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Reciprocity of Skew Hall-Littlewood-Schubert Series

Ron M. Adin, Tomer Bauer

Abstract

Carnevale, Schein and Voll proved self-reciprocity of the generalized Igusa functions, and Maglione and Voll did the same for the Hall-Littlewood-Schubert series. We introduce a simultaneous generalization and refinement of these two rational functions, and prove that it satisfies a self-reciprocity property. This answers a problem posed by Maglione and Voll. Our method of proof is elementary, avoiding the use of $p$-adic integration.

Reciprocity of Skew Hall-Littlewood-Schubert Series

Abstract

Carnevale, Schein and Voll proved self-reciprocity of the generalized Igusa functions, and Maglione and Voll did the same for the Hall-Littlewood-Schubert series. We introduce a simultaneous generalization and refinement of these two rational functions, and prove that it satisfies a self-reciprocity property. This answers a problem posed by Maglione and Voll. Our method of proof is elementary, avoiding the use of -adic integration.

Paper Structure

This paper contains 17 sections, 10 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Y-analogs
  4. Posets and chains
  5. Partitions, diagrams and tableaux
  6. Main definitions and their tableau interpretations
  7. Main definition
  8. Two alternative formulations
  9. Tableau interpretation
  10. Specializations
  11. The g = 1 case
  12. The general g case
  13. Proof of reciprocity
  14. Zeta and Möbius matrices
  15. Zeta and Möbius matrices over P_n,r
  16. ...and 2 more sections

Key Result

Theorem 1.4

For any ${\underline{n}}, {\underline{r}} \in {\mathbb N}_0^g$, the skew Hall--Littlewood--Schubert series satisfies where $N=\sum_{i=1}^{g}(n_{i} + r_{i})$ and

Figures (1)

  • Figure 1: Hasse diagrams of $P_{2,2}$. On the left the elements are $3$-tuples, and on the right the elements are the corresponding sub-multisets of $E_{2,2}$.

Theorems & Definitions (44)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Theorem 1.4: Skew ${\operatorname{HLS}}$ reciprocity
  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Example 3.5
  • Definition 3.6
  • ...and 34 more