Affine Stanley symmetric functions for classical types

TL;DR

This paper extends affine Stanley symmetric functions to classical groups, providing new algebraic models for their (co)homology and establishing Pieri rules, with conjectures for even orthogonal groups.

Contribution

It introduces affine Stanley symmetric functions for orthogonal groups, develops non-commutative k-Schur functions, and offers Pieri rules, advancing the algebraic understanding of affine Grassmannian cohomology.

Findings

Established Hopf-algebra isomorphism for odd orthogonal groups.

Developed type B and D non-commutative k-Schur functions.

Provided Pieri rules for homology multiplication.

Abstract

We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the special linear and symplectic groups, respectively. For the odd orthogonal groups, a Hopf-algebra isomorphism is given, identifying (co)homology Schubert classes with symmetric functions. For the even orthogonal groups, we conjecture an approximate model of (co)homology via symmetric functions. In the process, we develop type B and type D non-commutative k-Schur functions as elements of the nilCoxeter algebra that model homology of the affine Grassmannian. Additionally, Pieri rules for multiplication by special Schubert classes in homology are given in both cases. Finally, we present a type-free interpretation of Pieri factors, used in the definition of noncommutative k-Schur functions or affine Stanley symmetric functions for any classical type.