Kronecker Coefficients and Harrison Centers of Green's Ring $\mathcal{R}(S_6)$

Michael Sunne, Chi Zhang, Haoran Zhu

Published 2024-07-25Version 1

If the laws and explanations for the combinatorial aspects of the Kronecker coefficients can be discovered through a large number of computations, then the solution to a long-standing open problem can be provided. The aim of our work is to compute the Kronecker coefficients for the representation ring of the symmetric group $S_6$. Specifically, the power formulas of irreducible representations of the symmetric group $S_6$ are computed using the character theory of finite groups. In addition, by decomposing tensor products of irreducible representations of $S_6$, we characterise the representation ring $\mathcal{R}(S_6)$, such as generators, its unit group, primitive idempotents and Casimir number. We also give another way, using Harrison center theory, to study the representation ring. Finally, we leave some open problems for future consideration.