Papers › Join operation for the Bruhat order and Verma modules
Join operation for the Bruhat order and Verma modules
Hankyung Ko, Volodymyr Mazorchuk, Rafael Mrđen
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We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.
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