Representation theory of 0-Hecke-Clifford algebras

TL;DR

This paper explores the representation theory of 0-Hecke-Clifford algebras, revealing their rich combinatorial structure and establishing connections with peak quasisymmetric functions and symmetric functions.

Contribution

It advances understanding of the projective supermodules of 0-Hecke-Clifford algebras and links them to peak algebra and Schur's Q-functions.

Findings

Grothendieck ring is isomorphic to peak quasisymmetric functions

Restriction rules for induced projective supermodules are established

Peak quasisymmetric functions form a free module over symmetric functions

Abstract

The representation theory of 0-Hecke-Clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning. Bergeron et al. have proved that the Grothendieck ring of the category of finitely generated supermodules of 0-Hecke-Clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by Stembridge. In this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups. In particular, two kinds of restriction rules for induced projective supermodules are obtained. After that, we consider the corresponding Heisenberg double and its Fock representation to prove that the ring of peak quasisymmetric functions is free over the subring of symmetric functions spanned by Schur's Q-functions.