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Identifier
Values
['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