## Departmental PhD Thesis Exam – Bruce Fontaine

Everyone welcome. Refreshments will be served in the Math Lounge before the exam. The thesis can be viewed at

DEPARTMENTAL PHD THESIS EXAM

Tuesday, April 3, 2012, 2:10 pm, in BA 6183, 40 St. George St.

PhD Candidate: Bruce Fontaine

Thesis Title: Bases for Invariant Spaces and Geometric Representation Theory

Thesis Abstract:

Let $G$ be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, a configuration space of points in the affine Grassmannian is constructed. This configuration space gives a natural way of calculating the invariant vectors coming from webs.

In the case of $G=\mathrm{SL}_3$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is $\mathrm{CAT}(0)$, is explained by the fact that affine buildings are $\mathrm{CAT}(0)$. In the case of $G=\mathrm{SL}_n$, a sufficient condition for a set of webs to yield a basis is given. Using this condition and a generalization of a technique by Westbury, a basis is constructed for $\mathrm{SL}_n$.

Due to the geometric Satake correspondence there exists another natural basis of invariants, the Satake basis. This basis arises from the underlying geometry of the affine Grassmannian. There is an upper unitriangular change of basis from the basis constructed above to the Satake basis. An example is constructed showing that the Satake, web and dual canonical basis of the invariant space are all different.

The natural action of rotation on tensor factors sends invariant space to invariant space. Since the rotation of web is still a web, the set of vectors coming from webs is fixed by this action. The Satake basis is also fixed, and an explicit geometric and combinatorial description of this action is developed.