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 Sn. We construct a family of categories Cλ (depending on a vector of variables λ = (λ1, λ2, … , λl), that may be specialised to values in the ground ring) which are module categories over \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right). \) The categories Cλ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to Sn-mod which are tensor-compatible with the functors \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right)\to {S}_n-\operatorname{mod}. \) We show that Cλ 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.