Seminar series
Date
Tue, 08 May 2007
17:00
17:00
Location
L1
Speaker
Prof. Bernard Leclerc
Organisation
Caen
Let G be a complex semisimple algebraic group of type A,D,E. Fomin and
Zelevinsky conjecture that the coordinate rings of many interesting varieties
attached to G have a natural cluster algebra structure. In a joint work with C.
Geiss and J. Schroer we realize part of this program by introducing a cluster
structure on the multi-homogeneous coordinate ring of G/P for any parabolic
subgroup P of G. This was previously known only for P = B a Borel
(Berenstein-Fomin-Zelevinsky) and when G/P is a grassmannian Gr(k,n) (J. Scott).
We give a classification of all pairs (G,P) for which this cluster algebra has
finite type. Our construction relies on a finite-dimensional algebra attached to
G, the preprojective algebra introduced in 1979 by Gelfand and Ponomarev. We use
the fact that the coordinate ring of the unipotent radical of P is "categorified"
in a natural way by a certain subcategory of the module category of the
preprojective algebra.