A PERMUTATION MODULE DELIGNE CATEGORY AND STABLE PATTERNS OF KRONECKER COEFFICIENTS

Abstract

Deligne’s category $\underset{\_}{\mathrm{Rep}}\left({S}_t\right)$ 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 C_λ (depending on a vector of variables λ = (λ₁, λ₂, … , λ_l), that may be specialised to values in the ground ring) which are module categories over $\underset{\_}{\mathrm{Rep}}\left({S}_t\right).$ The categories C_λ 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 $\underset{\_}{\mathrm{Rep}}\left({S}_t\right)\to {S}_n-\operatorname{mod}.$ We show that C_λ can be presented using the Kostant integral form of Lusztig’s universal enveloping algebra $\dot{U}\left({\mathfrak{gl}}_{\infty}\right),$ and exhibit a categorification of some stability properties of Kronecker coefficients.