Abstract
This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.
Highlights
This paper brings together, for the first time, the two oldest open problems in the representation theory of the symmetric groups and their quiver Hecke algebras
The first problem is to understand the structure of Specht modules and the second is to describe the decomposition of a tensor product of two Specht modules — the Kronecker problem
A new benchmark for the Kronecker positivity problem is a conjecture of Heide, Saxl, Tiep and Zalesskii [HSTZ13] that was inspired by their investigation of the square of the Steinberg character for simple groups of Lie type
Summary
This paper brings together, for the first time, the two oldest open problems in the representation theory of the symmetric groups and their quiver Hecke algebras. The tensor square of Dk(ρ) is again a projective module, and the square of ξρ is the character to a projective module This allows us to bring to bear the tools of modular and graded representation theory on the study of the Kronecker coefficients. For H−C1(n), we show that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a sum of graded simple modules. (the only common trait of these partitions is that they label semisimple Specht modules for H−C1(n).) We shall illustrate below that the property of being a Carter–Saxl pair is very easy to work with diagrammatically. Theorem B allows us to provide explicit lower bounds on the Kronecker coefficients g(ρ(k), ρ(k), λ) for new infinite families of Saxl constituents, where again the multiplicities are unbounded. As a guide towards finding a simple module DC(λ) whose tensor square contains all simples, one would try to find a suitable symmetric p-core for a small prime p
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