Discrete Geometry and Combinatorics Seminar

Gabriel FriedenUniversity of Michigan
Geometric lifting of affine crystal combinatorics

Monday, November 13, 2017 - 2:30pm
Malott 206

Semistandard tableaux of rectangular shape have two remarkable properties. Let $B(k,L)$ denote the set of semistandard tableaux of shape $(L^k)$, with entries in ${1, ..., n}$. First, Schutzenberger promotion on $B(k,L)$ has order $n$. Second, the product of two Schur functions labeled by rectangular partitions is multiplicity-free. This is equivalent to the fact that given a pair $(T,U)$ in $B(k,L) \times B(k',L')$, there is a unique pair $(U',T')$ in $B(k',L') \times B(k,L)$ such that $T*U = U'*T'$, where $*$ denotes the product of tableaux in the plactic monoid. These two properties are related to the affine type $A$ crystal structure on rectangular tableaux. Promotion is the ``combinatorial avatar'' of the rotation of the affine type $A$ Dynkin diagram. The map $(T,U) \mapsto (U',T')$ is the unique affine crystal isomorphism between the tensor products $B(k,L) \otimes B(k',L')$ and $B(k',L') \otimes B(k,L)$; in the language of crystal theory, it is called the combinatorial $R$-matrix.

In this talk, I will present rational maps on (products of) Grassmannians which are geometric analogues of promotion and the combinatorial $R$-matrix. These maps interact appropriately with a geometric analogue of the affine crystal structure on rectangular tableaux. I will explain how these maps are "geometric liftings": when expressed in a suitable toric chart, they tropicalize to piecewise-linear formulas for the combinatorial maps. Several properties of the combinatorial maps (such as the order of promotion) are transparent in the geometric setting.