Demazure keys, virtual keys and Schubert varieties
 
 
Description: 

Demazure modules, opposite and their intersection are in natural correspondence respectively with Schubert varieties, opposite and  Richardson varieties. The classical Borel-Weil theorem explains this correspondence. It provides a geometric definition of irreducible highest weight representations in terms of full flag varieties which goes further down to Schubert varieties and Demazure modules. The Kashiwara crystal, the crystal of Lakshmibai-Seshadri paths and certain remarkable subsets, called Demazure crystals, are combinatorial skeletons for those objects. Though the crystals are isomorphic they do not provide the same explicit information. For Kashiwara crystals we address the question: how to check whether a vertex of a Kashiwara crystal belongs to a given Demazure crystal. The answer is given by  Demazure keys which are manifest in LS paths but needed to be computed in the Kashiwara crystal. Focusing on the virtualization from the Cartan types \( C_n \) to \( A_{2n-1} \), we show that it preserves left and right Demazure keys which allows the use of known tools in type \( A_{2n-1} \) for their computation.

Date:  2024-06-12
Start Time:   15:00
Speaker:  Olga Azenhas (CMUC, Univ. Coimbra)
Institution:  CMUC, UC
Place:  Sala 2.4, DMUC
Research Groups: -Algebra and Combinatorics
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