Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 5, October 2023
Page(s) 385 - 391
DOI https://doi.org/10.1051/wujns/2023285385
Published online 10 November 2023

© Wuhan University 2023

Licence Creative CommonsThis 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 be the q-Schur algebras over 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 of as a "limit" of q-Schur algebras over , and they defined an epimorphism from the quantum enveloping algebra of to the q-Schur algebras . 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 , and 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 from the perspective of Poincaré-Birkhoff-Witt (PBW) basis which is helpful to investigate the presentation of . We briefly recall the definitions of q-Schur and little q-Schur algebras in Section 1. In Section 2, we prove that the set is the basis of little q-Schur algebra , and present the relations of .

1 Preliminary

Let v be an indeterminate and . For any integers with , let , and . Let for . Let be a field containing an l'-th primitive root of unity with . Let be defined by . In this paper, we put . Specializing to , will be viewed as an -module. When is specialized to , , and specialize to , and .

For quantum enveloping algebra of , defined on has been generated by the elements ,, , and satisfies the corresponding relations referred to Section 3.2 of Ref. [13]. Set and define the root vectors in as follows. If , then set ,. For , inductively set , . There is an isomorphism [14] defined by , , , . Then obviously .

Following Refs. [15,16], let (respectively,, , ) be the-subalgebra of generated by all ,, and (respectively, , , and ), where for , . Let

Let , we denote the image of , in by , , and . Let be the k-subalgebra of generated by the elements , , for all i, j. Let , , be the k-subalgebras of generated respectively by the elements , and by Ref. [16].

When is odd, the elements are the central in . They generate an ideal of . Let and we call the infinitesimal quantum group of . The presentation for was given in Ref. [10]. When is even, there is no definition for infinitesimal quantum group of . Let be the algebra over introduced in Ref. [3]. It is shown to be naturally isomorphic to the q-Schur algebra in Ref. [4]. Put , we shall call and q-Schur algebras. There is an algebra epimorphism which could be referred to Refs. [3]. Let and , for . Moreover, we have [4]. Let . Let be the root system of type :. Here the form the standard orthonormal basis of the Euclidean space . Let ( , ) denote the inner product on this space and define simple roots and positive roots .

Theorem 1[10] The q-Schur algebras are generated by the elements , , , subject to the following relations:

;

Let be the set of all matrices over with all off-diagonal entries in . Let be the subset of consisting of matrices with entries all in , and let to be the map sending a matrix to the sum of its entries. Then, for , the inverse image is the subset of whose entries sum to . For , let be the matrix with . Let be the set of all whose diagonal entries are zero. Let be the set of all such that for all and be the set of all whose diagonal entries are zero. Let be the set of all such that for with . Let , .

Similar to monomial basis for q-Schur algebra [10], we give the following definition. For , put

The orders in which the products and are taken are respectively defined as follows. Put

where

Let , , for and , then we have

Proposition 1[9] 1) For any , we have

and similar results for can be obtained by applying the isomorphism .

2) Let , then for .

Theorem 2[9] The set forms a basis for q-Schur algebra .

Let , hence naturally induces a surjective homomorphism

When is odd, since , naturally induces a map from to . We call the image the little q-Schur algebra and denote it by . When is even, by restriction, we also get a map , then . By abuse of notations, we shall continue to denote the images of the generators for by the same letters used for .

Let (resp,, (resp,, (resp,. For a positive integer , let , set

be the map defined by .

Let . For , define

Proposition 2[10] For ,

1) If there exist for all and , then where otherwise, .

2) If there exist for all and , then where ; otherwise, .

2 The Little q-Schur Algebra

Theorem 3   The set forms a basis for .

Proof   Fix satisfying for all . By the definition of , we have

In the q-Schur algebra , for any , if for some , then it lies in the span of in the Theorem 2. Then we have

By Theorem 2, the elements above are linearly independent, thus the set is linearly independent.

Assume that . Inspired by Ref.[17], we have the following commutation formulas in the q-Schur algebras .

Lemma 1   In the little q-Schur algebra , for , satisfying for , we have , where , and , if ; , if .

Proof   For where , and if . We may assume , for all . By the definition of and commutation formulas between positive root vectors, it is enough to show for all ,

Here, we no longer discuss the situations of in 2) of Proposition 2 alone.We consider , where , then

Case 1 . It is supposed that , according to the commutation formula , we get

Thus, there exist some such that .

Case 1-1 If , by Proposition 1(1), it follows ;

Case 1-2 If , i.e. , then we have

By 2) of Proposition 1 and , we also obtain

When , we have .

Case 2 , assuming , by the commutation formulas , and , we have

Similarly, there exist some such that .

Case 2-1 If , by Proposition 1 1), it follows that ;

Case 2-2 If , namely , then we have

then

When , we also find .

For where , the proof is similar to the above, thus we need not to give. Then we complete the proof of Lemma 1.

Theorem 4   The little q-Schur algebra can be generated by the elements , subject to the following relations:

;

;

;

;

;

(resp. ) for (resp. ) and satisfies that if with then ;

For , with for , then , where satisfies and if , if .

We define an algebra which satisfies the generators and relations given in Theorem 4. Set (resp. ,) be the subalgebra generated by (resp. ,). For , we give the root vectors, which satisfy the commutation formulas in the same way. According to the relations (LR3)-(LR7) in Theorem 4, we know that can be spanned by all the elements , .

Proposition 3   The set is a spanning set for .

Proof   Fixing , to prove this proposition, it suffices to show that if with , satisfying for some , then lies in the span of . Here is a linear combination of the elements in and is a constant.

Case 1 If and , assuming , for , by the commutation formulas -, we have

Therefore, we only need to consider .

By Lemma 1, we have when , then we obtain that is a linear combination of the elements in .

Case 2 If , there is an assumption that and . Then, we only need to consider . By the commutation formulas and , we have

By Lemma 1, when , we have . Hence,

When appears in , we only require to exchange (resp. ) and other (resp. ) into the Case 1 via commutation formulas. Finally, we give the proof of Theorem 4.

The proof of Theorem 4 By the definition of and Theorem 1, it is clear that satisfies the relations . Hence, there is a surjective algebra homomorphism from to . On the foundations of Proposition 1, Proposition 2 and Lemma 1, we have the that satisfies the relations in Theorem 4. And according to the definition of , there is a surjective algebra homomorphism from to satisfying , , . On the other hand, by Proposition 3, we see that the map sends the spanning set of to the basis of . Hence is an isomorphism.

Remark 1   For the presentation of little q-Schur algebra in this paper, if we only consider for some j, then the relation (LR9) in Theorem 4 is more general than the relations , and in Ref.[12,3]. As becomes increasingly great, we need to consider several at the same time. Then there may be other more complex relations. For example, when the relation cannot be obtained by the relations (LR1)-(LR9). In conclusion, it is more difficult to study the generators and relations for .

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