Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux

By Arun Ram. 2006

72 citations recorded by the source corpus.

Together, Sections 2 and 5 of this paper form a self contained treatment of the theory of crystals and the path model. It is my hope that this will be useful to the many people who, over the years, have told me that they wished they understood crystals but have found the existing literature too daunting. One goal of the presentation here is to clarify the relationship between the general path model and the crystal operators of Lascoux and Schutzenberger used in the type A case [LS]. More specifically, Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model which is designed for readers whose only background is the material in Section 2. These two sections can be read independently of Sections 3 and 4.

View Paper: https://www.semanticscholar.org/paper/58b9c197fd06458bd95c5f18f7014f520a62ed40

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1. Martha Yip
2. Arun Ram
3. James Haglund
4. Stephen Griffeth
5. I. Gordon

Citations in the corpus

• A combinatorial formula for Macdonald polynomials (2008, background): “The papers [3,9,10,15] explain how the combinatorics of alcove walks is almost equivalent to the combinatorics of crystal bases and Kashiwara operators (at least for the positively folded alcove walks of maximal dimension).”