The irreducible representations of the Symmetric group are a classical subject, seemingly well understood. Yet, the multiplicities in the irreducible decomposition of the tensor product of two irreducibles, the Kronecker coefficients, present an 80+ long standing open problem. The problem has seen its revival in the context of Geometric Complexity Theory for the separation of complexity classes, the algebraic analogues of P vs NP.
In this talk we will review some recent results on the asymptotics and complexity of Kronecker coefficients and the underlying characters of the Symmetric group. Based on a series of joint works with Christian Ikenmeyer and Igor Pak.