Coulomb branch Hilbert series and Hall-Littlewood polynomials

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...

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

Combinatorial Formula for Modified Hall-Littlewood Polynomials

We obtain new combinatorial formulae for modified Hall–Littlewood polynomials, for matrix elements of the transition matrix between the elementary symmetric polynomials and Hall-Littlewood’s ones, and for the number of rational points over the finite field of unipotent partial flag variety. The definitions and examples of generalized mahonian statistic on the set of transport matrices and dual ...