# SCIENTIFIC PROGRAMS AND ACTIVITIES

September 20, 2014

## April 1-2, 2011 SOUTHERN ONTARIO GROUPS AND GEOMETRY MEETING to be held at the Fields Institute (map)

Organizing Committee:
Megumi Harada (McMaster University), Joel Kamnitzer (University of Toronto), Kevin Purbhoo (University of Waterloo)

### Abstracts

_______________________________
Jonah Blasiak (University of Michigan)
A crystal basis for two-row Kronecker

The Kronecker coefficient $g_{\lambda \mu \nu}$ is the multiplicity of the $GL(V)\times GL(W)$-irreducible $V_\lambda \otimes W_\mu$ in the restriction of the $GL(X)$-irreducible $X_\nu$ via the natural map $GL(V)\times GL(W) \to GL(V \otimes W)$, where  $V, W$ are $\mathbb{C}$-vector spaces and $X = V \otimes W$. A difficult open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We construct a global crystal basis for $X_\nu$ as a $U_q(\mathfrak{gl}(V)) \otimes U_q(\mathfrak{gl}(W))$-module in the $\dim V = \dim W = 2$ case and obtain nice formulas for two-row Kronecker coefficients by counting highest weight basis elements. We'll also discuss how this gives a basis for the coordinate ring $\mathbb{C}[V \otimes W \otimes Z]$, $V = W = \mathbb{C}^2, Z = \mathbb{C}^4$, and its connection to the geometry of $V \otimes W \otimes Z$ as a $GL(V)\times GL(W) \times GL(Z)$-variety. This work is joint with Ketan Mulmuley and Milind Sohoni.

_______________________________
Bruce Fontaine (University of Toronto)
Webs and the Affine Grassmannian

By the geometric Satake correspondence a basis of invariant vectors of a tensor product of minuscule representations of a reductive group G is given by the set of irreducible components of a certain variety constructed from the Affine Grassmannian of the Langlands dual of G. Alternatively, when G=SL_3, a basis of invariant vectors can also be calculated from the web diagrams of Greg Kuperberg. We will show a general method of calculating the invariant vector associated to a web via geometric means and that this basis is not the one coming from geometric Satake. Work joint with Joel Kamnitzer and Greg Kuperberg.

_______________________________
Jenna Rajchgot (Cornell University)
Compatibly split subvarieties of the Hilbert scheme of points in the plane

Consider the Hilbert scheme of n points in the affine plane and the divisor "at least one point is on a coordinate axis". One can intersect the components of this divisor, decompose the intersection, intersect the new components, and so on to stratify the Hilbert scheme by a collection of reduced (indeed, "compatibly Frobenius split") subvarieties. This may prompt one to ask, "What are these subvarieties?" or, better, "What are all of the compatibly split subvarieties?"

I'll discuss how one can use an algorithm of Allen Knutson, Thomas Lam and David Speyer (that is particularly motivated in the Frobenius split case) to study the second question and I'll present the answer for a few small values of n. Following this, I'll focus on an affine patch of the Hilbert scheme where the stratification by compatibly split subvarieties seems to be more understandable.

I won't assume any knowledge of Frobenius splitting.

_______________________________
Ilya Shapiro (University of Waterloo)
Characteristic classes from Lie algebra cohomology.

I will explain a general method, due to Kontsevich, of obtaining characteristic classes via formal geometry. More precisely, certain natural classes in the Lie algebra cohomology of formal vector fields (and their variants) give rise to many well known (and new) characteristic classes. One may consider manifolds, foliations, orbifolds, symplectic structures and various mixtures of the above from this perspective.

Joint with Xiang Tang.

_______________________________
Greg Smith (Queens University)
Vanishing theorems and equations of embedded varieties

Understanding the relationship between the algebraic equations that cut out a variety Y in X and the geometric features of the embedded variety Y lies at the heart of algebraic geometry. In this talk, we will discuss the key theorems when the ambient variety X is projective space. We'll then motivate and present new results designed for other ambient varieties.

_______________________________
David Speyer (University of Michigan)
Projected Richardson Varieties

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold G/P are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. We show that the projected Richardsons are the only compatibly split subvarieties of G/P (for the standard splitting). In the minuscule case, we describe Groebner degenerations of projected Richardsons. The theory is especially elegant in the case of the Grassmannian, where we obtain the "positroid" varieties, whose combinatorics can be described in terms of juggling patterns.

Joint work with Allen Knutson and Thomas Lam.

______________________________
Mike Zabrocki (York University)
k-Schur functions indexed by a maximal rectangle

Luc Lapointe and Jennifer Morse defined k-Schur functions and, along with Thomas Lam and Mark Shimozono, have developed connections with the geometry and combinatorics of the affine symmetric group. I will show how finding a Littlewood-Richardson rule is related to developing explicit formulas for k-Schur functions within the affine nil-Coxeter algebra of type A. This is joint work with Chris Berg, Nantel Bergeron and Hugh Thomas.