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

Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $\Lambda^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $\lambda\in\Lambda^+$, there is an embedding $\Delta(\lambda)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(\lambda)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful tilting $A$-module, then $A$ has the double centralizer property with respect to $T$. 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 $T$ over $A$ for which $A=\End_{\End_A(T)}(T)$. We also establish a Schur-Weyl duality between the symplectic Schur algebra $S^{sy}(m,n)$ and $\bb_{n}/\mathfrak{B}_{n}^{(f)}$ on $V^{\otimes n}/V^{\otimes n}\mathfrak{B}_{n}^{(f)}$ when $\cha K>\min\{n-f+m,n\}$, where $V$ is a $2m$-dimensional symplectic space over $K$, $\mathfrak{B}_{n}^{(f)}$ is the two-sided ideal of the Brauer algebra $\bb_{n}(-2m)$ generated by $e_1e_3\cdots e_{2f-1}$ with $1\leq f\leq [\frac{n}{2}]$.


Introduction
Let k be a field. Let A be a finite dimensional k-algebra with identity element. Let A-mod be the category of finite dimensional left A-modules. For any M ∈ A-mod, we use add(M ) to denote the full subcategory of direct summands of finite direct sums of M .
Let T ∈ A-mod. We define A ′ := End A (T ). Then T ∈ A ′ -mod. We next define A ′′ := End A ′ (T ). Then there is a canonical algebra homomorphism A → A ′′ . Similarly, we define A ′′′ := End A ′′ (T ). It is well-known that the canonical algebra homomorphism A ′ → A ′′′ is an isomorphism. Definition 1.1. Let T ∈ A-mod. We say A has the double centraliser property with respect to T if the canonical algebra homomorphism A → A ′′ is surjective. Example 1.2. Let A A be the left regular A-module. Then A has the double centraliser property with respect to A A. In fact, A ′ = A op and A ′′ = A. Example 1.3. If P ∈ A-mod is a progenerator, then A has the double centraliser property with respect to P .
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 GL(V ) and the symmetric group S r on the r-tensor space V ⊗r ( [69], [10], [17]) implies that the Schur algebra S(n, r) has the double centralizer property with respect to V ⊗r . Similarly, the Schur-Weyl duality between the symplectic group Sp(V ) (resp., orthogonal group O(V )) and the specialized Brauer algebra B n (− dim V ) (resp., B n (dim V )) on the n-tensor space V ⊗n ( [7], [8], [17], [20], [31]) implies that the symplectic Schur algebra (resp., the orthogonal Schur algebra) has the double centralizer property with respect to V ⊗n . For quantized version of these classical Schur-Weyl dualities, we refer the readers to [11], [32], [33], [40], [44] and [49]. The combinatorial V-functor (due to Soergel [64]) plays a crucial role in the study of the principal blocks of the BGG category O 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 O of the rational Cherednik algebras [37]. 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 [9], [12] and [19].
If T ∈ A-mod is a faithful A-module, then the double centralizer property of T is often closely related to the fully faithfulness of the hom functor Hom A (T, −) on projectives. Recall that the hom functor Hom A (T, −) is said to be fully faithful on projectives if for any projective modules P 1 , P 2 ∈ A-mod, the natural map θ : Hom A (P 1 , P 2 ) → Hom (EndA(T )) op (Hom A (T, P 1 ), Hom A (T, P 2 )) f → θ f : h → f • h. is an isomorphism.
For a faithful A-module T , it is well-known that A has the double centraliser property with respect to T if and only if the hom functor Hom A (T, −) is fully faithful on injectives. The following result relates the double centralizer property of T to the fully faithfulness of the hom functor Hom A (T, −) on projectives and we leave its proof to the readers. Let T ∈ A-mod be a faithful A-module. When T is not semisimple, it is often difficult to check the double centralizer property of A with respect to T (i.e., whether A = End End A (T ) (T ) or not) directly. König, Slungård and Xi in [50] studied the double centralizer property using the notion of dominant dimension. To state their result, we recall the following definition. 2) Let 1 ≤ r ∈ N. Since there is a Morita equivalence between End A (T ) with End A (T ⊕r ) which sends the End A (T )-module T to the End A (T ⊕r )-module T ⊕r , it follows that Then the dominant dimension of X relative to T is the supremum of all n ∈ N such that there exists an exact sequence The following theorem gives a necessary and sufficient condition for which A has the double centraliser property with respect to a faithful A-module T . In particular, the above condition means that there exists an injective left add(T )-approximation of A and the T -dominant dimension of A is at least two. In general, it is relatively easy to make δ into an add(T )-approximation, but it is hard to show that the cokernel of the map δ can be embedded into T ⊕s for some s ∈ N. 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 δ. In many examples of double centralizer property arising in algebraic Lie theory, T 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. Theorem 1.9. Let A be a finite dimensional standardly stratified algebra in the sense of [14]. Let T ∈ A-mod be a tilting module. Suppose that there is an integer r ∈ N such that for any λ ∈ Λ + , there is an embedding ι λ : ∆(λ) ֒→ T ⊕r as well as an epimorphism π λ : T ⊕r ։ ∇(λ) as A-modules, then T is a faithful module over A and A has the double centraliser property with respect to T . That is, Note that any quasi-hereditary algebra over a field is an example of standardly stratified algebras. Our second and the 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 T for which A has the double centralizer with respect to T . Theorem 1.10. Let A be a quasi-hereditary algebra with a simple preserving duality •. Let T be a faithful tilting module in A-mod. Then A has the double centralizer property with respect to T . In particular, the T -dominant dimension of A is at least two.
By [57,Corollary 2.4], there exists a faithful basic tilting module T ∈ A-mod such that A = End EndA(T ) (T ). The following theorem is the third main result of this paper, which affirmatively answer a question of Mazorchuk and Stroppel (see [57,Remark 2.5]) on the existence of minimal basic tilting module T for which A has the double centralizer property. Theorem 1.11. Let A be a quasi-hereditary algebra with a simple preserving duality. Then there exists a unique faithful basic tilting module T ∈ A-mod such that 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. Theorem 1.12. Suppose that char K > min{n − f + 2m, n}. Then there is an exact sequence of S sy K (m, n)-module homomorphisms: 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 [56, 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 T ∈ A-mod such that any other faithful tilting module T ′ ∈ A-mod must have T 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 S sy (m, n) and B n /B (f ) n on the space V ⊗n /V ⊗n B (f ) n of dual partially harmonic tensors, where V is a 2m-dimensional symplectic space over K, and B (f ) n is the two-sided ideal of the Brauer algebra B n (−2m) generated by e 1 e 3 · · · e 2f −1 The aim is to prove the surjectivity of the natural map from S sy (m, n) to the endomorphism algebra of the space V ⊗n /V ⊗n B (f ) n as a B n -module. The fourth main result Theorem 1.12 of this paper proves this surjectivity under the assumption char K > min{n − f + 2m, n}. Another surjection from B n /B (f ) n to the endomorphism algebra of the space V ⊗n /V ⊗n B (f ) n as a KSp(V )-module is established in an earlier work [45] by the first author of this paper.

Acknowledgements
The first author was support by the National Natural Science Foundation of China. The second author is supported by the NSF of Fujian Province (Grant No. 2018J01002) and the National NSF of China (Grant No. 11871107). Both authors wish to thank the referee for his/her substantial and insightful comments which significantly improves the final presentation of this article.

Standardly stratified algebras and their tilting modules
The purpose of this section is to gives a sufficient condition for the double centralizer property with respect to a tilting module over a finite dimensional standardly stratified algebra.
Let K be a field and A be a finite dimensional K-algebra with identity element. Let {L(λ)|λ ∈ Λ + } be a complete set of representatives of isomorphic classes of simple modules in A-mod. We always assume that A is split over K in the sense that End A (L(λ)) = K for any λ ∈ Λ + . For each λ ∈ Λ + , let P (λ) ∈ A-mod be the projective cover of L(λ) and I(λ) ∈ A-mod the injective hull of L(λ). For any M, N ∈ A-mod, we define the trace Tr M (N ) of M in N as the sum of the images of all A-homomorphisms from M to N .
is surjective and there are embeddings ι 1 : M 1 ֒→ T ⊕a , ι 2 : M 2 ֒→ T ⊕b as A-modules for some a, b ∈ N. Then there is an embedding ι 0 : M ֒→ T ⊕a+b as A-modules.
Proof. By assumption, it is clear that the induced natural map ι a, * : Hom A (M, T ⊕a ) → Hom A (M 1 , T ⊕a ) is surjective too. Hence there existsι 1 ∈ Hom A (M, T ⊕a ) such that ι a, * (ι 1 ) = ι 1 . In other words, We now define a map ι 0 : M → T ⊕a+b = T ⊕a ⊕ T ⊕b as follows: It is easy to check that ι 0 is an injective homomorphism in A-mod. This completes the proof of the lemma.
This follows from diagram chasing. Now we can give the proof of the first main result of this paper.
Proof of Theorem 1.9: Since A is a projective left A-module, A ∈ F (∆) by definition. Applying Lemmata 2.1 and 2.2, we can get an integer a ∈ N and an embedding δ 0 : A ֒→ T ⊕a as A-modules. In particular, T ⊕a and hence T is a faithful A-module.
The surjection of Hom A (T ⊕r ′ , ∇(λ)) → Hom A (A, ∇(λ)) for any λ ∈ Λ + and the fact that 1. Now applying Lemma 2.2 and using the assumption that for any µ ∈ Λ + , ∆(µ) ֒→ T ⊕k for some k ∈ N, we can deduce that there is an integer s ′ ∈ N such that T ⊕r ′ /A ֒→ T ⊕s ′ . In other words, the injective left add(T )- Finally, using Theorem 1.8, we prove the theorem.
Corollary 2.5. Let A be a finite dimensional standardly stratified algebra such that the injective hull of ∆(λ) is projective and the projective cover of ∇(λ) is injective for every λ ∈ Λ + . If T is a projectiveinjective generator, then A has the double centraliser property with respect to T .

Quasi-hereditary algebra with a simple preserving duality
In this section we shall focus on the finite dimensional quasi-hereditary algebras over a field with a simple preserving duality. We shall give the proof of the second and third main results (Theorem 1.10, Theorem 1.11) of this paper for this class of algebras.
Definition 3.1. We say that A has a simple preserving duality if there exists an exact, involutive and contravariant equivalence • : A-mod → A-mod which preserves the isomorphism classes of simple modules.
Let A be a finite dimensional standardly stratified algebra with a simple preserving duality. Then for each λ ∈ Λ + , we have In particular, A • is an injective left A-module. (1) the kernel of the canonical epimorphism ∆(λ) ։ L(λ) has a filtration with subquotients L(µ), µ ≺ λ; (2) ∆(λ) has a filtration with subquotients ∆(λ), then we shall call A a properly stratified algebra.
Let M ∈ A-mod. We define dim F (∆) M to be the minimal integer j such that there is an exact sequence of the form Proof. Applying the duality functor •, we get another short exact sequence Suppose that K / ∈ F (∆). Then we must have that dim F (∆) K = 1. Applying Lemma 3.3, we get that We have the following long exact sequence of homomorphisms: Applying Lemma 2.1 and noting that P is projective, we can deduce that Ext 1 On the other hand, from (3.5) we can get another long exact sequence of homomorphisms: which is a contradiction. This proves that K ∈ F (∆). The following proposition plays a crucial role in the proof of the second and the third main results of this paper.
Proposition 3.6. Let A be a quasi-hereditary algebra with a simple preserving duality •. Let T be a tilting module in A-mod. Suppose there is an embedding ι : A ֒→ T in A-mod. Then we have that T /A ∈ F (∆) and, for any λ ∈ Λ + , there exists a surjective homomorphism T ⊕m λ ։ ∇(λ) as well as an injective homomorphism ∆(λ) ֒→ T ⊕m λ , where m λ := dim ∇(λ).
Proof. Applying Lemma 3.4 to the short exact sequence 0 → A ι ֒→ T ։ T /A → 0, we get T /A ∈ F (∆). Applying Lemma 2.1, we get that Ext 1 A (T /A, ∇(λ)) = 0 for any λ ∈ Λ + . Thus we have an exact sequence of homomorphisms: . Now we define a map ρ : T ⊕m λ → ∇(λ) as follows: It is clear that ρ is a left A-module homomorphism and v j ∈ Image(ρ) for each 1 ≤ j ≤ m λ . This implies that ρ is a surjective A-module homomorphism. By taking duality, we also get an injective homomorphism ∆(λ) ֒→ T ⊕m λ . This completes the proof of the proposition. Now we can give the proof of our second main result of this paper.
Proof of Theorem 1.10: By assumption, T is a faithful module over A. Then there exists a natural number r and an embedding δ : A ֒→ T ⊕r as A-modules. By Remark 1.6, δ is an injective left add(T )approximation of A. Applying Proposition 3.6, we can find a natural number m := max{m λ |λ ∈ Λ + }, and a surjective homomorphism T ⊕rm ։ ∇(λ). Taking duality, we get an embedding ∆(λ) ֒→ Thus the theorem follows from Theorem 1.9.
Using Theorem 1.10, we can easily recover many known double centralizer properties or simplify the proof of the corresponding Schur-Weyl dualities in non-semisimple or even integral situation.
Example 3.7. Let n, r ∈ N and V be an n-dimensional vector space over an arbitrary field K. Let q ∈ K × . Let H q (S n ) be the Iwahori-Hecke algebra associated to the symmetric group with Hecke parameter q. There is a natural right action of H q (S n ) on V ⊗n ( [21]). Let S q (n, r) := End Hq(Sn) (V ⊗r ) be the Dipper-James q-Schur algebra over K ( [21]). Then V ⊗r is a faithful tilting module over S q (n, r). As a cellular algebra, S q (n, r) has an anti-involution * which sends its semistandard basis ϕ ST to ϕ T S (cf. [54,Proposition 4.13]). It follows from Theorem 1.10 that S q (n, r) has the double centralizer property with respect to V ⊗r . In particular, the V ⊗r -dominant dimension of S q (n, r) is at least two ([50, Theorem 3.6]). Specializing q to 1, we also get the double centralizer property of the classical Schur algebra S(n, r).
Let Λ be a subset of the set C n of all multicompositions of n which have ℓ components such that if λ ∈ Λ and µ is a multipartition of n such that µ λ, then µ ∈ Λ, where is the dominance partial order on C n defined in [54]. Let Λ + be the set of multipartitions in Λ. We define the cyclotomic tensor space M := λ∈Λ m λ H n , and define the cyclotomic q-Schur algebra S n := End Hn (M ).
By [22], we know that S n is a cellular and quasi-hereditary algebra. By [51,Lemma 4.8], M is a faithful tilting module over S n . Applying Theorem 1.10, we get that S n has the double centralizer property with respect to M . That is, End End Sn (M) (M ) = S n . This recovers the result [51, Theorem 4.10].
Example 3.9. Let m, n ∈ N and q ∈ K × . Let U v (sp 2m ) be Drinfeld-Jimbo's quantized enveloping algebra of the symplectic Lie algebra sp 2m (C) over the rational functional field Q(v). Let U q (sp 2m ) : -algebra by specializing v to q. Let V ∼ = L(ε 1 ) be the 2m-dimensional natural representation of U q (sp 2m ). Let B n,q = B n (−q 2m+1 , q) be the specialized Birman-Murakami-Wenzl algebra over K. We refer the readers to [46] for its precise definition. There is a natural right action of B n,q on V ⊗n which commutes with the natural left action of U q (sp 2m ). Let S sy q (m, n) := End Bn,q (V ⊗n ) be associated the symplectic q-Schur algebra. By [59], we know that S sy q (m, n) is cellular and quasi-hereditary. Moreover, V ⊗n is a faithful tilting module over S sy q (m, n). Applying Theorem 1.10, we get that S sy q (m, n) has the double centralizer property with respect to V ⊗n . That is, In particular, the V ⊗n -dominant dimension of S sy q (m, n) is at least two. Specializing q to 1, we also get the double centralizer property of the classical symplectic Schur algebra S sy (m, n) ( [58]).
The following example is an easier case of the double centralizer property which was already well known before (cf. [50, Section 2.3]).
Example 3.10. Let g be an arbitrary complex semisimple Lie algebra with root system Φ and the set Π of simple roots. Let g = n − ⊕ h ⊕ n + be a triangular decomposition of g and W the Weyl group of g. Let O be the Bernstein-Gelfand-Gelfand (BGG) category which consists of finitely generated U (g)-modules which are h-semisimple and locally U (n)-finite ( [47]). For any w ∈ W, λ ∈ h * , we define w · λ = w(λ + ρ) − ρ, where ρ is the half sum of all positive roots. Let I ⊂ Π be a subset of Π and p := p I the associated standard parabolic subalgebra of g. Let p I = l I ⊕ u I be the Levi decomposition of p. Let O p denote the corresponding parabolic BGG category [62] which is a full subcategory of O consisting of modules which are U (l I )-semisimple and locally U (u I )-finite. For each λ ∈ h * , let O λ be the Serre subcategory of O which is generated by L(x · λ) for all x ∈ W , let O p λ := O λ ∩ O p . The simple module L(λ) lies in O p if and only if λ ∈ Λ + I := {λ ∈ h * | λ, α ∨ ∈ Z ≥0 , ∀ α ∈ I}. For λ ∈ Λ + I , let M I (λ) denote the parabolic Verma module with highest weight λ and P I (λ) denote the projective cover of L(λ) in O p . Suppose that λ ∈ Λ + I is an integral weight. Let P I,λ := w∈W,w·λ∈Λ + It is well-known that A p λ is a quasi-hereditary basic algebra equipped with an anti-involution * (see [47,Chapter 1]). In particular, A p λ -mod has a simple preserving duality and each indecomposable projective over A p λ can be generated by an * -fixed primitive idempotent.
A weight w · λ ∈ Λ + I is called socular if L(w · λ) occurs in the socle of some parabolic Verma module M I (x · λ). By a result of Irving, P I (w · λ) is injective if and only if w · λ is socular. We define Q I,λ := w∈W w · λ is socular P I (w · λ), Since each P I (w ·λ) has a standard filtration, each simple module in the socle of P I (w ·λ) is labelled by a socular weight. It follows that the injective hull of each P I (w · λ) is a projective-injective module. Thus the basic algebra A p λ can be embedded into a direct sum of some copies of the basic projective-injective (hence tilting) module Hom O (P I,λ , Q I,λ ). In particular, Hom O (P I,λ , Q I,λ ) is a faithful tilting module over A p λ . Applying Corollary 2.5, we get that A p λ has the double centralizer property with respect to Hom O (P I,λ , Q I,λ ). In particular, by Lemma 1. Using Theorem 1.10, it is possible to simplify the proof of Schur-Weyl dualities in many nonsemisimple situations. General speaking, we have two algebras A, B and an (A, B)-bimodule M . By a Schur-Weyl duality between A and B on the bimodule M we mean that the following two canonical maps: are both surjective. Suppose that there is a Schur-Weyl duality between A and B on the bimodule M . That says, both ϕ and ψ are surjective. Then it is obvious that A has the double centralizer property with respect to M and B has the double centralizer property with respect to M .
Conversely, if we can show that the image of ϕ in End B (M ) is a quasi-hereditary algebra with a simple preserving duality, then the surjectivity of ϕ will follow from the surjectivity of ψ and applying Theorem 1.10. This is because in that case we have This is indeed the case as in many examples of Schur-Weyl dualities, where A often has a highest weight theory with M being a tilting module over A, and B is a diagrammatic algebra (symmetric or cellular). It is usually easier to handle the endomorphism algebra End A (M ) than to handle the endomorphism algebra End B (M ).
Example 3.12. Let G be a classical group over an algebraically closed field K with the natural module V . Following [39] and [28, §2.2], we define S r (G) := A r (G) * , where A r (G) is the coefficient space of V ⊗r (which is a coalgebra) in the coordinate algebra K[G]. The K-algebra S r (G) is isomorphic to the image of KG in End K (V ⊗r ) and hence acts faithfully on V ⊗r . When the set of dominant weights in V ⊗r is saturated in the sense of [27, A3], then S r (G) is a generalised Schur algebra. In particular it is quasi-hereditary. If furthermore V ⊗r is a faithful tilting module over S r (G) then we can apply Theorem 1.10.
In the type A case, let G = GL(V ), the general linear group on V , and n := dim V . The Schur-Weyl duality between KGL(V ) and the symmetric group algebra KS r on V ⊗r means that we have the following two surjective homomorphisms: In this case, the set of dominant weights in V ⊗r is Λ + (n, r) = {λ = λ 1 ε 1 + · · · + λ n ε n | n i=1 λ i = r, λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 0, λ i ∈ Z, ∀ i}, which is saturated, V ⊗r is a faithful tilting module over the image of ϕ.
In the type C case, let V be a 2m-dimensional symplectic space, G = Sp(V ), the symplectic group on V . The Schur-Weyl duality between KSp(V ) and the specialized Brauer B n,K (−2m) on V ⊗n means that we have the following two surjective homomorphisms: In this case, the set of dominant weights in V ⊗n is which is saturated, V ⊗n is a faithful tilting module over the image of ϕ.
In the type D case, we assume char K = 2 and m ≥ 2. Let V be a 2m-dimensional orthogonal space, G = SO(V ), the special orthogonal group on V . In this case, Donkin ([28, §2.5]) has shown that the set of dominant weights in V ⊗n is again saturated, and V ⊗n is a faithful tilting module over the image of KSO(V ) in End K (V ⊗n ). In particular, we see the image has the double centralizer property with respect to V ⊗n by Theorem 1.10. That is, we have the following natural surjective homomorphism: Note that the image of KO(V ) in End K (V ⊗n ) is not necessarily equal to the image of KSO(V ), and End KSO(V ) (V ⊗n ) in general does not coincide with End KO(V ) (V ⊗n ) in this case.
In the type B case, we assume char K = 2 and m ≥ 2. Let V be a 2m + 1-dimensional orthogonal space, G = SO(V ), the special orthogonal group on V . In this case, the set of dominant weights in V ⊗n is in general not saturated. Donkin ([28, §2.5]) has given a sufficient condition in [28, Page 108,(H)] under which the image of KSO(V ) in End K (V ⊗n ) is quasi-hereditary. However, in this case, O(V ) is generated by SO(V ) and an involution θ, and θ acts as − id on V ⊗n . So the image of KO(V ) in End K (V ⊗n ) coincides with the image of KSO(V ). Moreover, End KSO(V ) (V ⊗n ) = End KO(V ) (V ⊗n ). Thus by [31,Theorem 1.2] we can still get the following natural surjective homomorphism: Remark 3.13. Let g be a complex semisimple Lie algebra. Let U (g) be the universal enveloping algebra of g over Q. Let U Z (g) be the Kostant Z-form of U (g). For any field K, we define U K (g) := K⊗ Z U Z (g). The discussion in the above example should also work if we replace KG by U K (g), see [28, §3]. Furthermore, we remark that the argument of [28, §3] should also work if we replace U K (g) with the Lusztig's Z[v, v −1 ]form of the Drinfeld-Jimbo quantized enveloping algebra of g. In that case, the idea of using Theorem 1.10 should be able to simplify the lengthy argument in [44] and to provide a proof of the quantized integral Schur-Weyl dualities in the orthogonal cases as well. Details will be appeared elsewhere.
Let {T (λ)|λ ∈ Λ + } be a complete set of pairwise non-isomorphic indecomposable tilting modules over A. Recall that the tilting module ⊕ λ∈Λ + T (λ) is called the characteristic tilting module over A. By a well-known result of Ringel, we know that A has the double centraliser property with respect to the characteristic tilting module. Note that the characteristic tilting module is a basic tilting module in the sense of the following definition.
Definition 3.14. Let T ∈ A-mod be a tilting module. If T is a direct sum of some pairwise nonisomorphic indecomposable tilting modules, then we say that T is a basic tilting module. In general, if T = ⊕ λ∈Λ + T (λ) ⊕r λ , where r λ ∈ N for each λ, then we define In [57, Remark 2.5], Mazorchuk and Stroppel proposed a question about whether there exists a minimal basic tilting module with respect to which one has the double centraliser property. In the rest of this section we shall give the proof of Theorem 1.11, which affirmatively answers this question. By [57, Proposition 2.1], R maps tilting modules to projective modules, and maps projective modules to tilting modules and R defines an equivalence of subcategories R : F (∆ A ) ∼ = F (∆ R(A) ), where the ∆ A means the standard objects in A-mod and ∆ R(A) means the standard objects in R(A)-mod.
Let P be the projective cover of T in R(A)-mod. Then R −1 ( P ) is a tilting module in A-mod. We define T := R −1 ( P ) basic . Let r ′ ∈ N such that R −1 ( P ) is a direct summand of T ⊕r ′ . Let φ : P ⊕r ։ T be a surjective homomorphism in R(A)-mod. Since P ⊕r ′ , T ∈ F (∆ R(A) ), it follows that Ker φ ∈ F (∆ R(A) ).
is an exact sequence in F (∆ R(A) ). Applying the inverse of the Ringel dual functor R, we get that is an exact sequence in F (∆ A ). By definition, T := R −1 ( P ) basic . Thus there exist N ∈ A-mod and r ′ ≤ r ∈ N, such that We use ι 0 : A ֒→ T ⊕r to denote the composition of R −1 (π) with the natural injection (R −1 ( P )) ⊕r ′ ֒→ T ⊕r . Thus T ⊕r and hence T must be faithful tilting modules over A. Now applying Theorem 1.10 and Remark 1.6, we can deduce that A = End EndA(T ) (T ). Now suppose that T ′ is another faithful tilting module satisfying A = End End A (T ′ ) (T ′ ). Let s ∈ N such that A ֒→ (T ′ ) ⊕s . Applying Proposition 3.6, we can get an exact sequence in F (∆ A ): Applying the Ringel dual functor R, we get that is an exact sequence in F (∆ R(A) ). Note that R(T ′ ) is a projective module in R(A)-mod. The surjectivity of h in the above exact sequence implies that R(T ′ ) ⊕s must contains P as its direct summand. That says, R(T ′ ) ⊕s ∼ = P ⊕ P ′ . Applying the inverse of the Ringel dual functor R, we get that (T ′ ) ⊕s ∼ = R −1 ( P ) ⊕ R −1 ( P ′ ). By construction, T := R −1 ( P ) basic . This implies that T must be isomorphic to a direct summand of T ′ . This completes the proof of the theorem.
Corollary 3.15. Let A be a quasi-hereditary algebra with a simple preserving duality. Then there exists a unique minimal faithful basic tilting module T ∈ A-mod such that any other faithful tilting module T ′ ∈ A-mod must have T as a direct summand. Remark 3.16. In fact, the same argument can be used to show that Theorems 1.10 and 1.11 are true for properly stratified algebra A which has a simple preserving duality and that every tilting A-module is cotilting.

Brauer-Schur-Weyl duality for dual partially harmonic spaces
In this section, we shall apply the results in last Section to the study of Brauer-Schur-Weyl duality for dual partially harmonic spaces.
The notion of Brauer algebra was first introduced in [7] when Richard Brauer studied the decomposition of symplectic tensor spaces and orthogonal tensor spaces into direct sums of irreducible modules. Since then there have been a lot of study on the structure and representation of Brauer algebras, see [15,16,30,41,42,43,53,60,63,68] and references therein. In this section we only concern about these Brauer algebras with special parameters which play a role in the setting of Brauer-Schur-Weyl duality of type C. Let m, n ∈ Z ≥1 . The Brauer algebra B n,Z = B n (−2m) Z with parameter −2m over Z is a unital associative Z-algebra with generators s 1 , . . . , s n−1 , e 1 , . . . , e n−1 and relations (see [34]): It is well-known that B n,Z is a free Z-module of rank (2n − 1)!! = (2n − 1) · (2n − 3) · · · 3 · 1. For any field K, we define B n,K := K ⊗ Z B n,Z .
Alternatively, the Brauer algebra B n,K can be defined in a diagrammatic manner [7]. Recall that a Brauer n-diagram is a graph with 2n vertices arranged in two rows (each of n vertices) and n edges such that each vertex is incident to exactly one edge. Then B n,K can be defined as the K-linear space with basis the set Bd n of all the Brauer n-diagrams. The multiplication of two Brauer n-diagrams D 1 and D 2 is defined by the concatenation of D 1 and D 2 as follows: placing D 1 above D 2 , identifying the vertices in the bottom row of D 1 with the vertices in the top row of D 2 , removing the interior loops in the concatenation and obtaining the composite Brauer n-diagram D 1 • D 2 , writing n(D 1 , D 2 ) the number of interior loops, we then define the multiplication For a Brauer n-diagram, we label the vertices in the top row by 1, 2, . . . , n from left to right and the vertices in the bottom row by 1, 2, . . . , n also from left to right. The two definitions of Brauer algebra B n,K can be identified as follows: and e i = · · · · · · i i i + 1 i + 1 Let K be an infinite field. By [7,17,20], there is a Brauer-Schur-Weyl duality between the symplectic group Sp 2m (K) and the Brauer algebra B n,K on certain tensor space. To recall the result we need some more notations. Let V Z be a free Z-module of rank 2m. For each integer 1 ≤ i ≤ 2m, set i ′ := 2m + 1 − i.
For each integer 1 ≤ i ≤ 2m, we define There is a natural right action of B n,K on V ⊗n which is defined on generators by where for any i, j ∈ 1, 2, · · · , 2m , The above right action of B n,K on V ⊗n commutes with the natural left diagonal action of the symplectic group Sp(V ) ∼ = Sp 2m (K). Let k ∈ N. A partition of k is a non-increasing sequence of non-negative integers λ = (λ 1 , λ 2 , · · · ) which sum to k. We write λ ⊢ k. If λ ⊢ k then we set ℓ(λ) := max{t ≥ 1|λ t = 0}. The following results are often referred as Brauer-Schur-Weyl duality of type C.  [7,17,20]) Assume K is an algebraically closed field. The following two natural homomorphisms are both surjective: If m ≥ n then ϕ K is an isomorphism. Furthermore, if K = C, then there is a (CSp(V ), B n,C )-bimodule decomposition: where ∆(λ) and D(f, λ) denote the irreducible CSp(V )-module corresponding to λ and the irreducible B n,C -module corresponding to (f, λ) respectively. Definition 4.3. We call the endomorphism algebra S sy K (m, n) := End Bn,K (V ⊗n ) the symplectic Schur algebra.
By [25,26,58], we know that the symplectic Schur algebra is a quasi-hereditary algebra over K. Applying Theorem 1.8, we can get the following corollary. where r ∈ N, which can be continued to an exact sequence 0 → S sy K (m, n) δ → (V ⊗n ) ⊕r ε → (V ⊗n ) ⊕s for some s ∈ N. In particular, the V ⊗n -dominant dimension of S sy K (m, n) is at least two. Alternatively, the above corollary can also be deduced as a direct consequence of Theorem 1.9 where Stokke has proved in [65] that each Weyl module ∆(λ) can be embedded into V ⊗n .
There is another version of Brauer-Schur-Weyl duality for dual partially harmonic tensors which was investigated in [45]. Henceforth, we assume that K is an algebraically closed field unless otherwise stated. For each integer f with 0 ≤ f ≤ [n/2], let B (f ) n,K be the two-sided ideal of B n,K generated by e 1 e 3 · · · e 2f −1 . By convention, B Set This space is called (cf. [38], [52]) the space of partially harmonic tensors of valence f and plays an important role in the study of invariant theory of symplectic groups. It was proved in [45, 1.6] that there is a (KSp(V ), B n,K /B   Then the map ψ f,K is surjective.
One of our original starting point of this work is our attempt to the proof of the above Conjecture 4.7. First, we can make some reduction of the above conjecture. It is clear that  [20], the image of KSp(V ) in End K (V ⊗n ) is just S sy K (m, n) = End Bn,K V ⊗n . Let π f,K : End Bn,K (V ⊗n ) → End Bn,K V ⊗n /V ⊗n B (f ) n,K be the natural homomorphism. By construction, we have the following commutative diagram: n,K , where the top horizontal map and the left vertical map are both surjective, and the bottom horizontal map is injective. As a result, the map π f,K gives rise to a surjection (4.10) π f,K : S sy K (m, n) ։ S sy f,K (m, n). The advantage of working with S sy K (m, n) lies in that we can now allow K to be an arbitrary (not necessarily infinite) field or even an integral domain. We use ψ ′ f,K to denote the composition of π f,K with the natural inclusion S sy f,K (m, n) ֒→ End Bn, n,K ). It is easy to see that Conjecture 4.7 is a consequence of the following conjecture.
Let K be the algebraic closure of K. It is clear that ψ ′ f,K is surjective if and only if ψ ′ f,K is surjective. Suppose that char K = 0 or char K > n. Then by [36,Lemma 5.16], V ⊗n Therefore, it is easy to see that Conjecture 4.11 holds in this case. Next we shall show that Conjecture 4.11 also holds when char K > n − f + 2m.
Let U Z (sp 2m ) be the Kostant Z-form of the universal enveloping algebra of the symplectic Lie algebra sp 2m (C). For any field K, we define U K (sp 2m ) := K ⊗ Z U Z (sp 2m ). By the main result of [44], we have two surjective algebra homomorphisms: . As a result, S sy K (m, n) has the double centralizer property with respect to V ⊗n K . Now applying Theorem 1.8, we have an exact sequence of S sy K (m, n)-module homomorphisms: where r, s ∈ N, and δ K is a left add(V ⊗n K )-approximation of S sy K (m, n). Let 1 K be the unit element of S sy K (m, n). We can write n,K ). It follows that δ K (a) = au 1 ⊕ · · · ⊕ au r . Now π f,K (a) = 0 means that aV ⊗n This shows that δ K induces a well-defined homomorphism n,K ) ⊕r . Lemma 4.13. With the notations as above, the integers r, s, the map δ K and the elements u 1 , · · · , u r ∈ V ⊗n K can be chosen such that the map δ f,K is an injective left add(V ⊗n n,K )-approximation of S sy f,K (m, n). Proof. The integer r, the map δ K and u 1 , · · · , u r can be chosen such that u 1 , · · · , u r is a K-linear generator of V ⊗n K . In view of Remark 1.6, it suffices to show that δ f,K is injective. Suppose that (au 1 , · · · , au r ) = δ K (a) = δ f,K (a) = 0, where a ∈ S sy K (m, n). Since u 1 , · · · , u r is a K-linear generator of V ⊗n K , it follows that aV ⊗n n,K , which implies that π f,K (a) = 0 and hence ι f,K (a) = π f,K (a) = 0. This proves a = 0 as required.

It follows that
which is a contradiction. This proves our claim which is the following lemma 2 . Recall that V ⊗n K /V ⊗n K B (f ) n,K * ֒→ V ⊗n K * ∼ = V ⊗n K . Henceforth, we shall use this embedding to identify V ⊗n K /V ⊗n K B (f ) n,K * as a K-subspace of V ⊗n K . Since the isomorphism V ⊗n K * ∼ = V ⊗n K is a right B n,K -module isomorphism, it follows that for any x ∈ V ⊗n K , x ∈ V ⊗n K /V ⊗n K B  In particular, (f ) n,K * = dim V ⊗n K , it is clear that the second part of the lemma follows from the first part of the lemma.
Since B (f ) n,K . As a consequence, we see that ε K induces a S sy K (m, n)-module homomorphism ε f,K : n,K ) ⊕s . Now we can give the proof of the fourth main result of this paper.
Proof of Theorem 1.12: By Lemma 4.13, we know that δ f,K is injective. It follows from ε K • δ K = 0 that ε f,K • δ f,K = 0. We claim that Ker ε f,K = Image(δ f,K ). t suffices to show that Ker ε f,K ⊆ Image(δ f,K ).
It follows that On the other hand, since V ⊗n is filtered by some ∇(µ) with µ ∈ Λ + f . It follows from that n,K = 0. Hence by Lemma 4.17 we must have that ε K (y) = 0. Applying Corollary 4.4 and (4.12), we can deduce that y = δ K (a) for some a ∈ S sy K (m, n). It follows that x + V ⊗n K B