Deligne categories and reduced Kronecker coefficients

Abstract

The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions $${\lambda }, \mu , \tau $$ź,μ,ź of n, the multiplicity of $$\lambda $$ź in $$\mu \otimes \tau $$μźź is called the Kronecker coefficient $$g^{{\lambda }}_{\mu , \tau }$$gμ,źź. When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood---Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories $$\underline{Rep}(S_t)$$Repź(St). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories $$\underline{Rep}(S_t)$$Repź(St) and derive new combinatorial identities.