
SCIENTIFIC PROGRAMS AND ACTIVITIES 

July 26, 2014  
Abstracts_______________________________ 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 tworow 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. _______________________________ 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. _______________________________ 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?" _______________________________ _______________________________ 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. _______________________________ 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. ______________________________ Luc Lapointe and Jennifer Morse defined kSchur 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 LittlewoodRichardson rule is related to developing explicit formulas for kSchur functions within the affine nilCoxeter algebra of type A. This is joint work with Chris Berg, Nantel Bergeron and Hugh Thomas.

