Gabriel Frieden, UQAM
Title: Crystal invariant theory
Abstract: Many classical constructions with semistandard Young tableaux can be described by formulas consisting of the operations of addition, subtraction, and taking the minimum of several numbers. By de-tropicalizing these formulas鈥攖hat is, by replacing the operations min,+,- with addition, multiplication, and division鈥攐ne obtains subtraction-free rational maps with remarkable properties.
In this talk, we consider de-tropicalizations of several families of combinatorial maps connected to the representation theory of GL_n, namely, crystal operators and combinatorial R-matrices. We study the invariants of various subsets of these maps, and describe (conjectural) generating sets in each case. We view these invariants as crystal-theoretic analogues of the invariants of various combinations of SL_m, SL_n, S_m, and S_n acting on a polynomial ring in an m times n matrix of variables. This is based on joint work with Ben Brubaker, Pavlo Pylyavskyy, and Travis Scrimshaw.