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Odd Shifted Parking Functions

Zachary Hamaker, Jesse Kim

TL;DR

This work resolves Stanley's open problem by giving a concrete $V$-basis expansion of the shifted parking function symmetric function $SH_n$ in the odd subalgebra $SymP$, via the introduction of odd shifted parking functions. It provides a combinatorial realization using garages and a representation-theoretic shiftification realized through exterior and Clifford algebras, yielding a spin-character interpretation and a $t$-graded extension. The main result expresses $SH_n$ as $SH_n= obreak \sum_{ ho ext{ odd}} ext{OKrew}( ho) obreak V_ ho$, connecting to the Kreweras numbers and the structure of the $V$-basis. The paper also links these constructions to Schröder-path combinatorics and Haglund's $(q,t)$-Schröder theory, suggesting new avenues in diagonal harmonics and related algebraic combinatorics.

Abstract

Stanley recently introduced the shifted parking function symmetric function $SH_n$, which is the shiftification of Haiman's parking function symmetric function $PF_n$. The function $SH_n$ lives in the subalgebra of symmetric functions generated by odd power sums. Stanley showed how to expand $SH_n$ into the $V-$basis of this algebra, which is indexed by partitions with all parts odd and is analogous to the complete homogeneous (or elementary) basis of symmetric functions. We introduce odd shifted parking functions to give combinatorial and representation-theoretic realizations of the $V-$expansion of $SH_n$, resolving the main open problem in his paper. Further, we present two representation-theoretic realizations of shiftification allowing us to interpret $SH_n$ as the spin character of a projective representation. We conclude with further directions, including a relationship between $SH_n$ and Haglund's $(q,t)-$Schröder theorem.

Odd Shifted Parking Functions

TL;DR

This work resolves Stanley's open problem by giving a concrete -basis expansion of the shifted parking function symmetric function in the odd subalgebra , via the introduction of odd shifted parking functions. It provides a combinatorial realization using garages and a representation-theoretic shiftification realized through exterior and Clifford algebras, yielding a spin-character interpretation and a -graded extension. The main result expresses as , connecting to the Kreweras numbers and the structure of the -basis. The paper also links these constructions to Schröder-path combinatorics and Haglund's -Schröder theory, suggesting new avenues in diagonal harmonics and related algebraic combinatorics.

Abstract

Stanley recently introduced the shifted parking function symmetric function , which is the shiftification of Haiman's parking function symmetric function . The function lives in the subalgebra of symmetric functions generated by odd power sums. Stanley showed how to expand into the basis of this algebra, which is indexed by partitions with all parts odd and is analogous to the complete homogeneous (or elementary) basis of symmetric functions. We introduce odd shifted parking functions to give combinatorial and representation-theoretic realizations of the expansion of , resolving the main open problem in his paper. Further, we present two representation-theoretic realizations of shiftification allowing us to interpret as the spin character of a projective representation. We conclude with further directions, including a relationship between and Haglund's Schröder theorem.
Paper Structure (11 sections, 23 theorems, 62 equations, 1 figure)

This paper contains 11 sections, 23 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Parking functions
  4. Symmetric Functions
  5. Frobenius Characters and Spin Characters
  6. Main results
  7. Garages and naive shifted parking functions
  8. Odd shifted parking functions
  9. Problems and Extensions
  10. Enumerative aspects
  11. Algebraic aspects

Key Result

Proposition 1.1

Let $M$ be an $\mathfrak{S}_n$--module with Frobenius character $f$. Then the Frobenius character of $M \otimes \bigwedge(\mathbb{C}^n)$ is $\mathtt{sh}(f)$.

Figures (1)

  • Figure 1: The lattice path $L$ associated to the garage in Example \ref{['ex:lattice-path']}. Steps are labelled by $\upsilon$ with $\overline{\sigma}$ in the superscripts. The matching $\tau(L)$ is shown in red.

Theorems & Definitions (45)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Theorem 2.2: stanley2024shifted
  • Proposition 2.3: stembridge
  • proof
  • proof : Proof of Proposition \ref{['p:shiftification']}
  • Corollary 2.5
  • Lemma 2.6
  • ...and 35 more