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This purpose of this chapter is to introduce Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The presentation roughly follows three lectures I gave at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.

The purpose of this chapter is to outline some of the results and open problems related to k-Schur functions, mostly in the setting of symmetric function theory. This chapter roughly follows the outline of several talks given by Luc Lapointe and Jennifer Morse at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

This chapter discusses how k-Schur and dual k-Schur functions can be defined for all types. This is done via some combinatorial problems that come from the geometry of a very large family of generalized flag varieties. They apply to the expansion of products of Schur functions, k-Schur functions and their dual basis, and Schubert polynomials. Despite the geometric origin of these problems, concrete algebraic models will be given for the relevant cohomology rings and their Schubert bases.

The Jack and Macdonald symmetric polynomials enter, respectively, in the wave functions of the quantum integrable Calogero—Sutherland (CS) and Ruijsenaars—Schneider (RS) models. This paper is devoted to recent advances in the combinatorial theory of these functions. The emphasis is put on the creation operators that allow us to construct the excited state wave functions of the CS and RS models from their ground state wave functions.