In their famous book, James and Kerber stated in 1981: "The inner tensor product of two ordinary irreducible representations of a symmetric group is in general reducible, so the question arises how to evaluate its decomposition into irreducible constituents." This is equivalent to a central problem of algebraic combinatorics, namely to decompose the Kronecker product of Schur s-functions, which are a fundamental basis of the algebra of symmetric functions.
In a more intricate vein, such problems for representations of the double covers of the symmetric group are linked to Schur P- and Q-functions.
The search for an algorithmic description of the Kronecker coefficients was initiated already about 80 years ago, but positive combinatorial formulae are known so far only in very special cases. In recent years this problem has gained further attention due to connections with quantum information theory and geometric complexity theory.
In the talk, we will focus on some classification problems for Kronecker products and related problems for skew characters and skew symmetric functions. In particular, the questions when the decomposition in the natural basis of the corresponding context is homogeneous or multiplicity-free will be considered.