From Anderson to zeta

详细信息    查看全文

• 作者：Marko Thiel ^{markth@math.uzh.ch" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E10 ; 05E18 ; 17B22 ; 52C35
• 刊名：Advances in Applied Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：81
• 期：Complete
• 页码：156-201
• 全文大小：827 K

文摘

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$\mathcal{A}$ from the set ${\tilde{W}}^p$ of affine Weyl group elements with no inversions of height p   to the finite torus Q^∨/pQ^∨$Q^{\vee }/p{Q}^{\vee }$. Here Q^∨$Q^{\vee }$ 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${\mathcal{A}}_{GMV}$ defined by Gorsky, Mazin and Vazirani when Φ is of type A_n−1$A_{n-1}$.

Specialising to 2432de947" title="Click to view the MathML source">p=mh+1$p=mh+1$, we use A$\mathcal{A}$ to define a uniform W-set isomorphism ζ   from the finite torus Q^∨/(mh+1)Q^∨$Q^{\vee }/(mh+1)Q^{\vee }$ to the set of m  -nonnesting parking functions ${\mathsf{Park}}_{\mathrm{\Phi }}^{(m)}$ of Φ. The map ζ is equivalent to the zeta map  ζ_HL${\zeta }_{HL}$ of Haglund and Loehr when m=1$m=1$ and Φ is of type A_n−1$A_{n-1}$.