Complexity of the usual torus action on Kazhdan–Lusztig varieties
Maria Donten-Bury, Laura Escobar, Irem Portakal
Source abstract
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.
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