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A geometric and generating function approach to plethysm

Álvaro Gutiérrez, Rosa Orellana, Franco Saliola, Anne Schilling, Mike Zabrocki

Abstract

Plethysm coefficients $\mathsf{a}_{μ[ν]}^λ$ are the structure coefficients of the plethysm of Schur functions $s_μ[s_ν] = \sum_λ \mathsf{a}_{μ[ν]}^λs_λ$. We study a bivariate generating function of plethysm coefficients when $λ$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients.

A geometric and generating function approach to plethysm

Abstract

Plethysm coefficients are the structure coefficients of the plethysm of Schur functions . We study a bivariate generating function of plethysm coefficients when has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is we give an explicit geometric algorithm to compute it using -Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the -plethysm coefficients.

Paper Structure

This paper contains 19 sections, 34 theorems, 160 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Geometric picture
  3. Preliminaries
  4. $\textup{SL}_2$-plethysm coefficients
  5. The integer-point enumerator and Schur polynomials
  6. Quantum Ehrhart polynomials
  7. MacMahon's combinatory analysis
  8. Combinatorial interpretation for plethysm coefficients
  9. Generating functions, cones, and partitions
  10. Recurrences for plethysm coefficients
  11. An explicit expression for $A_\mu(z,q)$
  12. A conjecture for the denominator
  13. A theorem: the denominator divides the expression
  14. Reciprocity
  15. A conjecture of Mulmuley
  16. ...and 4 more sections

Key Result

Lemma 2.1

The integer-point enumerator of $\mathsf{Ch}^{\textup{fine}}(\gamma)$ at height $h$ is the monomial quasisymmetric polynomial $M_{\alpha^\gamma}(x_0,\ldots,x_h)$.

Figures (4)

  • Figure 1: Let $w=3$. On the left, the integral points of the cube $9\square$ divided into its six chambers. On the right, a generic slice by a plane perpendicular to $(1,1,1)$, overlaid with the braid hyperplane arrangement.
  • Figure 2: On the left, the $\mathsf{Ch}^{\textup{fine}}(-)$ faces of $\square$ for $w=3$; in the middle, the $\mathsf{Ch}(-)$ chambers; on the right, the $\mathsf{Ch}^{\textup{coar}}(-)$ subdivision.
  • Figure 3: A generic slice of $[0,1]^4$ projected to a $3$-dimensional sphere. Illustrated is the front part of the sphere, containing $12$ chambers, indexed by permutations of $S_4$. Each color represents a different coarse chamber $\mathsf{Ch}^{\textup{coar}}(-)$.
  • Figure 4: Let $\tilde{P} = [-1/2,1/2]^3$. The $\mathsf{PT}^{q}$ operator applied to $\mathsf{qEhr}^u_{\tilde{P}}(z,q)$ with $u = (1,\ldots,1)$ is the $q$-Ehrhart series of the displayed sub-polytope of $\tilde{P}$, with respect to the grading $u$.

Theorems & Definitions (67)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 57 more