A tensor-cube version of the Saxl conjecture (Q6042823)

The Specht modules are important objects in the group representation theory and algebraic combinatorics. They are closely related to the Kronecker coefficients of symmetric groups and the modules of Schur algebras. Let \(n\) be a positive integer. The complex Specht module labeled by the staircase partition \((n, n-1,\ldots, 1)\) is denoted by \(S^{\rho_n}\). The main result of this paper shows that each complex Specht module of the symmetric group on \(\binom{n+1}{ 2}\) letters is isomorphic to an irreducible direct summand of the tensor-cube \(S^{\rho_n}\otimes S^{\rho_n}\otimes S^{\rho_n}\), which is the tensor-cube version of the Saxl conjecture. This paper contains two proofs of the main result that are well organized and the main result is novel and interesting.