Asymptotic Semigroups and Two-sided Weak Orders
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Publication:6319061
DOI10.1016/J.JALGEBRA.2022.07.021arXiv1905.08316MaRDI QIDQ6319061
Publication date: 20 May 2019
Abstract: Various partial orders related to the structures of dual canonical monoids are investigated. It is shown that the nilpotent variety of a dual canonical monoid is equidimensional; its dimension is found. It is shown in type A that certain intervals of the Putcha poset of a dual canonical monoid are isomorphic to the Renner monoids of matrices. The notion of a two-sided weak order on a normal reductive monoid is introduced. A criterion, in terms of type maps, for the covering relations in a two-sided weak order to have degree 2 is found. It is shown that, for the unique equivariant divisor of a dual canonical monoid (the asymptotic semigroup), the covering relations of the two-sided weak order are always of degree 1. These computations provide new insights for the two-sided weak orders on Coxeter groups. In type A, some enumerative results for the covering relations are presented.
Partial orders, general (06A06) Homogeneous spaces and generalizations (14M17) General structure theory for semigroups (20M10) Algebraic monoids (20M32)
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