Rational Shi tableaux and the skew length statistic

TL;DR

This paper introduces two refined statistics on core partitions, linking them to rational Shi tableaux and affine symmetric groups, and proves their key properties including injectivity and invariance under conjugation.

Contribution

It defines new refinements of skew length, connects them to rational Shi tableaux, and extends these concepts to Weyl groups with conjectured injectivity.

Findings

Refined skew length is invariant under conjugation.

Rational Shi tableaux uniquely determine dominant p-stable affine permutations.

Generalisation of rational Shi tableaux to Weyl groups with conjectured injectivity.

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.