Tropical geometry and the representation theory of SLn(C)Algebra & Discrete Mathematics
|Chris Manon, UC Berkeley
|Fri, Jan 28 2011, 2:10PM
The (m,n) dissimilarity vector is a combinatorial invariant of a metric tree which appears in Mathematical Biology as a useful tool for analyzing phylogenies between n-taxa. It was conjectured by Cools that the m-dissimilarity vector of a metric tree with n leaves satisfies the combinatorial relations coming from the tropicalizing of the (m,n) Plucker ideal and is therefore a point on the (m,n) tropical Grassmannian. We will discuss our proof of this conjecture and connections between dissimilarity vectors, tropical geometry, and combinatorial aspects of the representation theory of the special linear group. Time permitting, we will also discuss the relationship between tropical geometry and branching problems from the representation theory of reductive groups.