Combinatorial Hopf algebra structure on packed square matrices

TL;DR

This paper introduces a new bigraded Hopf algebra based on square matrices with entries from a finite set, generalizing several known algebraic structures and revealing new properties and subalgebras.

Contribution

It constructs a novel Hopf algebra on packed square matrices, generalizing existing structures, and explores its algebraic properties, including freeness, self-duality, and bidendriform structure.

Findings

The Hopf algebra is free and self-dual.

It admits a bidendriform bialgebra structure.

A new subalgebra indexed by alternating sign matrices is identified.

Abstract

We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet $\{0, 1, ..., k\}$, $k \geq 1$, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of $k$-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics.