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Rational Shi tableaux and the skew length statistic
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<span title="2016-09-15">2016</span>
<i >
arXiv
</i>
<span class="release-stage" >pre-print</span>
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<span title="2015-12-14">2015</span>
<span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.04320v1">arXiv:1512.04320v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lbouaehypna3lcc3oecxlbswai">fatcat:lbouaehypna3lcc3oecxlbswai</a> </span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.04320v1">arXiv:1512.04320v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lbouaehypna3lcc3oecxlbswai">fatcat:lbouaehypna3lcc3oecxlbswai</a> </span>
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
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<a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.04320v2">arXiv:1512.04320v2</a>
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... 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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