Stanley Symmetric Functions and Peterson Algebras

These are (mostly) expository notes for lectures on affine Stanley symmetric functions given at the Fields Institute in 2010. We focus on the algebraic and combinatorial parts of the theory. The notes contain a number of exercises and open problems. Stanley symmetric functions are a family {Fw | w ∈ Sn} of symmetric functions indexed by permutations. They were invented by Stanley [Sta] to enumerate the reduced words of elements of the symmetric group. The most important properties of the Stanley symmetric functions are their symmetry, established by Stanley, and their Schur positivity, first proven by Edelman and Greene [EG], and by Lascoux and Schützenberger [LSc82]. Recently, a generalization of Stanley symmetric functions to affine permutations was developed in [Lam06]. These affine Stanley symmetric functions turned out to have a natural geometric interpretation [Lam08]: they are pullbacks of the cohomology Schubert classes of the affine flag variety LSU(n)/T to the affine Grassmannian (or based loop space) ΩSU(n) under the natural map ΩSU(n)→ LSU(n)/T . The combinatorics of reduced words and the geometry of the affine homogeneous spaces are connected via the nilHecke ring of Kostant and Kumar [KK], together with a remarkable commutative subalgebra due to Peterson [Pet]. The symmetry of affine Stanley symmetric functions follows from the commutativity of Peterson’s subalgebra, and the positivity in terms of affine Schur functions is established via the relationship between affine Schubert calculus and quantum Schubert calculus [LS10, LL]. The affine-quantum connection was also discovered by Peterson. The affine generalization also connects Stanley symmetric functions with the theory of Macdonald polynomials [Mac] – my own involvement in this subject began when I heard a conjecture of Mark Shimozono relating the Lapointe-Lascoux-Morse k-Schur functions [LLM] to the affine Grassmannian. While the definition of (affine) Stanley symmetric functions does not easily generalize to other (affine) Weyl groups (see [BH, BL, FK96, LSS10, Pon]), the algebraic and geometric constructions mentioned above do. This article introduces Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The goal is to develop the theory (with the exception of positivity) without appealing to geometric reasoning. The notes are aimed at an audience with some familiarity with symmetric functions, Young tableaux and Coxeter groups/root systems. Date: July 16, 2010. The author was supported by NSF grants DMS-0652641 and DMS-0901111, and by a Sloan Fellowship.