Universal Gysin formulas for the universal Hall-Littlewood functions

TL;DR

This paper introduces universal Hall-Littlewood functions and provides Gysin formulas for them in complex cobordism, extending classical results for Schur polynomials to a broader, universal setting.

Contribution

It develops a universal analogue of Hall-Littlewood polynomials and establishes Gysin formulas for these functions in complex cobordism theory.

Findings

Universal Hall-Littlewood functions are defined.

Gysin formulas are derived for these functions in complex cobordism.

Universal analogues of Schur polynomials are introduced and formulas established.

Abstract

It is known that the usual Schur $S$- and $P$-polynomials can be described via the Gysin homomorphisms for flag bundles in the ordinary cohomology theory. Recently, P. Pragacz generalized these Gysin formulas to the Hall-Littlewood polynomials. In this paper, we introduce a {\it universal} analogue of the Hall-Littlewood polynomials, which we call the {\it universal Hall-Littlewood functions}, and give Gysin formulas for various flag bundles in the complex cobordism theory. Furthermore, we give two kinds of the {\it universal} analogue of the schur polynomials, and some Gysin formulas for these functions are established.