A Permutation Module Deligne Category and Stable Patterns of Kronecker Coefficients

Abstract

Deligne's category $\underline{\mathrm{Rep}}(S_t)$ is a tensor category depending on a parameter $t$ "interpolating" the categories of representations of the symmetric groups $S_n$. We construct a family of categories ${\mathcal{C}}_{\lambda }$ (depending on a vector of variables $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_l)$, that may be specialised to values in the ground ring) which are module categories over $\underline{\mathrm{Rep}}(S_t)$. The categories ${\mathcal{C}}_{\lambda }$ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to $S_n$-mod which are tensor-compatible with the functors $\underline{\mathrm{Rep}}(S_t)\to S_n$-mod. We show that ${\mathcal{C}}_{\lambda }$ can be presented using the Kostant integral form of Lusztig's universal enveloping algebra $\dot{U}(\mathfrak{g}{\mathfrak{l}}_{\infty })$, and exhibit a categorification of some stability properties of Kronecker coefficients.

30 pages