Multiparameter colored partition category and the product of the reduced Kronecker coefficients

TL;DR

This paper introduces a multiparameter colored partition category extending the partition category, providing new algebraic structures, formulas for reduced Kronecker coefficients, and analogues of the Robinson--Schensted correspondence.

Contribution

It constructs a new multiparameter colored partition category and derives explicit formulas for reduced Kronecker coefficients using this framework.

Findings

Category $ ext{CPar}( extbf{x})$ is generically semisimple.

Provides a closed formula for reduced Kronecker coefficients.

Classifies Green's relations via colored Robinson--Schensted correspondences.

Abstract

We introduce and study a multiparameter colored partition category $\mathcal{CPar}(\textbf{x})$ by extending the construction of the partition category, over an algebraically closed field $\Bbbk$ of characteristic zero and for a multiparameter $\textbf{x}\in \Bbbk^{r}$. The morphism spaces in $\mathcal{CPar}(\textbf{x})$ have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group $C_r$. We show that the endomorphism spaces of $\mathcal{CPar}(\textbf{x})$ and additive Karoubi envelope of $\mathcal{CPar}(\textbf{x})$ are generically semisimple. The category $\mathcal{CPar}(\textbf{x})$ is rigid symmetric strict monoidal and we give a presentation of $\mathcal{CPar}(\textbf{x})$ as a monoidal category. The path algebra of $\mathcal{CPar}(\textbf{x})$ admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups $G(r,n)$. We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of $\mathcal{CPar}(\textbf{x})$ in two ways. Among other things, this gives a closed formula for the product of the reduced Kronecker coefficients in terms of the Littlewood--Richardson coefficients for $G(r,n)$ and certain Kronecker coefficients for the wreath product $(C_r \times C_r)\wr S_n$. For $r=1$, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson--Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.