Tensor and wreath products of symmetric groups

The Kronecker and plethysm coefficients describe the decompositions of products of symmetric functions into their simple constituents. They are as ubiquitous across mathematics as the notion of symmetry itself. The Kronecker coefficients have been described as "perhaps the most challenging, deep and mysterious objects in algebraic combinatorics''.

They play an important role in the theory of symmetric functions and in the representation theory of general linear and symmetric groups. Despite 80-years of study, "frustratingly little is known'' about these coefficients.

I hope to understand the whole blueprint for these Kronecker and plethysm coefficients by first considering what they looks like "generically" or "up to a finite instability". I have recently pioneered a new approach to understanding the (stable) blueprints of these coefficients in the context of the partition algebra and hence completely described one half of the stable blueprint for Kronecker coefficients and inductively described the stable blueprint for plethysm coefficients.

Building on this success, this proposal seeks to completely understand the stable Kronecker and plethysm coefficients and to solve old and new conjectures concerning the positivity properties of non-stable Kronecker and plethysm coefficients.