Honeycombs for Hall Polynomials

Nonnegative Hall Polynomials

The number of subgroups of type n and cotype v in a finite abelian p-group of type A is a polynomial g$v(p) with integral coefficients. We prove g u v ( p ) has nonnegative coefficients for all partitions p and v if and only if no two parts of A differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certa...

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 ...

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...

Abacus-tournament models for Hall-Littlewood polynomials

Symmetric Functions and Hall Polynomials