Poincare Series of Coxeter Groups

introduction

• Poincaré Series of a Coxeter Group $W$ (Poincare series for ring of coinvariants)

$$ P_{W}(q)=\sum_{w\in W}q^{\ell(w)} $$

• for finite $W$,

$$ P_{W}(q)=\prod_{\alpha>0}\frac{q^{\operatorname{ht}(\alpha)+1}-1}{q^{\operatorname{ht}(\alpha)}-1}=\prod_{i=1}^{k}\begin{bmatrix} d_i \end{bmatrix}_{q} $$ where $d_i$'s are degrees

example

• $A_2$ example
• degree : 2,3
• $W$ has 6 elements : $1,s_1,s_2,s_1s_2,s_2s_1,s_1s_2s_1$
• $P_{W}(q)=1 + 2 q + 2 q^2 + q^3=(1 + q) (1 + q + q^2)$
• using heights of roots

$$ \prod_{\alpha>0}\frac{q^{\operatorname{ht}(\alpha)+1}-1}{q^{\operatorname{ht}(\alpha)}-1}=\frac{q^2-1}{q-1}\frac{q^2-1}{q-1}\frac{q^3-1}{q^2-1}=(1 + q) (1 + q + q^2) $$

• using degrees

$$ \prod_{i=1}^{k}\begin{bmatrix} d_i \end{bmatrix}_{q}=\begin{bmatrix} 2 \end{bmatrix}_{q}\begin{bmatrix} 3 \end{bmatrix}_{q}=(1+q)(1+q+q^2) $$

related items

• Macdonald constant term conjecture
• Degrees and exponents

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxbF9YQmxvNjBFYzQ/edit

books

• Richard Kane
  • 144p, 219p, 236p

articles

• Macdonald, I. G. 1972. “The Poincaré Series of a Coxeter Group.” Mathematische Annalen 199 (3) (September 1): 161–174.

doi:10.1007/BF01431421