A categorification of the Malvenuto--Reutenauer algebra via a tower of groups

TL;DR

This paper constructs a categorification of the Malvenuto--Reutenauer Hopf algebra using a tower of p-groups, linking its basis to supercharacter theory, and providing new insights into its algebraic structure.

Contribution

It introduces a novel categorification of the Malvenuto--Reutenauer algebra via supercharacter theory and a tower of p-groups, connecting combinatorial bases to representation theory.

Findings

Realizes the Hopf structure through functors on p-group representations

Establishes correspondence between fundamental basis and supercharacter basis

Provides a new framework for understanding the algebra's positivity and bases

Abstract

There is a long tradition of categorifying combinatorial Hopf algebras by the modules of a tower of algebras (or even better via the representation theory of a tower of groups). From the point of view of combinatorics, such a categorification supplies canonical bases, inner products, and a natural avenue to prove positivity results. Recent ideas in supercharacter theory have made fashioning the representation theory of a tower of groups into a Hopf structure more tractable. This paper applies such a program to the Malvenuto--Reutenauer Hopf algebra. In particular, we design functors on the representation theory of a tower of $p$-groups that realize the Hopf structure of the Malvenuto--Reutenauer algebra in such a way that its well-known fundamental basis corresponds to a supercharacter basis.