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.

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:336756Code

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**searching the database

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