Affine Stanley symmetric functions

Abstract

Abstract. We define a new familyFw(X) of generating functions for w ∈Sn which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalise the (dual of the) k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Conjecturally, affine Stanley symmetric functions should be related to the cohomology of the affine flag variety.

Schubert polynomials for the affine Grassmannian

“…Our main theorem (Theorem 7.1) identifies the Schubert classes σ w ∈ H * (Gr) and σ w ∈ H * (Gr) as explicit symmetric functions under this isomorphism. These symmetric functions are combinatorially defined: in homology the σ w ∈ H * (Gr) are represented by Lascoux-LapointeMorse's k-Schur functions s (k) w (x) ∈ Λ n and in cohomology the σ w ∈ H * (Gr) are represented by the dual k-Schur functions (or affine Schur functions)F w (x) ∈ Λ n ; see [21,24,17]. Our theorem was originally conjectured by Mark Shimozono (the conjecture was made explicit in the cohomology case by Jennifer Morse).…”

“…These algebras are tied together by the study of elements s (k) w ∈ B, introduced in [17], called non-commutative k-Schur functions. We show that they have a trio of descriptions:…”

Schubert polynomials for the affine Grassmannian of the symplectic group

“…For G = SL n (C), Lam [15] identified the Schubert basis of H * (Gr SLn(C) ) with symmetric functions, called k-Schur functions, of Lapointe, Lascoux and Morse [18]; these arose in the study of Macdonald polynomials. The Schubert basis of H * (Gr SLn(C) ) are the dual k-Schur functions [19] which are generalized by the affine Stanley symmetric functions [14]. In [16] Pieri rules were given for the multiplication of Bott's generators on the Schubert bases of Bott's realization of H * (Gr SLn(C) ) and H * (Gr SLn(C) ).…”

“…The symmetric functions Q (n) w are type C analogues of the affine Stanley symmetric functions studied in [14]. Some examples for the type C affine Stanley symmetric functions are given in Appendix B.…”

The center of the affine nilTemperley–Lieb algebra

“…The affine nilCoxeter algebra is closely connected with affine Schur functions, k-Schur functions, and the affine Stanley symmetric functions, which are related to reduced word decompositions in the affine symmetric group (see e.g. [14,15]). The nilCoxeter algebra U N has generators u i , 1 ≤ i ≤ N − 1, which satisfy the same relations as they do in U N .…”