Kronecker products, characters, partitions, and the tensor square conjectures

Abstract

We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups S_n contain all irreducibles as their constituents. Our main result is that they contain representations corresponding to hooks and two row Young diagrams. For that, we develop a new sufficient condition for the positivity of Kronecker coefficients in terms of characters, and use combinatorics of rim hook tableaux combined with known results on unimodality of certain partition functions. We also present connections and speculations on random characters of S_n.

23 pages, 5 figures

References in corpus (6)

• Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH
• Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry
• Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type
• Kronecker coefficients for one hook shape
• The partition algebra and the Kronecker coefficients
• Kronecker multiplicities in the $(k,\ell )$ hook are polynomially bounded

Cited by in corpus (6)

• Strict unimodality of q-binomial coefficients
• On the complexity of computing Kronecker coefficients
• A diagrammatic approach to Kronecker squares
• Bounds on the Kronecker coefficients
• Unimodality via Kronecker products
• The Saxl Conjecture and the Dominance Order