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.


Crystals via Mirković-Vilonen cycles

by Brad Hannigan-Daley | University of Toronto
Time: 15:00  (Monday, Nov. 28, 2011)
Location: BA6180, Bahen Center, 40 St. George St.
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 perverse-sheaves side of things, and it turns out to be visible by way of certain subspaces of $Gr$ called semi-infinite orbits. In particular, the weight decomposition of an irreducible representation $V(\lambda)$ can be described in terms of subvarieties called Mirković-Vilonen (MV) cycles. Moreover, the corresponding set of MV cycles can be equipped with a crystal structure, thereby realizing the highest-weight 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 Hannigan-Daley)
· Monday, Nov. 28, 2011: Crystals via Mirković-Vilonen cycles (Brad Hannigan-Daley)
· Monday, Dec. 05, 2011: MV polytope model for crystals (Joel Kamnitzer)