##### Department of Mathematics,

University of California San Diego

****************************

### Math 269 - Combinatorics

## Mark Shimozono

#### Virginia Tech

## Affine insertion and Schubert calculus on the affine Grassmannian

##### Abstract:

By Bott's work on loop groups on simple Lie groups and a result of Quillen, the homology of the affine Grassmannian Gr of SL(k+1) is a ring that is naturally embedded in the ring of symmetric functions. Since Gr is a Kac-Moody G/P it is natural to consider the Schubert bases of its homology and cohomology. The Schubert structure constants of the homology of Gr are particularly interesting, as they include the genus zero Gromov-Witten invariants for Grassmannians, or equivalently the structure constants for the fusion tensor product for the Weiss-Zumino-Witten conformal field theory model. In the summer of 2004 I conjectured that the Schubert basis of homology of Gr is given by the k-Schur functions of Lascoux, Lapointe, and Morse. This was recently proved by Thomas Lam. Lapointe and Morse also had a dual basis called the dual k-Schur functions or affine Schur functions; these are the Schubert basis of cohomology of Gr. The dual k-Schur functions may be realized as generating functions for k-tableaux or weak tableaux, which are defined using the weak order on the affine symmetric group. We introduce strong tableaux, whose definition is based on the strong Bruhat order on the affine symmetric group. The generating functions of strong tableaux are symmetric functions, which we prove are equal to the k-Schur functions. We give an algorithm called affine insertion, which maps certain biwords to pairs of tableaux, one strong and one weak. This bijection proves an analogue of the Cauchy identity coming from the pairing between the homology and cohomology of Gr. As k goes to infinity, weak and strong tableaux both converge to semistandard tableaux and our bijection converges to the usual row insertion Robinson-Schensted-Knuth correspondence. As corollaries we obtain Pieri rules for the Schubert calculus of both the homology and cohomology of the affine Grassmannian. Both of these are new. This is joint work with Thomas Lam, Luc La

Hosts: Mike Zabrocki and Allen Knutson

### May 22, 2006

### 11:00 AM

### AP&M 6218

****************************