Double Centralizer Properties, Dominant Dimension, and Tilting Modules

Abstract

Double centralizer properties play a central role in many parts of algebraic Lie theory. Soergel's double centralizer theorem relates the principal block of the Bernstein ᎐Gelfand᎐Gelfand category O O of a semisimple complex Lie algebra with Ž the coinvariant algebra i.e., the cohomology algebra of the corresponding flag . manifold . Schur᎐Weyl duality relates the representation theories of general linear Ž . and symmetric groups in defining characteristic, or via the quantized version in nondefining characteristic. In this paper we exhibit algebraic structures behind these double centralizer properties. We show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective᎐injective or tilting modules. General arguments which combine methods Ž . from ring theory QF-3 rings and dominant dimension with tools from representa-Ž . tion theory approximations, tilting modules then imply the validity of these double centralizer properties as well as new ones. In contrast to the traditional Ž . proofs e.g., by the fundamental theorems of invariant theory no computations are necessary.

Schur–Weyl duality for higher levels

“…By Theorem 4.8 and Corollaries 4.9 and 4.6, P c ⊗ V ⊗d is a direct sum of all the self-dual projective indecomposable modules in O d c (λ), each appearing with some non-zero multiplicity. Applying [KSX,Theorem 2.10] as in the proof of [KSX,Theorem 3.2], the proof of the first statement reduces to checking that there exists an exact sequence 0 → P 0 → P 1 → P 2 in O d c (λ) such that P 0 is a projective generator and P 1 , P 2 are self-dual projectives. To see this, there is for…”

“…Theorem B should be compared with Soergel's Endomorphismensatz from [S]; the connection with Whittaker modules in that case is due to Backelin [Ba]. As observed originally by Stroppel [S1, Theorem 10.1] (as an application of [KSX,Theorem 2.10]), there is also a version of Soergel's Struktursatz for our parabolic setup: the functor Hom g (P ⊗ V ⊗d , ?) is fully faithful on projective objects.…”

Schur functors and dominant dimension

“…Then by classical results of Morita, Tachikawa and others (see [30]) there is a double centralizer property between A and its centralizer subalgebra eAe: End A (Ae) = eAe and End eAe (Ae) = A. In [19] it has been shown that classical Schur-Weyl duality between the Schur algebra S(n, r) (for n ≥ r) and the group algebra kΣ r of the symmetric group as well as Soergel's 'Struktursatz' for the Bernstein-Gelfand-Gelfand category O, providing a double centralizer property between a block and a subalgebra of the coinvariant algebra, both are special cases of this situation. In particular, those Schur algebras and also blocks of O have dominant dimension at least two.…”

“…As examples, (quantized) Schur algebras S q (n, r) with n ≥ r and block algebras of Bernstein-Gelfand-Gelfand category O are in A ; see [13,15,19]. Indeed, each such algebra has a set of idempotents fixed by the involution defining the duality and generating the ideals in a heredity chain.…”

“…This means that A has an injective coresolution 0 → A → I 0 → I 1 → I 2 → · · · such that I 0 and I 1 are also projective. In [19] it has been shown that this point of view is also feasible in a practical way; indeed, Schur-Weyl duality and Soergel's result have been reproved there in a brief and general way by checking that the dominant dimension of the respective algebra A is at least two.…”