Recorded 08 January 2024. Victor Ostrik of the University of Oregon, Mathematics, presents "Growth in tensor powers" at IPAM's Symmetric Tensor Categories and Representation Theory Workshop. Abstract: This talk is based on joint work with K. Coulembier, P. Etingof, D. Tubbenhauer. Let ?? be any group and let V be a nite dimensional representation of ?? over arbitrary eld. We consider tensor powers V n of V and their decompositions into indecomposable summands. Let bn(V ) be the total number of indecomposable summands in V n. We prove that lim n!1 n p bn(V ) = dim(V ): Similarly let dn(V ) be the number of indecomposable summands of V n with dimension not divisible by the characteristic of the eld. Then we dene (V ) := lim n!1 n p dn(V ): The real number (V ) is an interesting invariant of the representation V . Using theory of tensor categories we show that this invariant is ad- ditive (under direct sums), multiplicative and takes values in algebraic integers. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/symmetric-tensor-categories-and-representation-theory/?tab=overview