The stability of the Kronecker products of Schur functions

2009/07/31 by Emmanuel Briand, Rosa Orellana, Mercedes Rosas · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics #Computer science #Geometry #Hadamard product #Kronecker delta #Kronecker product #Mathematical analysis #Mathematics #Physics #Product (mathematics) #Pure mathematics #Schur complement #Schur decomposition #Schur product theorem #Schur's theorem #Stability (learning theory) #math.CO #math.RT #msc:05E05 #msc:20C30

paper · pdf · doi:10.1016/j.jalgebra.2010.12.026

16 pages

arxiv created 2009/08/06 · openalex publication_date 2011/01/22 · crossref created 2011/01/22 · crossref issued 2011/04/01 · crossref published 2011/04/01 · crossref published-print 2011/04/01 · arxiv updated 2020/04/14 · crossref deposited 2024/04/03 · openalex created_date 2025/10/10 · crossref indexed 2026/03/02 · openalex updated_date 2026/08/05

Abstract

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.

Citations

• Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
• Quantum marginal problem and representations of the symmetric group
• Geometric Complexity III: on deciding positivity of Littlewood-Richardson coefficients
• Nonzero Kronecker Coefficients and What They Tell us about Spectra

Cited by

• On the asymptotics of Kronecker coefficients, 2
• Representation theory of the symmetric group [wikipedia]

Related

• Stability of coefficients in the Kronecker product of a hook and a rectangle
• Stability of the Heisenberg Product on Symmetric Functions
• The Kronecker product of Schur functions indexed by two-row shapes or hook shapes
• Kronecker coefficients and noncommutative super Schur functions
• The Newton polytope of the Kronecker product