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arXiv · 1512.04320

Rational Shi tableaux and the skew length statistic

Robin Sulzgruber
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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.

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Robin Sulzgruber. 2015-12-14. Rational Shi tableaux and the skew length statistic. https://arxiv.org/abs/1512.04320

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