Poincare Series of Coxeter Groups
imported>Pythagoras0님의 2013년 7월 5일 (금) 06:49 판 (새 문서: ==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 $W$, $$ P_{W}(q)=\prod_{\alp...)
introduction
- Poincaré Series of a Coxeter Group $W$ (Poincare series for ring of coinvariants)
$$ 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) $$
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.