3.30 pm, Seminar Hall
Schur-Weyl duality and a new tensor product for representations of the general linear group
Joint work with Shraddha Srivastava and K.V. Subrahamanyam
Upendra Kulkarni
Chennai Mathematical Institute.
06-07-17

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Abstract

I will start by recalling Schur-Weyl duality, arising from commuting actions of the general linear group GL(V) and the symmetric group S_d on the d-fold tensor power of V. This setup intimately connects the representation theories of GL(V) and S_d. The bridge is the Schur functor, which produces a representation of S_d from a degree d representation of GL(V).

Recently a new tensor structure was introduced by H. Krause on polynomial representations of GL(V) of a FIXED degree d. I will discuss (i) how this new structure maps via the Schur functor to the usual tensor product of S_d-modules, (ii) how the new structure is a substantial enrichment of the old structure when working over an arbitrary commutative ring, in particular over a field of positive characteristic, and (iii) calculation of the new tensor product for some interesting classes of representations.

I will show how to define the new tensor structure in two ways: (i) using a variant of the notion of a bialgebra, and (ii) Krause's original approach using Day convolution of polynomial functors. The second approach is more transparent, but it requires some category theory. I will also mention existing/possible applications of the new structure to representation theory, Koszul duality and functor homology.