A Permutation Module Deligne Category and Stable Patterns of Kronecker Coefficients

2019/09/09 by Ryba, Christopher
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1909.04100

Abstract

Deligne's category \underline\rm 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 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 \underline\rm Rep(S_t). 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 \underline\rm Rep(S_t) → S_n-mod. We show that C_λ can be presented using the Kostant integral form of Lusztig's universal enveloping algebra U(\mathfrakgl_∞), and exhibit a categorification of some stability properties of Kronecker coefficients.

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