Representation theory of 0-Hecke–Clifford algebras

Abstract

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-HeckeClifford 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.

Quasisymmetric functions and Heisenberg doubles

“…Therefore, H + = Peak * is free over H + proj , where H + proj is the subring of symmetric functions spanned by Schur's Q-functions. This recovers Proposition 4.2.2 in [10]. In [8], the algebra HS n is defined to be the subalgebra of End(CS n ) generated by both sets of operators from C[S n ] and H n (0).…”

“…(Tower of 0-Hecke-Clifford algebras)Let A = ⊕ n∈N HCl n (0), where HCl n (0) is the 0-Hecke-Clifford algebra of degree n. Then H + = G(A) and H − = K(A) form a dual pair of Hopf algebras. There are two main ideas used in[10] (section 3.3) to prove that (G(A), K(A)) is a dual pair of Hopf algebras. First, the Mackey property of 0-Hecke-Clifford algebras guarantees that G(A) and K(A) are Hopf algebras.…”

Toward a polynomial basis of the algebra of peak quasisymmetric functions

“…The author would like to thank the anonymous referee who reviewed [16] for valuable comments and the partial support of NSFC (Grant No. 11501214) for this work.…”

“…There were many meaningful attempts to solve the Ditters conjecture, and the first rigorous proof was given by Hazewinkel in [10], where he first dealt with a p-adic version of the Ditters conjecture then completed the case over the integers. Later, another more direct proof appeared in [12] using the technique of lambda rings; see also [11,16.71], [13, §6]. In fact, Hazewinkel gave a nice structure theorem for QSym, which constructs a polymonial basis of QSym and implies that QSym is free over its subring Λ of symmetric functions.…”

“…In [19], Savage and Yacobi proved the freeness of QSym over its subring Λ of symmetric functions, alternatively using the technique of representation theory, namely, Heisenberg doubles arising from the tower of 0-Hecke algebras. Later, in [16] we applied such method to the case of PQSym by the tower of 0-Hecke-Clifford algebras, in order to prove the freeness of PQSym over its subring Γ spanned by Schur's Q-functions. Consequently, it is natural to ask for a more straightforward approach.…”