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Identifier
Values
=>
Cc0022;cc-rep
['A',1]=>1 ['A',2]=>3 ['B',2]=>6 ['G',2]=>15 ['A',3]=>16 ['B',3]=>68 ['C',3]=>68 ['A',4]=>125 ['B',4]=>1138 ['C',4]=>1138 ['D',4]=>315 ['F',4]=>7560 ['A',5]=>1296 ['B',5]=>25218 ['C',5]=>25218 ['D',5]=>7712 ['A',6]=>16807 ['B',6]=>695860 ['C',6]=>695860 ['D',6]=>228055 ['E',6]=>846720 ['A',7]=>262144 ['B',7]=>22985512 ['C',7]=>22985512 ['D',7]=>7932816 ['E',7]=>221714415 ['A',8]=>4782969 ['C',2]=>6
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Description
The number of subsets of the positive roots that form a basis of the associated vector space.
For the group $W$ and an associated set of positive roots $\Phi^+ \subseteq V$ this counts the number of subsets $S \subseteq \Phi^+$ that form a basis of $V$.
This is also the number of subsets of the reflections $R \subseteq W$ that form a minimal set of generators of a reflection subgroup of full rank.
The Coxeter permutahedron can be defined as the Minkowski sum of the line segments $[- \frac{\alpha}{2}, \frac{\alpha}{2}]$ for $\alpha \in \Phi^+$. As a zonotope this polytope can be decomposed into a (disjoint) union of (half-open) parallel epipeds [1]. This also counts the number of full dimensional parallel epipeds among this decomposition.
References
[1] Shephard, G. C. Combinatorial properties of associated zonotopes MathSciNet:0362054 DOI:10.4153/CJM-1974-032-5
Created
Dec 13, 2021 at 13:33 by Dennis Jahn
Updated
May 04, 2022 at 12:03 by Dennis Jahn