Affine Stanley Symmetric Functions

We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions to the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. In [Sta84], Stanley introduced a family {Fw(X)} of symmetric functions now known as Stanley symmetric functions. He used these functions to study the number of reduced decompositions of permutations w ∈ Sn. Later, the functions Fw(X) were found to be stable limits of Schubert polynomials. Another fundamental property of Stanley symmetric functions is the fact that they are Schur-positive ([EG, LS]). This extended abstract describes work in progress on an analogue of Stanley symmetric functions for the affine symmetric group S̃n which we call affine Stanley symmetric functions. Our first main theorem is that these functions F̃w(X) are indeed symmetric functions. Most of the other main properties of Stanley symmetric functions established in [Sta84] also have analogues in the affine setting. Our definition of affine Stanley symmetric functions is motivated by relations with two other classes of symmetric functions which have received attention lately. Lapointe, Lascoux and Morse [LLM] initiated the study of k-Schur functions, denoted s (k) λ (X), in their study of Macdonald polynomial positivity. Lapointe and Morse have more recently connected k-Schur functions with the Verlinde algebra of SL(n). Separately, cylindric Schur functions were defined by Postnikov [Pos] in connection with the quantum cohomology of the Grassmannian (see also [GK]). We shall connect these two classes of symmetric functions via affine Stanley symmetric functions. More precisely, we show that when w ∈ S̃n is a “Grassmannian” affine permutation then F̃w(X) is “dual” to the k-Schur functions s (k) λ (X). We call these functions F̃w(X) affine Schur functions. Affine Schur functions were earlier defined by Lapointe and Morse who called them dual k-Schur functions. In analogy with the usual Stanley symmetric function case, conjecture that all affine Stanley symmetric functions expand positively in terms of affine Schur functions. We then show that cylindric Schur functions are special cases of skew affine Schur functions and correspond to 321-avoiding affine permutations. The non-affine case suggests that our work may be connected with the affine flag variety and objects that might be called “affine Schubert polynomials”. Shimozono has conjectured a precise relationship between k-Schur functions and the homology of the affine Grassmannian. The dual conjecture ([MS]) is that affine Schur functions represent Schubert classes in the cohomology H(G/P) of the affine Grassmannian. In section 1, we establish some notation for permutations and affine permutations, and for symmetric functions. In section 2 we recall the definition of Stanley symmetric functions, give their main properties and explain the relationship with Schubert polynomials. In section 3, we define affine Stanley symmetric functions and prove that they are symmetric. In section 4, Date: November, 2004; revised February, 2005. 1