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 of a semisimple complex Lie algebra with the coinvariant algebra (i.e., the cohomology algebra of the corresponding flag manifold). Schur–Weyl duality relates the representation theories of general linear 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 from ring theory (QF-3 rings and dominant dimension) with tools from representation theory (approximations, tilting modules) then imply the validity of these double centralizer properties as well as new ones. In contrast to the traditional proofs (e.g., by the fundamental theorems of invariant theory) no computations are necessary.