The Kronecker coefficients of the Symmetric group S_n count the multiplicities of one irreducible representation of S_n in the tensor product of two other irreducibles. It has been a 76-year long standing question to find a combinatorial interpretation for these nonnegative integers, in the sense of enumerating some discrete feasible objects. Recently, the Kronecker coefficients found a new role in the Geometric complexity Theory with the conjectures (Mulmuley) that deciding their positivity is in P, and computing them in NP.

In this talk we will elaborate on this topic, explain certain methods from algebraic combinatorics used to handle the problems and discuss some recent results on both the combinatorial/algebraic aspect and the issues on complexity.