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arXiv · 2402.18716

Temperley-Lieb Crystals

Son Nguyen·Pavlo Pylyavskyy
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Abstract

Elements of Lusztig's dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis - called Temperley-Lieb immanants - we give a generalization of Littlewood-Richardson rule: we provide a combinatorial interpretation for the coefficient of a particular Schur function in the evaluation of a particular Temperley-Lieb immanant on a particular Jacobi-Trudi matrix. For this we introduce shuffle tableaux, and apply Stembridge's axioms to show that certain graphs on shuffle tableaux are type $A$ Kashiwara crystals.

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Son Nguyen, Pavlo Pylyavskyy. 2024-02-28. Temperley-Lieb Crystals. https://arxiv.org/abs/2402.18716

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