Hall-Littlewood polynomials and a Hecke action on ordered set partitions

Non-symmetric Hall–littlewood Polynomials

Using the action of the Yang–Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall–Littlewood polynomials are a subfamily of one of them. For q = 0, these bases specialize to the two families of classical Key polynomials (i.e., Demazure characters for type A). We give a scalar product for which the two bases are adjoint to each other.

Noncommutative Symmetric Hall-Littlewood Polynomials

Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric funct...

A Gysin formula for Hall-Littlewood polynomials

Let E → X be a vector bundle of rank n over a nonsingular variety X over an algebraically closed field. Denote by π : G(E) → X the Grassmann bundle parametrizing rank q quotients of E. Let π∗ : A(G (E)) → A(X) be the homomorphism of the Chow groups of algebraic cycles modulo rational equivalence, induced by pushing-forward cycles (see [3, Chap. 1]). There exists an analogous map of cohomology g...

Hall-Littlewood polynomials and fixed point enumeration

We resolve affirmatively some conjectures of Reiner, Stanton, and White [12] regarding enumeration of various classes of matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms of the bicyclic sieving phenomenon introduced by Barcelo, Reiner, and Stanton [1]. The proofs of our results use various tools from symmetric function theory such as t...

Abacus-tournament models for Hall-Littlewood polynomials