A geometric and generating function approach to plethysm

2025/11/04 by Gutiérrez, Álvaro, Orellana, Rosa, Saliola, Franco +2 · 2 citations
Mathematics · #05E05 (Secondary) #05E10 (Primary) 05A15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Graph theory and applications

paper · doi:10.48550/arxiv.2511.02649

openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

Abstract

Plethysm coefficients a_μ[ν]^λ are the structure coefficients of the plethysm of Schur functions s_μ[s_ν] = ∑_λ 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 SL₂-plethysm coefficients.

Citations

• A combinatorial interpretation for certain plethysm and Kronecker coefficients
• A Closer Look at Chapoton's q-Ehrhart Polynomials
• Weighted Ehrhart functions
• Harmonics and graded Ehrhart theory
• Computational Complexity in Algebraic Combinatorics
• Weighted quasisymmetric enumerator for generalized permutohedra
• Obstructions to combinatorial formulas for plethysm
• Hopf Algebras in Combinatorics
• Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry
• The Face Semigroup Algebra of a Hyperplane Arrangement
• A Fast Algorithm for MacMahon's Partition Analysis
• An Invitation to the Generalized Saturation Conjecture
• MacMahon's Partition Analysis VI: A New Reduction Algorithm

Cited by

• Ehrhart Theory over Abelian Group Rings
• A combinatorial interpretation for certain plethysm and Kronecker coefficients

Related

• The classification of multiplicity-free plethysms of Schur functions
• The first term of plethysms
• Stability Properties Of The Plethysm: A Combinatorial Approach
• Plethysms of symmetric functions and highest weight representations
• A combinatorial proof of a plethystic Murnaghan--Nakayama rule