Presenting Little q-Schur Superalgebra uk(2|1,3)

Xiaoxue GE; Li ZHANG

doi:10.1051/wujns/2026313263

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 31, Number 3, June 2026


Page(s)		263 - 268
DOI		https://doi.org/10.1051/wujns/2026313263
Published online		24 June 2026

Wuhan University Journal of Natural Sciences, 2026, Vol.31 No.3, 263-268

Mathematics

CLC number: O152.5

Presenting Little q-Schur Superalgebra u_k(2|1,3)

小q-Schur超代数 u_k(2|1,3)的展示

Xiaoxue GE (葛小雪) and Li ZHANG (张立)^†

College of Mathematics and Physics, China Three Gorges University, Yichang 443002, Hubei, China (三峡大学　数理学院，湖北　宜昌　443002)

E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 February 2025

Abstract

The purpose of this paper is to investigate the generators and relations for little q-Schur superalgebra $u_{k} (2 | 1,3)$ Mathematical equation . The basis of $u_{k} (2 | 1,3)$ is obtained through the Poincaré-Birkhoff-Witt (PBW) basis of q-Schur superalgebra $S_{v} (m | n, r)$ , then we present a new set for generators and relations for $u_{k} (2 | 1,3)$ by PBW basis.

摘要

本文研究小q-Schur超代数 $u_{k} (2 | 1,3)$ Mathematical equation 的生成元与关系式。由q-Schur超代数 $S_{v} (m | n, r)$ 的Poincaré-Birkhoff-Witt(PBW)基得到了小q-Schur超代数 $u_{k} (2 | 1,3)$ 的基，并由此给出了小q-Schur超代数 $u_{k} (2 | 1,3)$ 一组新的生成元与关系式。

Key words: q-Schur superalgebra / little q-Schur superalgebra / Poincaré-Birkhoff-Witt (PBW) basis / generators and relations

关键字 : q-Schur超代数 / 小q-Schur超代数 / Poincaré-Birkhoff-Witt(PBW)基 / 生成元与关系式

Cite this article: GE Xiaoxue, ZHANG Li. Presenting Little q-Schur Superalgebra u_k(21,3)[J]. Wuhan Univ J of Nat Sci, 2026, 31(3): 263-268.

Biography: GE Xiaoxue, female, Master candidate, research direction: algebraic groups, quantum groups. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Foundation item: Supported by the Natural Research Project of Yichang(A23-026)

© Wuhan University 2026

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Let $U (n, r)$ Mathematical equation be the q-Schur algebras over $Q (v)$ introduced by Dipper and James^[1-2], who have showed that q-Schur algebras play an important role in the investigation of presentation theory of the finite general linear groups. The presentation of the q-Schur algebras could be referred to Refs. [3-4]. Gao and Liu^[5] gave a set of primitives for the center of q-Schur algebras $U (2, r)$ Mathematical equation . Du et al^[6] studied the representation of little q-Schur algebra and gave a classification of finite representations of little q-Schur algebras under odd unit roots. And they pointed out that there is no nice and simple presentation for $u_{k} (n, r)$ . Then, Bian and Liu^[7] gave the presentation of little q-Schur algebras $u_{k} (2, r)$ Mathematical equation .

Little q-Schur superalgebras have been an important approach to studying q-Schur superalgebras. Turkey and Kujawa^[8] gave a presentation of the Schur superalgebra and its quantum analogue. Chen^[9] generalized algebra and quantum supergroup to the infinitesimal case, and finally obtained the infinitesimal theory of q-Schur superalgebras and the Beilinson-Lusztig-MacPherson (BLM) realization of integral quantum supergroups. Then, on one hand, we want to get deep understanding of the structure of the little q-Schur superalgebra $u_{k} (2 | 1, r)$ Mathematical equation by investigating the representation of $u_{k} (2 | 1,3)$ in this paper; on the other hand, since little q-Schur superalgebras are closely related to q-Schur superalgebras, the research results on $u_{k} (2 | 1,3)$ may be generalized to the study of more general little q-Schur superalgebras and even q-Schur superalgebras.

This paper studies the generators and relations for little q-Schur superalgebra $u_{k} (2 | 1,3)$ Mathematical equation .We briefly recall the definitions of q-Schur superalgebras $S_{v} (m | n, r)$ in Section 1. In Section 2, we prove that the set $W_{k}$ is the basis of $u_{k} (2 | 1,3)$ and present the generators and relations of $u_{k} (2 | 1,3)$ .

1 Preliminary

In the case of superalgebras, define a parity function, a mapping $\hat{} : {1,2, \dots, m + n} \to ℤ_{2}$ Mathematical equation , on the set ${1,2, \dots, m + n}$ , for any $a \in {1,2, \dots, m + n}$ , there is

$\hat{a} = {\begin{array}{l} \bar{0}, 1 \leq a \leq m; \\ \bar{1}, m + 1 \leq a \leq m + n . \end{array}$ Mathematical equation

Note $U (m | n) = U_{v} (g l_{m | n})$ Mathematical equation is the quantum enveloping superalgebra of the Lie superalgebra $g l_{m | n}$ ^[10]. Let $g = g l_{m | n}$ . As a vector space, $g$ is the set of $m + n$ by $m + n$ matrices. For $1 \leq i, j \leq m + n$ , we set $E_{i, j}$ to be the matrix unit with a 1 in $i$ -th row and $j$ Mathematical equation -th column. Then the set of matrix units forms a homogeneous basis for $g$ . The degree of $E_{i, j}$ is $\bar{i} + \bar{j}$ . For any $X, Y \in U (m | n)$ , $\hat{X}$ is the parity of $X$ , and the square bracket $[-, -]$ is the super-Lie bracket, it means $[X, Y] = X Y - {(- 1)}^{\hat{X} \hat{Y}} Y X$ Mathematical equation . From now on, we mark $ν_{a} = ν^{{(- 1)}^{\hat{a}}}$ for convenience.

Definition 1^[8] The quantum supergroup $U (m | n) = U_{v} (g l_{m | n})$ Mathematical equation is generated by the elements

$K_{a}^{\pm 1}, E_{h, h + 1}, E_{h + 1, h} (1 \leq a \leq m + n, 1 \leq h \leq m + n)$ Mathematical equation

subject to the following relations:

(Q1) $K_{a} K_{b} = K_{b} K_{a}, K_{a} K_{a}^{- 1} = 1$ Mathematical equation ;

(Q2) $K_{a} E_{h, h + 1} = v_{a}^{δ_{a, h} - δ_{a, h + 1}} E_{h, h + 1} K_{a}, K_{a} E_{h + 1, h} = v_{a}^{δ_{a, h + 1} - δ_{a, h}} E_{h + 1, h} K_{a};$ Mathematical equation

(Q3) $[E_{a, a + 1}, E_{b, b + 1}] = δ_{a, b} \frac{K_{a} K_{a + 1}^{- 1} - K_{a}^{- 1} K_{a + 1}}{ν_{a} - ν_{a}^{- 1}}$ Mathematical equation ;

(Q4) $[E_{a, a + 1}, E_{b, b + 1}] = 0, [E_{a + 1, a}, E_{b + 1, b}] = 0, f o r | a - b | > 1;$ Mathematical equation

(Q5) if $h \neq m$ Mathematical equation ,

$\begin{array}{l} E_{h, h + 1}^{2} E_{h + 1, h + 2} - (v_{h} + v_{h}^{- 1}) E_{h, h + 1} E_{h + 1, h + 2} E_{h, h + 1} \\ + E_{h - 1, h} E_{h, h + 1}^{2} = 0; \end{array}$ Mathematical equation

$\begin{array}{l} E_{h, h + 1}^{2} E_{h - 1, h} - (v_{h} + v_{h}^{- 1}) E_{h, h + 1} E_{h - 1, h} E_{h, h + 1} \\ + E_{h - 1, h} E_{h, h + 1}^{2} = 0; \end{array}$ Mathematical equation

$\begin{array}{l} E_{h, h + 1}^{2} E_{h + 2, h + 1} - (v_{h} + v_{h}^{- 1}) E_{h + 1, h} E_{h + 2, h + 1} E_{h + 1, h} \\ + E_{h + 2, h + 1} E_{h + 1, h}^{2} = 0; \end{array}$ Mathematical equation

$\begin{array}{l} E_{h + 1, h}^{2} E_{h, h - 1} - (v_{h} + v_{h}^{- 1}) E_{h + 1, h} E_{h, h - 1} E_{h + 1, h} \\ + E_{h, h - 1} E_{h + 1, h}^{2} = 0; \end{array}$ Mathematical equation

(Q6) $[E_{m, m + 1}, E_{m - 1, m + 2}] = [E_{m + 1, m}, E_{m + 2, m - 1}] = 0 .$ Mathematical equation

Let v be an indeterminate and $A = Z [v, v^{- 1}]$ Mathematical equation , and Gauss symmetric polynomials will be introduced here . For any integers $a \geq b \geq 0$ , let $[a] = \frac{v^{a} - v^{- a}}{v - v^{- 1}}$ , ${[a]}^{!} = [1] [2] \dots [a]$

and $[\begin{matrix} a \\ b \end{matrix}] = \prod_{s = 1}^{b} \frac{v^{a - s + 1} - v^{- a + s - 1}}{v^{s} - v^{- s}} = \frac{{[a]}^{!}}{{[b]}^{!} {[a - b]}^{!}}$ Mathematical equation . For any integers $1 \leq a, b \leq m + n$ and $t, c \in Z$ , $t \geq 0$ , we write $K_{a, b} = K_{a} K_{b}^{- 1}, [\begin{matrix} K_{a}; c \\ t \end{matrix}] = \prod_{s = 1}^{t} \frac{K_{a} v_{a}^{^{c - s + 1}} - K_{a}^{- 1} v_{a}^{- c + s - 1}}{v^{s} - v^{- s}}, [\begin{matrix} K_{a, b}; c \\ t \end{matrix}] = \prod_{s = 1}^{t} \frac{K_{a, b} v_{a}^{^{c - s + 1}} - K_{a, b}^{- 1} v_{a}^{- c + s - 1}}{v_{a}^{s} - v_{a}^{- s}}, E^{(m)} = \frac{E^{m}}{{[m]}^{!}}, F^{(m)} = \frac{F^{m}}{{[m]}^{!}},$ Mathematical equation and define the root vectors in $U (m | n)$ as follows. If $j - i = 1$ , then set $E_{i, j} = E_{i}$ , $F_{j, i} = F_{i}$ . For $j - i > 1$ , inductively set $E_{i, j} = E_{i} E_{i + 1, j} - v^{- 1} E_{i + 1, j} E_{i}$ , $F_{i, j} = F_{j, i + 1} F_{i} - v F_{i} F_{j, i + 1}$ Mathematical equation . Besides for short, let $[\begin{matrix} K_{a} \\ t \end{matrix}] = [\begin{matrix} K_{a}; 0 \\ t \end{matrix}]$ . Specializing $v$ to $ε$ , $k$ will be viewed as an $A$ -module. When $v$ is specialized to $ε$ , $[c]$ , ${[t]}^{!}$ and $[\begin{matrix} c \\ t \end{matrix}]$ Mathematical equation specialize to ${[c]}_{ε}$ , ${[t]}^{!}_{ε}$ and ${[\begin{matrix} c \\ t \end{matrix}]}_{ε}$ . And we define $σ_{i} (A) = a_{i, i} + \sum_{1 \leq j < i} (a_{i, j} + a_{j, i})$ .

Theorem 1^[8] The q-Schur superalgebras $S_{v} (m | n, r)$ Mathematical equation are generated by the elements

$K_{a}^{\pm 1}, E_{h, h + 1}, E_{h + 1, h} (1 \leq a \leq m + n, 1 \leq h \leq m + n)$ Mathematical equation

subject to the following relations:

(Q1)-(Q6) in Definition 1.

(Q7) $K_{1} K_{2} \dots K_{m} K_{m + 1}^{- 1} \dots K_{m + n}^{- 1} = v^{r};$ Mathematical equation

(Q8) $\prod_{i = 0}^{r} (K_{a} - v_{a}^{i}) = 0$ Mathematical equation , $1 \leq a \leq m + n$ .

By Theorem 1, we can obtain the algebraic full homomorphism ${ζ_{r}}^{'} : U (m | n) \to S_{v} (m | n, r)$ Mathematical equation , such that there is ${ζ_{r}}^{'} (K_{a}^{\pm 1}) = K_{a}^{\pm 1}$ , ${ζ_{r}}^{'} (K_{h, h + 1}) = K_{h, h + 1}$ , ${ζ_{r}}^{'} (K_{h + 1, h}) = K_{h + 1, h}$ for $1 \leq a \leq m + n$ and $1 \leq h \leq m + n$ .

Theorem 2^[11] There is an algebra homomorphism ${ζ_{r}}^{'} : U (m | n) \to S_{v} (m | n, r)$ Mathematical equation satisfying

${ζ_{r}}^{'} (K_{h, h + 1}) = K_{h, h + 1} (0, r), {ζ_{r}}^{'} (K_{h + 1, h}) = K_{h + 1, h} (0, r), {ζ_{r}}^{'} (K_{a}) = O (e_{a}, r),$ Mathematical equation

where $e_{a} = (0, \dots, 0,1, 0, \dots, 0) \in ℕ^{m + n}$ Mathematical equation (1 in position a).

Lemma 1^[9] 1) The set ${k_{λ} | λ \in Λ (m | n, r)}$ Mathematical equation is the orthogonal idempotent set of $S_{v} (m | n, r)$ ;

2) For any $λ \in Λ (m | n, r)$ Mathematical equation and $1 \leq a \leq m + n$ , $k_{a} k_{λ} = ν_{a}^{λ_{a}} k_{λ}$ .

Theorem 3^[12] For all $i$ Mathematical equation , the set

$W : = {e^{c (A^{+})} P_{\bar{λ}} f^{c (A^{-})} | λ \in Λ (m | n, r), λ_{i} \geq σ_{i} (A), A \in Γ (m {| n)}^{\pm}}$ Mathematical equation forms a $Q$ (q)-basis for q-Schur superalgebras $S_{v} (m | n, r)$ .

Let $A \in Γ {(m | n, r)}^{\pm}, \bar{λ} \in \bar{Λ (m | n, r)}$ Mathematical equation and $j \in Z^{m + n}$ , define

$P_{\bar{λ}} = {\begin{matrix} \sum_{μ \in Λ (m | n, r), \bar{μ} = \bar{λ}} k_{μ}, & i f \bar{λ} \in \bar{Λ (m | n, r)}; \\ 0, & o t h e r w i s e . \end{matrix}$ Mathematical equation

Let $Γ$ Mathematical equation be the set of all $A = (a_{i, j}) \in \tilde{Ξ} (2 1)$ such that $a_{i, j} < l$ for all $i \neq j$ and $Γ^{\pm}$ be the set of all $A \in Γ$ whose diagonal entries are zero. According to Theorem 1, the following results can be obtained.

Lemma 2^[8] For $1 \leq a \leq m + n, t \in Z, c \in Z, λ \in$ Mathematical equation

$Λ (m | n, r)$ Mathematical equation and $μ \in Λ (m | n, r)$ , we have

1) $K_{a}^{\pm 1} P_{λ} = ν_{a}^{\pm λ_{a}} P_{λ}, [\begin{matrix} K_{a}; c \\ t \end{matrix}] P_{λ} = [\begin{matrix} λ_{a} + c \\ t \end{matrix}] P_{λ}$ Mathematical equation ;

2) $K_{μ} P_{λ} = λ_{μ} P_{λ}$ Mathematical equation , where $λ_{μ} = \prod_{a} [\begin{array}{l} λ_{a} \\ μ_{a} \end{array}]$ ;

3) $K_{μ} = \sum_{λ \in Λ (m | n, r)} λ_{μ} P_{λ}$ Mathematical equation .

Proposition 1^[9] For $A \in Γ^{\pm} (m | n, r)$ Mathematical equation ,

1) If there exist $μ \in Λ (m | n, r), μ_{i} \geq σ_{i} (A^{+})$ Mathematical equation for all $i$ and $\bar{μ} = \bar{λ}$ , then $e^{c (A^{+})} P_{\bar{λ}} = P_{{\bar{λ}}^{'}} e^{c (A^{+})}$ , where $\bar{λ^{'}} = \bar{λ - c o (A^{+}) + r o (A^{+})}$ ; otherwise, $e^{c (A^{+})} P_{\bar{λ}} = 0$ ;

2) If there exist $μ \in Λ (m | n, r), μ_{i} \geq σ_{i} (A^{-})$ Mathematical equation for all $i$ and $\bar{μ} = \bar{λ}$ , then $P_{\bar{λ}} f^{c (A^{-})} = f^{c (A^{-})} P_{{\bar{λ}}^{'}}$ , where $\bar{λ^{'}} = \bar{λ + c o (A^{+}) - r o (A^{+})}$ ; otherwise, $P_{\bar{λ}} f^{c (A^{-})} = 0$ , where $c o (A^{+})$ and $r o (A^{+})$ denote the column sum and row sum of the matrix $A^{+}$ Mathematical equation , respectively.

2 The Little q-Schur Superalgebra $u_{k} (2 | 1,3)$

Let $\tilde{u_{k}}$ Mathematical equation be the subalgebra of $U_{k}$ which is generated by all $K_{b}^{\pm 1}$ , $E_{a, a + 1}, E_{a + 1, a} (1 \leq a \leq m + n, 1 \leq b \leq m + n)$ .

According to (Q2), $K_{a}^{l} - 1$ Mathematical equation ( $1 \leq a \leq m + n$ ) is the central element of $U_{k}$ (and therefore also $\tilde{u_{k}}$ ). Let $〈 K_{a}^{l} - 1 | 1 \leq a \leq m + n 〉$ be the ideal generated by $K_{a}^{l} - 1$ ( $1 \leq a \leq m + n$ ) in $\tilde{u_{k}}$ and $u_{k} = \tilde{u_{k}} / 〈 K_{a}^{l} - 1 | 1 \leq a \leq m + n 〉$ Mathematical equation .

Thus quotient (super) algebra $u_{k}$ Mathematical equation is called $g l_{m | n}$ infinitesimal quantum supergroup. The image of $\tilde{u_{k}}$ under $ζ_{r, k}$ is denoted as $u_{k} (m | n, r)$ . $u_{k} (m | n, r)$ is called little q-Schur superalgebra related to $m$ , $n$ , $r$ and $l$ .

Theorem 4 The set

$W_{k} : = {e^{c (A^{+})} P_{\bar{λ}} f^{c (A^{-})} | λ \in Λ (2 | 1,3), λ_{i} \geq σ_{i} (A), f o r i = 1,2,$ Mathematical equation

3, $A \in Γ^{\pm} (2 | 1)}$ Mathematical equation forms a basis for $u_{k} (2 | 1,3)$ .

Proof Fix $A \in Γ {(2 | 1)}^{\pm}$ Mathematical equation satisfying $λ_{i} \geq σ_{i} (A)$ for all $i$ . By the definition of $P_{\bar{λ}}$ , we have

$e^{c (A^{+})} P_{\bar{λ}} f^{c (A^{-})} = \sum_{\bar{λ} = \bar{μ}, \forall i, μ_{i} \geq σ_{i} (A)} e^{c (A^{+})} k_{μ} f^{c (A^{-})} + \sum_{\bar{λ} = \bar{μ^{'}}, \exists i, μ_{i}^{'} < σ_{i} (A)} e^{c (A^{+})} k_{μ^{'}} f^{c (A^{-})} .$ Mathematical equation

In the q-Schur superalgebra $S_{v} (2 | 1,3)$ Mathematical equation , for any $e^{c (A^{+})} k_{μ^{'}} f^{c (A^{-})}$ , if $μ_{i}^{'} < σ_{i} (A)$ for some $i$ , then it lies in the span of $W$ in Theorem 3. Then we have

$\sum_{\bar{λ} = \bar{μ^{'}}, \exists i, μ_{i}^{'} < σ_{i} (A)} e^{c (A^{+})} k_{μ^{'}} f^{c (A^{-})} = \sum_{\begin{matrix} B \in Γ^{\pm} (2 | 1) \\ σ_{i} (B) < σ_{i} (A), μ_{i}^{'} \geq σ_{i} (B) \end{matrix}} f_{B, A} e^{c (B^{+})} k_{μ^{'}} f^{c (B^{-})} (f_{B, A} \in k) .$ Mathematical equation

By Theorem 3, the elements above are linearly independent, thus the set $W_{k}$ Mathematical equation is linearly independent.

Inspired by Ref. [8], we have the following commutation formulas in the $u_{k} (2 | 1,3)$ Mathematical equation .

Proposition 2 For $1 \leq i \leq 2, M \in Z_{+}, M \geq 1$ Mathematical equation , $e_{i, i + 1} = e_{i}$ , $f_{i + 1, i} = f_{i}$ , we define:

${\begin{array}{l} e_{1,3} = e_{1,2} e_{2,3} - ε^{- 1} e_{2,3} e_{1,2}; \\ f_{3,1} = f_{3,2} f_{2,1} - ε f_{2,1} f_{3,2} . \end{array}$ Mathematical equation

(2.1a) $e_{1,3} e_{2,3} = - ε^{- 1} e_{2,3} e_{1,3}$ Mathematical equation ;

(2.1b) $e_{1,3} f_{3,1} = - f_{3,1} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} [λ_{1} + λ_{3}] P_{\bar{λ}}$ Mathematical equation ;

(2.1c) $e_{1,2}^{(M)} f_{3,2} = f_{3,2} e_{1,2}^{(M)}$ Mathematical equation ;

(2.1d) $e_{1,3} f_{3,2} = - f_{3,2} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{2} + λ_{3}} P_{\bar{λ}} e_{1,2}$ Mathematical equation ;

(2.1e) $e_{1,2}^{(M)} f_{3,1} = f_{3,1} e_{1,2}^{(M)} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 2 - M} P_{\bar{λ}} e_{1,2}^{(M - 1)}$ Mathematical equation ;

(2.1f) $e_{1,2}^{(M)} e_{1,3} = ε^{M} e_{1,3} e_{1,2}^{(M)}$ Mathematical equation ;

(2.1g) $e_{1,2}^{(M)} e_{2,3} = ε^{- M} e_{2,3} e_{1,2}^{(M)} + e_{1,3} e_{1,2}^{(M - 1)}$ Mathematical equation .

Proof For (2.1a), by $e_{1,3} e_{2,3} = e_{1,2} e_{2,3}^{2} - ε^{- 1} e_{2,3} e_{1,2} e_{2,3} = - ε^{- 1} e_{2,3} e_{1,2} e_{2,3} = - ε^{- 1} e_{2,3} e_{1,3}$ Mathematical equation .The equation holds.

For (2.1b), we can obtain $e_{1,3} f_{3,1} = - f_{3,1} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} [λ_{1} + λ_{3}] P_{\bar{λ}}$ Mathematical equation by

$\begin{array}{l} e_{1,3} f_{3,1} = e_{1,2} e_{2,3} f_{3,1} - ε^{- 1} e_{2,3} e_{1,2} f_{3,1} \\ = - f_{3,1} e_{1,3} + (f_{3,2} e_{2,3} + e_{2,3} f_{3,2}) \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2}} P_{\bar{λ}} \\ + (e_{1,2} f_{2,1} - f_{2,1} e_{1,2}) \sum_{λ \in Λ (2 | 1,3)} ε^{- λ_{2} - λ_{3}} P_{\bar{λ}} \\ = - f_{3,1} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2}} [λ_{2} + λ_{3}] P_{\bar{λ}} \\ + \sum_{λ \in Λ (2 | 1,3)} ε^{- λ_{2} - λ_{3}} [λ_{1} - λ_{2}] P_{\bar{λ}} \\ = - f_{3,1} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} [λ_{1} + λ_{3}] P_{\bar{λ}} . \end{array}$ Mathematical equation

For (2.1c), in order to prove $e_{1,2}^{(M)} f_{3,2} = f_{3,2} e_{1,2}^{(M)}$ Mathematical equation , we need following steps:

Step 1 When $M = 1$ Mathematical equation ,we can easily prove $e_{1,2} f_{3,2} = f_{3,2} e_{1,2}$ .

Step 2 When $M \geq 1$ Mathematical equation , we assume $e_{1,2}^{(M)} f_{3,2} = f_{3,2} e_{1,2}^{(M)}$ holds, then we will discuss the following situation: $\begin{array}{l} e_{1,2}^{(M + 1)} f_{3,2} = {[M + 1]}^{- 1} e_{1,2} e_{1,2}^{(M)} f_{3,2} = {[M + 1]}^{- 1} e_{1,2} f_{3,2} e_{1,2}^{(M)} \\ = {[M + 1]}^{- 1} f_{3,2} e_{1,2} e_{1,2}^{(M)} = f_{3,2} e_{1,2}^{(M + 1)} . \end{array}$ Mathematical equation

For (2.1d), we can obtain $e_{1,3} f_{3,2} = - f_{3,2} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{2} + λ_{3}} P_{\bar{λ}} e_{1,2}$ Mathematical equation by (2.1c)

$\begin{array}{l} e_{1,3} f_{3,2} = e_{1,2} e_{2,3} f_{3,2} - ε^{- 1} e_{2,3} e_{1,2} f_{3,2} \\ = - e_{1,2} f_{3,2} e_{2,3} + e_{1,2} \sum_{λ \in Λ (2 | 1,3)} [λ_{2} + λ_{3}] P_{\bar{λ}} \\ + ε^{- 1} f_{3,2} e_{2,3} e_{1,2} - ε^{- 1} \sum_{λ \in Λ (2 | 1,3)} [λ_{2} + λ_{3}] P_{\bar{λ}} e_{1,2} \\ = - f_{3,2} e_{1,3} + \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{2} + λ_{3}} P_{\bar{λ}} e_{1,2} . \end{array}$ Mathematical equation

For (2.1e), in order to prove $e_{1,2}^{(M)} f_{3,1} = f_{3,1} e_{1,2}^{(M)} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 2 - M} P_{\bar{λ}} e_{1,2}^{(M - 1)}$ Mathematical equation , we need the following steps:

Step 3 When $M = 1$ Mathematical equation ,we can prove $e_{1,2} f_{3,1} = f_{3,1} e_{1,2} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 1} P_{\bar{λ}} b y (2.1 c),$

$\begin{array}{l} e_{1,2} f_{3,1} = e_{1,2} f_{3,2} f_{2,1} - ε e_{1,2} f_{2,1} f_{3,2} \\ = f_{3,1} e_{1,2} + f_{3,2} \sum_{λ \in Λ (2 | 1,3)} [λ_{1} - λ_{2}] P_{\bar{λ}} \\ - ε \sum_{λ \in Λ (2 | 1,3)} [λ_{1} - λ_{2}] P_{\bar{λ}} f_{3,2} \\ = f_{3,1} e_{1,2} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 1} P_{\bar{λ}} . \end{array}$ Mathematical equation

Step 4 When $M \geq 1$ Mathematical equation , we assume $e_{1,2}^{(M)} f_{3,1} = f_{3,1} e_{1,2}^{(M)} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 2 - M} P_{\bar{λ}} e_{1,2}^{(M - 1)}$ holds, then we will discuss the following situation:

$\begin{array}{l} e_{1,2}^{(M + 1)} f_{3,1} \\ = {[M + 1]}^{- 1} e_{1,2} (f_{3,1} e_{1,2}^{(M)} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 2 - M} P_{\bar{λ}} e_{1,2}^{(M - 1)}) \\ = f_{3,1} e_{1,2}^{(M + 1)} - {[M + 1]}^{- 1} f_{3,2} \sum_{λ \in Λ (2 | 1,3)} (ε^{λ_{1} - λ_{2} + 1} + ε^{λ_{1} - λ_{2} - M} [M]) P_{\bar{λ}} e_{1,2}^{(M)} \\ = f_{3,1} e_{1,2}^{(M + 1)} - f_{3,2} \sum_{λ \in Λ (2 | 1,3)} ε^{λ_{1} - λ_{2} + 1 - M} P_{\bar{λ}} e_{1,2}^{(M)} . \end{array}$ Mathematical equation

For (2.1f, 2.1g), the proof is similar to the above.

Theorem 5 The algebra $T$ Mathematical equation is generated by the elements $e_{i}, f_{i},$ $P_{\bar{λ}} (λ \in Λ (2 | 1,3), (1 \leq i \leq 2))$ , subject to the following relations:

$(L R 1) e_{i}^{l} = 0, f_{i}^{l} = 0$ Mathematical equation when $i \neq 2$ ;

$(L R 2) \sum_{λ \in Λ (2 | 1,3)} P_{\bar{λ}} = 1, P_{\bar{λ}} P_{\bar{μ}} = δ_{λ μ} P_{\bar{λ}}$ Mathematical equation ;

$(L R 3) [e_{i}, f_{j}] = δ_{i j} \sum_{λ \in Λ (2 | 1,3)} [λ_{i} - {(- 1)}^{\hat{e_{i}} \hat{f_{j}}} λ_{i + 1}]_{ε} P_{\bar{λ}}$ Mathematical equation ;

$(L R 4) e_{i}^{2} e_{j} - (ε + ε^{- 1}) e_{i} e_{j} e_{i} + e_{j} e_{i}^{2} = 0$ Mathematical equation when $| i - j | = 1$ ;

$(L R 5) f_{i}^{2} f_{j} - (ε + ε^{- 1}) f_{i} f_{j} f_{i} + f_{j} f_{i}^{2} = 0$ Mathematical equation when $| i - j | = 1$ ;

$(L R 6) [e_{a}, e_{b}] = 0, [f_{a}, f_{b}] = 0$ Mathematical equation when $| a - b | > 1$ ;

$(L R 7) e_{i} P_{\bar{λ}} = {\begin{array}{l} P_{\bar{μ + α_{i}}} e_{i}, & i f \exists \bar{μ} = \bar{λ}, μ_{i + 1} \geq 1 \\ 0, & o t h e r w i s e \end{array},$ Mathematical equation

$P_{\bar{λ}} e_{i} = {\begin{array}{l} e_{i} P_{\bar{μ - α_{i}}}, & i f \exists \bar{μ} = \bar{λ}, μ_{i} \geq 1 \\ 0, & o t h e r w i s e \end{array}$ Mathematical equation ;

$(L R 8) f_{i} P_{\bar{λ}} = {\begin{array}{l} P_{\bar{μ - α_{i}}} f_{i}, & i f \exists \bar{μ} = \bar{λ}, μ_{i} \geq 1 \\ 0, & o t h e r w i s e \end{array},$ Mathematical equation

$P_{\bar{λ}} f_{i} = {\begin{array}{l} f_{i} P_{\bar{μ + α_{i}}}, & i f \exists \bar{μ} = \bar{λ}, μ_{i + 1} \geq 1 \\ 0, & o t h e r w i s e \end{array}$ Mathematical equation ;

$(L R 9)$ Mathematical equation $e^{c (A_{j}^{+})} P_{\bar{λ}} = 0$ (resp. $P_{\bar{λ}} f^{c (A_{j}^{-})} = 0$ ) for $A_{j}^{+} \in Γ_{j}^{+} (2 | 1,3)$ (resp. $A_{j}^{-} \in Γ_{j}^{-} (2 | 1,3)$ ) and $λ \in Λ (2 | 1,3)$ satisfying that if $μ \in Λ (2 | 1,3) w i t h \bar{μ} = \bar{λ} t h e n μ_{j} < σ_{j} (A);$ Mathematical equation

$(L R 10) e_{1} P_{(\bar{2}, \bar{1}, \bar{0)}} f_{1} = 0, e_{2} P_{(\bar{0}, \bar{2}, \bar{1})} f_{2} = 0 .$ Mathematical equation

The key to prove Theorem 5 is that in $u_{k} (2 | 1,3)$ Mathematical equation , the relation $(L R 10)$ holds.

Proposition 3 In little q-Schur superalgebra $u_{k} (2 | 1,3)$ Mathematical equation , we have $e_{i} P_{\bar{λ}} f_{i} = 0$ .

If $λ_{j} < 3, 1 \leq j \neq i + 1 \leq 3, λ_{j} = 2$ Mathematical equation and $λ_{i + 1} = 1$ .

Proof $\begin{array}{l} e_{1,2} P (2,1, 0) f_{2,1} = e_{1,2} k (2,1, 0) f_{2,1} \\ = k (3,0, 0) e_{1,2} f_{2,1} = k (3,0, 0) f_{2,1} e_{1,2} = 0; e_{2,3} P (0,2, 1) f_{3,2} = e_{2,3} k (0,2, 1) f_{3,2} = k (0,3, 0) e_{2,3} f_{3,2} \\ = - k (0,3, 0) f_{3,2} e_{2,3} = 0 . \end{array}$ Mathematical equation

From the above conclusions, it can be seen that little q-Schur superalgebra $u_{k} (2 | 1,3)$ Mathematical equation satisfies the relation $(L R 10)$ of Theorem 5.

We define an algebra $T$ Mathematical equation which satisfies the generators and relations given in Theorem 5. Set $T^{+}$ (resp. $T^{-}$ , $T^{0}$ ) be the subalgebra generated by $e_{i}$ (resp. $f_{i}$ , $P_{\bar{λ}}$ ). For $T$ , we give the root vectors in the same way, which satisfy the commutation formulas (2.1a)-(2.1g). According to the relations (LR3)-(LR7) in Theorem 5, we know that $T = T^{+} T^{0} T^{-}$ Mathematical equation can be spanned by all the elements $W_{A, \bar{λ}} : = e^{c (A^{+})} P_{\bar{λ}} f^{c (A^{-})}$ , $A \in Γ^{\pm} (2 | 1,3)$ .

Proposition 4 The set

$W_{k} = {e^{c (A^{+})} P_{\bar{λ}} f^{c (A^{-})} | λ \in Λ (2 | 1,3), λ_{i} \geq σ_{i} (A), i = 1,2, 3, A \in$ Mathematical equation

$Γ^{\pm} (2 | 1)}$ Mathematical equation is a spanning set for $T$ .

Proof Fixing $B \in Γ^{\pm} (2 | 1)$ Mathematical equation , to prove this proposition, it suffices to show that if $λ^{'} \in Λ (2 | 1,3)$ with $\bar{λ^{'}} = \bar{λ}$ , satisfying $λ_{j}^{'} < σ_{j} (B)$ for some $j$ , then $W_{B, {\bar{λ}}^{'}} = e^{c (B^{+})} P_{{\bar{λ}}^{'}} f^{c (B^{-})}$ lies in the span of $W_{k}$ Mathematical equation . Here $W_{k_{a}} (a \in N^{+})$ is a linear combination of the elements in $W_{k}$ and $g_{a} (a \in N^{+})$ is a constant.

Case 1 If $λ_{2}^{'} < σ_{2} (B)$ Mathematical equation and $λ_{3}^{'} \geq σ_{3} (B)$ , assuming $λ_{2}^{'} = σ_{2} (B) - 1$ . For $e^{c (B^{+})} P_{\bar{λ^{'}}} f^{c (B^{-})}$ , we only need to consider $e_{1,2}^{(b_{1,2})} P_{{\bar{λ}}^{'}} f_{2,1}^{(b_{2,1})}$ .

By Theorem 5 $(L R 9)$ Mathematical equation , when $b_{1,2} = 0$ (or $b_{2,1} = 0$ ), $P_{{\bar{λ}}^{'}} f_{2,1}^{(b_{2,1})}$ (or $e_{1,2}^{(b_{1,2})} P_{{\bar{λ}}^{'}}$ ) is a linear combination of the elements in $W_{k}$ .Then we consider $e_{1,2}^{(b_{1,2})} P_{{\bar{λ}}^{'}} f_{2,1}^{(b_{2,1})}$ , where $1 \leq b_{1,2}, b_{2,1} \leq 2$ Mathematical equation .

Case 1-1 If $λ_{2}^{'} = 1$ Mathematical equation and $σ_{2} (B) = 2$ , for $e_{1,2} P_{(\bar{λ_{1}^{'}}, \bar{1}, \bar{λ_{3}^{'}})} f_{2,1}$ ,

1) When $0 \leq λ_{1}^{'} \leq 1$ Mathematical equation , by Theorem 5 $(L R 3)$ ,

$e_{1,2} P_{(\bar{λ_{1}^{'}}, \bar{1}, \bar{λ_{3}^{'}})} f_{2,1} = P_{(\bar{λ_{1}^{'} + 1}, \bar{0}, \bar{λ_{3}^{'}})} e_{1,2} f_{2,1} = P_{(\bar{λ_{1}^{'} + 1}, \bar{0}, \bar{λ_{3}^{'}})} f_{2,1} e_{1,2} + W_{k_{1}}$ Mathematical equation

where $W_{k_{1}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

2) When $λ_{1}^{'} = 2$ Mathematical equation , by Theorem 5 $(L R 10)$ , we can obtain $e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} f_{2,1} = 0$ .

Case 1-2 If $λ_{2}^{'} = 2$ Mathematical equation and $σ_{2} (B) = 3$ , for $e_{1,2}^{(b_{1,2})} P_{(\bar{λ_{1}^{'}}, \bar{2}, \bar{λ_{3}^{'}})} f_{2,1}^{(b_{2,1})}$ , where $b_{1,2} = 1, b_{2,1} = 2$ , by Theorem 5 $(L R 3)$ ,

$\begin{array}{l} e_{1,2} P_{(\bar{λ_{1}^{'}}, \bar{2}, \bar{λ_{3}^{'}})} f_{2,1}^{(2)} = {[2]}_{ε}^{- 1} e_{1,2} f_{2,1} P_{(\bar{λ_{1}^{'} + 1}, \bar{1}, \bar{λ_{3}^{'}})} f_{2,1} \\ = {[2]}_{ε}^{- 1} f_{2,1} e_{1,2} P_{(\bar{λ_{1}^{'} + 1}, \bar{1}, \bar{λ_{3}^{'}})} f_{2,1} + W_{k_{2}} . \end{array}$ Mathematical equation

Then,

1) When $λ_{1}^{'} = 0$ Mathematical equation ,

$\begin{array}{l} e_{1,2} P_{(\bar{0}, \bar{2}, \bar{1})} f_{2,1}^{(2)} = {[2]}_{ε}^{- 1} f_{2,1} e_{1,2} P_{(\bar{1}, \bar{1}, \bar{1})} f_{2,1} + W_{k_{2}} \\ = {[2]}_{ε}^{- 1} f_{2,1}^{2} e_{1,2} P_{(\bar{2}, \bar{0}, \bar{1})} + {[2]}_{ε}^{- 1} {[2]}_{ε} P_{(\bar{1}, \bar{1}, \bar{1})} f_{2,1} + W_{k_{2}} \\ = W_{k_{2}} + W_{k_{3}} . \end{array}$ Mathematical equation

2) When $λ_{1}^{'} = 1$ Mathematical equation , by Theorem 5 $(L R 10)$ , we can obtain

$e_{1,2} P_{(\bar{λ_{1}^{'}}, \bar{2}, \bar{λ_{3}^{'}})} f_{2,1}^{(2)} = {[2]}_{ε}^{- 1} f_{2,1} e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} f_{2,1} + W_{k_{2}}$ Mathematical equation

where $W_{k_{2}}$ Mathematical equation and $W_{k_{3}}$ are linear combinations of the elements in $W_{k}$ .

For $e_{1,2}^{(b_{1,2})} P_{(\bar{λ_{1}^{'}}, \bar{2}, \bar{λ_{3}^{'}})} f_{2,1}^{(b_{2,1})}$ Mathematical equation , when $b_{1,2} = 2, b_{2,1} = 1$ , the proof is similar to the above.

Case 1-3 If $λ_{2}^{'} = 3$ Mathematical equation and $σ_{2} (B) = 4$ , for $e_{1,2}^{(b_{1,2})} P_{(\bar{0}, \bar{3}, \bar{0})} f_{2,1}^{(b_{2,1})}$ ,

When $b_{1,2} = 1$ Mathematical equation (or $b_{2,1} = 1$ ), by Theorem 5 $(L R 1)$ , we can obtain $e_{1,2}^{(b_{1,2})} P_{(\bar{0}, \bar{3}, \bar{0})} f_{2,1}^{(b_{2,1})} = 0$ .

When $b_{1,2} = b_{2,1} = 2$ Mathematical equation , by Theorem 5 $(L R 3, L R 10)$ ,

$\begin{array}{l} e_{1,2}^{(2)} P_{(\bar{0}, \bar{3}, \bar{0})} f_{2,1}^{(2)} = {[2]}_{ε}^{- 2} e_{1,2} P_{(\bar{1}, \bar{2}, \bar{0})} e_{1,2} f_{2,1}^{2} \\ = {[2]}_{ε}^{- 2} e_{1,2} f_{2,1} P_{(\bar{2}, \bar{1}, \bar{0})} e_{1,2} f_{2,1} - {[2]}_{ε}^{- 2} e_{1,2} P_{(\bar{1}, \bar{2}, \bar{0})} f_{2,1} \\ = {[2]}_{ε}^{- 2} f_{2,1} e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} f_{2,1} e_{1,2} + {[2]}_{ε}^{- 2} e_{1,2} P_{(\bar{1}, \bar{2}, \bar{0})} f_{2,1} \\ - {[2]}_{ε}^{- 2} P_{(\bar{2}, \bar{1}, \bar{0})} \\ = {[2]}_{ε}^{- 2} f_{2,1} e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} f_{2,1} e_{1,2} + W_{k_{4}}, \end{array}$ Mathematical equation

where $W_{k_{4}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

Case 2 If $λ_{3}^{'} < σ_{3} (B)$ Mathematical equation and $λ_{2}^{'} \geq σ_{2} (B)$ , assuming $λ_{3}^{'} = σ_{3} (B) - 1$ , for $e^{c (B^{+})} P_{\bar{λ^{'}}} f^{c (B^{-})}$ , by the commutation formulas (2.1a), (2.1f) , we have

$\begin{array}{l} e^{c (B^{+})} P_{\bar{λ^{'}}} f^{c (B^{-})} = e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} e_{1,2}^{(b_{1,2})} P_{\bar{λ^{'}}} f_{2,1}^{(b_{2,1})} f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})} \\ = ε^{g_{3}} e_{1,2}^{(b_{1,2})} (e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} P_{\bar{λ^{'}}} f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})}) f_{2,1}^{(b_{2,1})} + W_{k_{5}}, \end{array}$ Mathematical equation

where $W_{k_{5}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .Therefore, we only need to consider $e^{c (B_{3}^{+})} P_{{\bar{λ}}^{'}} f^{c (B_{3}^{-})}$ .

By Theorem 5 $(L R 9)$ Mathematical equation , when $b_{1,3} + b_{2,3} = 0$ (or $b_{3,2} + b_{3,1} = 0$ ), $P_{{\bar{λ}}^{'}} f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})}$ (or $e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} P_{{\bar{λ}}^{'}}$ ) is a linear combination of the elements in $W_{k}$ . We consider $e^{c (B_{3}^{+})} P_{{\bar{λ}}^{'}} f^{c (B_{3}^{-})}$ Mathematical equation , where $b_{1,3} + b_{2,3} \neq 0$ and $b_{3,2} + b_{3,1} \neq 0$ .

Case 2-1 If $λ_{3}^{'} = 1$ Mathematical equation and $σ_{3} (B) = 2$ , we need to consider $e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})}$ ,

When $b_{2,3} = b_{3,2} = 1$ Mathematical equation , for $e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2}$ ,

1) When $0 \leq λ_{2}^{'} \leq 1$ Mathematical equation , by Theorem 5 $(L R 3)$ ,

$e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2} = P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} e_{2,3} f_{3,2} = - P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} f_{3,2} e_{2,3} + W_{k_{6}}$ Mathematical equation

where $W_{k_{6}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

2) When $λ_{2}^{'} = 2$ Mathematical equation , by Theorem 5 $(L R 10)$ , we can obtain $e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2} = e_{2,3} P_{(\bar{0}, \bar{2}, \bar{1})} f_{3,2} = 0$ .

When $b_{2,3} = b_{3,1} = 1$ Mathematical equation , for $e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,1}$ ,

1) When $0 \leq λ_{1}^{'} \leq 1$ Mathematical equation , by Theorem 5 (LR3, LR9) and the commutation formulas (2.1c),

$\begin{array}{l} e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,1} \\ = P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} e_{2,3} f_{3,2} f_{2,1} - ε P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} e_{2,3} f_{2,1} f_{3,2} \\ = - P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} f_{3,2} e_{2,3} f_{2,1} + {[λ_{2}^{'} + 1]}_{ε} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} f_{2,1} \\ - ε f_{2,1} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{0})} e_{2,3} f_{3,2} \\ = ε f_{2,1} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{0})} f_{3,2} e_{2,3} + ({[λ_{2}^{'} + 1]}_{ε} - ε {[λ_{2}^{'}]}_{ε}) P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{0})} f_{2,1} \\ = ε f_{2,1} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{0})} f_{3,2} e_{2,3} + W_{k_{7}}, \end{array}$ Mathematical equation

where $W_{k_{7}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

2) When $λ_{1}^{'} = 2$ Mathematical equation , by Theorem 5 $(L R 3, L R 9, L R 10)$ ,

$\begin{array}{l} e_{2,3} P_{(\bar{2}, \bar{0}, \bar{1})} f_{3,1} \\ = e_{2,3} P_{(\bar{2}, \bar{0}, \bar{1})} f_{3,2} f_{2,1} - ε e_{2,3} P_{(\bar{2}, \bar{0}, \bar{1})} f_{2,1} f_{3,2} \\ = e_{2,3} P_{(\bar{2}, \bar{0}, \bar{1})} f_{3,2} f_{2,1} \\ = - P_{(\bar{2}, \bar{1}, \bar{0})} f_{3,2} e_{2,3} f_{2,1} + P_{(\bar{2}, \bar{1}, \bar{0})} f_{2,1} \\ = - P_{(\bar{2}, \bar{1}, \bar{0})} f_{3,2} e_{2,3} f_{2,1} + W_{k_{8}}, \end{array}$ Mathematical equation

where $W_{k_{8}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

When $b_{1,3} = b_{3,1} = 1$ Mathematical equation , for $e_{1,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,1}$ ,

1) When $0 \leq λ_{1}^{'} \leq 1$ Mathematical equation , by the commutation formulas (2.1b), we have $e_{1,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,1} = P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{0})} e_{1,3} f_{3,1} = - P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{0})} f_{3,1} e_{1,3} + {[λ_{1}^{'} + 1]}_{ε} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{0})} = W_{k_{9}}$ Mathematical equation ,

where $W_{k_{9}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

2) When $λ_{1}^{'} = 2$ Mathematical equation , by theorem 5 $(L R 10, L R 9)$ and the commutation formulas (2.1d), we have

$\begin{array}{l} e_{1,3} P_{(\bar{2}, \bar{0}, \bar{1})} f_{3,1} \\ = e_{1,2} e_{2,3} P_{(\bar{2}, \bar{0}, \bar{1})} f_{3,1} - ε^{- 1} e_{2,3} e_{1,2} P_{(\bar{2}, \bar{0}, \bar{1})} f_{3,1} \\ = e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} e_{2,3} f_{3,1} \\ = - e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} f_{3,1} e_{2,3} + e_{1,2} P_{(\bar{2}, \bar{1}, \bar{0})} f_{2,1} = 0 . \end{array}$ Mathematical equation

For $e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})}$ Mathematical equation , when $b_{1,3} = b_{3,2} = 1$ , the proof is similar to the above.

Case 2-2 If $λ_{3}^{'} = 2$ Mathematical equation and $σ_{3} (B) = 3$ , for $e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{2})}$

$f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})}$ Mathematical equation , when $b_{1,3} = 0$ , for $e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{2})} f_{3,2} f_{3,1}$ ,by Theorem 5 $(L R 3)$ and the commutation formulas (2.1d),

$\begin{array}{l} e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{2})} f_{3,2} f_{3,1} \\ = - f_{3,2} e_{2,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{1})} f_{3,1} + {[λ_{2}^{'} + 2]}_{ε} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{1})} f_{3,1} \\ = ε {[λ_{2}^{'} + 1]}_{ε} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{1})} f_{3,1} - ε^{- λ_{2}^{'}} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{1})} f_{2,1} f_{3,2} = W_{k_{10}}, \end{array}$ Mathematical equation

where $W_{k_{10}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

When $b_{2,3} = 0$ Mathematical equation , by the commutation formulas (2.1b , 2.1d ), we have

$\begin{array}{l} e_{1,3} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{2})} f_{3,2} f_{3,1} \\ = P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{1})} e_{1,3} f_{3,2} f_{3,1} = - P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2} e_{1,3} f_{3,1} \\ + ε^{λ_{2}^{'} + 1} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{1})} e_{1,2} f_{3,1} \\ = - f_{3,2} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'} + 1}, \bar{0})} e_{1,3} f_{3,1} + ε^{λ_{2}^{'} + 1} e_{1,2} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{1})} f_{3,1} \\ = f_{3,2} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'} + 1}, \bar{0})} f_{3,1} e_{1,3} - {[λ_{1}^{'} + 1]}_{ε} P_{(\bar{λ_{1}^{'} + 1}, \bar{λ_{2}^{'}}, \bar{1})} f_{3,2} \end{array}$ Mathematical equation

$+ ε^{λ_{2}^{'} + 1} e_{1,2} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'} + 1}, \bar{1})} f_{3,1} = W_{k_{11}},$ Mathematical equation

where $W_{k_{11}}$ Mathematical equation is a linear combination of the elements in $W_{k}$ .

For $e_{1,3}^{(b_{1,3})} e_{2,3}^{(b_{2,3})} P_{(\bar{λ_{1}^{'}}, \bar{λ_{2}^{'}}, \bar{2})} f_{3,2}^{(b_{3,2})} f_{3,1}^{(b_{3,1})}$ Mathematical equation , when $b_{3,1} = 0$ and $b_{3,2} = 0$ , the proof is similar to the above.

Case 2-3 If $λ_{3}^{'} = 3$ Mathematical equation and $σ_{3} (B) = 4$ ,we need to consider $e_{1,3} e_{2,3} P_{(\bar{0}, \bar{0}, \bar{3})} f_{3,2} f_{3,1}$ , by Theorem 5 ( $L R 9$ , $L R 10$ ) and the commutation formulas (2.1b , 2.1d ), we have

$\begin{array}{l} e_{1,3} e_{2,3} P_{(\bar{0}, \bar{0}, \bar{3})} f_{3,2} f_{3,1} \\ = - P_{(\bar{1}, \bar{1}, \bar{1})} e_{1,3} f_{3,2} e_{2,3} f_{3,1} + {[3]}_{ε} e_{1,3} P_{(\bar{0}, \bar{1}, \bar{2})} f_{3,1} \\ = P_{(\bar{1}, \bar{1}, \bar{1})} f_{3,2} f_{3,1} e_{1,3} e_{2,3} - P_{(\bar{1}, \bar{1}, \bar{1})} f_{3,2} e_{2,3} \\ - ε^{- 2} P_{(\bar{1}, \bar{1}, \bar{1})} e_{1,3} f_{3,2} f_{2,1} + W_{k_{12}} \\ = e_{2,3} P_{(\bar{1}, \bar{0}, \bar{2})} f_{3,2} - {[2]}_{ε} P_{(\bar{1}, \bar{1}, \bar{1})} - ε^{- 2} e_{1,3} P_{(\bar{0}, \bar{1}, \bar{2})} f_{3,1} \\ - ε^{- 1} e_{1,3} P_{(\bar{0}, \bar{1}, \bar{2})} f_{2,1} f_{3,2} + W_{k_{12}} = W_{k_{13}}, \end{array}$ Mathematical equation

where $W_{k_{12}}$ Mathematical equation and $W_{k_{13}}$ are the linear combination of the elements in $W_{k}$ . We complete the proof of Proposition 4.

Proof of Theorem 5 By the definition of $T$ Mathematical equation and Theorem 1, it is clear that $T$ satisfies the relations (Q1-Q6) in Definition 1. Hence, there is a surjective algebra homomorphism from $T$ to $u_{k} (2 | 1,3)$ . On the basis of Theorem 3, Propositions 1 and 4, we have the $u_{k} (2 | 1,3)$ that satisfies the relations in Theorem 5. According to the definition of $u_{k} (2 | 1,3)$ Mathematical equation , there is a surjective algebra homomorphism $ψ$ from $T$ to $u_{k} (2 | 1,3)$ satisfying $ψ (e_{i}) = e_{i}$ , $ψ (f_{i}) = f_{i}$ , $ψ (P_{\bar{λ}}) = P_{\bar{λ}}$ . On the other hand, by Proposition 1, we see that the map $ψ$ sends the spanning set of $T$ to the basis of $u_{k} (2 | 1,3)$ Mathematical equation . Hence $ψ$ is an isomorphism.

References

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[R1] Dipper R, James G. The q-Schur algebra[J]. Proceedings of the London Mathematical Society, 1989, s3-59(1): 23-50. [Google Scholar]

[R2] Dipper R, James G. Q-tensor space and q-Weyl modules[J]. Transactions of the American Mathematical Society, 1991, 327(1): 251-282. [MathSciNet] [Google Scholar]

[R3] Doty S, Giaquinto A. Presenting Schur algebras[J]. International Mathematics Research Notices, 2002, 2002(36): 1907-1944. [Google Scholar]

[R4] Du J, Parshall B. Monomial bases for q-Schur algebras[J]. Transactions of the American Mathematical Society, 2003, 355(4): 1593-1620. [MathSciNet] [Google Scholar]

[R5] Gao W T, Liu M Q. The center of q-Schur algebra U(2, r)[J]. Communications in Algebra, 2021, 49(12): 5031-5040. [Google Scholar]

[R6] Du J, Fu Q, Wang J P. Representations of little q-Schur algebras[J]. Pacific Journal of Mathematics, 2012, 257(2): 343-378. [CrossRef] [MathSciNet] [Google Scholar]

[R7] Bian Z H, Liu M Q. Presenting little q-Schur algebras u_k(2; r)[J]. Algebra Colloquium, 2017, 24(2): 297-308. [CrossRef] [MathSciNet] [Google Scholar]

[R8] El Turkey H, Kujawa J. Presenting schur superalgebras[J]. Pacific Journal of Mathematics, 2013, 262(2): 285-316. [Google Scholar]

[R9] Chen X Y. The Infinitesimal Theory of q-Schur Superalgebra and the BLM Implementation of Integer Quantum Supergroup[D]. Shanghai: East China Normal University, 2014(Ch). [Google Scholar]

[R10] Zhang R B. Finite dimensional irreducible representations of the quantum supergroup[J]. Journal of Mathematical Physics, 1993, 34(3): 1236-1254. [Google Scholar]

[R11] Du J, Gu H X. A realization of the quantum supergroup[J]. Journal of Algebra, 2014, 404: 60-99. [Google Scholar]

[R12] Du J, Rui H B. Quantum Schur superalgebras and Kazhdan-lusztig combinatorics[J]. Journal of Pure and Applied Algebra, 2011, 215(11): 2715-2737. [Google Scholar]