Crystals Learning Seminar
In this seminar we will learn about crystals, which are nice combinatorial models for representations of Lie algebras. The first half of the seminar will focus on quantum groups and construction of crystal bases from this perspective (Kashiwara's "grand loop argument"). The second half will focus on combinatorial (Young tableaux, Littelmann path model) and geometric (quiver varieties, MV polytopes) constructions.
Details
Crystals via MirkoviÄ‡Vilonen cycles
by
Brad HanniganDaley

University of Toronto
Time: 15:00 (
Monday, Nov. 28, 2011)
Location: BA6180, Bahen Center, 40 St. George St.
Abstract:
For $G$ a reductive algebraic group with Langlands dual $G^\vee$, the geometric Satake correspondence is an equivalence between the representation category of $G^\vee$ and the category of $G(\mathbb{C}((t)))$equivariant perverse sheaves on $Gr$, the affine Grassmannian for $G$. One may then ask how the weight decomposition of representations looks on the perversesheaves side of things, and it turns out to be visible by way of certain subspaces of $Gr$ called semiinfinite orbits. In particular, the weight decomposition of an irreducible representation $V(\lambda)$ can be described in terms of subvarieties called MirkoviÄ‡Vilonen (MV) cycles. Moreover, the corresponding set of MV cycles can be equipped with a crystal structure, thereby realizing the highestweight crystal $B(\lambda)$.
Dates in this series
 · Monday, Sep. 26, 2011:
Introduction to quantum groups II (Iva Halacheva)
 · Monday, Nov. 21, 2011:
The affine Grassmannian and geometric Satake correspondence (Brad HanniganDaley)
 · Monday, Nov. 28, 2011:
Crystals via MirkoviÄ‡Vilonen cycles (Brad HanniganDaley)
 · Monday, Dec. 05, 2011:
MV polytope model for crystals (Joel Kamnitzer)