Chennai Mathematical Institute

Seminars




12.00 noon
Doing the twist by dimers

Jeanne Scott
Institute of Mathematical Sciences, Chennai.
17-11-10


Abstract

The 'twist' map $\tau$ is special kind of birational automorphism arising in Lie theory and defined, in particular, for any partial flag variety $\mathcal{F}$ attached to a complex semi-simple algebraic group. Such an automorphism $\tau$ is related to the attending cluster algebra structure of the partial flag variety and, in types A-D-E, corresponds to an Auslander-Reiten translation functor of a module subcategory determined by $\mathcal{F}$ within the associated type A-D-E preprojective algbra.

In this talk I plan to describe a certain atlas of coordinate charts for the Grassmannian $\text{Gr}_{k,n}$ and compute Laurent expansions for 'twisted' Pl\"ucker coordinates --- Pl\"ucker coordinates precomposed with the twist automorphism --- with respect to any of these charts. These expansions, which are predicted using the theory of cluster algebras, can be explicitly tabulated using dimer partitions functions subordinate to a given chart.





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