ALGEBRAIC COMBINATORICS

no. 3
Complexity of the usual torus action on Kazhdan–Lusztig varieties
Donten-Bury, Maria1; Escobar, Laura2; Portakal, Irem3
1 University of Warsaw Institute of Mathematics Banacha 2 02-097 Warszawa Poland
2 Department of Mathematics and Statistics Washington University in St. Louis One Brookings Drive St. Louis, Missouri 63130 U.S.A.
3 Technische Universität München Lehrstuhl für Mathematische Statistik 85748 Garching b. München Boltzmannstr. 3
Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 835-861.

We investigate the class of Kazhdan–Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety X w ¯ as X w ¯=Y w ×ℂ d (where d is maximal possible), we show that Y w can be of complexity-k exactly when k≠1. Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan–Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations v and w, the complexity of Kazhdan–Lusztig variety indexed by (v,w) is the same as the complexity of the Richardson variety indexed by (v,w). Finally, we use this description to compute the complexity of certain Kazhdan–Lusztig varieties.

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.279
Classification: 14M15, 14M25, 52B20, 05E10, 05C20
Keywords: Schubert variety, Kazhdan–Lusztig variety, weight cone, torus action, toric variety, $T$-variety, edge cone, directed graph.
Donten-Bury, Maria 1; Escobar, Laura 2; Portakal, Irem 3

1 University of Warsaw Institute of Mathematics Banacha 2 02-097 Warszawa Poland
2 Department of Mathematics and Statistics Washington University in St. Louis One Brookings Drive St. Louis, Missouri 63130 U.S.A.
3 Technische Universität München Lehrstuhl für Mathematische Statistik 85748 Garching b. München Boltzmannstr. 3
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Donten-Bury, Maria; Escobar, Laura; Portakal, Irem. Complexity of the usual torus action on Kazhdan–Lusztig varieties. Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 835-861. doi : 10.5802/alco.279. https://alco.centre-mersenne.org/articles/10.5802/alco.279/

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