The Saxl Conjecture and the Dominance Order

2014/10/24 by Christian Ikenmeyer, Ikenmeyer, Christian · 2 citations
Mathematics · #20C30 #20G05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1410.6549

openalex publication_date 2014/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 2012 Jan Saxl conjectured that all irreducible representations of the symmetric group occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. We make progress on this conjecture by proving the occurrence of all those irreducibles which correspond to partitions that are comparable to the staircase partition in the dominance order. Moreover, we use our result to show the occurrence of all irreducibles corresponding to hook partitions. This generalizes results by Pak, Panova, and Vallejo from 2013.

Citations

• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
• On the complexity of computing Kronecker coefficients

Cited by

• The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture
• Introduction to finite Coxeter groups and their representations

Related

• Staircase Minimality and a Proof of Saxl's Conjecture
• A Tensor-Cube Version of the Saxl Conjecture
• The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture
• Saxl Conjecture for triple hooks
• Kronecker products, characters, partitions, and the tensor square conjectures