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Extending the parking space

Andrew Berget, Brendon Rhoades

Abstract

The action of the symmetric group $S_n$ on the set $Park_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $Park_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.

Extending the parking space

Abstract

The action of the symmetric group on the set of parking functions of size has received a great deal of attention in algebraic combinatorics. We prove that the action of on extends to an action of . More precisely, we construct a graded -module such that the restriction of to is isomorphic to . We describe the -Frobenius characters of the module in all degrees and describe the -Frobenius characters of in extreme degrees. We give a bivariate generalization of our module whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.

Paper Structure

This paper contains 5 sections, 8 theorems, 32 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and Main Results
  3. Proofs
  4. Concluding Remarks
  5. Acknowledgements

Key Result

Theorem 2

Embed $S_n$ into $S_{n+1}$ by letting $S_n$ act on the first $n$ letters. We have that where the direct sum is over all Dyck paths of size $n$ and area $k$. In particular, by Equation eq:cosetdecomposition we have that

Figures (6)

  • Figure 1: A Dyck path of size $6$.
  • Figure 2: The four slim subgraphs of $K_3$.
  • Figure 3: The $5$ Dyck paths of size $3$. From left to right, their contributions to the graded Frobenius character $\mathrm{grFrob}(\mathrm{Res}^{S_4}_{S_3}(V_3); q)$ are $h_{(3)}q^0, h_{(2,1)} q^1, h_{(2,1)} q^2, h_{(2,1)}q^2,$ and $h_{(1,1,1)} q^3$.
  • Figure 4: A $(2,2)$-Dyck path of size $3$.
  • Figure 5: A Dyck path $D$ of size $5$ and the associated subgraph $G(D)$ of $K_6$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1
  • Theorem 2
  • Example 3
  • Example 4
  • Theorem 5
  • Definition 6
  • Theorem 7
  • Example 8
  • Lemma 9
  • proof
  • ...and 9 more