Deligne categories and reduced Kronecker coefficients

The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups; namely, given three partitions $\lambda, \mu, \tau$ of $n$, the multiplicity of $\lambda$ in $\mu \otimes \tau$ is called the Kronecker coefficient $g^{\lambda}_{\mu, \tau}$. 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 the Deligne categories $\underline{Rep}(S_t)$. This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of the categories $\underline{Rep}(S_t)$.