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Algebra Seminar
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Matthew Dyer
University of Notre Dame

Shellability and highest weight representations

Friday 4th, April 12:05-12:55pm, Stephen Roberts.

Shellable simplicial complexes are built from the empty complex by adding in the closures of the maximal faces one at a time in a recursively controlled way; for such a complex, one obtains detailed explicit information about the homotopy type, cohomology etc. In this talk, a definition and some consequences of an analogous notion of shellability of highest weight representation categories will be described. For a special class of representation categories associated to a pure simplicial complex, this notion reduces to the shellability of the complex. It is expected that many natural categories arising in Lie theory (e.g. category O of representations of a semisimple complex Lie algebra or associated quantized enveloping algebra) come from shellable ones; if true this would provide a deeper, uniform explanation and stronger versions of several important known common properties of these categories, such as factorizations of Shapovalov determinants.