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The universal factorial Hall-Littlewood $P$- and $Q$-functions

Masaki Nakagawa, Hiroshi Naruse

TL;DR

The paper develops factorial and universal analogues of Hall-Littlewood P- and Q-polynomials within the complex cobordism framework, defined via the universal formal group law and encoded by formulas that respect a vanishing property. Central to the work are Gysin-map characterizations on flag bundles, Pieri-type and hook-length-type results, and generating-function techniques that produce explicit coefficient extractions and Pfaffian formulas. These constructions enable stable, deformation-aware descriptions of symmetric functions in cobordism and their specializations, with applications to torus-equivariant and K-theoretic settings. The results lay groundwork for torus-equivariant cobordism of flag varieties associated with complex reflection groups and point toward further connections with dual Grothendieck polynomials and related Pfaffian structures.

Abstract

In this paper, we introduce {\it factorial} analogues of the ordinary Hall--Littlewood $P$- and $Q$-polynomials, which we call the {\it factorial Hall--Littlewood $P$- and $Q$-polynomials}. Using the {\it universal} formal group law, we further generalize these polynomials to the {\it universal factorial Hall--Littlewood $P$- and $Q$-functions}. We show that these functions satisfy the {\it vanishing property} which the ordinary factorial Schur $S$-, $P$-, and $Q$-polynomials have. By the vanishing property, we derive the Pieri-type formula and a certain generalization of the classical hook formula. We then characterize our functions in terms of Gysin maps from flag bundles in the complex cobordism theory. Using this characterization and Gysin formulas for flag bundles, we can obtain generating functions for the universal factorial Hall--Littlewood $P$- and $Q$-functions. Using our generating functions, we can show that our factorial Hall--Littlewood $P$- and $Q$-polynomials have a certain {\it cancellation property}. Further applications such as Pfaffian formulas for $K$-theoretic factorial $Q$-polynomials are also given.

The universal factorial Hall-Littlewood $P$- and $Q$-functions

TL;DR

The paper develops factorial and universal analogues of Hall-Littlewood P- and Q-polynomials within the complex cobordism framework, defined via the universal formal group law and encoded by formulas that respect a vanishing property. Central to the work are Gysin-map characterizations on flag bundles, Pieri-type and hook-length-type results, and generating-function techniques that produce explicit coefficient extractions and Pfaffian formulas. These constructions enable stable, deformation-aware descriptions of symmetric functions in cobordism and their specializations, with applications to torus-equivariant and K-theoretic settings. The results lay groundwork for torus-equivariant cobordism of flag varieties associated with complex reflection groups and point toward further connections with dual Grothendieck polynomials and related Pfaffian structures.

Abstract

In this paper, we introduce {\it factorial} analogues of the ordinary Hall--Littlewood - and -polynomials, which we call the {\it factorial Hall--Littlewood - and -polynomials}. Using the {\it universal} formal group law, we further generalize these polynomials to the {\it universal factorial Hall--Littlewood - and -functions}. We show that these functions satisfy the {\it vanishing property} which the ordinary factorial Schur -, -, and -polynomials have. By the vanishing property, we derive the Pieri-type formula and a certain generalization of the classical hook formula. We then characterize our functions in terms of Gysin maps from flag bundles in the complex cobordism theory. Using this characterization and Gysin formulas for flag bundles, we can obtain generating functions for the universal factorial Hall--Littlewood - and -functions. Using our generating functions, we can show that our factorial Hall--Littlewood - and -polynomials have a certain {\it cancellation property}. Further applications such as Pfaffian formulas for -theoretic factorial -polynomials are also given.

Paper Structure

This paper contains 21 sections, 15 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation, conventions, and preliminary results
  4. Lazard ring $\mathbb L$ and the universal formal group law $F_{\mathbb L}$
  5. Gysin formula for a projective bundle in complex cobordism
  6. Universal factorial Hall--Littlewood $P$- and $Q$-functions
  7. Universal factorial Hall--Littlewood $P$- and $Q$-functions
  8. Definition of the universal factorial Hall--Littlewood $P$- and $Q$-functions
  9. Factorial Hall--Littlewood $P$- and $Q$-polynomials
  10. Characterization of the universal factorial Hall--Littlewood $P$- and $Q$-functions
  11. Vanishing properties of factorial Hall--Littlewood $P$- and $Q$-polynomials
  12. Pieri-type formula and Hook formula
  13. Generating functions for the universal factorial Hall--Littlewood $P$- and $Q$-functions
  14. Generating function for $HP^{\mathbb L}_{\lambda}(\bm{x}_{n}; t|\bm{b})$
  15. Generating function for $HQ^{\mathbb L}_{\lambda}(\bm{x}_{n} ; t |\bm{b})$
  16. ...and 6 more sections

Key Result

Theorem 1.2

The universal factorial Hall--Littlewood $P$-function $HP^{\mathbb L}_{\lambda}(\bm{x}_{n}; t|\bm{b})$ is the coefficient of $\bm{u}^{-\lambda} = u_{1}^{-\lambda_{1}} u_{2}^{-\lambda_{2}} \cdots u_{r}^{-\lambda_{r}}$ in $\widetilde{\mathcal{HP}}^{\mathbb L, (n)}_{\lambda}(u_{1}, u_{2}, \ldots, u_{

Theorems & Definitions (32)

  • Definition 1.1: Definition \ref{['def:DefinitionUFH-LPQ']}, cf. Naruse Naruse2017
  • Theorem 1.2: Theorem \ref{['thm:GFUFH-LP']}
  • Proposition 2.1
  • Definition 3.1: Universal factorial Hall--Littlewood $P$- and $Q$-functions
  • Definition 3.2: Factorial Hall--Littlewood $P$- and $Q$-polynomials
  • Example 3.3
  • Remark 3.4
  • Proposition 3.5: Characterization of the universal factorial Hall--Littlewood $P$- and $Q$-functions
  • Remark 3.6
  • Proposition 3.7: Vanishing property
  • ...and 22 more