"Poincare Series of Coxeter Groups"의 두 판 사이의 차이

2014년 7월 3일 (목) 11:25 판

introduction

틀:수학노트}

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Poincaré Series of a Coxeter Group $W$ (Poincare series for ring of coinvariants)
+* {{수학노트|url=콕세터_군의_푸앵카레_급수}}}
-$$
-P_{W}(q)=\sum_{w\in W}q^{\ell(w)}
-$$
-* 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
-$$
-\prod_{i=1}^{k}\begin{bmatrix} d_i \end{bmatrix}_{q}=\begin{bmatrix} 2 \end{bmatrix}_{q}\begin{bmatrix} 3 \end{bmatrix}_{q}=(1+q)(1+q+q^2)
-$$
@@ 32번째 줄: / 6번째 줄: @@
 * [[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:[http://dx.doi.org/10.1007/BF01431421 10.1007/BF01431421]

"Poincare Series of Coxeter Groups"의 두 판 사이의 차이

2014년 7월 3일 (목) 11:25 판

introduction

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