Matroid-minor Hopf algebra: a cancellation-free antipode formula and other applications of sign-reversing involutions

Abstract

In this paper, we give a cancellation-free antipode formula for the matroid-minor Hopf algebra. We then explore applications of this formula. For example, the cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions. We also prove that all matroids over any hyperfield (in the sense of Baker and Bowler) have cancellation-free antipode formulas; furthermore, the cancellations in the antipode are independent of the hyperfield structure and only depend on the underlying matroid.

On the cancellation-free antipode formula for the Malvenuto-Reutenauer Hopf Algebra

“…However, this antipode formula usually contains a great number of cancellations in its alternating sum. Recently, significant attention has been dedicated to developing cancellationfree antipode formulas for various combinatorial Hopf algebras, including the Hopf algebra of graphs, matroids, simplicial complexes, set operads and posets, among others [3,4,5,6,8,9,11,14,15]. Notably, Benedetti and Sagan [7] employed the approach of sign-reversing involution to establish cancellation-free antipode formulas for nine different combinatorial Hopf algebras.…”

On the Hopf algebra of multi-complexes

“…In their groundbreaking work [AA17], Aguiar and Ardila provided an elegant unified way to find a cancellation-free and grouping-free antipode formula for various classes of combinatorial Hopf algebras by reducing the question to the case of generalized permutahedra (or polymatroids). In [BEJM18], Bucher, Eppolito, Jun, and Matherne also employed the idea of sign-reversing involution, which was introduced in [BS17] by Benedetti and Sagan, and provided a cancellation-free antipode formula for the matroid-minor Hopf algebra. This approach can be also used to provide a cancellation-free antipode formula for Hopf algebras, defined by Eppolito, Jun, and Szczesny in [EJS17], arising from matroids over hyperfields as in Baker-Bowler [BB19].…”