# Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

## Abstract

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if is a quasi-hereditary algebra with a simple preserving duality and is a faithful tilting then -module, has the double centralizer property with respect to This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module . over for which As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra . and the Brauer algebra on the space of dual partially harmonic tensors under certain condition.

## 1. Introduction

Let be a field. Let be a finite dimensional with identity element. Let -algebra be the category of finite dimensional left For any -modules. we use , to denote the full subcategory of direct summands of finite direct sums of .

Let We define . Then . We next define . Then there is a canonical algebra homomorphism . Similarly, we define . It is well-known that the canonical algebra homomorphism . is an isomorphism.

The double centralizer property plays a central role in many part of the representation theory in algebraic Lie theory. For example, the Schur-Weyl duality between the general linear group and the symmetric group on the space -tensor (Reference 65, Reference 8, Reference 15) implies that the Schur algebra has the double centralizer property with respect to Similarly, the Schur-Weyl duality between the symplectic group . (resp., orthogonal group and the specialized Brauer algebra ) (resp., on the ) space -tensor (Reference 5, Reference 6, Reference 15, Reference 17, Reference 27) implies that the symplectic Schur algebra (resp., the orthogonal Schur algebra) has the double centralizer property with respect to For quantized version of these classical Schur-Weyl dualities, we refer the readers to .Reference 9, Reference 28, Reference 29, Reference 36, Reference 40 and Reference 45. The combinatorial (due to Soergel -functorReference 60) plays a crucial role in the study of the principal blocks of the BGG category of any semisimple Lie algebras. The key property of this functor relies on the double centralizer property of the corresponding basic projective-injective module. A similar idea is used in the study of the category of the rational Cherednik algebras Reference 33. For more examples and applications of the double centralizer property in higher Schur-Weyl duality, quantum affine Schur-Weyl duality, etc., we refer the readers to Reference 7, Reference 10 and Reference 16.

If is a faithful then the double centralizer property of -module, is often closely related to the fully faithfulness of the hom functor on projectives. Recall that the hom functor is said to be fully faithful on projectives if for any projective modules the natural map ,

is an isomorphism.

For a faithful -module it is well-known that , has the double centraliser property with respect to if and only if the hom functor is fully faithful on injectives. The following result relates the double centralizer property of to the fully faithfulness of the hom functor on projectives and we leave its proof to the readers.

Let be a faithful When -module. is not semisimple, it is often difficult to check the double centralizer property of with respect to (i.e., whether or not) directly. König, Slungård and Xi in Reference 46 studied the double centralizer property using the notion of dominant dimension. To state their result, we recall the following definition.

The following theorem gives a necessary and sufficient condition for which has the double centraliser property with respect to a faithful -module .

In particular, the above condition means that there exists an injective left of -approximation and the dimension of -dominant is at least two. In general, it is relatively easy to make into an but it is hard to show that the cokernel of the map -approximation, can be embedded into for some By the way, the above theorem actually holds for any finitely generated algebra over a commutative noetherian domain, though we only concentrate on the finite dimensional algebras over a field in this paper. .

The starting point of this work is to look for a simple and effective way to verify the above-mentioned embedding property of the cokernel of the map In many examples of double centralizer property arising in algebraic Lie theory, . is often a tilting module over a finite dimensional quasi-hereditary algebra or even a standardly stratified algebra. The following theorem, which gives a sufficient condition for the double centralizer property with respect to a tilting module over a finite dimensional standardly stratified algebra, is the first main result of this paper.

Note that any quasi-hereditary algebra over a field is an example of standardly stratified algebras. Our second and third main results focus on the finite dimensional quasi-hereditary algebra with a simple preserving duality. The second main result of this paper gives a simple criterion on for which has the double centralizer with respect to .

By Reference 53, Corollary 2.4, there exists a faithful basic tilting module such that The following theorem is the third main result of this paper, which affirmatively answer a question of Mazorchuk and Stroppel (see .Reference 53, Remark 2.5) on the existence of minimal basic tilting module for which has the double centralizer property.

The fourth main result of this paper deals with a concrete situation of Brauer-Schur-Weyl duality related to the space of dual partially harmonic tensors. We refer the readers to Section 4 for unexplained notations below.

The content of the paper is organised as follows. In Section 2, we first recall the notions of standardly stratified algebras and their basic properties and then give the first main result Theorem 1.9 of this paper. In Section 3, we shall focus on the quasi-hereditary algebra with a simple preserving duality. Proposition 3.6 is a key step in the proof of the second main result (Theorem 1.10) of this paper. The proof of Proposition 3.6 makes use of a homological result Reference 52, Corollary 6 of Mazorchuk and Ovsienko for properly stratified algebras. The proof of the third main result Theorem 1.11 is also given in this section. As a remarkable consequence of Theorem 1.11, we obtained in Corollary 3.15 that the existence of a unique minimal faithful basic tilting module such that any other faithful tilting module must have as a direct summand. In Section 4, we use the tool of dominant dimension to study the Schur-Weyl duality between the symplectic Schur algebra and on the space dual partially harmonic tensors, where of is a symplectic space over -dimensional and , is the two-sided ideal of the Brauer algebra generated by with The aim is to prove the surjectivity of the natural map from . to the endomorphism algebra of the space as a The fourth main result Theorem -module.1.12 of this paper proves this surjectivity under the assumption Another surjection from . to the endomorphism algebra of the space as a is established in an earlier work -moduleReference 41 by the first author of this paper.

## 2. Standardly stratified algebras and their tilting modules

The purpose of this section is to give a sufficient condition for the double centralizer property with respect to a tilting module over a finite dimensional standardly stratified algebra.

Let be a field and be a finite dimensional with identity element. Let -algebra be a complete set of representatives of isomorphic classes of simple modules in We always assume that . is split over in the sense that for any For each . let , be the projective cover of and the injective hull of For any . we define the trace , of in as the sum of the images of all from -homomorphisms to .

Let be a finite dimensional standardly stratified algebraFootnote^{1} in the sense of Reference 12. That means, there is a partial *preorder* “ on ” and if set (for any , )

^{1}

Another slightly different class of standardly stratified algebras was introduced and studied in Reference 1Reference 2 under the same name.

then

- (1)
the kernel of the canonical surjection has a filtration with subquotients where , and ;

- (2)
the kernel of the canonical surjection has a filtration with subquotients where , .

We call the **standard module** corresponding to Note that . is the maximal quotient of such that for all In particular, . We define the .**proper standard module** to be

which is the maximal quotient of

Similarly, let **proper costandard module**

under the canonical epimorphism **costandard module**

which is the maximal submodule of

We use