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Statistic identifier: St001790

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Collection: Finite Cartan types

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Description: The number of reflection subgroups of the associated Weyl group.
Let $\mathcal{R} \subseteq W$ be the set of reflections in the Weyl group $W$.
A (possibly empty) subset $X \subseteq \mathcal{R}$ generates a subgroup of $W$ that is again a reflection group. This is the number of all pairwise different subgroups of $W$ obtained this way (including the trivial subgroup).
If $\Phi^+$ is an associated set of positive roots, then this also is the number of subsets $Y \subseteq \Phi^+$ such that $Y$ is a simple system of some type (including the empty system for type $A_0$).
Such a subset $Y$ is identified as simple system if for all $x \neq y \in Y$ we have $\langle x,y \rangle \leq 0$.

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References: 

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Code:


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Statistic values:

['A',1] => 2
['A',2] => 5
['B',2] => 8
['G',2] => 13
['A',3] => 15
['B',3] => 38
['C',3] => 38
['A',4] => 52
['B',4] => 218
['C',4] => 218
['D',4] => 75
['F',4] => 637
['A',5] => 203
['B',5] => 1430
['C',5] => 1430
['D',5] => 428
['A',6] => 877
['B',6] => 10514
['C',6] => 10514
['D',6] => 2781
['E',6] => 5079
['A',7] => 4140
['B',7] => 85202
['C',7] => 85202
['D',7] => 20093
['E',7] => 107911
['A',8] => 21147
['B',8] => 751982
['C',8] => 751982
['D',8] => 159340
['E',8] => 7591975
['C',2] => 8

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Created: May 03, 2022 at 13:32 by Dennis Jahn

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Last Updated: May 04, 2022 at 11:00 by Dennis Jahn