콕세터 군의 푸앵카레 급수

introduction

• 콕세터 군 $W$의 푸앵카레 급수

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

정리 (Chevalley-Solomon)

유한반사군 $W$에 대하여, 다음이 성립한다 $$ P_{W}(q)=\prod_{i=1}^{k}\begin{bmatrix} d_i \end{bmatrix}_{q} $$ 여기서 $d_i$는 $W$의 기본차수

• 다음이 성립한다

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

예

• $A_2$
• 차수 : 2,3
• $W$는 6개의 원소를 가짐 : $1,s_1,s_2,s_1s_2,s_2s_1,s_1s_2s_1$
• $P_{W}(q)=1 + 2 q + 2 q^2 + q^3=(1 + q) (1 + q + q^2)$
• 차수를 이용하여 다음을 얻는다

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

• 루트시스템에 정의되는 height를 이용하여 다음을 얻을 수 있다

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

관련된 항목들

• 콕세터 군의 차수와 지수 (degrees and exponents)

computational resource

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

메모

• 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
• Richard Kane
  • 144p, 219p, 236p

리뷰, 에세이, 강의노트

• Reiner, Victor. Notes on Poincaré series of finite and affine Coxeter groups

관련논문

• Macdonald, I. G. 1972. “The Poincaré Series of a Coxeter Group.” Mathematische Annalen 199 (3) (September 1): 161–174. doi:10.1007/BF01431421
• Solomon, Louis. 1966. “The Orders of the Finite Chevalley Groups.” Journal of Algebra 3 (3): 376–93. doi:10.1016/0021-8693(66)90007-X.
• Solomon, Louis. 1963. “Invariants of Finite Reflection Groups.” Nagoya Mathematical Journal 22: 57–64.
• Bott, Raoul. “An Application of the Morse Theory to the Topology of Lie-Groups.” Bulletin de La Société Mathématique de France 84 (1956): 251–81.
• Chevalley, C. “Sur certains groupes simples.” Tohoku Mathematical Journal 7, no. 1–2 (1955): 14–66. doi:10.2748/tmj/1178245104.