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Identifier
Values
=>
Cc0022;cc-rep
['A',1]=>1 ['A',2]=>2 ['B',2]=>2 ['G',2]=>2 ['A',3]=>6 ['B',3]=>8 ['C',3]=>8 ['A',4]=>22 ['B',4]=>46 ['C',4]=>46 ['D',4]=>30 ['F',4]=>94 ['A',5]=>101 ['B',5]=>340 ['C',5]=>340 ['D',5]=>212 ['A',6]=>573 ['B',6]=>3210 ['C',6]=>3210 ['D',6]=>1924 ['E',6]=>3662 ['A',7]=>3836 ['B',7]=>36336 ['C',7]=>36336 ['D',7]=>21280 ['E',7]=>131046 ['A',8]=>29228 ['B',8]=>484636 ['C',8]=>484636 ['D',8]=>277788 ['E',8]=>18210722
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Description
The largest coefficient in the PoincarĂ© polynomial of the Weyl group of given Cartan type.
The PoincarĂ© polynomial of a Weyl group $W$ is
$$\sum_{w\in W} q^{\ell(w)} = \prod_i [d_i]_q,$$
where $\ell$ denotes the Coxeter length, $d_1,\dots$ are the degrees (or exponents) of $W$ and $[n]_q=1 +\dots+q^{n-1}$ is the $q$-integer.
Thus, this statistic records the frequency of the most common length in the group.
References
[1] Gaichenkov, M. The growth of maximum elements for the reflection group $D_n$ MathOverflow:336756
Code
def statistic(C):
from sage.combinat.q_analogues import q_int
return max(prod(q_int(d, q) for d in WeylGroup(C).degrees()).list())


Created
Jul 22, 2019 at 22:51 by Martin Rubey
Updated
Aug 07, 2019 at 11:03 by Martin Rubey