A PERMUTATION MODULE DELIGNE CATEGORY AND STABLE PATTERNS OF KRONECKER COEFFICIENTS

Deligne’s category $$ \underset{\_}{\mathrm{Rep}}\left({S}_t\right) is a tensor depending on parameter t “interpolating” the categories of representations symmetric groups Sn. We construct family Cλ (depending vector variables λ = (λ1, λ2, … , λl), that may be specialised to values in ground ring) which are module over \underset{\_}{\mathrm{Rep}}\left({S}_t\right). The defined any ring and constructed by interpolating permutation representations. Further, they admit specialisation functors Sn-mod tensor-compatible with \underset{\_}{\mathrm{Rep}}\left({S}_t\right)\to {S}_n-\operatorname{mod}. show can presented using Kostant integral form Lusztig’s universal enveloping algebra \dot{U}\left({\mathfrak{gl}}_{\infty}\right), exhibit categorification some stability properties Kronecker coefficients.