Identifier
Identifier
Values
['A',1] => 1
['A',2] => 8
['B',2] => 12
['G',2] => 20
['A',3] => 60
['B',3] => 152
['C',3] => 152
['A',4] => 482
['B',4] => 2148
['C',4] => 2148
['D',4] => 892
['F',4] => 8920
['A',5] => 4268
['B',5] => 35070
['C',5] => 35070
['D',5] => 14874
['A',6] => 41934
['B',6] => 679152
['C',6] => 679152
['D',6] => 287438
['E',6] => 846476
['A',7] => 457782
Description
The number of pairs in the Weyl group of given type with mu-coefficient of the Kazhdan Lusztig polynomial being non-zero.
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
 Vogan, D. Number of pairs of permutation in $S_n$ whose µ-coefficient (of their Kazhdan Lusztig polynomial) is non-zero MathOverflow:298028
 Warrington, G. S. Equivalence classes for the µ-coefficient of Kazhdan-Lusztig polynomials in $S_n$ MathSciNet:2859901
Code
def statistic(C):
"""
sage: statistic(CartanType(["A", 4]))
482
"""
W = CoxeterGroup(C, implementation='coxeter3')
r = 0
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)
if p[ldiff//2] != 0:
r += 1
return r


Created
Apr 18, 2018 at 22:32 by Martin Rubey
Updated
Apr 18, 2018 at 22:32 by Martin Rubey