{n-i}}q_{n-i}c(\mathcal{S}_{i-1}),

(3.22)

for $i=1,2,\cdots,n-1$ , which are the quantum Whitney relations of the equivariant quantum cohomology ring $QH^{*}_{T}(\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N))$ and are equivalent to the ring relations proved in [13].

When $\beta\neq 0$ , let $X^{(m)}_{a}=1+\beta\sigma^{(m)}_{a}$ , then Eq.(3.5)-(3.7) can be written as

		$\displaystyle\prod_{\alpha=1}^{k_{m-1}}\left(X^{(m)}_{a}-X^{(m-1)}_{\alpha}\right)\prod_{l=1}^{k_{m}}\left(X^{(m)}_{l}\right)$		(3.23)
	$\displaystyle=$	$\displaystyle(-1)^{k_{m}-1}q_{m}\beta^{k_{m-1}-k_{m+1}}\prod_{s=1}^{k_{m-1}}\left(X^{(m-1)}_{s}\right)\prod_{\gamma=1}^{k_{m+1}}\left(X^{(m+1)}_{\gamma}-X^{(m)}_{a}\right)\left(X^{(m)}_{a}\right)^{k_{m}-k_{m+1}}$		(3.23)

for $a=1,\cdots,k_{m},m=1,2,\cdots,n-1$ , where $X^{(0)}_{i}=1+\beta t_{i},k_{0}=N,k_{n}=0$ . Eq.(3.23) implies that $X^{(m)}_{a},a=1,\cdots,k_{m},$ are roots of the polynomial $\sum_{l=0}^{k_{m-1}}a_{l}x^{l}$ , where

	$\displaystyle a_{k_{m-1}-l}=$	$\displaystyle(-1)^{l}e_{l}(X^{(m-1)})e_{k_{m}}(X^{(m)})$
		$\displaystyle+(-1)^{k_{m-1}-k_{m+1}+l}q_{m}\beta^{k_{m-1}-k_{m+1}}e_{k_{m-1}}(X^{(m-1)})e_{l-k_{m-1}+k_{m}}(X^{(m+1)}).$

Assume the roots of $\sum_{l=0}^{k_{m-1}}a_{l}x^{l}$ are $X^{(m)}_{a},a=1,\cdots,k_{m},$ and $Y^{(m)}_{b},b=1,\cdots,k_{m-1}-k_{m}$ , then Vieta’s formula yields

		$\displaystyle e_{k_{m}}(X^{(m)})\sum_{i+j=l}e_{i}(X^{(m)})e_{j}(Y^{(m)})$		(3.24)
	$\displaystyle=$	$\displaystyle e_{k_{m}}(X^{(m)})e_{l}(X^{(m-1)})+q_{m}(-\beta)^{k_{m-1}-k_{m+1}}e_{k_{m-1}}(X^{(m-1)})e_{l-k_{m-1}+k_{m}}(X^{(m+1)}).$		(3.24)

Taking $l=k_{m-1}$ in Eq.(3.24), we get

e_{k_{m-1}}(X^{(m-1)})=e_{k_{m}}(X^{(m)})e_{k_{m-1}-k_{m}}(Y^{(m)}).

(3.25)

Eq.(3.24) and (3.25) together give us the relation

\sum_{i+j=l}e_{i}(X^{(m)})e_{j}(Y^{(m)})=e_{l}(X^{(m-1)})+q_{m}(-\beta)^{k_{m-1}-k_{m+1}}e_{l-k_{m-1}+k_{m}}(X^{(m+1)})e_{k_{m-1}-k_{m}}(Y^{(m)}).

(3.26)

In the case $\beta=-1$ , (3.26) was interpreted as the quantum Whitney relation of the equivariant quantum K-theory ring of the flag variety in [12]. The idea is to identify $e_{i}(X^{(m)})$ with $\wedge^{i}\mathcal{S}_{n-m}$ and make the following identification

e_{l}(Y^{(m)})=\left\{\begin{array}[]{ll}\wedge^{l}(\mathcal{S}_{n-m+1}/\mathcal{S}_{n-m}),&l<k_{m-1}-k_{m},\\ \frac{1}{1-q_{m}}\det(\mathcal{S}_{n-m+1}/\mathcal{S}_{n-m}),&l=k_{m-1}-k_{m}.\end{array}\right.

Then Eq.(3.26) for $\beta=-1$ can be written in terms of $\lambda_{y}$ class as⁵⁵5For a vector bundle $\mathcal{E}$ , the $\lambda_{y}$ class is defined as $\lambda_{y}(\mathcal{E})=\sum_{i=0}^{\mathrm{rank}(\mathcal{E})}y^{i}\wedge^{i}\mathcal{E}$ in the Grothendieck group.

\lambda_{y}(\mathcal{S}_{i})*\lambda_{y}(\mathcal{S}_{i+1}/\mathcal{S}_{i})=\lambda_{y}(\mathcal{S}_{i+1})+\frac{q_{n-i}}{1-q_{n-i}}y^{k_{n-i-1}-k_{n-i}}\left(\lambda_{y}(\mathcal{S}_{i-1})-\lambda_{y}(\mathcal{S}_{i})\right)*\det(\mathcal{S}_{i+1}/\mathcal{S}_{i})

(3.27)

for $i=1,2,\cdots,n-1$ , which are the quantum Whitney relations of the equivariant quantum K-theory ring $QK_{T}(\mathrm{Fl}(k_{n-1},\cdots,k_{2},k_{1};N))$ proposed in [12].

Remark 1.

Notice that in [12], the analogue of the Bethe ansatz equations were derived from the 3d $\mathcal{N}=2$ gauged linear sigma model (GLSM) compactified on a circle. The GLSMs for the flag varieties are quiver gauge theories, whose quiver diagrams have the following form in our convention [14]

where circles represent $U(k_{i})$ gauge groups, square represents $U(N)$ global symmetry and arrows represent chiral fields in bifundamental representations. The Higgs branch of the GLSM is a nonlinear sigma model with target space being the flag variety, the equations for vacuum states on the Coulomb branch are exactly the Bethe ansatz equations we obtained above in the case of $\beta=-1$ . Though ring relations of quantum K-theory ring of general partial flag varieties have not been rigorously established, it was checked in [12] that the ring relations obtained from the GLSMs match those of the special cases, such as incidence varieties and full flag varieties, in which exact results have been proved.

Remark 2.

Theorem 3.1 shows that the Bethe ansatz equations of the quantum integrable model defined by the R-matrix (2.3) and monodromy matrix (2.17) reduce to the vacuum equations on the Coulomb branch of the 2d GLSM for flag varieties when $\beta=0$ , and reduce to the vacuum equations on the Coulomb branch of the 3d GLSM for flag varieties when $\beta=-1$ . Therefore, we have established a Bethe/Gauge correspondence in the sense of [15, 16].

4 Relationship with the double $\beta$ -Grothendieck polynomials

In this section, we consider the case of full flag variety, i.e. $n=N$ and $k_{N-i}=i$ for $i=1,\cdots,N-1$ . We will show that the Bethe ansatz states generate the double $\beta$ -Grothendieck polynomials (Theorem 4.6 and 4.11). The double $\beta$ -Grothendieck polynomials $\mathcal{G}^{(\beta)}_{w}(\bm{x};\bm{y})$ for $w\in S_{N}$ and $\bm{x}=(x_{1},\cdots,x_{N}),\bm{y}=(y_{1},\cdots,y_{N})$ are defined in the appendix, see Eq.(A.1)-(A.3).

In addition to Eq.(2.2), let us also define

\ominus t:=\frac{-t}{1+\beta t}

(4.1)

and $\ominus\bm{t}:=(\ominus t_{1},\cdots,\ominus t_{N})$ for $\bm{t}=(t_{1},\cdots,t_{N})$ . We use $\omega_{N}$ to denote the permutation of $N$ elements with maximal length, and $l(w)$ denotes the length of the permutation $w$ .

From Eq.(2.4), (2.17) and the fact that $B_{k}(u)=[{T^{(n)}_{a}(u)}]_{0,k}$ (Eq.(2.22)), it is easy to compute

		$\displaystyle B_{k}(u)\ket{n_{1},n_{2},\cdots,n_{N}}=$		(4.2)
		$\displaystyle=\sum_{m_{1}=0}^{N-1}\cdots\sum_{m_{N-1}=0}^{N-1}{R_{aN}(u,t_{N})}_{0,m_{N-1}}\cdots{R_{a2}(u,t_{2})}_{m_{2},m_{1}}{R_{a1}(u,t_{1})}_{m_{1},k}\ket{n_{1},n_{2},\cdots,n_{N}}$
		$\displaystyle=\sum_{m_{1}=0}^{N-1}\cdots\sum_{m_{N-1}=0}^{N-1}\bigotimes_{i=1}^{N}\left[\delta_{m_{i},n_{i}}(1+\delta_{m_{i}>m_{i-1}}\beta(u\ominus t_{i}))\ket{m_{i-1}}+\delta_{m_{i}>n_{i}}\delta_{m_{i},m_{i-1}}(u\ominus t_{i})\ket{n_{i}}\right],$

where $m_{0}=k,m_{N}=0$ . The state (4.2) vanishes unless $m_{i}\geq n_{i}$ for all $1\leq i\leq N$ , therefore it vanishes when $n_{N}\neq 0$ . Also, $m_{i}=m_{i-1}$ if $m_{i}>n_{i}$ .

Refer to caption — Figure 1: The left action of $B_{k}(u)$ on $\ket{n_{1},n_{2},\cdots,n_{N}}$ for $N=12$ and a specific choice of $(m_{1},\cdots,m_{N-1})$ in the expansion (4.2).

The left action of $B_{k}(u)$ on $\ket{n_{1},n_{2},\cdots,n_{N}}$ can then be depicted as Fig.1 (for a specific $(m_{1},\cdots,m_{N-1})$ ). We see for any $\bm{m}=(m_{1},\cdots,m_{N})$ in the expansion (4.2), the effect of the operator $B_{k}(u)$ is to annihilate a $\ket{0}$ state, create a $\ket{k}$ state, and permute $n_{i}$ among the sites according to $\bm{m}$ . If we write the expansion (4.2) as $B_{k}(u)\ket{\bm{n}}=\sum_{\bm{m}}C_{\bm{m}}\ket{\psi_{\bm{m}}}$ , then for any $n_{i}<m_{i}$ , there is a factor $(u\ominus t_{i})$ in $C_{\bm{m}}$ , and for any $m_{i}>m_{i-1}$ , there is a factor $(1+\beta(u\ominus t_{i}))$ in $C_{\bm{m}}$ .

For $w\in S_{N}$ , let us define

\ket{w}:=\ket{w(012\cdots N-1)},

(4.3)

then, from (4.2) and the discussion above, we have the following

Lemma 4.1.

For any $\pi\in S_{N}$ , we have the expansion

B_{\pi(1)}(u_{1})B_{\pi(2)}(u_{2})\cdots B_{\pi(N-1)}(u_{N-1})\ket{\Omega^{(0)}}=\sum_{w\in S_{N}}C^{\pi}_{w}(u_{1},\cdots,u_{N-1})\ket{w},

where $C^{\pi}_{w}\in\mathbb{C}[u_{1},\cdots,u_{N-1},t_{1},\cdots,t_{N}]$ and $\ket{\Omega^{(0)}}$ is defined by Eq.(3.2).

Similarly, the right action of $B_{k}(u)$ on the dual state $\bra{n_{1},n_{2},\cdots,n_{N}}$ is

		$\displaystyle\bra{n_{1},n_{2},\cdots,n_{N}}B_{k}(u)$		(4.4)
		$\displaystyle=\sum_{m_{2}=0}^{N-1}\cdots\sum_{m_{N}=0}^{N-1}\bra{n_{1},n_{2},\cdots,n_{N}}{R_{aN}(u,t_{N})}_{0,m_{N}}\cdots{R_{a2}(u,t_{2})}_{m_{3},m_{2}}{R_{a1}(u,t_{1})}_{m_{2},k}$
		$\displaystyle=\sum_{m_{2}=0}^{N-1}\cdots\sum_{m_{N}=0}^{N-1}\bigotimes_{i=1}^{N}\left[\delta_{m_{i},n_{i}}(1+\delta_{m_{i}<m_{i+1}}\beta(u\ominus t_{i}))\bra{m_{i+1}}+\delta_{m_{i}>n_{i}}\delta_{m_{i},m_{i+1}}(u\ominus t_{i})\bra{n_{i}}\right],$

where $m_{1}=k,m_{N+1}=0$ . The state (4.4) vanishes unless $m_{i}\geq n_{i}$ for all $1\leq i\leq N$ and there is some $n_{a}$ such that $n_{a}=k=m_{a}$ and $m_{j}>n_{j}$ for all $j<a$ . Moreover, $m_{i}=m_{i+1}$ if $m_{i}>n_{i}$ .

The right action of $B_{k}(u)$ on $\bra{n_{1},n_{2},\cdots,n_{N}}$ can therefore be depicted as Fig.2 (for a specific choice of $(m_{2},\cdots,m_{N})$ ). We see for any $\bm{m}$ in the expansion (4.4), the effect of the operator $B_{k}(u)$ is to annihilate a $\bra{k}$ state, create a $\bra{0}$ state on the $N$ th site, and permute $n_{i}$ among the sites according to $\bm{m}$ . If we write the expansion (4.4) as $\bra{\bm{n}}B_{k}(u)=\sum_{\bm{m}}\bra{\psi_{\bm{m}}}C_{\bm{m}}$ , then for any $n_{i}<m_{i}$ , there is a factor $(u\ominus t_{i})$ in $C_{\bm{m}}$ , and there is a factor $(1+\beta(u\ominus t_{i}))$ in $C_{\bm{m}}$ for any $m_{i}<m_{i+1}$ .

For $w\in S_{N}$ , define

\bra{w}:=\bra{w(012\cdots N-1)},

(4.5)

then, from the expansion Eq.(4.4), one can prove

Lemma 4.2.

If $(n_{1},n_{2},\cdots,n_{N-1})$ satisfies: (i) $0\leq n_{i}<k$ ; (ii) $n_{l+1}\geq n_{l+2}\geq\cdots\geq n_{N-1}$ for some $l\geq 0$ , then

\bra{n_{\pi(1)},n_{\pi(2)},\cdots,n_{\pi(l)},k,n_{l+1},\cdots,n_{N-1}}B_{k}(u)=\prod_{i=1}^{l}(u\ominus t_{i})\bra{n_{\pi(1)},\cdots,n_{\pi(l)},n_{l+1},\cdots,n_{N-1},0}

for any $\pi\in S_{l}$ .

Proof.

Given the constraints on $n_{i}$ , the only possible $(m_{2},m_{3},\cdots,m_{N})$ in the expansion (4.4) is

m_{i}=\left\{\begin{array}[]{ll}k,&2\leq i\leq l+1,\\ n_{i},&l+2\leq i\leq N.\end{array}\right.

Since $n_{\pi(i)}<m_{i}$ for $1\leq i\leq l$ , we have the factor $\prod_{i=1}^{l}(u\ominus t_{i})$ from (4.4). ∎

Let us define

\rho_{k}:=s_{1}s_{2}\cdots s_{k}\in S_{N},

(4.6)

where $s_{i}$ is the simple transposition of the $i$ th and $(i+1)$ th elements, and $\rho_{0}$ is defined to be the identity permutation. Let $\partial^{\beta}_{i}$ be the $\beta$ -divided difference operator defined by Eq.(A.1):

\partial^{\beta}_{i}f(x_{1},\cdots,x_{N}):=\frac{(1+\beta x_{i+1})f(x_{1},\cdots,x_{N})-(1+\beta x_{i})f(x_{1},\cdots,x_{i+1},x_{i},\cdots,x_{N})}{x_{i}-x_{i+1}}.

Generally, for $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l(w)}}$ , we have the definition⁶⁶6In the following, if a polynomial $f(\bm{x};\bm{y})$ depends on two sets of variables $\bm{x}$ and $\bm{y}$ as in the case of double $\beta$ -Grothendieck polynomials, $\partial^{\beta}_{w}$ in $\partial^{\beta}_{w}f(\bm{x};\bm{y})$ is understood to act on the first set of variables, see Eq.(A.1). The same convention applies to the operator $\bar{\partial}^{\beta}_{w}$ defined by Eq.(4.31).

\partial^{\beta}_{w}:=\partial^{\beta}_{i_{1}}\partial^{\beta}_{i_{2}}\cdots\partial^{\beta}_{i_{l(w)}}.

(4.7)

Now we can use Lemma 4.2 to prove the following

Lemma 4.3.

\bra{1,2,\cdots,k,0,k+1,\cdots,N-1}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})=\mathcal{G}^{(\beta)}_{\rho^{-1}_{k}\omega_{N}}(\bm{\sigma};\ominus\bm{t})\bra{\Omega^{(0)}}.

Proof.

By repeated use of Lemma 4.2, one can compute

		$\displaystyle\bra{1,2,\cdots,k,0,k+1,\cdots,N-1}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})$		(4.8)
	$\displaystyle=$	$\displaystyle\prod_{a=1}^{N-k-1}\prod_{i=1}^{N-a}(\sigma_{a}\ominus t_{i})\prod_{b=N-k}^{N-2}\prod_{j=1}^{N-b-1}(\sigma_{b}\ominus t_{j})\bra{\Omega^{(0)}}.$		(4.8)

Notice that $\sigma\ominus t=\frac{\sigma-t}{1+\beta t}=\sigma+\ominus t+\beta\sigma(\ominus t)$ , Eq.(A.2) then yields

\prod_{i+j\leq N}(\sigma_{i}\ominus t_{j})=\prod_{i+j\leq N}(\sigma_{i}+\ominus t_{j}+\beta\sigma_{i}(\ominus t_{j}))=\mathcal{G}^{(\beta)}_{\omega_{N}}(\bm{\sigma};\ominus\bm{t}).

Therefore, from (4.8) and the definition of the double $\beta$ -Grothendieck polynomials Eq.(A.1)-(A.3), we get

		$\displaystyle\bra{1,2,\cdots,k,0,k+1,\cdots,N-1}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})$
	$\displaystyle=$	$\displaystyle\frac{\mathcal{G}^{(\beta)}_{\omega_{N}}(\bm{\sigma};\ominus\bm{t})}{(\sigma_{N-1}\ominus t_{1})(\sigma_{N-2}\ominus t_{2})\cdots(\sigma_{N-k}\ominus t_{k})}\bra{\Omega^{(0)}}$
	$\displaystyle=$	$\displaystyle\partial^{\beta}_{N-1}\partial^{\beta}_{N-2}\cdots\partial^{\beta}_{N-k}\mathcal{G}^{(\beta)}_{\omega_{N}}(\bm{\sigma};\ominus\bm{t})\bra{\Omega^{(0)}}$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{\omega_{N}s_{N-k}\cdots s_{N-2}s_{N-1}}(\bm{\sigma};\ominus\bm{t})\bra{\Omega^{(0)}}$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{s_{k}\cdots s_{2}s_{1}\omega_{N}}(\bm{\sigma};\ominus\bm{t})\bra{\Omega^{(0)}}=\mathcal{G}^{(\beta)}_{\rho^{-1}_{k}\omega_{N}}(\bm{\sigma};\ominus\bm{t})\bra{\Omega^{(0)}},$

where in the second to last step, we used $\omega_{N}s_{N-i}=s_{i}\omega_{N}$ . ∎

Lemma 4.4.

If $\pi\in S_{N}$ and $l(s_{i}\pi)>l(\pi)$ , then

		$\displaystyle(\bra{s_{i}\pi}+\beta\bra{\pi})B_{N-1}(\sigma_{1})B_{N-2}(\sigma_{2})\cdots B_{i+1}(\sigma_{N-i-1})B_{i}(\sigma_{N-i})B_{i-1}(\sigma_{N-i+1})$
	$\displaystyle=$	$\displaystyle\bra{\pi}B_{N-1}(\sigma_{1})B_{N-2}(\sigma_{2})\cdots B_{i+1}(\sigma_{N-i-1})B_{i-1}(\sigma_{N-i})B_{i}(\sigma_{N-i+1}).$

Proof.

First we show, for

\bra{\psi}=\bra{n_{1},\cdots,n_{a-1},i-1,n_{a+1},\cdots,n_{b-1},i,n_{b+1},\cdots,n_{N}}

(4.9)

and

\bra{s_{i}\psi}=\bra{n_{1},\cdots,n_{a-1},i,n_{a+1},\cdots,n_{b-1},i-1,n_{b+1},\cdots,n_{N}},

(4.10)

where $n_{s}<i-1$ for $s\neq a,b$ , we have

(\bra{s_{i}\psi}+\beta\bra{\psi})B_{i}(x)B_{i-1}(y)=\bra{\psi}B_{i-1}(x)B_{i}(y).

(4.11)

We have the following expansions in the natural basis according to (4.4)

		$\displaystyle\bra{\psi}B_{i-1}(x)=\sum_{\bm{m}\in M_{i-1}}C_{\bm{m}}^{(i-1)}\bra{\psi^{(i-1)}_{\bm{m}}},$
		$\displaystyle\bra{\psi}B_{i}(x)=\sum_{\bm{m}\in M_{i}}C_{\bm{m}}^{(i)}\bra{\psi^{(i)}_{\bm{m}}},$
		$\displaystyle\bra{s_{i}\psi}B_{i}(x)=\sum_{\bm{m}\in\bar{M}_{i}}\bar{C}^{(i)}_{\bm{m}}\bra{\bar{\psi}^{(i)}_{\bm{m}}},$

where $M_{i-1},M_{i}$ and $\bar{M}_{i}$ are the sets in which $\bm{m}=(m_{1},\cdots,m_{N})$ takes value such that the corresponding coefficients $C_{\bm{m}}^{(i-1)}$ , $C_{\bm{m}}^{(i)}$ and $\bar{C}^{(i)}_{\bm{m}}$ in (4.4) are nonzero, and $\bra{\psi^{(i-1)}_{\bm{m}}}$ , $\bra{\psi^{(i)}_{\bm{m}}}$ and $\bra{\bar{\psi}^{(i)}_{\bm{m}}}$ are members of the natural basis (3.1). Because $m_{a}=i-1,m_{b}=i$ for $\bm{m}\in M_{i-1}$ and $m_{a}=i,m_{b}=i-1$ for $\bm{m}\in\bar{M}_{i}$ , there is a map $\phi:M_{i-1}\rightarrow\bar{M}_{i}$ defined by

\phi(\bm{m})_{j}=\left\{\begin{array}[]{ll}i,&\mathrm{if}~{}m_{j}=i-1,\\ i-1,&\mathrm{if}~{}m_{j}=i,\\ m_{j},&\mathrm{if}~{}m_{j}\neq i~{}\mathrm{and}~{}m_{j}\neq i-1.\end{array}\right.

$\phi$ is a bijection since $n_{s}<i-1$ for $s\neq a,b$ . Moreover, for $\bm{m}\in M_{i}$ , we have $n_{j}<m_{j}$ for $1\leq j<b$ and $m_{b}=i$ , therefore we have an injective map $\tau:M_{i}\rightarrow\bar{M}_{i}$ defined by

\tau(\bm{m})_{j}=\left\{\begin{array}[]{ll}i,&1\leq j\leq a,\\ i-1,&a+1\leq j\leq b,\\ m_{j},&j>b.\end{array}\right.

For $\bm{m}\in\mathrm{im}\tau$ , i.e. $m_{s}=i,m_{t}=i-1$ for $1\leq s\leq a,a+1\leq t\leq b$ , since $n_{a}=m_{a}>m_{a+1}$ and $\phi^{-1}(\bm{m})_{a}<\phi^{-1}(\bm{m})_{a+1},\tau^{-1}(\bm{m})_{a}>n_{a}$ , Eq.(4.4) implies

C^{(i-1)}_{\phi^{-1}(\bm{m})}=(1+\beta(x\ominus t_{a}))\bar{C}^{(i)}_{\bm{m}},\quad C^{(i)}_{\tau^{-1}(\bm{m})}=(x\ominus t_{a})\bar{C}^{(i)}_{\bm{m}}.

(4.12)

On the other hand, if $\bm{m}\not\in\mathrm{im}\tau$ , then $m_{a+1}<i-1$ , therefore

C^{(i-1)}_{\phi^{-1}(\bm{m})}=\bar{C}^{(i)}_{\bm{m}}.

(4.13)

Consequently,

		$\displaystyle\bra{\psi}B_{i-1}(x)=\sum_{\phi^{-1}(\bm{m})\in M_{i-1}}C_{\phi^{-1}(\bm{m})}^{(i-1)}\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}$		(4.14)
		$\displaystyle\quad\quad\quad\quad\quad=\sum_{\bm{m}\in\mathrm{im}\tau}(1+\beta(x\ominus t_{a}))\bar{C}^{(i)}_{\bm{m}}\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}+\sum_{\bm{m}\not\in\mathrm{im}\tau}\bar{C}^{(i)}_{\bm{m}}\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}},$
		$\displaystyle\bra{\psi}B_{i}(x)=\sum_{\bar{\bm{m}}\in M_{i}}C^{(i)}_{\bar{\bm{m}}}\bra{\psi^{(i)}_{\bar{\bm{m}}}}=\sum_{\bar{\bm{m}}\in M_{i}}(x\ominus t_{a})\bar{C}^{(i)}_{\tau(\bar{\bm{m}})}\bra{\psi^{(i)}_{\bar{\bm{m}}}}=\sum_{\bm{m}\in\mathrm{im}\tau}(x\ominus t_{a})\bar{C}^{(i)}_{\bm{m}}\bra{\psi^{(i)}_{\tau^{-1}(\bm{m})}},$
		$\displaystyle\bra{s_{i}\psi}B_{i}(x)=\sum_{\bm{m}\in\bar{M}_{i}}\bar{C}_{\bm{m}}^{(i)}\bra{\bar{\psi}^{(i)}_{\bm{m}}}=\sum_{\bm{m}\in\mathrm{im}\tau}\bar{C}^{(i)}_{\bm{m}}\bra{\bar{\psi}^{(i)}_{\bm{m}}}+\sum_{\bm{m}\not\in\mathrm{im}\tau}\bar{C}^{(i)}_{\bm{m}}\bra{\bar{\psi}^{(i)}_{\bm{m}}},$

where, according to (4.4),

		$\displaystyle\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}=\bra{n_{1},\cdots,n_{a-1},i,n_{a+1},\cdots,n_{b-1},n^{\prime}_{s_{1}},\cdots,n^{\prime}_{s_{N-b+1}}},$
		$\displaystyle\bra{\psi^{(i)}_{\tau^{-1}(\bm{m})}}=\bra{n_{1},\cdots,n_{a-1},i-1,n_{a+1},\cdots,n_{b-1},n^{\prime}_{s_{1}},\cdots,n^{\prime}_{s_{N-b+1}}},$
		$\displaystyle\bra{\bar{\psi}^{(i)}_{\bm{m}}}=\bra{n_{1},\cdots,n_{a-1},i-1,n_{a+1},\cdots,n_{b-1},n^{\prime}_{s_{1}},\cdots,n^{\prime}_{s_{N-b+1}}},$

with some $n^{\prime}_{s_{1}},\cdots,n^{\prime}_{s_{N-b+1}}$ determined by $\bm{m}$ . From the equations above, we see
$\bra{\psi^{(i)}_{\tau^{-1}(\bm{m})}}=\bra{\bar{\psi}^{(i)}_{\bm{m}}}$ and $\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}B_{i}(y)=\bra{\bar{\psi}^{(i)}_{\bm{m}}}B_{i-1}(y)$ , from which, together with Eq.(4.14), one can compute

		$\displaystyle(\bra{s_{i}\psi}+\beta\bra{\psi})B_{i}(x)B_{i-1}(y)$
	$\displaystyle=$	$\displaystyle\sum_{\bm{m}\not\in\mathrm{im}\tau}\bar{C}^{(i)}_{\bm{m}}\bra{\bar{\psi}^{(i)}_{\bm{m}}}B_{i-1}(y)+\sum_{\bm{m}\in\mathrm{im}\tau}\bar{C}^{(i)}_{\bm{m}}\bra{\bar{\psi}^{(i)}_{\bm{m}}}B_{i-1}(y)+\beta\sum_{\bm{m}\in\mathrm{im}\tau}(x\ominus t_{a})\bar{C}^{(i)}_{\bm{m}}\bra{\bar{\psi}^{(i)}_{\bm{m}}}B_{i-1}(y)$
	$\displaystyle=$	$\displaystyle\sum_{\bm{m}\not\in\mathrm{im}\tau}\bar{C}^{(i)}_{\bm{m}}\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}B_{i}(y)+\sum_{\bm{m}\in\mathrm{im}\tau}(1+\beta(x\ominus t_{a}))\bar{C}^{(i)}_{\bm{m}}\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}B_{i}(y)$
	$\displaystyle=$	$\displaystyle\sum_{\phi^{-1}(\bm{m})\in M_{i-1}}C^{(i-1)}_{\phi^{-1}(\bm{m})}\bra{\psi^{(i-1)}_{\phi^{-1}(\bm{m})}}B_{i}(y)$
	$\displaystyle=$	$\displaystyle\bra{\psi}B_{i-1}(x)B_{i}(y).$

Thus Eq.(4.11) is proved.

Now assume, for $k\geq i+1$ and $\bra{\varphi}=\bra{n_{1},\cdots,n_{N}}$ with $\{n_{1},\cdots,n_{N}\}=\{1,2,\cdots,k-1,0,\cdots,0\}$ , we have

(\bra{s_{i}\varphi}+\beta\bra{\varphi})B_{k-1}(\sigma_{1})\cdots B_{i}(\sigma_{k-i})B_{i-1}(\sigma_{k-i+1})=\bra{\varphi}B_{k-1}(\sigma_{1})\cdots B_{i-1}(\sigma_{k-i})B_{i}(\sigma_{k-i+1}),

(4.15)

where $\bra{\varphi}$ and $\bra{s_{i}\varphi}$ are defined in the same way as Eq.(4.9) and (4.10). The $k=i+1$ case of (4.15) is valid due to Eq.(4.11). Now let $\bra{\psi}$ be a state of the form

\bra{\psi}=\bra{n_{1},\cdots,n_{a-1},i-1,n_{a+1},\cdots,n_{b-1},i,n_{b+1},\cdots,n_{N}}

with $\{n_{1},\cdots,n_{N}\}=\{1,2,\cdots,k,0,\cdots,0\}$ , where $n_{a}=i-1,n_{b}=i$ . We expand the following states in the natural basis according to (4.4):

	$\displaystyle\bra{\psi}B_{k}(u)=\sum_{\bm{m}\in M_{1}}C_{\bm{m}}\bra{\psi_{\bm{m}}}+\sum_{\bm{m}\in M_{2}}C^{\prime}_{\bm{m}}\bra{\psi^{\prime}_{\bm{m}}},$		(4.16)
	$\displaystyle\bra{s_{i}\psi}B_{k}(u)=\sum_{\bm{m}\in\bar{M}_{1}}\bar{C}_{\bm{m}}\bra{\bar{\psi}_{\bm{m}}},$		(4.17)

where $M_{2}$ contains $\bm{m}$ ’s satisfying $i-1=n_{a}<m_{a}=m_{a+1}=\cdots=m_{b}=i$ , and $M_{1}$ consists of all the other $\bm{m}$ ’s such that the corresponding coefficients are nonzero. One important difference between these two sets is that $i-1$ remains to the left of $i$ in $\bra{\psi_{\bm{m}}}$ , while $i$ is moved to the left of $i-1$ in $\bra{\psi^{\prime}_{\bm{m}}}$ . Since it is not possible to have $i=n_{a}<m_{a}=m_{a+1}=\cdots=m_{b}=i-1$ in the expansion (4.17), we have a one-to-one correspondence between $M_{1}$ and $\bar{M}_{1}$ given by $\tilde{\phi}:M_{1}\rightarrow\bar{M}_{1}$ with $\tilde{\phi}$ defined by

\tilde{\phi}(\bm{m})_{j}=\left\{\begin{array}[]{ll}i,&\mathrm{if}~{}m_{j}=i-1,\\ i-1,&\mathrm{if}~{}m_{j}=i,\\ m_{j},&\mathrm{if}~{}m_{j}\neq i~{}\mathrm{and}~{}m_{j}\neq i-1.\end{array}\right.

Since $\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}$ differs from $\bra{\psi_{\bm{m}}}$ by exchange of $i-1$ and $i$ , it is easy to see

\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}=\bra{s_{i}\psi_{\bm{m}}}.

(4.18)

Notice that $n_{s}<i-1$ or $n_{s}>i$ for $s\neq a,b$ , Eq.(4.4) implies

C_{\bm{m}}=\left\{\begin{array}[]{ll}(1+\beta(u\ominus t_{a}))\bar{C}_{\tilde{\phi}(\bm{m})},&\mathrm{if}~{}m_{a}=i-1,m_{b}=i,n_{s}<m_{s}=m_{b}~{}\mathrm{for}~{}a<s<b,\\ \bar{C}_{\tilde{\phi}(\bm{m})},&\mathrm{otherwise}.\end{array}\right.

(4.19)

Moreover, there is an injective map $\tilde{\tau}:M_{2}\rightarrow M_{1}$ given by

\tilde{\tau}(\bm{m})_{j}=\left\{\begin{array}[]{ll}i-1,&\mathrm{if}~{}m_{j}=i~{}\mathrm{and}~{}j\leq a,\\ m_{j},&\mathrm{otherwise}.\end{array}\right.

Notice that $\bm{m}\in\mathrm{im}\tilde{\tau}$ is equivalent to $m_{a}=i-1,m_{b}=i,n_{s}<m_{s}=m_{b}$ for $a<s<b$ , so Eq.(4.19) can be rewritten as

C_{\bm{m}}=\left\{\begin{array}[]{ll}(1+\beta(u\ominus t_{a}))\bar{C}_{\tilde{\phi}(\bm{m})},&\mathrm{if}~{}\bm{m}\in\mathrm{im}\tilde{\tau},\\ \bar{C}_{\tilde{\phi}(\bm{m})},&\mathrm{otherwise}.\end{array}\right.

(4.20)

In addition, if $\bm{m}\in\mathrm{im}\tilde{\tau}$ , then

\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}=\bra{\psi^{\prime}_{\tilde{\tau}^{-1}(\bm{m})}},

(4.21)

because on both sides, $i-1$ is moved to the $a$ th site, $i$ is moved to the $c$ th site, where $c=\mathrm{max}\{l<a~{}|~{}n_{l}=m_{l}\}$ , and the other $n_{s}$ ’s are moved in the same way, and

C^{\prime}_{\tilde{\tau}^{-1}(\bm{m})}=(u\ominus t_{a})\bar{C}_{\tilde{\phi}(\bm{m})},

(4.22)

because $\tilde{\phi}(\bm{m})_{a}=n_{a}=i$ , $\tilde{\tau}^{-1}(\bm{m})_{a}>n_{a}=i-1$ , and $n_{s}<i-1$ or $n_{s}>i$ for $s\neq a,b$ . Therefore

	$\displaystyle\sum_{\bm{m}\in M_{1}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}+\beta\sum_{\bm{m}\in M_{2}}C^{\prime}_{\bm{m}}\bra{\psi^{\prime}_{\bm{m}}}$	(4.23)
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}\backslash\mathrm{im}\tilde{\tau}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}+\sum_{\bm{m}\in\mathrm{im}\tilde{\tau}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}+\beta\sum_{\bm{m}\in M_{2}}C^{\prime}_{\bm{m}}\bra{\psi^{\prime}_{\bm{m}}}$
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}\backslash\mathrm{im}\tilde{\tau}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{s_{i}\psi_{\bm{m}}}+\sum_{\bm{m}\in\mathrm{im}\tilde{\tau}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{s_{i}\psi_{\bm{m}}}+\beta(u\ominus t_{a})\sum_{\bm{m}\in\mathrm{im}\tilde{\tau}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{s_{i}\psi_{\bm{m}}}$
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}}C_{\bm{m}}\bra{s_{i}\psi_{\bm{m}}},$

where the second equality is due to Eq.(4.18), (4.21) and (4.22), and in the last equality we used Eq.(4.20). Consequently,

	$\displaystyle(\bra{s_{i}\psi}+\beta\bra{\psi})B_{k}(u)$	(4.24)
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}}\bar{C}_{\tilde{\phi}(\bm{m})}\bra{\bar{\psi}_{\tilde{\phi}(\bm{m})}}+\beta\sum_{\bm{m}\in M_{2}}C^{\prime}_{\bm{m}}\bra{\psi^{\prime}_{\bm{m}}}+\beta\sum_{\bm{m}\in M_{1}}C_{\bm{m}}\bra{\psi_{\bm{m}}}$
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}}C_{\bm{m}}(\bra{s_{i}\psi_{\bm{m}}}+\beta\bra{\psi_{\bm{m}}}),$

where the second equality is due to (4.23). From the induction hypothesis Eq.(4.15), we have

(\bra{s_{i}\psi_{\bm{m}}}+\beta\bra{\psi_{\bm{m}}})B_{k-1}(\sigma_{1})\cdots B_{i}(\sigma_{k-i})B_{i-1}(\sigma_{k-i+1})=\bra{\psi_{\bm{m}}}B_{k-1}(\sigma_{1})\cdots B_{i-1}(\sigma_{k-i})B_{i}(\sigma_{k-i+1}),

(4.25)

which results in

	$\displaystyle(\bra{s_{i}\psi}+\beta\bra{\psi})B_{k}(u)B_{k-1}(\sigma_{1})\cdots B_{i}(\sigma_{k-i})B_{i-1}(\sigma_{k-i+1})$	(4.26)
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}}C_{\bm{m}}(\bra{s_{i}\psi_{\bm{m}}}+\beta\bra{\psi_{\bm{m}}})B_{k-1}(\sigma_{1})\cdots B_{i}(\sigma_{k-i})B_{i-1}(\sigma_{k-i+1})$
$\displaystyle=$	$\displaystyle\sum_{\bm{m}\in M_{1}}C_{\bm{m}}\bra{\psi_{\bm{m}}}B_{k-1}(\sigma_{1})\cdots B_{i-1}(\sigma_{k-i})B_{i}(\sigma_{k-i+1})$
$\displaystyle=$	$\displaystyle\bra{\psi}B_{k}(u)B_{k-1}(\sigma_{1})\cdots B_{i-1}(\sigma_{k-i})B_{i}(\sigma_{k-i+1}),$

where the first equality is due to (4.24), the second equality is due to (4.25), and in the last step we used Eq.(4.16) and the fact that, for $\bm{m}\in M_{2}$ , $i$ is to the left of $i-1$ in $\bra{\psi^{\prime}_{\bm{m}}}$ , so (notice that $B_{l}$ with $l>i$ cannot move $i-1$ to the left of $i$ because the only way to do this is to have $i<m_{s}=m_{t}=i-1$ for some $s<t$ , which is impossible)

\bra{\psi^{\prime}_{\bm{m}}}B_{k-1}(\sigma_{1})\cdots B_{i-1}(\sigma_{k-i})B_{i}(\sigma_{k-i+1})=0,\quad\bm{m}\in M_{2},

because it is impossible to have $i<m_{s}=m_{s+1}=\cdots=m_{t}=i-1$ when $B_{i-1}(\sigma_{k-i})$ acts on the state.

Combining Eq.(4.11), (4.15) and (4.26), the lemma is proved by induction. ∎

In order to relate the quantum states of the integrable system to the double $\beta$ -Grothendieck polynomials, we need the commutation relations among the different $B_{i}$ operators. From Eq.(2.4) and (2.25), we have the commutation relations

	$\displaystyle B_{i}(x)B_{i}(y)=B_{i}(y)B_{i}(x),$		(4.27)
	$\displaystyle B_{i}(x)B_{j}(y)=\frac{B_{j}(y)B_{i}(x)-B_{j}(x)B_{i}(y)}{y\ominus x}=(1+\beta x)\frac{B_{j}(x)B_{i}(y)-B_{j}(y)B_{i}(x)}{x-y},\quad i<j.$		(4.28)

Let us define

\bar{\partial}^{\beta}_{i}=(1+\beta x_{i})\partial_{i},

(4.29)

where $\partial_{i}$ is the ordinary divided difference operator defined by $\partial^{\beta}_{i}|_{\beta=0}$ . It is easy to check that

\bar{\partial}^{\beta}_{i}=\partial^{\beta}_{i}+\beta.

(4.30)

Similar to Eq.(4.7), for $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l(w)}}$ , we define

\bar{\partial}^{\beta}_{w}:=\bar{\partial}^{\beta}_{i_{1}}\bar{\partial}^{\beta}_{i_{2}}\cdots\bar{\partial}^{\beta}_{i_{l(w)}}.

(4.31)

From Eq.(4.28), (4.29) and Lemma 4.4, we can prove

Lemma 4.5.

For $\pi\in S_{N}$ , if $l(s_{i}\pi)>l(\pi)$ , then

\bra{s_{i}\pi}B_{N-1}(\sigma_{1})\cdots B_{i}(\sigma_{N-i})B_{i-1}(\sigma_{N-i+1})=\partial^{\beta}_{N-i}\bra{\pi}B_{N-1}(\sigma_{1})\cdots B_{i}(\sigma_{N-i})B_{i-1}(\sigma_{N-i+1}).

Proof.

		$\displaystyle(\bra{s_{i}\pi}+\beta\bra{\pi})B_{N-1}(\sigma_{1})\cdots B_{i}(\sigma_{N-i})B_{i-1}(\sigma_{N-i+1})$
	$\displaystyle=$	$\displaystyle\bra{\pi}B_{N-1}(\sigma_{1})\cdots B_{i-1}(\sigma_{N-i})B_{i}(\sigma_{N-i+1})$
	$\displaystyle=$	$\displaystyle(\partial^{\beta}_{N-i}+\beta)\bra{\pi}B_{N-1}(\sigma_{1})\cdots B_{i}(\sigma_{N-i})B_{i-1}(\sigma_{N-i+1}),$

where the first equality is due to Lemma 4.4, and the second equality is due to Eq.(4.28), (4.29) and (4.30). ∎

Theorem 4.6.

For $\pi\in S_{N}$ ,

B_{N-1}(\sigma_{1})B_{N-2}(\sigma_{2})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}=\sum_{w\in S_{N}}\mathcal{G}^{(\beta)}_{w}(\bm{\sigma};\ominus\bm{t})\ket{\omega_{N}w^{-1}},

where $\omega_{N}$ is the permutation in $S_{N}$ with maximal length.

Proof.

Lemma 4.1 suggests that we only need to show

\bra{\omega_{N}w^{-1}}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}=\mathcal{G}^{(\beta)}_{w}(\bm{\sigma};\ominus\bm{t}).

(4.32)

Lemma 4.3 shows Eq.(4.32) is valid for $\omega_{N}w^{-1}=\rho_{k},k=0,1,\cdots,N-1$ . Because any permutation $w\in S_{N}$ can be written as $w=\pi\rho_{k}$ for some $\pi\in S_{N-1}$ acting on the last $N-1$ elements and some $k=0,\cdots,N-1$ , it remains to show the validity of Eq.(4.32) for $\omega_{N}w^{-1}=\pi\rho_{k}$ with $\pi\in S_{N-1}$ acting on $\{1,2,\cdots,N-1\}$ . We show this by induction on the lenghth of $\pi$ . Let us assume

\bra{\tilde{\pi}\rho_{k}}B_{N-1}(\sigma_{1})\cdots B_{2}(\sigma_{N-2})B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}=\mathcal{G}^{(\beta)}_{(\tilde{\pi}\rho_{k})^{-1}\omega_{N}}(\bm{\sigma};\ominus\bm{t})

for $\tilde{\pi}\in S_{N-1}$ with $l(\tilde{\pi})\leq m-1$ for some $m\geq 1$ . Now, for $\pi\in S_{N-1}$ with $l(\pi)=m$ , there exists $\tilde{\pi}\in S_{N-1}$ with $l(\tilde{\pi})=m-1$ , such that $\pi=s_{i}\tilde{\pi}$ . Then

		$\displaystyle\bra{\pi\rho_{k}}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}$
	$\displaystyle=$	$\displaystyle\bra{s_{i}\tilde{\pi}\rho_{k}}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}$
	$\displaystyle=$	$\displaystyle\partial^{\beta}_{N-i}\bra{\tilde{\pi}\rho_{k}}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1})\ket{\Omega^{(0)}}$
	$\displaystyle=$	$\displaystyle\partial^{\beta}_{N-i}\mathcal{G}^{(\beta)}_{\rho_{k}^{-1}\tilde{\pi}^{-1}\omega_{N}}(\bm{\sigma};\ominus\bm{t})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{\rho_{k}^{-1}\tilde{\pi}^{-1}\omega_{N}s_{N-i}}(\bm{\sigma};\ominus\bm{t})=\mathcal{G}^{(\beta)}_{\rho_{k}^{-1}\tilde{\pi}^{-1}s_{i}\omega_{N}}(\bm{\sigma};\ominus\bm{t})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{(\pi\rho_{k})^{-1}\omega_{N}}(\bm{\sigma};\ominus\bm{t}),$

where the second equality is due to Lemma 4.5. ∎

Lemma 4.7.

For any $w\in S_{N-1}$ ,

B_{w(1)}(\sigma_{1})\cdots B_{w(N-1)}(\sigma_{N-1})=\bar{\partial}^{\beta}_{w^{-1}\omega_{N-1}}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1}).

Proof.

Let $w=\pi\omega_{N-1}$ , then

B_{w(1)}(\sigma_{1})\cdots B_{w(N-1)}(\sigma_{N-1})=B_{\pi(N-1)}(\sigma_{1})\cdots B_{\pi(1)}(\sigma_{N-1}).

(4.33)

Let $\pi=s_{i_{1}}s_{i_{2}}\cdots s_{i_{p}}$ , where $p=l(\pi)$ . Then we can apply the transpositions $s_{N-i_{p}-1},\cdots,s_{N-i_{2}-1}$ , $s_{N-i_{1}-1}$ successively to the subindices of the right hand side of (4.33) to bring them to the strictly descending order, where $s_{j}$ exchanges the $j$ th and $(j+1)$ th subindices at each step. Since we need at least $p$ transpositions to complete the ordering, every transposition must bring a smaller subindex to the right of a neighboring larger subindex. According to (4.28), the application of $s_{j}$ must be accompanied by the action of $\bar{\partial}^{\beta}_{j}$ to the right hand side of (4.33) in order to keep the equality. Therefore,

B_{\pi(N-1)}(\sigma_{1})\cdots B_{\pi(1)}(\sigma_{N-1})=\bar{\partial}^{\beta}_{N-i_{p}-1}\cdots\bar{\partial}^{\beta}_{N-i_{2}-1}\bar{\partial}^{\beta}_{N-i_{1}-1}B_{N-1}(\sigma_{1})\cdots B_{1}(\sigma_{N-1}).

Because of the identity

s_{N-i_{p}-1}\cdots s_{N-i_{2}-1}s_{N-i_{1}-1}=\omega_{N-1}s_{i_{p}}\cdots s_{i_{2}}s_{i_{1}}\omega_{N-1}=\omega_{N-1}\pi^{-1}\omega_{N-1}=w^{-1}\omega_{N-1},

the proof is thus complete. ∎

Remark 3.

Lemma 4.7 allows us to deduce the matrix elements of $B_{w(1)}(\sigma_{1})\cdots B_{w(N-1)}(\sigma_{N-1})$ from Theorem 4.6.

Example 1.

For $N=3$ , we have, by Eq.(4.2) or Theorem 4.6,

		$\displaystyle B_{2}(\sigma_{1})B_{1}(\sigma_{2})\ket{000}$
	$\displaystyle=$	$\displaystyle\ket{210}+(\sigma_{1}\ominus t_{1})\ket{120}+[(\sigma_{1}\ominus t_{1})+(\sigma_{2}\ominus t_{2})+\beta(\sigma_{1}\ominus t_{1})(\sigma_{2}\ominus t_{2})]\ket{201}$
		$\displaystyle+(\sigma_{1}\ominus t_{1})(\sigma_{1}\ominus t_{2})\ket{102}+(\sigma_{1}\ominus t_{1})(\sigma_{2}\ominus t_{1})\ket{021}+(\sigma_{1}\ominus t_{1})(\sigma_{1}\ominus t_{2})(\sigma_{2}\ominus t_{1})\ket{012}$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{123}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})\ket{210}+\mathcal{G}^{(\beta)}_{213}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})\ket{120}+\mathcal{G}^{(\beta)}_{132}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})\ket{201}$
		$\displaystyle+\mathcal{G}^{(\beta)}_{312}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})\ket{102}+\mathcal{G}^{(\beta)}_{231}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})\ket{021}+\mathcal{G}^{(\beta)}_{321}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})\ket{012},$

while

	$\displaystyle B_{1}(\sigma_{1})B_{2}(\sigma_{2})\ket{000}$	(4.34)
$\displaystyle=$	$\displaystyle(1+\beta(\sigma_{1}\ominus t_{1}))\ket{120}+[(\sigma_{2}\ominus t_{1})(1+\beta(\sigma_{1}\ominus t_{2}))+(\sigma_{1}\ominus t_{2})(1+\beta(\sigma_{1}\ominus t_{1}))]\ket{102}$
	$\displaystyle+(\sigma_{1}\ominus t_{1})(\sigma_{2}\ominus t_{1})(1+\beta(\sigma_{1}\ominus t_{2}))\ket{012}.$

It is easy to verify that Eq.(4.34) is consistent with Lemma 4.7 since we have

		$\displaystyle\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{213}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=1+\beta(\sigma_{1}\ominus t_{1}),$
		$\displaystyle\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{321}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=(\sigma_{1}\ominus t_{1})(\sigma_{2}\ominus t_{1})(1+\beta(\sigma_{1}\ominus t_{2})),$
		$\displaystyle\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{312}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=(\sigma_{2}\ominus t_{1})(1+\beta(\sigma_{1}\ominus t_{2}))+(\sigma_{1}\ominus t_{2})(1+\beta(\sigma_{1}\ominus t_{1})),$
		$\displaystyle\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{231}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{132}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{123}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=0.$

Now we would like to apply Theorem 4.6 and Lemma 4.7 to the Bethe ansatz state (3.4). First, we show the following

Theorem 4.8.

For any $w\in S_{n}$ , we have the following form of the Cauchy identity

\mathcal{G}^{(\beta)}_{w}(\bm{x};\ominus\bm{t})=\sum_{v\in S_{n}}\mathcal{G}^{(\beta)}_{v}(\bm{x};\ominus\bm{z})\bar{\partial}^{\beta}_{v}\mathcal{G}^{(\beta)}_{w}(\bm{z};\ominus\bm{t}).

Proof.

The generalized Cauchy identity Eq.(A.7) can be rewritten as (notice that $\ominus\ominus z=z$ )

\mathcal{G}^{(\beta)}_{w}(\ominus\bm{t};\bm{x})=\sum_{v\in S_{n}}\mathcal{G}^{(\beta)}_{v}(\bm{x};\ominus\bm{z})\mathcal{G}^{(\beta)}_{v^{-1},w}(\ominus\bm{t};\bm{z}).

(4.35)

From Eq.(A.4), (A.5) and (A.6), we get

\mathcal{G}^{(\beta)}_{v^{-1},w}(\ominus\bm{t};\bm{z})=\bar{\partial}^{\beta}_{v,\bm{z}}\mathcal{G}^{(\beta)}_{1,w}(\ominus\bm{t};\bm{z})=\bar{\partial}^{\beta}_{v}\mathcal{G}^{(\beta)}_{w^{-1}}(\bm{z};\ominus\bm{t}).

Plugging the identity above in (4.35), we arrive at

\mathcal{G}^{(\beta)}_{w^{-1}}(\bm{x};\ominus\bm{t})=\sum_{v\in S_{n}}\mathcal{G}^{(\beta)}_{v}(\bm{x};\ominus\bm{z})\bar{\partial}^{\beta}_{v}\mathcal{G}^{(\beta)}_{w^{-1}}(\bm{z};\ominus\bm{t}).

Then the theorem is proved by replacing $w^{-1}$ with $w$ . ∎

Let $L_{n}(\bm{t},\beta)$ be the submodule of $\mathbb{C}(\bm{t},\beta)[x_{1},\cdots,x_{n}]$ defined by

L_{n}(\bm{t},\beta)=\mathrm{span}_{\mathbb{C}(\bm{t},\beta)}\{x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}~{}|~{}0\leq i_{j}\leq n-j\}.

(4.36)

Corollary 4.9.

For any $F(\bm{x};\bm{t},\beta)\in L_{n}(\bm{t},\beta)$ , we have

F(\bm{x};\bm{t},\beta)=\sum_{v\in S_{n}}\mathcal{G}^{(\beta)}_{v}(\bm{x};\ominus\bm{z})\bar{\partial}^{\beta}_{v}F(\bm{z};\bm{t},\beta).

(4.37)

Proof.

Since $\{\mathcal{G}^{(\beta)}_{w}(\bm{x};\ominus\bm{t})~{}|~{}w\in S_{n}\}$ is a $\mathbb{C}(\bm{t},\beta)$ -basis of $L_{n}(\bm{t},\beta)$ (as are the Schubert and Grothendieck polynomials [20, Proposition 2.7]), Eq.(4.37) follows from Theorem 4.8 and linearity of $\bar{\partial}^{\beta}_{v}$ . ∎

In section 3, we have seen that $\sigma^{(N-i)}_{a}$ ( $\beta=0$ ) or $1-\sigma^{(N-i)}_{a}$ ( $\beta=-1$ ), $a=1,\cdots,i$ , can be identified with the Chern roots of $\mathcal{S}_{i}$ when $\sigma^{(N-i)}_{a}$ ’s satisfy the Bethe ansatz equations Eq.(3.5)-(3.7). We will show that the Bethe ansatz state (3.4) generates the double $\beta$ -Grothendieck polynomials when this identification is implemented. For this purpose, we need the following

Lemma 4.10.

For $\pi\in S_{n}$ , if $e_{l}(\sigma_{1},\cdots,\sigma_{n-1})$ is identified with $e_{l}(x_{1},\cdots,x_{n-1})$ for every $l=1,\cdots,n-1$ , then

		$\displaystyle\sum_{w\in S_{n-1}}\mathcal{G}^{(\beta)}_{w}(x_{1},\cdots,x_{n-2};\ominus\sigma_{1},\cdots,\ominus\sigma_{n-2})\cdot\bar{\partial}^{\beta}_{w}\mathcal{G}^{(\beta)}_{\pi}(\sigma_{1},\cdots,\sigma_{n-1};\ominus t_{1},\cdots,\ominus t_{n-1})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{\pi}(x_{1},\cdots,x_{n-1};\ominus t_{1},\cdots,\ominus t_{n-1}).$

Proof.

Let $L_{n}(\bm{t},\beta)$ be the submodule defined by (4.36). Then, since $\{\mathcal{G}^{(\beta)}_{w}(\bm{x};\ominus\bm{t})~{}|~{}w\in S_{n}\}$ and
$\{e_{i_{1}}(x_{1})e_{i_{2}}(x_{1},x_{2})\cdots e_{i_{n-1}}(x_{1},\cdots,x_{n-1})~{}|~{}0\leq i_{j}\leq j\}$ are both $\mathbb{C}(\bm{t},\beta)$ -bases of $L_{n}(\bm{t},\beta)$ , we have the expansion

		$\displaystyle\mathcal{G}^{(\beta)}_{\pi}(\sigma_{1},\cdots,\sigma_{n-1};\ominus t_{1},\cdots,\ominus t_{n-1})$
	$\displaystyle=$	$\displaystyle\sum_{i_{1},i_{2},\cdots,i_{n-1}}C^{\pi}_{i_{1}\cdots i_{n-1}}(\bm{t},\beta)e_{i_{1}}(\sigma_{1})e_{i_{2}}(\sigma_{1},\sigma_{2})\cdots e_{i_{n-1}}(\sigma_{1},\cdots,\sigma_{n-1}),$

where $C^{\pi}_{i_{1}\cdots i_{n-1}}(\bm{t},\beta)\in\mathbb{C}(\bm{t},\beta)$ .

Therefore

		$\displaystyle\sum_{w\in S_{n-1}}\mathcal{G}^{(\beta)}_{w}(x_{1},\cdots,x_{n-2};\ominus\sigma_{1},\cdots,\ominus\sigma_{n-2})\cdot\bar{\partial}^{\beta}_{w}\mathcal{G}^{(\beta)}_{\pi}(\sigma_{1},\cdots,\sigma_{n-1};\ominus t_{1},\cdots,\ominus t_{n-1})$
	$\displaystyle=$	$\displaystyle\sum_{w\in S_{n-1}}\mathcal{G}^{(\beta)}_{w}(x_{1},\cdots,x_{n-2};\ominus\sigma_{1},\cdots,\ominus\sigma_{n-2})$
		$\displaystyle\quad\quad\quad\cdot\bar{\partial}^{\beta}_{w}\left[\sum_{i_{1},i_{2},\cdots,i_{n-1}}C^{\pi}_{i_{1}\cdots i_{n-1}}(\bm{t},\beta)e_{i_{1}}(\sigma_{1})e_{i_{2}}(\sigma_{1},\sigma_{2})\cdots e_{i_{n-1}}(\sigma_{1},\cdots,\sigma_{n-1})\right]$
	$\displaystyle=$	$\displaystyle\sum_{i_{1},i_{2},\cdots,i_{n-1}}C^{\pi}_{i_{1}\cdots i_{n-1}}(\bm{t},\beta)e_{i_{n-1}}(\sigma_{1},\cdots,\sigma_{n-1})$
		$\displaystyle\cdot\left[\sum_{w\in S_{n-1}}\mathcal{G}^{(\beta)}_{w}(x_{1},\cdots,x_{n-2};\ominus\sigma_{1},\cdots,\ominus\sigma_{n-2})\cdot\bar{\partial}^{\beta}_{w}\left(e_{i_{1}}(\sigma_{1})e_{i_{2}}(\sigma_{1},\sigma_{2})\cdots e_{i_{n-2}}(\sigma_{1},\cdots,\sigma_{n-2})\right)\right]$
	$\displaystyle=$	$\displaystyle\sum_{i_{1},i_{2},\cdots,i_{n-1}}C^{\pi}_{i_{1}\cdots i_{n-1}}(\bm{t},\beta)e_{i_{1}}(x_{1})e_{i_{2}}(x_{1},x_{2})\cdots e_{i_{n-2}}(x_{1},\cdots,x_{n-2})e_{i_{n-1}}(\sigma_{1},\cdots,\sigma_{n-1})$
	$\displaystyle=$	$\displaystyle\sum_{i_{1},i_{2},\cdots,i_{n-1}}C^{\pi}_{i_{1}\cdots i_{n-1}}(\bm{t},\beta)e_{i_{1}}(x_{1})e_{i_{2}}(x_{1},x_{2})\cdots e_{i_{n-2}}(x_{1},\cdots,x_{n-2})e_{i_{n-1}}(x_{1},\cdots,x_{n-1})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{\pi}(x_{1},\cdots,x_{n-1};\ominus t_{1},\cdots,\ominus t_{n-1}),$

where the second equality is due to $\bar{\partial}^{\beta}_{i}(f\cdot g)=f(\bar{\partial}^{\beta}_{i}g)$ if $f$ is symmetric in $(\sigma_{1},\cdots,\sigma_{n-1})$ , the third equality is due to Corollary 4.9, and $e_{i_{n-1}}(\sigma_{1},\cdots,\sigma_{n-1})$ is identified with $e_{i_{n-1}}(x_{1},\cdots,x_{n-1})$ in the fourth equality. ∎

Example 2.

Notice that $\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus\sigma_{1})=1,\mathcal{G}^{(\beta)}_{21}(x_{1};\ominus\sigma_{1})=x_{1}\ominus\sigma_{1}$ . In the case $n=3$ , following Example 1, one can check

		$\displaystyle\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus\sigma_{1})\mathcal{G}^{(\beta)}_{321}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})+\mathcal{G}^{(\beta)}_{21}(x_{1};\ominus\sigma_{1})\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{321}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})$
	$\displaystyle=$	$\displaystyle(\sigma_{1}\ominus t_{1})(\sigma_{2}\ominus t_{1})(x_{1}\ominus t_{2})=(x_{1}\ominus t_{1})(x_{2}\ominus t_{1})(x_{1}\ominus t_{2})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{321}(x_{1},x_{2};\ominus t_{1},\ominus t_{2}),$

		$\displaystyle\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus\sigma_{1})\mathcal{G}^{(\beta)}_{231}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=(\sigma_{1}\ominus t_{1})(\sigma_{2}\ominus t_{1})=(x_{1}\ominus t_{1})(x_{2}\ominus t_{1})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{231}(x_{1},x_{2};\ominus t_{1},\ominus t_{2}),$

		$\displaystyle\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus\sigma_{1})\mathcal{G}^{(\beta)}_{312}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})+\mathcal{G}^{(\beta)}_{21}(x_{1};\ominus\sigma_{1})\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{312}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})$
	$\displaystyle=$	$\displaystyle\frac{x_{1}(\sigma_{1}+\sigma_{2}-t_{1}-t_{2})+(t_{1}t_{2}-\sigma_{1}\sigma_{2})}{(1+\beta t_{1})(1+\beta t_{2})}=(x_{1}\ominus t_{1})(x_{2}\ominus t_{2})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{312}(x_{1},x_{2};\ominus t_{1},\ominus t_{2}),$

		$\displaystyle\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus\sigma_{1})\mathcal{G}^{(\beta)}_{132}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})=\frac{x_{1}+x_{2}-t_{1}-t_{2}+\beta(x_{1}x_{2}-t_{1}t_{2})}{(1+\beta t_{1})(1+\beta t_{2})}$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{132}(x_{1},x_{2};\ominus t_{1},\ominus t_{2}),$

		$\displaystyle\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus\sigma_{1})\mathcal{G}^{(\beta)}_{213}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})+\mathcal{G}^{(\beta)}_{21}(x_{1};\ominus\sigma_{1})\bar{\partial}^{\beta}_{1}\mathcal{G}^{(\beta)}_{213}(\sigma_{1},\sigma_{2};\ominus t_{1},\ominus t_{2})$
	$\displaystyle=$	$\displaystyle(\sigma_{1}\ominus t_{1})+(x_{1}\ominus\sigma_{1})(1+\beta(\sigma_{1}\ominus t_{1}))=x_{1}\ominus t_{1}$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{213}(x_{1},x_{2};\ominus t_{1},\ominus t_{2}),$

where the following identifications have been made

e_{1}(\sigma_{1},\sigma_{2})=e_{1}(x_{1},x_{2}),\quad e_{2}(\sigma_{1},\sigma_{2})=e_{2}(x_{1},x_{2}).

Finally, we show that the Bethe state satisfies the following

Theorem 4.11.

When $e_{l}(\sigma^{(i)}_{1},\cdots,\sigma^{(i)}_{N-i})$ is identified with $e_{l}(x_{1},\cdots,x_{N-i})$ for all $i=1,\cdots,N-1$ and $l=1,\cdots,N-i$ , the Bethe ansatz state (3.4) (in the case $k_{i}=N-i$ ) has the expansion

\ket{\psi^{(0)}}=\sum_{w\in S_{N}}\mathcal{G}^{(\beta)}_{w}(x_{1},\cdots,x_{N-1};\ominus t_{1},\cdots,\ominus t_{N-1})\ket{\omega_{N}w^{-1}}.

(4.38)

Proof.

We prove by induction on the number of sites. For a two-site system, it is easy to check that, when $\sigma^{(1)}_{1}=x_{1}$ ,

		$\displaystyle\bra{01}B_{1}(\sigma^{(1)}_{1})\ket{00}=\sigma^{(1)}_{1}\ominus t_{1}=x_{1}\ominus t_{1}=\mathcal{G}^{(\beta)}_{21}(x_{1};\ominus t_{1}),$
		$\displaystyle\bra{10}B_{1}(\sigma^{(1)}_{1})\ket{00}=1=\mathcal{G}^{(\beta)}_{12}(x_{1};\ominus t_{1}),$

so Eq.(4.38) holds for $N=2$ . Now assume Eq.(4.38) holds for $N=2,3,\cdots,m$ with $m\geq 2$ , then we have, from Eq.(3.14) and (3.15), for $w_{l}\in S_{l}$ (notice that $\ket{\psi^{(N-s)}}$ is the Bethe state of a GL $(s)$ system with $s$ sites),

	$\displaystyle\braket{\omega_{2}w_{2}^{-1}}{\psi^{(N-2)}}$	$\displaystyle=\mathcal{G}^{(\beta)}_{w_{2}}(x_{1};\ominus\sigma_{1}^{(N-2)}),$
	$\displaystyle\braket{\omega_{3}w_{3}^{-1}}{\psi^{(N-3)}}$	$\displaystyle=\mathcal{G}^{(\beta)}_{w_{3}}(x_{1},x_{2};\ominus\sigma_{1}^{(N-3)},\ominus\sigma_{2}^{(N-3)}),$
		$\displaystyle~{}~{}\vdots$
	$\displaystyle\braket{\omega_{m}w_{m}^{-1}}{\psi^{(N-m)}}$	$\displaystyle=\mathcal{G}^{(\beta)}_{w_{m}}(x_{1},\cdots,x_{m-1};\ominus\sigma_{1}^{(N-m)},\cdots,\ominus\sigma_{m-1}^{(N-m)}).$

Eq.(3.14) and (3.15) yield

\ket{\psi^{(N-m-1)}}=\sum_{v_{m}\in S_{m}}\braket{v_{m}}{\psi^{(N-m)}}B^{(N-m-1)}_{v_{m}(1)}(\sigma^{(N-m)}_{1})\cdots B^{(N-m-1)}_{v_{m}(m)}(\sigma^{(N-m)}_{m})\ket{\Omega^{(N-m-1)}}.

(4.39)

It is easy to verify that

		$\displaystyle\bra{\omega_{m+1}w_{m+1}^{-1}}B^{(N-m-1)}_{v_{m}(1)}(\sigma^{(N-m)}_{1})\cdots B^{(N-m-1)}_{v_{m}(m)}(\sigma^{(N-m)}_{m})\ket{\Omega^{(N-m-1)}}$
	$\displaystyle=$	$\displaystyle\bar{\partial}^{\beta}_{v^{-1}_{m}\omega_{m}}\bra{\omega_{m+1}w_{m+1}^{-1}}B^{(N-m-1)}_{m}(\sigma^{(N-m)}_{1})\cdots B^{(N-m-1)}_{1}(\sigma^{(N-m)}_{m})\ket{\Omega^{(N-m-1)}}$
	$\displaystyle=$	$\displaystyle\bar{\partial}^{\beta}_{v^{-1}_{m}\omega_{m}}\mathcal{G}^{(\beta)}_{w_{m+1}}(\sigma^{(N-m)}_{1},\cdots,\sigma^{(N-m)}_{m};\ominus\sigma_{1}^{(N-m-1)},\cdots,\ominus\sigma_{m}^{(N-m-1)}),$

where the first equality is due to Lemma 4.7 and the second equality is due to Theorem 4.6. As $\braket{v_{m}}{\psi^{(N-m)}}=\mathcal{G}^{(\beta)}_{v_{m}^{-1}\omega_{m}}(x_{1},\cdots,x_{m-1};\ominus\sigma^{(N-m)}_{1},\cdots,\ominus\sigma^{(N-m)}_{m-1})$ from the induction hypothesis, Lemma 4.10 and Eq.(4.39) then yield

		$\displaystyle\braket{\omega_{m+1}w_{m+1}^{-1}}{\psi^{(N-m-1)}}$
	$\displaystyle=$	$\displaystyle\sum_{v_{m}\in S_{m}}\braket{v_{m}}{\psi^{(N-m)}}\bra{\omega_{m+1}w_{m+1}^{-1}}B^{(N-m-1)}_{v_{m}(1)}(\sigma^{(N-m)}_{1})\cdots B^{(N-m-1)}_{v_{m}(m)}(\sigma^{(N-m)}_{m})\ket{\Omega^{(N-m-1)}}$
	$\displaystyle=$	$\displaystyle\sum_{v_{m}\in S_{m}}\mathcal{G}^{(\beta)}_{v^{-1}_{m}\omega_{m}}(x_{1},\cdots,x_{m-1};\ominus\sigma^{(N-m)}_{1},\cdots,\ominus\sigma^{(N-m)}_{m-1})$
		$\displaystyle\quad\quad\quad\cdot\bar{\partial}^{\beta}_{v_{m}^{-1}\omega_{m}}\mathcal{G}^{(\beta)}_{w_{m+1}}(\sigma^{(N-m)}_{1},\cdots,\sigma^{(N-m)}_{m};\ominus\sigma^{(N-m-1)}_{1},\cdots,\ominus\sigma^{(N-m-1)}_{m})$
	$\displaystyle=$	$\displaystyle\mathcal{G}^{(\beta)}_{w_{m+1}}(x_{1},\cdots,x_{m};\ominus\sigma^{(N-m-1)}_{1},\cdots,\ominus\sigma^{(N-m-1)}_{m}).$

Notice that $\sigma^{(0)}_{i}=t_{i}$ , the proof is thus complete by induction. ∎

Remark 4.

As discussed in Section 3, when the Bethe ansatz equations Eq.(3.5)-(3.7) are taken into account, we should interpret $\sigma^{(N-i)}_{a}$ ( $\beta=0$ ) or $1-\sigma^{(N-i)}_{a}$ ( $\beta=-1$ ), $a=1,\cdots,i$ , as the Chern roots of $\mathcal{S}_{i}$ . Therefore, $x_{i}$ should be interpreted as the first Chern class of $\mathcal{S}_{i}/\mathcal{S}_{i-1}$ when $\beta=0$ in the cohomology ring, and $1-x_{i}$ should be interpreted as the K-theory class of $\mathcal{S}_{i}/\mathcal{S}_{i-1}$ when $\beta=-1$ in the K-theory ring.

Remark 5.

More generally, for $\beta\neq 0$ , we expect $1+\beta x_{i}$ to represent $\mathcal{S}_{i}/\mathcal{S}_{i-1}$ in the connective K-theory ring of the flag variety [21].

Remark 6.

In the case of full flag variety, $i_{a}\neq i_{b}$ for $a\neq b$ in Eq.(3.14). In the case of general partial flag variety $\mathrm{Fl}(k_{n-1},k_{n-2},\cdots,k_{1};N)$ , it is possible that $i_{a}=i_{b}$ for some $a\neq b$ . Because of $[B_{i}(x),B_{i}(y)]=0$ (Eq.(4.27)), we expect the expansion coefficients of the Bethe ansatz state in the natural basis to be given by the $\beta$ -Grothendieck polynomials indexed by $S_{N}/W$ , where $W$ is generated by the simple transpositions $s_{i}$ for $i\not\in\{k_{n-1},k_{n-2},\cdots,k_{1}\}$ , i.e. $\beta$ -Grothendieck polynomials symmetric in $(x_{k_{s}+1},\cdots,x_{k_{s-1}})$ , which can be identified with the Chern roots of $\mathcal{S}_{n-s+1}/\mathcal{S}_{n-s}$ , for all $s=1,\cdots,n$ .

Acknowledgement

The author would like to thank Xiang-Mao Ding, Leonardo Mihalcea and Eric Sharpe for useful comments. This work is partially supported by the National Natural Science Foundation of China (Grant No. 12475005), the Natural Science Foundation of Shanghai (Grant No. 24ZR1468600), and the Fundamental Research Funds for the Central Universities.

Appendix A Quantum cohomology/K-theory of flag varieties and the double $\beta$ -Grothendieck polynomials

In this appendix, we give a brief review of the quantum cohomology/K-theory ring of flag varieties, and the double $\beta$ -Grothendieck polynomials. The reader may refer to e.g. [13, 17, 18, 20, 22, 23, 24] and references therein for more information.

Given $E=\mathbb{C}^{N}$ and $\bm{a}=(a_{1},a_{2},\cdots,a_{m})$ such that $0<a_{1}<a_{2}<\cdots<a_{m}<N$ , the flag variety $\mathrm{Fl}(\bm{a};N)=\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N)$ is defined to be the set of the flags

V_{1}\subset V_{2}\subset\cdots\subset V_{m}\subset E

such that $\dim(V_{i})=a_{i}$ for all $i$ .

There is a sequence of vector bundles on $\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N)$ (the tautological sequence):

\mathcal{S}_{1}\subset\mathcal{S}_{2}\subset\cdots\subset\mathcal{S}_{m}\subset E

with $\mathrm{rank}(\mathcal{S}_{i})=a_{i}$ . Let $S_{N}$ be the group of permutations of $N$ elements, and let $l(w)$ be the length⁷⁷7The length of a permutation $w\in S_{N}$ is the number of pairs $(i,j)$ such that $i<j$ and $w(i)>w(j)$ . of a permutation $w\in\mathrm{S}_{N}$ . The Schubert varieties are subvarieties of $\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N)$ indexed by the set $S_{N}(\bm{a})=S_{N}/W_{\bm{a}}$ , where $W_{\bm{a}}\subset S_{N}$ is the subgroup generated by the simple transpositions $s_{i}=(i,i+1)$ for $i\not\in\{a_{1},\cdots,a_{m}\}$ . The codimension of the Schubert variety indexed by the representative $w\in S_{N}$ with shortest length is equal to $l(w)$ .

Let $\Omega^{(\bm{a})}_{w}$ denote the cohomology class Poincaré dual to the Schubert variety indexed by $w$ , then the Schubert classes $\Omega^{(\bm{a})}_{w}$ form a basis for the cohomology ring $H^{*}(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{l};N),\mathbb{Z})$ .

Let $\mathrm{Fl}(N):=\mathrm{Fl}(1,2,\cdots,N-1;N)$ denote the full flag variety. The cohomology of $\mathrm{Fl}(N)$ has the following presentation

H^{*}(\mathrm{Fl}(N))=\mathbb{Z}[x_{1},\cdots,x_{N}]/(e^{N}_{1},\cdots,e^{N}_{N}),

where $e^{N}_{i}=e_{i}(x_{1},\cdots,x_{N})$ is the $i$ th elementary symmetric polynomial in $N$ variables. In this presentation, $x_{i}$ is identified with the Chern class $c_{1}(\mathcal{S}_{i}/\mathcal{S}_{i-1})$ and the Schubert class $\Omega_{w}$ is represented by the Schubert polynomial $X_{w}(x_{1},\cdots,x_{N})$ , which means that the multiplication rules of the Schubert classes are the same as those of the Schubert polynomials modulo $e^{N}_{i},i=1,\cdots,N$ .

The quantum elementary polynomials $E^{N}_{i}$ are defined via the Givental-Kim determinant. Let

\Gamma_{N}=\left(\begin{array}[]{ccccc}x_{1}&q_{1}&0&\cdots&0\\ -1&x_{2}&q_{2}&\cdots&0\\ 0&-1&x_{3}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&x_{N}\end{array}\right),

then $E^{N}_{i}$ is defined as the coefficient of $\lambda^{i}$ in the characteristic polynomial $\det(1+\lambda\Gamma_{N})$ . The quantum cohomology ring of the full flag variety can be presented as

QH^{*}(\mathrm{Fl}(N))=\mathbb{Z}[x_{1},\cdots,x_{N}]/(E^{N}_{1},\cdots,E^{N}_{N}).

Ring relations of the quantum cohomology of partial flag varieties $QH^{*}(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ can be presented similarly, see [13, 22]. The structure constants of $QH^{*}(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ encode the Gromov-Witten invariants of the flag variety.

The K-theory ring of the flag variety $K(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ is the Grothendieck group of the coherent sheaves of $\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N)$ . A basis of $K(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ can be chosen as the structure sheaves of the Schubert varieties $\{\mathcal{O}_{w}|w\in S_{n}(\bm{a})\}$ , which can be represented by the Grothendieck polynomials $G_{w}(x_{1},\cdots,x_{N})$ . Ring relations can be found in e.g. [23]. One way to present the ring relations of the (quantum) K-theory ring of $\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N)$ is to encapsulate them in the (quantum) Whitney relations [12]. The structure constants of the quantum K-theory ring $QK(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ encode the K-theoretic Gromov-Witten invariants of the flag variety.

For the $T=(\mathbb{C}^{*})^{N}$ action on $E=\mathbb{C}^{N}$ with characters (equivariant parameters) $(t_{1},\cdots,t_{N})$ , we can define the equivariant quantum cohomology ring $QH_{T}^{*}(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ and equivariant quantum K-theory ring $QK_{T}(\mathrm{Fl}(a_{1},a_{2},\cdots,a_{m};N))$ . The representatives of the Schubert classes in the equivariant cohomology and K-theory rings are given by the double Schubert polynomials $X_{w}(x_{1},\cdots,x_{N};t_{1},\cdots,t_{N})$ and the double Grothendieck polynomials $G_{w}(x_{1},\cdots,x_{N};t_{1},\cdots,t_{N})$ respectively.

The double Schubert and double Grothendieck polynomials are special cases of a one-parameter family of polynomials called the double $\beta$ -Grothedieck polynomials $\mathcal{G}^{(\beta)}_{w},w\in S_{N}$ [18]. Let $\mathbb{K}$ be a field of zero characteristic and $\beta$ be a formal variable, $\bm{x}=(x_{1},\cdots,x_{N}),\bm{y}=(y_{1},\cdots,y_{N})$ , define the $\beta$ -divided-difference operator $\partial^{\beta}_{i}$ acting on $\mathbb{K}(\beta)[x_{1},\cdots,x_{N},y_{1},\cdots,y_{N}]$ by

\partial^{\beta}_{i}f(x_{1},\cdots,x_{N};\bm{y}):=\frac{(1+\beta x_{i+1})f(x_{1},\cdots,x_{N};\bm{y})-(1+\beta x_{i})f(x_{1},\cdots,x_{i+1},x_{i},\cdots,x_{N};\bm{y})}{x_{i}-x_{i+1}}.

(A.1)

If $w=\omega_{N}$ is the permutation in $S_{N}$ with maximal length, then we define

\mathcal{G}^{(\beta)}_{\omega_{N}}(\bm{x};\bm{y})=\prod_{i+j\leq N}(x_{i}+y_{j}+\beta x_{i}y_{j}),

(A.2)

and for other $w\in\mathrm{S}_{N}$ , $\mathcal{G}^{(\beta)}_{w}$ can be defined recursively by

\mathcal{G}^{(\beta)}_{ws_{i}}(\bm{x};\bm{y})=\partial^{\beta}_{i}\mathcal{G}^{(\beta)}_{w}(\bm{x};\bm{y})

(A.3)

if $l(ws_{i})=l(w)-1$ .

The double $\beta$ -Grothendieck polynomial $\mathcal{G}^{(\beta)}_{w}(\bm{x};\bm{y})$ reduces to the double Schubert polynomial $X_{w}(\bm{x};\bm{y})$ when $\beta=0$ , and reduces to the double Grothendieck polynomial $G_{w}(\bm{x};\bm{y})$ when $\beta=-1$ .

The double $\beta$ -Grothendieck polynomials satisfy (see Lemma 5.8 of [21])

\mathcal{G}^{(\beta)}_{w}(\bm{x};\bm{y})=\mathcal{G}^{(\beta)}_{w^{-1}}(\bm{y};\bm{x}).

(A.4)

The dual double $\beta$ -Grothendieck polynomials $\mathcal{H}^{(\beta)}_{w}(\bm{x};\bm{y})$ can be defined recursively by setting $\mathcal{H}^{(\beta)}_{w_{N}}(\bm{x};\bm{y})=\mathcal{G}^{(\beta)}_{w_{N}}(\bm{x};\bm{y})$ and

\mathcal{H}^{(\beta)}_{ws_{i}}(\bm{x};\bm{y})=\bar{\partial}^{\beta}_{i}\mathcal{H}^{(\beta)}_{w}(\bm{x};\bm{y})

if $l(ws_{i})=l(w)-1$ [19], where $\bar{\partial}^{\beta}_{i}=\partial^{\beta}_{i}+\beta$ .

More generally, for $v,w\in S_{N}$ , one can define the biaxial double $\beta$ -Grothendieck polynomials $\mathcal{G}^{(\beta)}_{v,w}(\bm{x};\bm{y})$ as follows (see Definition 2.6 of [9]):

\mathcal{G}^{(\beta)}_{1,w}(\bm{x};\bm{y})=\mathcal{G}^{(\beta)}_{w}(\bm{x};\bm{y}).

(A.5)

If $l(vs_{i})=l(v)+1$ , set

\mathcal{G}^{(\beta)}_{vs_{i},w}(\bm{x};\bm{y})=\bar{\partial}^{\beta}_{i,\bm{y}}\mathcal{G}^{(\beta)}_{v,w}(\bm{x};\bm{y}),

(A.6)

where $\bar{\partial}^{\beta}_{i,\bm{y}}$ is the $\bar{\partial}^{\beta}_{i}$ operator acting on the set of variables $\bm{y}=(y_{1},y_{2},\cdots,y_{N})$ . We have the following generalized Cauchy identity:

Theorem A.1 (Theorem 7.1 of [9]).

For any $w\in S_{N}$ ,

\mathcal{G}^{(\beta)}_{w}(\bm{x};\bm{y})=\sum_{v\in S_{N}}\mathcal{G}^{(\beta)}_{v}(\bm{y};\bm{z})\mathcal{G}^{(\beta)}_{v^{-1},w}(\bm{x};\ominus\bm{z}),

(A.7)

where $\ominus z_{i}=-z_{i}/(1+\beta z_{i})$ .

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Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double β\beta-Grothendieck polynomials

Abstract

1 Introduction

2 The GL(n)(n) asymmetric five vertex model

Theorem 2.1.

Proof.

3 Bethe ansatz equations and the equivariant quantum cohomology/K-theory ring of flag varieties

Theorem 3.1.

Proof.

Remark 1.

Remark 2.

4 Relationship with the double β\beta-Grothendieck polynomials

Lemma 4.1.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Theorem 4.6.

Proof.

Lemma 4.7.

Proof.

Remark 3.

Example 1.

Theorem 4.8.

Proof.

Corollary 4.9.

Proof.

Lemma 4.10.

Proof.

Example 2.

Theorem 4.11.

Proof.

Remark 4.

Remark 5.

Remark 6.

Acknowledgement

Appendix A Quantum cohomology/K-theory of flag varieties and the double β\beta-Grothendieck polynomials

Theorem A.1 (Theorem 7.1 of [9]).

References

Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $\beta$ -Grothendieck polynomials

2 The GL $(n)$ asymmetric five vertex model

3 Bethe ansatz equations and the equivariant quantum
cohomology/K-theory ring of flag varieties

4 Relationship with the double $\beta$ -Grothendieck polynomials

Appendix A Quantum cohomology/K-theory of flag varieties and the double $\beta$ -Grothendieck polynomials