PUBLIC VIVA-VOCE NOTIFICATION
11.50 am, Seminar Hall
On internal tensor product of modules over Schur algebra and analogous new centralizer algebras
Chennai Mathematical Institute.
The classical Schur-Weyl duality induces a functor, namely the Schur functor, from the category of modules over Schur algebra to the category of modules over the group algebra of the symmetric group. For modules over Schur algebra, H. Krause defined a new internal tensor product using the language of strict polynomial functors. We show that this internal tensor product corresponds to the Kronecker product of modules over the group algebra of symmetric group via the Schur functor. This result is true even at the level of unbounded derived categories. We calculate the internal tensor product in several interesting cases involving classical functors (e.g. divided power functor, symmetric power functor and exterior power functor) and the Weyl functors. This is a joint work with U. Kulkarni and K.V. Subrahmanyam.
We observe that the classical double centralizer property remains true when we replace the role of symmetric group by any subgroup of it and as a result similar to Schur algebra new centralizer algebra arises. Given any subgroup of a symmetric group, we associate certain functors which are a natural generalization of strict polynomial functors. We also obtain that there exists an internal tensor product of modules over these new centralizer algebras. We show that Schur algebra and the new centralizer algebra are new explicit examples of sesquialgebras with pre-antipode in the sense of Tang, Weinstein and Zhu.