We investigate some applications of a certain technique (that we call Freese's technique), which is a tool for identifying certain lattices as sublattices of the congruence lattice of a given algebra. In particular we gave sufficient conditions for two families of lattices (called the rods and the snakes) to be admissible as sublattices of a variety generated by a given algebra, extending a result of R. Freese and P. Lipparini.
On some admissible sublattices of a congruence lattice
Tue, Mar. 12 2:30pm (MATH 3…
Lie Theory
Aram Bingham (Colorado School of Mines)
X
Given a reductive algebraic group G with Borel subgroup B, a spherical variety X is a G-variety such that B acts with finitely many orbits. Well-known examples include toric varieties, flag varieties, symmetric varieties and their embeddings, and so-called “wonderful varieties” which play an important role in the classification of spherical varieties. In many cases, Borel subgroup orbits can be combinatorially parametrized and equipped with poset structure induced by the closure-containment (or Bruhat) order. Shellability is a much-studied property in poset topology which implies that a simplicial complex associated to a poset has the homotopy type of a wedge of spheres. It has been known since the 1980s that Bruhat orders of flag varieties are shellable, and these results have been extended to other related settings more recently. We will discuss some results, open questions, and hunches relating these two areas, based on joint work with Néstor Díaz.
SOME SPECULATIONS ON SHELLABILITY AND SPHERICAL VARIETIES Sponsored by the Meyer Fund