From Anderson to Zeta

2015/04/28 by Marko Thiel, Thiel, Marko · 1 citation
Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1504.07363

openalex publication_date 2015/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an irreducible crystallographic root system Φ and a positive integer p relatively prime to the Coxeter number h of Φ, we give a natural bijection A from the set \widetildeW^p of affine Weyl group elements with no inversions of height p to the finite torus \checkQ/p\checkQ. Here \checkQ is the coroot lattice of Φ. This bijection is defined uniformly for all irreducible crystallographic root systems Φ and is equivalent to the Anderson map A_GMV defined by Gorsky, Mazin and Vazirani when Φ is of type A_n-1. Specialising to p=mh+1, we use A to define a uniform W-set isomorphism ζ from the finite torus \checkQ/(mh+1)\checkQ to the set of m-nonnesting parking functions Park_Φ^(m) of Φ. The map ζ is equivalent to the zeta map ζ_HL of Haglund and Loehr when m=1 and Φ is of type A_n-1.

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