Séminaire de Géométrie Tropicale
Mercredi 27 novembre, 16h15 salle 1525-502
Tropical Plücker functions and Kashiwara crystals
Résumé :
A finite Kashiwara crystal is a finite
edge-colored digraph (directed graph) in which each connected monochromatic
subgraph is a simple path and there are certain interrelations on the lengths
of such paths, described via coefficients of a Cartan matrix
(features of this matrix characterize the crystal type).
Crystals are of importance in representation theory.
I will explain a new model for free n-colored crystals of
type A: the vertices of the free crystal graph are identified with
integer-valued tropical Pl\"ucker functions on
the n-dimensional Boolean cube and the edges of color i (yielding the
crystal operation $\mathbf i$), i=1,...,n, are defined by use of the
restriction of TP-functions to a rhombus tiling which is adapted to this color.
The subgraph whose vertices correspond to the submodular
TP-functions is isomorphic to the Kashiwara crystal graph B(\infty) which serves a
`combinatorial skeleton' of the canonical basis introduced by
Lusztig.
This model of A-type crystals is symmetric with respect to the inversion of
colors. This allows us to get a transparent construction of crystals (free
crystals) of types B and C by considering the corresponding symmetric
TP-functions and rhombus tilings.
This is a joint work with V.Danilov and A.Karzanov