Rational Shi tableaux and the skew length statistic

Abstract

We define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. These rational Shi tableaux encode dominant $p$-stable elements in the affine symmetric group. We prove that the rational Shi tableau is injective, that is, each dominant $p$-stable affine permutation is determined uniquely by its Shi tableau. Moreover, we provide a uniform generalisation of rational Shi tableaux to Weyl groups, and conjecture injectivity in the general case.