"Poincare Series of Coxeter Groups"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
| 9번째 줄: | 9번째 줄: | ||
$$ | $$ | ||
where $d_i$'s are degrees | where $d_i$'s are degrees | ||
| − | http://mathoverflow.net/questions/28422/does-the-poincare-series-of-a-coxeter-group-always-describe-a-flag-variety?rq=1 | + | * http://mathoverflow.net/questions/28422/does-the-poincare-series-of-a-coxeter-group-always-describe-a-flag-variety?rq=1 |
| + | * Poincare series of crystallographic Coxeter groups are always Betti numbers of Kac-Moody flag varieties, and that Kumar's book covers this very well | ||
| + | |||
| + | |||
==example== | ==example== | ||
* $A_2$ example | * $A_2$ example | ||
2014년 7월 1일 (화) 19:01 판
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
- http://mathoverflow.net/questions/28422/does-the-poincare-series-of-a-coxeter-group-always-describe-a-flag-variety?rq=1
- Poincare series of crystallographic Coxeter groups are always Betti numbers of Kac-Moody flag varieties, and that Kumar's book covers this very well
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
$$ \prod_{i=1}^{k}\begin{bmatrix} d_i \end{bmatrix}_{q}=\begin{bmatrix} 2 \end{bmatrix}_{q}\begin{bmatrix} 3 \end{bmatrix}_{q}=(1+q)(1+q+q^2) $$
computational resource
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.