- FindStat

Identifier

St001144: Finite Cartan types ⟶ ℤ

Values

['A',1] => 1
['A',2] => 1
['B',2] => 1
['G',2] => 1
['A',3] => 1
['B',3] => 1
['C',3] => 1
['A',4] => 1
['B',4] => 1
['C',4] => 1
['D',4] => 1
['F',4] => 1
['A',5] => 1
['B',5] => 2
['C',5] => 2
['D',5] => 1
['A',6] => 1
['A',7] => 1
['A',8] => 1
['A',9] => 5
['A',10] => 28

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$F_{1} = q$

$F_{2} = 3\ q$

$F_{3} = 3\ q$

$F_{4} = 5\ q$

$F_{5} = 2\ q + 2\ q^{2}$

Description

The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Weyl group of given type.
The

$\mu$ -coefficient of the Kazhdan-Lusztig polynomial

$P_{u,w}(q)$ is the coefficient of

$q^{\frac{l(w)-l(u)-1}{2}}$ in

$P_{u,w}(q)$ .

References

[1] Warrington, G. S. Equivalence classes for the µ-coefficient of Kazhdan-Lusztig polynomials in $S_n$ MathSciNet:2859901

Code

def statistic(C):
    W = CoxeterGroup(C, implementation='coxeter3')
    r = []
    for u in W:
	U = (W(v) for v in W.bruhat_interval(u, W.long_element()))
        next(U)
        for v in U:
            ldiff = v.length()-u.length()-1
            if is_even(ldiff):
                p = W.kazhdan_lusztig_polynomial(u, v)
                r.append(p[ldiff//2])
    return max(r)

Created

Apr 18, 2018 at 22:49 by Martin Rubey

Updated

Apr 18, 2018 at 22:49 by Martin Rubey