Diagram categories and reduced Kronecker coefficients

Abstract: Partition algebras are a class of diagram algebras which naturally fit into a tower and the so called partition category provides a unified framework for the study of the algebras in the tower. The path algebra of the partition category admits a triangular decomposition similar to a triangular decomposition of the universal enveloping algebra of a finite dimensional complex semisimple Lie algebra. In such a decomposition, the direct sum of symmetric group algebras plays a role analogous to Cartan subalgebra and this provides a natural approach to the representation theory of the partition category. The tensor structure on the partition category induces a ring structure on the associated Grothendieck group. Reduced Kronecker coefficients for symmetric groups appear as structure constants in the Grothendieck ring.

In this talk, we discuss the partition category and its connection to reduced Kronecker coefficients (these are results of several authors). We introduce the multiparameter colored partition category where the Cartan subalgebra in the corresponding triangular decomposition is given by complex reflection groups of type $G(r,1,n)$. The multiparameter colored partition category also admits a tensor structure. If time permits, we also relate the associated Grothendieck ring for this category with the ring of symmetric functions. This talk is based on joint work with Volodymyr Mazorchuk.

combinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

( video )

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ARCSIN - Algebra, Representations, Combinatorics and Symmetric functions in INdia

Series comments: Timings may vary depending on the time zone of the speakers.

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