**Identifier**

Identifier

Values

['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

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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