Identifier
Values
=>
Cc0022;cc-rep
['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
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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 $P_{u,w}(q)$.
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] Vogan, D. Number of pairs of permutation in $S_n$ whose µ-coefficient (of their Kazhdan Lusztig polynomial) is non-zero MathOverflow:298028
[2] Warrington, G. S. Equivalence classes for the µ-coefficient of Kazhdan-Lusztig polynomials in $S_n$ MathSciNet:2859901
[2] 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
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