Richard P. Stanley Seminar in Combinatorics: Fibers of maps to totally nonnegative spaces

We will discuss the fibers of some intriguing maps from representation theory, maps whose images were previously studied by Fomin-Shapiro, Lusztig, and several others.  These fibers encode the nonnegative real relations amongst exponentiated Chevalley generators.   These maps $f_{(i_1,\dots ,i_d)}$  are each specified by a reduced or nonreduced word $(i_1,\dots ,i_d)$  for an element in a finite Coxeter group.  We prove that the stratification on each fiber of $f_{(i_1,\dots ,i_d)}$ induced by the natural stratification of $\RR_{\ge 0}^d$ is a cell decomposition of the fiber, doing so by providing a parametrization for each cell.  We also prove that the face poset for this cell decomposition is the face poset of a regular CW complex, namely of the interior dual block complex of a subword complex.  This talk will focus on examples, background, some motivations, and some of the key ingredients in the proofs.  This is joint work with Jim Davis and Ezra Miller.

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