A lift of Schur's Q-functions to the peak algebra

Abstract

Let B n denote the centralizer of a fixed-point free involution in the symmetric group S 2n. Each of the four one-dimensional representations of B n induces a multiplicity-free representation of S 2n , and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of S n. Schur's Q-functions are a family of symmetric polynomials Q L (x 1 , x 2 ,...) indexed by partitions A with distinct parts. They were originally defined in Schur's 1911 paper [16] as the Pfaffians of certain skew-symmetric matrices. The main point of Schur's paper was to prove that the Q-functions "encode" the characters of the irreducible projective representations of symmetric groups, in the same sense that Schur's S-functions encode the ordinary irreducible characters of symmetric groups. In the past 10 years, there have been a number of developments showing that Schur's Q-functions arise naturally in several seemingly unrelated areas, just as Schur's S-functions arise as the answer to a number of natural algebraic and geometric questions. In particular, (1) Sergeev [17] has proved that the Q-functions Q L (x 1 ,...,x m) are (aside from scalar factors) the characters of the irreducible tensor representations of a certain Lie superalgebra Q(m); (2) Pragacz [15] has proved that the cohomology ring of the isotropic Grassmanian Sp 2n /U n is a homomorphic image of the ring generated by Q-functions, and furthermore, this homomorphism maps Q-functions to Schubert cycles; and (3) You [22] has proved that the Q-functions are the polynomial solutions of the BKP hierarchy *Partially supported by NSF Grants DMS-8807279 and DMS-9057192.

Representation theory of 0-Hecke–Clifford algebras

“…Recently, Berg et al in [1] construct a noncommutative lift of Schur functions, called the immaculate basis, and then in [2] find the correspondence of its dual basis to the category of finitely generated modules of 0-Hecke algebras under the Frobenius isomorphism defined in [17]. Inspired by their work, we construct a lift of Schur's Q-functions to the peak algebra and thus extract a new basis from it in [15]. Dually we define a new basis, called the quasisymmetric Schur's Q-functions, in the Stembridge algebra, whose expansion in the peak functions is expected to be positive based on concrete examples [15, Conjecture 4.15].…”

“…Inspired by their work, we construct a lift of Schur's Q-functions to the peak algebra and thus extract a new basis from it in [15]. Dually we define a new basis, called the quasisymmetric Schur's Q-functions, in the Stembridge algebra, whose expansion in the peak functions is expected to be positive based on concrete examples [15, Conjecture 4.15]. It implies the chance for a representation theoretical meaning of such basis on the supermodule category of 0-Hecke-Clifford algebras.…”

The shifted Poirier–Reutenauer algebra

“…That is, the shifted analog is constructed inside the Malvenuto-Reutenauer algebra ZS of permutations. Recently Schur P-functions have been successively lifted onto the peak algebra Peak in [10]. Our current construction gives a higher lift of Schur P-functions onto the Poirier-Reutenauer algebra PR ′ as shown in diagram (4.25).…”

Schur functions in noncommuting variables

“…However, the question of the existence of a basis of NCSym with properties analogous to classical Schur functions, which maps to (scaled) Schur functions under the projection map remained open. This was despite the flourishing area of Schur-like functions pioneered by the quasisymmetric Schur functions of Haglund, Luoto, Mason and the third author [16], and followed by discoveries of row-strict quasisymmetric Schur functions [23], Young quasisymmetric Schur functions [21], noncommutative Schur functions [7] and immaculate functions [2,19], quasisymmetric Schur Q-functions [17], quasi-Grothendieck polynomials [24] and quasisymmetric Macdonald polynomials [9].…”