A variation on the tableau switching and a Pak-Vallejo's conjecture

Abstract

Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the Littlewood-Richardson coefficients c μ, = c ,μ. They have exhibited four fundamental symmetry maps and conjectured that they are all identical (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and is shown, in this paper, to be identical to the first one defined by the involution property of the Benkart-Sottile- Stroomer tableau switching. The proof of this equivalence exhibits switching as an operation satisfying the interlacing property between normal shapes of the pairs of tableaux and pairs of subtableaux. That property leads to a jeu de taquin-like on Littlewood-Richardson tableaux which explains the mentioned nonstandard operations and provides a variation of the tableau switching on Littlewood-Richardson tableau pairs.