© Wuhan University 2023

0 Introduction

Let $\mathit{U}(n,r)$ be the q-Schur algebras over $\mathbb{Q}(v)$ 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. Using a geometric setting for q-Schur algebras, Beilinson-Lusztig-MacPherson^[3] realized the quantum enveloping algebra $\mathit{U}(n)$ of $\mathrm{g}{\mathrm{l}}_n$ as a "limit" of q-Schur algebras over $\mathbb{Q}(v)$, and they defined an epimorphism ${\zeta }_r$ from the quantum enveloping algebra $\mathit{U}(n)$ of $\mathrm{g}{\mathrm{l}}_n$ to the q-Schur algebras $\mathit{U}(n,r)$. Then, Du^[4] proved that this epimorphism can be applied to any field. For little q-Schur algebras, the definition at odd roots has been obtained, which could be referred to Ref. [5]. Moreover, some results of little q-Schur algebras have been classified in Ref. [6], like simple modules, little q-Schur algebras and finite representation type in the odd roots of unity case. Fu^[7] gave the definition of little q-Schur algebras at even unit roots, and explored the related basis and dimension formulas. Therefore, for little q-Schur algebras at any unit roots, the complete classification of representation type has been given by Bian and Liu in Ref. [8].

In the theory of quantum group, it is an important issue to study the presentation of algebras, in other words, to give a set of generators and relations. In addition, the presentation of q-Schur algebras has been obtained in Refs. [9,10]. It is a natural question to present the little q-Schur algebras. Therefore, the presentations obtained from the perspective of monomial basis for little q-Schur algebras $u_k(\mathrm{2},r)$, $u_k(\mathrm{3,3})$ and $u_k(\mathrm{3,4})$ could be respectively referred to Refs. [11,12]. However, with the increase of r, the relations may become increasingly complex.

The goal of this paper is to study the generators and relations for little q-Schur algebra $u_k(\mathrm{3,5})$ from the perspective of Poincaré-Birkhoff-Witt (PBW) basis which is helpful to investigate the presentation of $u_k(\mathrm{3},r)$. We briefly recall the definitions of q-Schur and little q-Schur algebras in Section 1. In Section 2, we prove that the set ${{W}}_k$ is the basis of little q-Schur algebra $u_k(\mathrm{3,5})$, and present the relations of $u_k(\mathrm{3,5})$.

1 Preliminary

Let v be an indeterminate and ${A}=\mathbf{\mathbb{Z}}[v,v^{-\mathrm{1}}]$. For any integers $c,t$ with $t\ge \mathrm{1}$, let $[c]=\frac{v^c-v^{-c}}{v-v^{-\mathrm{1}}}\in {A}$, ${[t]}^{!}=[\mathrm{1}][\mathrm{2}]\cdots [t]$ and $\left[\begin{array}{c}c\\ t\end{array}\right]={\displaystyle \prod _{s=\mathrm{1}}^t}\frac{v^{c-s+\mathrm{1}}-v^{-c+s-\mathrm{1}}}{v^s-v^{-s}}={[c]}^{!}/{[t]}^{!}{[c-t]}^{!}\in {A}$. Let $\left[\begin{array}{c}c\\ t\end{array}\right]=\mathrm{0}$ for $t>c\ge \mathrm{0}$. Let $k$ be a field containing an l'-th primitive root $\epsilon$ of unity with $l^{\text{'}}>\mathrm{3}$. Let $l>\mathrm{1}$ be defined by $l=\{\begin{array}{c}{l}^{\text{'}},\hspace{1em} \mathrm{i}\mathrm{f}{l}^{\text{'}}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{d}\mathrm{d}\\ l^{\text{'}}/\mathrm{2},\mathrm{i}\mathrm{f}{l}^{\text{'}}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\end{array}$. In this paper, we put $l=\mathrm{3}$. Specializing $v$ to $\epsilon$, $k$ will be viewed as an ${A}$-module. When $v$ is specialized to $\epsilon$, $[c]$, ${[t]}^{!}$ and $\left[\begin{array}{c}c\\ t\end{array}\right]$ specialize to ${[c]}_{\epsilon }$, ${{[t]}^{!}}_{\epsilon }$ and ${\left[\begin{array}{c}c\\ t\end{array}\right]}_{\epsilon }$.

For quantum enveloping algebra of $\mathrm{g}{\mathrm{l}}_n$, $\mathit{U}(n)$ defined on $\mathbb{Q}(v)$ has been generated by the elements $E_i$,$F_i(\mathrm{1}\le i\le n-\mathrm{1})$, $K_j$, $K_j^{-\mathrm{1}}(\mathrm{1}\le j\le n)$ and satisfies the corresponding relations referred to Section 3.2 of Ref. [13]. Set $K_{ij}=K_i{K}_j^{-\mathrm{1}}(\mathrm{1}\le i\ne j\le n)\in {\mathit{U}}^{\mathrm{0}}(n)$ and define the root vectors in $\mathit{U}(n)$ as follows. If $j-i=\mathrm{1}$, then set $E_{i,j}=E_i$,$F_{j,i}=F_i$. For $j-i>\mathrm{1}$, inductively set $E_{i,j}=v^{-\mathrm{1}}{E}_i{E}_{i+\mathrm{1},j}-E_{i+\mathrm{1},j}{E}_i$, $F_{j,i}=v{F}_{j,i+\mathrm{1}}{F}_i-F_i{F}_{j,i+\mathrm{1}}$. There is an isomorphism $\mathrm{\Omega }:\mathit{U}(n)\to \mathit{U}{(n)}^{\mathrm{o}\mathrm{p}\mathrm{p}}$^[14] defined by $\mathrm{\Omega }(E_i)=F_i$, $\mathrm{\Omega }(F_i)=E_i$, $\mathrm{\Omega }(K_i)=K_i^{-\mathrm{1}}$, $\mathrm{\Omega }(v)=v^{-\mathrm{1}}$. Then obviously $\mathrm{\Omega }(E_{i,j})=F_{j,i}$.

Following Refs. [15,16], let $U_{{A}}(n)$ (respectively,$U_{{A}}^{+}(n)$, $U_{{A}}^{-}(n)$, $U_{{A}}^{\mathrm{0}}(n)$) be the${A}$-subalgebra of $\mathit{U}(n)$ generated by all $E_i^{(m)}$,$F_i^{(m)}$,$K_j$ and $\left[\begin{array}{c}{K}_j;\mathrm{0}\\ t\end{array}\right]$ (respectively, $E_i^{(m)}$, $F_i^{(m)}$, $K_j$ and $\left[\begin{array}{c}{K}_j;\mathrm{0}\\ t\end{array}\right]$), where for $m,t\in \mathbb{N}$, $c\in \mathbb{Z}$. Let

$E_i^{{}^{(m)}}=\frac{E_i^m}{{[m]}^{!}},F_i^{{}^{(m)}}=\frac{F_i^m}{{[m]}^{!}},\left[\begin{array}{c}{K}_j;c\\ t\end{array}\right]={\displaystyle \prod _{s=\mathrm{1}}^t}\frac{K_j{v}^{c-s+\mathrm{1}}-K_j^{-\mathrm{1}}{v}^{-c+s-\mathrm{1}}}{v^s-v^{-s}}.$

Let $U_k(n)=U_{{A}}(n){\otimes }_{{A}}k$, we denote the image of $E_i\otimes \mathrm{1}$, $F_i\otimes \mathrm{1},\mathrm{a}\mathrm{n}\mathrm{d}$$K_j\otimes \mathrm{1}$ in $U_k(n)$ by $E_i$, $F_i$, and $K_j$. Let ${\tilde{u}}_k(n)$ be the k-subalgebra of $U_k(n)$ generated by the elements $E_i$, $F_i$, $K_j^{\pm \mathrm{1}}$ for all i, j. Let ${\tilde{u}}_k{(n)}^{+}$, ${\tilde{u}}_k{(n)}^{\mathrm{0}}$, ${\tilde{u}}_k{(n)}^{-}$ be the k-subalgebras of ${\tilde{u}}_k(n)$ generated respectively by the elements $E_i$, $K_j^{\pm \mathrm{1}}$ and $F_i$ by Ref. [16].

When $l^{\text{'}}$ is odd, the elements $K_i^l-\mathrm{1}$ are the central in ${\tilde{u}}_k(n)$. They generate an ideal $<K_{\mathrm{1}}^l-\mathrm{1},\cdots ,K_n^l-\mathrm{1}>$ of ${\tilde{u}}_k(n)$. Let $u_k(n)={\tilde{u}}_k(n)/<K_{\mathrm{1}}^l-\mathrm{1},\cdots ,K_n^l-\mathrm{1}>$ and we call $u_k(n)$ the infinitesimal quantum group of $\mathrm{g}{\mathrm{l}}_n$. The presentation for $u_k(n)$ was given in Ref. [10]. When $l^{\text{'}}$ is even, there is no definition for infinitesimal quantum group of $\mathrm{g}{\mathrm{l}}_n$. Let $U_{{A}}(n,r)$ be the algebra over ${A}$ introduced in Ref. [3]. It is shown to be naturally isomorphic to the q-Schur algebra in Ref. [4]. Put $\mathit{U}(n,r)=U_{{A}}(n,r){\otimes }_{{A}}\mathbb{Q}(v)$, we shall call $U_{{A}}(n,r)$ and $\mathit{U}(n,r)$q-Schur algebras. There is an algebra epimorphism ${\zeta }_r:\mathit{U}(n)\twoheadrightarrow \mathit{U}(n,r)$ which could be referred to Refs. [3]. Let $\left[\begin{array}{c}{k}_i;c\\ t_i\end{array}\right]={\zeta }_r\left(\left[\begin{array}{c}{K}_i;c\\ t_i\end{array}\right]\right)$ and ${\mathit{k}}_t={\displaystyle \prod _{i=\mathrm{1}}^n}\left[\begin{array}{c}{k}_i;\mathrm{0}\\ t_i\end{array}\right]$, for $t=(t_{\mathrm{1}},\cdots ,t_n)\in {\mathbf{\mathbb{N}}}^n$. Moreover, we have ${\zeta }_r(U_{{A}}(n))=U_{{A}}(n,r)$^[4]. Let $\mathrm{\Lambda }(n,r)=\{\lambda \in {\mathbf{\mathbb{N}}}^n|{\lambda }_{\mathrm{1}}+{\lambda }_{\mathrm{2}}+\cdots +{\lambda }_n=r\}$. Let $\Phi$ be the root system of type $A_{n-\mathrm{1}}$:$\Phi =\{{\epsilon }_i-{\epsilon }_j|\mathrm{1}\le i\ne j\le n\}$. Here the ${\epsilon }_i$ form the standard orthonormal basis of the Euclidean space ${\mathbb{R}}^n$. Let ( , ) denote the inner product on this space and define simple roots ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+\mathrm{1}}$ and positive roots ${\alpha }_{ij}={\epsilon }_i-{\epsilon }_j$.

Theorem 1^[10] The q-Schur algebras $\mathit{U}(n,r)$ are generated by the elements ${\mathit{e}}_i$, ${\mathit{f}}_i(\mathrm{1}\le i\le n-\mathrm{1})$, ${\mathit{k}}_{\lambda }(\lambda \in \mathrm{\Lambda }(n,r))$, subject to the following relations:

$(\mathrm{R}\mathrm{1})\sum _{\lambda \in \mathrm{\Lambda }(n,r)}{\mathit{k}}_{\lambda }=\mathrm{1},{\mathit{k}}_{\lambda }{\mathit{k}}_{\mu }={\delta }_{\lambda \mu }{\mathit{k}}_{\lambda };$

$(\mathrm{R}\mathrm{2}){\mathit{e}}_i{\mathit{f}}_j-{\mathit{f}}_j{\mathit{e}}_i={\delta }_{ij}\sum _{\lambda \in \mathrm{\Lambda }(n,r)}\left[{\lambda }_i-{\lambda }_{i+\mathrm{1}}\right]{\mathit{k}}_{\lambda };$

$(\mathrm{R}\mathrm{3}){\mathit{e}}_i^{\mathrm{2}}{\mathit{e}}_j-(v+v^{-\mathrm{1}}){\mathit{e}}_i{\mathit{e}}_j{\mathit{e}}_i+{\mathit{e}}_j{\mathit{e}}_i^{\mathrm{2}}=\mathrm{0},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}|i-j|=\mathrm{1};$

$(\mathrm{R}\mathrm{4}){\mathit{f}}_i^{\mathrm{2}}{\mathit{f}}_j-(v+v^{-\mathrm{1}}){\mathit{f}}_i{\mathit{f}}_j{\mathit{f}}_i+{\mathit{f}}_j{\mathit{f}}_i^{\mathrm{2}}=\mathrm{0},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}|i-j|=\mathrm{1};$

$(\mathrm{R}\mathrm{5}){\mathit{e}}_i{\mathit{k}}_{\lambda }=\{\begin{array}{ll}{\mathit{k}}_{\lambda +{\alpha }_i}{\mathit{e}}_i,& \mathrm{i}\mathrm{f}\lambda +{\alpha }_i\in \mathrm{\Lambda }(n,r)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array},{\mathit{k}}_{\lambda }{\mathit{e}}_i=\{\begin{array}{ll}{\mathit{e}}_i{\mathit{k}}_{\lambda -{\alpha }_i},& \mathrm{i}\mathrm{f}\lambda -{\alpha }_i\in \mathrm{\Lambda }(n,r)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$;

$(\mathrm{R}\mathrm{6}){\mathit{f}}_i{\mathit{k}}_{\lambda }=\{\begin{array}{ll}{\mathit{k}}_{\lambda -{\alpha }_i}{\mathit{f}}_i,& \mathrm{i}\mathrm{f}\lambda -{\alpha }_i\in \mathrm{\Lambda }(n,r)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array},{\mathit{k}}_{\lambda }{\mathit{f}}_i=\{\begin{array}{ll}{\mathit{f}}_i{\mathit{k}}_{\lambda +{\alpha }_i},& \mathrm{i}\mathrm{f}\lambda +{\alpha }_i\in \mathrm{\Lambda }(n,r)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$

Let $\tilde{\mathrm{\Xi }}(n)$ be the set of all $n\times n$ matrices over $\mathbb{Z}$ with all off-diagonal entries in $\mathbb{N}$. Let $\mathrm{\Xi }(n)=M_n(\mathbf{\mathbb{N}})$ be the subset of $\tilde{\mathrm{\Xi }}(n)$ consisting of matrices with entries all in $\mathbb{N}$, and let $\sigma :\mathrm{\Xi }(n)\to \mathbf{\mathbb{N}}$ to be the map sending a matrix to the sum of its entries. Then, for $r\in \mathbf{\mathbb{N}}$, the inverse image $\mathrm{\Xi }(n,r):={\sigma }^{-\mathrm{1}}(r)$ is the subset of $\mathrm{\Xi }(n)$ whose entries sum to $r$. For $\mathrm{1}\le i,j\le n$, let $E_{i,j}\in \mathrm{\Xi }(n)$ be the matrix $(a_{s,t})$ with $a_{s,t}={\delta }_{i,s}{\delta }_{j,t}$. Let $\mathrm{\Xi }{(n)}^{\pm }$ be the set of all $A\in \mathrm{\Xi }(n)$ whose diagonal entries are zero. Let $\mathrm{\Gamma }$ be the set of all $A=(a_{i,j})\in \mathrm{\Xi }(n)$ such that $a_{i,j}<l$ for all $i\ne j$ and ${\mathrm{\Gamma }}^{\pm }$ be the set of all $A\in \mathrm{\Gamma }$ whose diagonal entries are zero. Let ${\mathrm{\Gamma }}_m^{\pm }={\mathrm{\Gamma }}_m^{+}+{\mathrm{\Gamma }}_m^{-}$ be the set of all $A^{\pm }=A^{+}+A^{-}\in {\mathrm{\Gamma }}^{\pm }$ such that ${\sigma }_i(A)=\mathrm{0}$ for $i\ne m$ with ${\sigma }_i(A)=a_{i,i}+\sum _{\mathrm{1}\le j<i}(a_{i,j}+a_{j,i})$. Let $\mathrm{c}\mathrm{o}(A)=(\sum _i{a}_{i,\mathrm{1}},\sum _i{a}_{i,\mathrm{2}},\cdots ,\sum _i{a}_{i,n})$, $\mathrm{r}\mathrm{o}(A)=(\sum _j{a}_{\mathrm{1},j},\sum _i{a}_{\mathrm{2},j},\cdots ,\sum _i{a}_{n,j})$.

Similar to monomial basis for q-Schur algebra $\mathit{U}(n,r)$^[10], we give the following definition. For $A\in {\mathrm{\Xi }}^{\pm }(n)$, put

$\begin{array}{c}{E}^{c(A^{+})}=\prod _{\mathrm{1}\le i\le h<j\le n}{E}_{i,j}^{(a_{i,j})},\hspace{1em}{F}^{c(A^{-})}=\prod _{\mathrm{1}\le j\le h<i\le n}{F}_{j,i}^{(a_{j,i})}\end{array}$

The orders in which the products $E^{c(A^{+})}$ and $F^{c(A^{-})}$ are taken are respectively defined as follows. Put

$E^{c(A^{+})}=O_{\mathrm{2}}{O}_{\mathrm{3}}\cdots O_n,\hspace{1em}{F}^{c(A^{-})}=O_n^{\mathrm{\text{'}}}{O}_{n-\mathrm{1}}^{\mathrm{\text{'}}}\cdots O_{\mathrm{2}}^{\mathrm{\text{'}}},$

where $O_j=E_{\mathrm{1},j}^{(a_{\mathrm{1},j})}{E}_{\mathrm{2},j}^{(a_{\mathrm{2},j})}\cdots E_{j-\mathrm{1},j}^{(a_{j-\mathrm{1},j})},O_j^{\text{'}}=F_{j,j-\mathrm{1}}^{(a_{j,j-\mathrm{1}})}{F}_{j,j-\mathrm{2}}^{(a_{j,j-\mathrm{2}})}\cdots F_{j,\mathrm{1}}^{(a_{j,\mathrm{1}})}.$

Let ${\mathit{e}}_{i,j}={\zeta }_r(E_{i,j})$, ${\mathit{f}}_{j,i}={\zeta }_r(F_{j,i})$, $k_s={\zeta }_r(K_s)$ for $\mathrm{1}\le i\ne j\le n-\mathrm{1}$ and $\mathrm{1}\le s\le n$, then we have

${\mathit{e}}^{c(A^{+})}={\zeta }_r(E^{c(A^{+})}),{\mathit{f}}^{c(A^{-})}={\zeta }_r(F^{c(A^{-})}).$

Proposition 1^[9] 1) For any $\lambda \in \mathrm{\Lambda }(n,r)$, we have

$\begin{array}{l}{\mathit{e}}_{i,j}{\mathit{k}}_{\lambda }=\{\begin{array}{ll}{\mathit{k}}_{\lambda +{\alpha }_{ij}}{\mathit{e}}_{i,j},& \mathrm{i}\mathrm{f}\lambda +{\alpha }_{ij}\in \mathrm{\Lambda }(n,r)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\\ {\mathit{k}}_{\lambda }{\mathit{e}}_{i,j}=\{\begin{array}{ll}{\mathit{e}}_{i,j}{\mathit{k}}_{\lambda -{\alpha }_{ij}},& \mathrm{i}\mathrm{f}\lambda -{\alpha }_{ij}\in \mathrm{\Lambda }(n,r)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\end{array}$

and similar results for ${\mathit{f}}_{j,i}$ can be obtained by applying the isomorphism $\mathrm{\Omega }$.

2) Let $\lambda \in \mathrm{\Lambda }(n,r)$, then $k_i{\mathit{k}}_{\lambda }=v^{{\lambda }_i}{k}_{\lambda },\hspace{1em}\left[\begin{array}{c}{k}_i;c\\ t\end{array}\right]{\mathit{k}}_{\lambda }=\left[\begin{array}{c}{\lambda }_i+c\\ t\end{array}\right]{\mathit{k}}_{\lambda }$ for $\mathrm{1}\le i\le n$.

Theorem 2^[9] The set ${W}:=\{{\mathit{e}}^{c(A^{+})}{\mathit{k}}_{\lambda }{\mathit{f}}^{c(A^{-})}\mid \lambda \in \mathrm{\Lambda }(\mathrm{3,5}),{\lambda }_i\ge {\sigma }_i(A)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i,A\in {\mathrm{\Gamma }}^{\pm }\}$ forms a basis for q-Schur algebra $U_{{A}}(\mathrm{3,5})$.

Let $U_k(n,r)=U_{{A}}(n,r){\otimes }_{{A}}k$, hence ${\zeta }_r$ naturally induces a surjective homomorphism

${\zeta }_{r,k}:={\zeta }_r\otimes \mathrm{i}\mathrm{d}:U_k(n)=U_{{A}}(n){\otimes }_{{A}}k\twoheadrightarrow U_k(n,r)$

When $l^{\text{'}}$ is odd, since $k_i^l={\sum }_{{}_{\lambda \in \Lambda (n,r)}}{k}_i^l{\mathit{k}}_{\lambda }={\sum }_{{}_{\lambda \in \Lambda (n,r)}}{\epsilon }^{{\lambda }_{i}l}{\mathit{k}}_{\lambda }=\mathrm{1}$, ${\zeta }_r$ naturally induces a map from $u_k(n)$ to $U_k(n,r)$. We call the image ${\zeta }_{r,k}(u_k(n))$ the little q-Schur algebra and denote it by $u_k(n,r)$. When $l^{\text{'}}$ is even, by restriction, we also get a map ${\zeta }_{r,k}:{\tilde{u}}_k(n)\to U_k(n,r)$, then $u_k(n,r)={\zeta }_{r,k}({\tilde{u}}_k(n))$. By abuse of notations, we shall continue to denote the images of the generators $E_i,F_i,K_i$ for ${\tilde{u}}_k(n)$ by the same letters ${\mathit{e}}_i,{\mathit{f}}_i,{\mathit{k}}_i$ used for $U_k(n,r)$.

Let $u_k{(n,r)}^{+}={\zeta }_{r,k}({\tilde{u}}_k{(n)}^{+})$ (resp,${\zeta }_{r,k}(u_k{(n)}^{+}))$,$u_k{(n,r)}^{-}={\zeta }_{r,k}({\tilde{u}}_k{(n)}^{-})$ (resp,${\zeta }_{r,k}(u_k{(n)}^{-}))$,$u_k{(n,r)}^{\mathrm{0}}={\zeta }_{r,k}({\tilde{u}}_k{(n)}^{\mathrm{0}})$ (resp,${\zeta }_{r,k}(u_k{(n)}^{\mathrm{0}}))$. For a positive integer $m$, let ${\mathbb{Z}}_m=\mathbb{Z}/m\mathbb{Z}$, set

$\overline{}:{\mathbf{\mathbb{Z}}}^n\to ({\mathbf{\mathbb{Z}}}_{l^{\text{'}}}{)}^n$

be the map defined by $\overline{(j_{\mathrm{1}},j_{\mathrm{2}},\cdots ,j_n)}=(\overline{j_{\mathrm{1}}},\overline{j_{\mathrm{2}}},\cdots ,\overline{j_n})$.

Let $\overline{\mathrm{\Lambda }(n,r)}=\{\overline{\lambda }\in ({\mathbf{\mathbb{Z}}}_{l^{\text{'}}}{)}^n|\lambda \in \mathrm{\Lambda }(n,r)\}$. For $\overline{\lambda }\in ({\mathbf{\mathbb{Z}}}_{l^{\text{'}}}{)}^n$, define

${\mathit{P}}_{\overline{\lambda }}=\{\begin{array}{cc}\sum _{\mu \in \mathrm{\Lambda }(n,r),\overline{\mu }=\overline{\lambda }}{\mathit{k}}_{\mu },& \mathrm{i}\mathrm{f}\overline{\lambda }\in \overline{\mathrm{\Lambda }(n,r)}\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$

Proposition 2^[10] For $A\in {\mathrm{\Gamma }}^{\pm },\lambda \in \mathrm{\Lambda }(n,r)$,

1)　If there exist $\mathit{\mu }\in \mathbf{\Lambda }(\mathit{n},\mathit{r}),{\mathit{\mu }}_{\mathit{i}}\ge {\mathit{\sigma }}_{\mathit{i}}({\mathit{A}}^{+})$ for all $\mathit{i}$ and $\overline{\mathit{\mu }}=\overline{\mathit{\lambda }}$, then ${\mathit{e}}^{\mathit{c}({\mathit{A}}^{+})}{\mathit{P}}_{\overline{\mathit{\lambda }}}={\mathit{P}}_{{\overline{\mathit{\lambda }}}^{\mathit{\text{'}}}}{\mathit{e}}^{\mathit{c}({\mathit{A}}^{+})}$ where $\overline{{\mathit{\lambda }}^{\mathit{\text{'}}}}=\overline{\mathit{\lambda }-\mathbf{c}\mathbf{o}({\mathit{A}}^{+})+\mathbf{r}\mathbf{o}({\mathit{A}}^{+})}$ otherwise, ${\mathit{e}}^{\mathit{c}({\mathit{A}}^{+})}{\mathit{P}}_{\overline{\mathit{\lambda }}}=\mathrm{0}$.

2)　If there exist $\mu \in \mathrm{\Lambda }(n,r),{\mu }_i\ge {\sigma }_i(A^{-})$ for all $i$ and $\overline{\mu }=\overline{\lambda }$, then ${\mathit{P}}_{\overline{\lambda }}{\mathit{f}}^{c(A^{-})}={\mathit{f}}^{c(A^{-})}{\mathit{P}}_{{\overline{\lambda }}^{\text{'}}}$ where $\overline{{\lambda }^{\text{'}}}=\overline{\lambda -\mathrm{c}\mathrm{o}(A^{+})+\mathrm{r}\mathrm{o}(A^{+})}$; otherwise, ${\mathit{P}}_{\overline{\lambda }}{\mathit{f}}^{c(A^{-})}=\mathrm{0}$.

2 The Little q-Schur Algebra ${\mathit{u}}_{\mathit{k}}(\mathrm{3,5})$

Theorem 3   The set ${{W}}_k:=\{{\mathit{e}}^{c(A^{+})}{\mathit{P}}_{\overline{\lambda }}{\mathit{f}}^{c(A^{-})}\mid \lambda \in \mathrm{\Lambda }(\mathrm{3,5}),{\lambda }_i\ge {\sigma }_i(A)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i,A\in {\mathrm{\Gamma }}^{\pm }\}$ forms a basis for $u_k(\mathrm{3,5})$.

Proof   Fix $A\in {\mathrm{\Gamma }}^{\pm }$ satisfying ${\lambda }_i\ge {\sigma }_i(A)$ for all $i$. By the definition of ${\mathit{P}}_{\overline{\lambda }}$, we have

${\mathit{e}}^{c(A^{+})}{\mathit{P}}_{\overline{\lambda }}{\mathit{f}}^{c(A^{-})}=\sum _{\overline{\lambda }=\overline{\mu },\forall i,{\mu }_i\ge {\sigma }_i(A)}{\mathit{e}}^{c(A^{+})}{\mathit{k}}_{\mu }{\mathit{f}}^{c(A^{-})}+\sum _{\overline{\lambda }=\overline{{\mu }^{\text{'}}},\exists i,{\mu }_i^{\text{'}}<{\sigma }_i(A)}{\mathit{e}}^{c(A^{+})}{\mathit{k}}_{{\mu }^{\text{'}}}{\mathit{f}}^{c(A^{-})}$

In the q-Schur algebra $U_{{A}}(\mathrm{3,5})$, for any ${\mathit{e}}^{c(A^{+})}{\mathit{k}}_{{\mu }^{\text{'}}}{\mathit{f}}^{c(A^{-})}$, if ${\mu }_i^{\text{'}}<{\sigma }_i(A)$ for some $i$, then it lies in the span of ${W}$ in the Theorem 2. Then we have

$\sum _{\overline{\lambda }=\overline{{\mu }^{\text{'}}},\exists i,{\mu }_i^{\text{'}}<{\sigma }_i(A)}{\mathit{e}}^{c(A^{+})}{\mathit{k}}_{{\mu }^{\text{'}}}{\mathit{f}}^{c(A^{-})}=\sum _{\begin{array}{c}B\in {\mathrm{\Gamma }}^{\pm }\\ {\sigma }_i(B)<{\sigma }_i(A),{\mu }_i^{\text{'}}\ge {\sigma }_i(B)\end{array}}{f}_{B,A}{\mathit{e}}^{c(B^{+})}{\mathit{k}}_{{\mu }^{\text{'}}}{\mathit{f}}^{c(B^{-})}(f_{B,A}\in k)$

By Theorem 2, the elements above are linearly independent, thus the set ${{W}}_k$ is linearly independent.

Assume that $i<j,m<l$. Inspired by Ref.[17], we have the following commutation formulas in the q-Schur algebras $\mathit{U}(n,r)$.

$(\mathrm{2.1}\mathrm{a}){\mathit{e}}_{i,j}^{(M)}{\mathit{e}}_{m,l}^{(N)}=v^{-MN}{\mathit{e}}_{m,l}^{(N)}{\mathit{e}}_{i,j}^{(M)},i=m<j<l\mathrm{o}\mathrm{r}i<m<j=l$

$(\mathrm{2.1}\mathrm{b}){\mathit{e}}_{i,j}^{(M)}{\mathit{e}}_{m,l}^{(N)}={\displaystyle \sum _{t=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}(M,N)}}{v}^{(M-t)(N-t)+t}{\mathit{e}}_{m,l}^{(N-t)}{\mathit{e}}_{i,l}^{(t)}{\mathit{e}}_{i,j}^{(M-t)},j=m$

$(\mathrm{2.1}\mathrm{c}){\mathit{f}}_{j,i}^{(M)}{\mathit{f}}_{l,m}^{(N)}=v^{MN}{\mathit{f}}_{l,m}^{(N)}{\mathit{f}}_{j,i}^{(M)},i=m<j<l\mathrm{o}\mathrm{r}i<m<j=l$

$(\mathrm{2.1}\mathrm{d}){\mathit{f}}_{j,i}^{(M)}{\mathit{f}}_{l,m}^{(N)}={\displaystyle \sum _{t=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}(M,N)}}{v}^{-(M-t)(N-t)-t}{\mathit{f}}_{l,m}^{(N-t)}{\mathit{f}}_{l,i}^{(t)}{\mathit{f}}_{j,i}^{(M-t)},j=m$

$(\mathrm{2.1}\mathrm{e}){\mathit{e}}_{i,j}^{(M)}{\mathit{f}}_{l,m}^{(N)}={\displaystyle \sum _{t=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}(M,N)}}{(-\mathrm{1})}^t{v}^{t(M-t)}{\mathit{f}}_{l,j}^{(t)}{\mathit{f}}_{l,m}^{(N-t)}{k}_{ij}^{-t}{\mathit{e}}_{i,j}^{(M-t)},i=m<j<l$

$(\mathrm{2.1}\mathrm{f}){\mathit{e}}_{i,j}^{(M)}{\mathit{f}}_{j,i}^{(N)}={\displaystyle \sum _{t=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}(M,N)}}{\mathit{f}}_{j,i}^{(N-t)}\left[\begin{array}{c}{k}_{ij};\mathrm{2}t-M-N\\ t\end{array}\right]{\mathit{e}}_{i,j}^{(M-t)},i=m,j=l$

$(\mathrm{2.1}\mathrm{h}){\mathit{f}}_{j,i}^{(M)}{\mathit{e}}_{m,l}^{(N)}={\displaystyle \sum _{t=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}(M,N)}}{v}^{t(N-t-\mathrm{1})}{\mathit{e}}_{m,j}^{(N-t)}{k}_{mj}^t{\mathit{f}}_{j,i}^{(M-t)}{\mathit{f}}_{m,i}^{(t)},i<m<j=l$

$(\mathrm{2.1}\mathrm{g}){\mathit{f}}_{j,i}^{(M)}{\mathit{e}}_{i,j}^{(N)}={\displaystyle \sum _{t=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}(M,N)}}{\mathit{e}}_{i,j}^{(N-t)}\left[\begin{array}{c}{k}_{ij}^{-\mathrm{1}};\mathrm{2}t-M-N\\ t\end{array}\right]{\mathit{f}}_{j,i}^{(M-t)},i=m,j=l$

Lemma 1   In the little q-Schur algebra $u_k(\mathrm{3,5})$, for $A_j\in {\mathrm{\Gamma }}_j^{\pm }$, $\lambda \in \mathrm{\Lambda }(\mathrm{3,5})$ satisfying ${\lambda }_j={\sigma }_j(A)-\mathrm{1}$ for $i\ne j,{\lambda }_i<l^{\text{'}}$, we have ${\mathit{P}}_{\overline{\mu }}{\mathit{f}}^{c(A_j^{-})}{\mathit{e}}^{c(A_j^{+})}=\mathrm{0}$, where $\mu \in \mathrm{\Lambda }(\mathrm{3,5})$,${\mu }_j={\sigma }_j(A_j^{-})-\mathrm{1}$ and ${\mu }_i={\lambda }_i+a_{i,j}$, if $i<j$; ${\mu }_i<l^{\text{'}}$, if $i>j$.

Proof   For ${\mathit{e}}^{c(A_j)}{\mathit{P}}_{(\overline{{\lambda }_{\mathrm{1}}},\overline{{\lambda }_{\mathrm{2}}},\overline{{\lambda }_{\mathrm{3}}})}{\mathit{f}}^{c(A_j)}$ where ${\lambda }_j={\sigma }_j(A)-\mathrm{1}$, and ${\lambda }_i<l^{\text{'}}$ if $i\ne j$. We may assume $a_{i,j}>\mathrm{0}$, $a_{j,i}>\mathrm{0}$ for all $\mathrm{1}\le i<j\le \mathrm{3}$. By the definition of $u_k{(\mathrm{3,5})}^{\mathrm{0}}$ and commutation formulas between positive root vectors, it is enough to show for all $\mathrm{1}\le i<j\le \mathrm{3}$,