Friday, March 5, 2021
03/05/2021 - 1:00pm
In 1998, Allen Knutson and Terry Tao introduced a rational polyhedral cone with some amazing combinatorial properties in their proof of the saturation conjecture. The "Knutson-Tao hive cone" encodes the number of copies of a given irreducible representation of GL_n appearing in the tensor product of two others-- so it tells us how to rewrite a tensor product as a direct sum. Choosing the representations of interest slices the cone to give a bounded polytope, and counting the integral points in this bounded polytope gives the number we're looking for.
This cone (and plenty of others with the same wonderful combinatorial properties) can actually be obtained by completely general mirror symmetry considerations, without any representation theory at all. We'll get much more than the cone too. In this setting, integral points are elements of a canonical basis, and the combinatorial data is just the cardinality this basis. Moreover, this is a construction that in theory applies whenever you have a space equipped with the right sort of volume form, so the Knutson-Tao hive cone is part of a very broad framework when viewed in this way. I'll give an overview of how this all works.