Universal Gysin formulas for the universal Hall-Littlewood functions

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 universal analogue of the Hall-Littlewood polynomials, which we call the universal Hall-Littlewood functions, and give Gysin formulas for various flag bundles in the complex cobordism theory. Furthermore, we give two kinds of the universal analogue of the schur polynomials, and some Gysin formulas for these functions are established.

Generating functions for the universal factorial Hall-Littlewood $P$- and $Q$-functions

“…From this point of view, it is natural to ask a generalization of (1.1) and (1.2) to other complex-oriented cohomology theory, especially to the complex cobordism theory. That is one of the main motivation of the current work started from our previous paper [34]. In fact, we have introduced a "universal" analogue of the Hall-Littlewood P -polynomial denoted by H L λ (x n ; t) in [34,Definition 3.2].…”

“…That is one of the main motivation of the current work started from our previous paper [34]. In fact, we have introduced a "universal" analogue of the Hall-Littlewood P -polynomial denoted by H L λ (x n ; t) in [34,Definition 3.2]. We have named this function the universal Hall-Littlewood function, and established a formula which is a direct generalization of (1.1) to the complex cobordism theory ( [34,Corollary 4.11]).…”

“…In fact, we have introduced a "universal" analogue of the Hall-Littlewood P -polynomial denoted by H L λ (x n ; t) in [34,Definition 3.2]. We have named this function the universal Hall-Littlewood function, and established a formula which is a direct generalization of (1.1) to the complex cobordism theory ( [34,Corollary 4.11]). However, contrary to our expectation, the "t = 0 specialization" of H L λ (x n ; t) does not coincide with the universal Schur function s L λ (x n ) ([33, Definition 4.10]), which we thought a direct, universal analogue of the usual Schur polynomial s λ (x n ).…”

“…In our previous papers [33], [34], we introduced the universal factorial Schur functions s L λ (x n |b) and the new universal factorial Schur functions (or the universal factorial Schur functions of Damon type) S L λ (x n |b). In this section, motivated by the recent work due to Hudson-Matsumura [13], we shall introduce another "universal analogues" of the usual Schur functions, denoted by s L,KL λ (x n |b), which we call the universal factorial Schur functions of Kempf-Laksov type.…”

“…Consider the associated flag bundle τ r : F ℓ r,r−1,...,1 (E) −→ X. Then it follows immediately from the above definition (5.1) and a description of the Gysin homomorphism (τ r ) * as a certain symmetrizing operator (see Nakagawa-Naruse [34, Theorem 4.10]) that the following formula holds (see also [34,Corollary 4.12]):…”