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<p>We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group <i>W</i> act on these canonical bases by bijections up to lower-order terms. Examples of this phenomenon include the action of separable permutations on the Kazhdan–Lusztig basis of irreducible representations for the symmetric group, and the action of separable elements of <i>W</i> on dual canonical bases of weight zero in tensor product representations of a Lie algebra. Our methods arise from categorical representation theory, and in particular the study of the perversity of Rickard complexes acting on triangulated categories.</p>

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On the action of the Weyl group on canonical bases

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月18日

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  • Fern Gossow,
  • Oded Yacobi

摘要

We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group W act on these canonical bases by bijections up to lower-order terms. Examples of this phenomenon include the action of separable permutations on the Kazhdan–Lusztig basis of irreducible representations for the symmetric group, and the action of separable elements of W on dual canonical bases of weight zero in tensor product representations of a Lie algebra. Our methods arise from categorical representation theory, and in particular the study of the perversity of Rickard complexes acting on triangulated categories.