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Bijections between different combinatorial models for $q$-Whittaker and modified Hall-Littlewood polynomials

T V Ratheesh

Abstract

We consider the monomial expansion of the $q$-Whittaker and modified Hall-Littlewood polynomialsarising from specialization of the modified Macdonald polynomial. The two combinatorial formulas for the latter due to Haglund, Haiman, and Loehr and Ayyer, Mandelshtam and Martin give rise to two different parameterizing sets in each case. We produce bijections between the parameterizing sets which preserve the content and major index statistics. We identify the major index with the charge or cocharge of appropriate words and use descriptions of the latter due to Lascoux-Sch$\ddot{\text{u}}$tzenberger and Killpatrick to show that our bijections have the desired properties.

Bijections between different combinatorial models for $q$-Whittaker and modified Hall-Littlewood polynomials

Abstract

We consider the monomial expansion of the -Whittaker and modified Hall-Littlewood polynomialsarising from specialization of the modified Macdonald polynomial. The two combinatorial formulas for the latter due to Haglund, Haiman, and Loehr and Ayyer, Mandelshtam and Martin give rise to two different parameterizing sets in each case. We produce bijections between the parameterizing sets which preserve the content and major index statistics. We identify the major index with the charge or cocharge of appropriate words and use descriptions of the latter due to Lascoux-Schtzenberger and Killpatrick to show that our bijections have the desired properties.
Paper Structure (15 sections, 9 theorems, 34 equations)

This paper contains 15 sections, 9 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two descriptions of the charge and cocharge statistics
  4. Classical description of charge
  5. New description of charge
  6. Cocharge of a word
  7. Charge and cocharge words
  8. Bijection between the models for -Whittaker polynomials
  9. inv maximisers
  10. quinv maximisers
  11. The first main theorem
  12. The bijection in the modifed Hall-Littlewood case
  13. inv-zero fillings
  14. quinv-zero fillings
  15. The second main theorem

Key Result

Theorem 2.3

HHL01 Let $\lambda$ be a partition. The modified Macdonald polynomial can be written as

Theorems & Definitions (24)

  • Example 2.1
  • Example 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Conjecture 1
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8
  • Example 2.9
  • ...and 14 more