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 | 1 | initial version |
From the first definition in the OP, it is enough to compute the $a$-funcition on involutions. Here is a simple piece of code doing this in a verbose form for $S_3$, the group of permutations of the set ${1,2,3}$:
W = WeylGroup('A2', prefix='s')
print(f"W = {W}")
print(f"W has order |W| = {W.order()}\n")
R.<q> = LaurentPolynomialRing(QQ)
KL = KazhdanLusztigPolynomial(W, q)
involutions = [w for w in W if w.order() in (1, 2)]
for w in involutions:
lw = w.length()
pw = KL.P( W(1), w )
dw = pw.degree()
aw = lw - 2*dw
print(f"w = {w} = {w.to_permutation()} has:\n"
f"l(w) = {lw}\n"
f"d(w) = deg P(1,w) = deg( {pw} ) = {dw}\n"
f"a(w) = l(w) - 2d(w) = {aw}\n")
Results:
W = Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
W has order |W| = 6
w = 1 = (1, 2, 3) has:
l(w) = 0
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 0
w = s1 = (2, 1, 3) has:
l(w) = 1
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 1
w = s2 = (1, 3, 2) has:
l(w) = 1
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 1
w = s1*s2*s1 = (3, 2, 1) has:
l(w) = 3
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 3
(All K-L polynomials are trivial.) To do the same for the group $S_4$, using a less verbose output...
W = WeylGroup('A3', prefix='s')
print(f"W = {W}")
print(f"W has order |W| = {W.order()}\n")
R.<q> = LaurentPolynomialRing(QQ)
KL = KazhdanLusztigPolynomial(W, q)
involutions = [w for w in W if w.order() in (1, 2)]
for w in involutions:
lw = w.length()
pw = KL.P( W(1), w )
dw = pw.degree()
aw = lw - 2*dw
print(f"w = {w} = {w.to_permutation()} has:\n"
f"a(w) = l(w) - 2d(w) = {lw} - 2deg({pw}) = {aw}\n")
Results:
W = Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)
W has order |W| = 24
w = 1 = (1, 2, 3, 4) has:
a(w) = l(w) - 2d(w) = 0 - 2deg(1) = 0
w = s3 = (1, 2, 4, 3) has:
a(w) = l(w) - 2d(w) = 1 - 2deg(1) = 1
w = s2 = (1, 3, 2, 4) has:
a(w) = l(w) - 2d(w) = 1 - 2deg(1) = 1
w = s2*s3*s2 = (1, 4, 3, 2) has:
a(w) = l(w) - 2d(w) = 3 - 2deg(1) = 3
w = s2*s3*s1*s2 = (3, 4, 1, 2) has:
a(w) = l(w) - 2d(w) = 4 - 2deg(1 + q) = 2
w = s1 = (2, 1, 3, 4) has:
a(w) = l(w) - 2d(w) = 1 - 2deg(1) = 1
w = s3*s1 = (2, 1, 4, 3) has:
a(w) = l(w) - 2d(w) = 2 - 2deg(1) = 2
w = s1*s2*s3*s2*s1 = (4, 2, 3, 1) has:
a(w) = l(w) - 2d(w) = 5 - 2deg(1 + q) = 3
w = s1*s2*s1 = (3, 2, 1, 4) has:
a(w) = l(w) - 2d(w) = 3 - 2deg(1) = 3
w = s1*s2*s3*s1*s2*s1 = (4, 3, 2, 1) has:
a(w) = l(w) - 2d(w) = 6 - 2deg(1) = 6
I would like to have the KL-cells inside sage, but i do not know how to get them now.
Personal note: There was some years ago the chance to use gap3+chevie, then make sage know about them, then get the corresponding information from chevie. Since gap4 i no longer use gap. After repeated switches between linux distros and linux reinstallations, i decided that it is simpler to rewrite all pieces of code inside chevie to be inside sage. But i need time and some support... maybe this could be a project for this year.
Let's try to get the same numbers using the RSK correspondence. The code results are also formatted to be printed in an array...
W = WeylGroup('A3', prefix='s')
print(f"W = {W}")
print(f"W has order |W| = {W.order()}\n")
R.<q> = LaurentPolynomialRing(QQ)
KL = KazhdanLusztigPolynomial(W, q)
def a_involution(w):
lw = w.length()
pw = KL.P( W(1), w )
dw = pw.degree()
aw = lw - 2*dw
return aw
def a_page198(shape):
return sum([binomial(entry, 2) for entry in shape])
def transposed_shape(shape):
"""this should be a basic method, but i could not find it...
For the shape [3,2,2,1] we return [4,3,1] and conversely...
[3,2,2,1] is [4,3,1] is
XXX XXXX
XX XXX
XX X
X
"""
n = shape[0]
return [len([rowlength for rowlength in shape if rowlength > k]) for k in range(n)]
for w in W:
wp = w.to_permutation()
Q = RSK( list(wp) )[1]
sh = list(Q.shape())
sht = transposed_shape(sh)
if w.order() in (1, 2):
a_inv = a_involution(w)
else:
a_inv = ''
aw = a_page198(sht)
print(r"{} & {} & {} & {} & {} & {} & {}\\\hline"
.format(latex(w), wp, Q, sh, sht, a_inv, aw))
Results:
W = Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
W has order |W| = 6
1 & (1, 2, 3) & [[1, 2, 3]] & [3] & [1, 1, 1] & 0 & 0\\hline s_{1}s_{2} & (2, 3, 1) & [[1, 2], [3]] & [2, 1] & [2, 1] & & 1\\hline s_{2}s_{1} & (3, 1, 2) & [[1, 3], [2]] & [2, 1] & [2, 1] & & 1\\hline s_{1} & (2, 1, 3) & [[1, 3], [2]] & [2, 1] & [2, 1] & 1 & 1\\hline s_{2} & (1, 3, 2) & [[1, 2], [3]] & [2, 1] & [2, 1] & 1 & 1\\hline s_{1}s_{2}s_{1} & (3, 2, 1) & [[1], [2], [3]] & [1, 1, 1] & [3] & 3 & 3\\hline
In latex: $$ \begin{array}[|l|c|l|l|l|r|r|] \hline w\in W & w\in S_3 & Q &\text{shape}(Q) &\text{shape}(Q)^t & a\text{ if inv.} &a\\hline\hline 1 & (1, 2, 3) & [[1, 2, 3]] & [3] & [1, 1, 1] & 0 & 0\\hline s_{1}s_{2} & (2, 3, 1) & [[1, 2], [3]] & [2, 1] & [2, 1] & & 1\\hline s_{2}s_{1} & (3, 1, 2) & [[1, 3], [2]] & [2, 1] & [2, 1] & & 1\\hline s_{1} & (2, 1, 3) & [[1, 3], [2]] & [2, 1] & [2, 1] & 1 & 1\\hline s_{2} & (1, 3, 2) & [[1, 2], [3]] & [2, 1] & [2, 1] & 1 & 1\\hline s_{1}s_{2}s_{1} & (3, 2, 1) & [[1], [2], [3]] & [1, 1, 1] & [3] & 3 & 3\\hline \end{array} $$
Let us do the same also for the $A_3$ case. The code remains, we only change the type in the WeylGroup constructor to A3, the result is in a table...
$$ \begin{array}[|l|c|l|l|l|r|r|] \hline 1 & (1, 2, 3, 4) & [[1, 2, 3, 4]] & [4] & [1, 1, 1, 1] & 0 & 0\\hline s_{3} & (1, 2, 4, 3) & [[1, 2, 3], [4]] & [3, 1] & [2, 1, 1] & 1 & 1\\hline s_{3}s_{2} & (1, 4, 2, 3) & [[1, 2, 4], [3]] & [3, 1] & [2, 1, 1] & & 1\\hline s_{3}s_{2}s_{1} & (4, 1, 2, 3) & [[1, 3, 4], [2]] & [3, 1] & [2, 1, 1] & & 1\\hline s_{2} & (1, 3, 2, 4) & [[1, 2, 4], [3]] & [3, 1] & [2, 1, 1] & 1 & 1\\hline s_{2}s_{3} & (1, 3, 4, 2) & [[1, 2, 3], [4]] & [3, 1] & [2, 1, 1] & & 1\\hline s_{2}s_{3}s_{2} & (1, 4, 3, 2) & [[1, 2], [3], [4]] & [2, 1, 1] & [3, 1] & 3 & 3\\hline s_{2}s_{3}s_{2}s_{1} & (4, 1, 3, 2) & [[1, 3], [2], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline s_{2}s_{1} & (3, 1, 2, 4) & [[1, 3, 4], [2]] & [3, 1] & [2, 1, 1] & & 1\\hline s_{2}s_{3}s_{1} & (3, 1, 4, 2) & [[1, 3], [2, 4]] & [2, 2] & [2, 2] & & 2\\hline s_{2}s_{3}s_{1}s_{2} & (3, 4, 1, 2) & [[1, 2], [3, 4]] & [2, 2] & [2, 2] & 2 & 2\\hline s_{2}s_{3}s_{1}s_{2}s_{1} & (4, 3, 1, 2) & [[1, 4], [2], [3]] & [2, 1, 1] & [3, 1] & & 3\\hline s_{1} & (2, 1, 3, 4) & [[1, 3, 4], [2]] & [3, 1] & [2, 1, 1] & 1 & 1\\hline s_{3}s_{1} & (2, 1, 4, 3) & [[1, 3], [2, 4]] & [2, 2] & [2, 2] & 2 & 2\\hline s_{3}s_{1}s_{2} & (2, 4, 1, 3) & [[1, 2], [3, 4]] & [2, 2] & [2, 2] & & 2\\hline s_{3}s_{1}s_{2}s_{1} & (4, 2, 1, 3) & [[1, 4], [2], [3]] & [2, 1, 1] & [3, 1] & & 3\\hline s_{1}s_{2} & (2, 3, 1, 4) & [[1, 2, 4], [3]] & [3, 1] & [2, 1, 1] & & 1\\hline s_{1}s_{2}s_{3} & (2, 3, 4, 1) & [[1, 2, 3], [4]] & [3, 1] & [2, 1, 1] & & 1\\hline s_{1}s_{2}s_{3}s_{2} & (2, 4, 3, 1) & [[1, 2], [3], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline s_{1}s_{2}s_{3}s_{2}s_{1} & (4, 2, 3, 1) & [[1, 3], [2], [4]] & [2, 1, 1] & [3, 1] & 3 & 3\\hline s_{1}s_{2}s_{1} & (3, 2, 1, 4) & [[1, 4], [2], [3]] & [2, 1, 1] & [3, 1] & 3 & 3\\hline s_{1}s_{2}s_{3}s_{1} & (3, 2, 4, 1) & [[1, 3], [2], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline s_{1}s_{2}s_{3}s_{1}s_{2} & (3, 4, 2, 1) & [[1, 2], [3], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline s_{1}s_{2}s_{3}s_{1}s_{2}s_{1} & (4, 3, 2, 1) & [[1], [2], [3], [4]] & [1, 1, 1, 1] & [4] & 6 & 6\\hline \end{array} $$
| | 2 | No.2 Revision |
From the first definition in the OP, it is enough to compute the $a$-funcition on involutions. Here is a simple piece of code doing this in a verbose form for $S_3$, the group of permutations of the set ${1,2,3}$:
W = WeylGroup('A2', prefix='s')
print(f"W = {W}")
print(f"W has order |W| = {W.order()}\n")
R.<q> = LaurentPolynomialRing(QQ)
KL = KazhdanLusztigPolynomial(W, q)
involutions = [w for w in W if w.order() in (1, 2)]
for w in involutions:
lw = w.length()
pw = KL.P( W(1), w )
dw = pw.degree()
aw = lw - 2*dw
print(f"w = {w} = {w.to_permutation()} has:\n"
f"l(w) = {lw}\n"
f"d(w) = deg P(1,w) = deg( {pw} ) = {dw}\n"
f"a(w) = l(w) - 2d(w) = {aw}\n")
Results:
W = Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
W has order |W| = 6
w = 1 = (1, 2, 3) has:
l(w) = 0
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 0
w = s1 = (2, 1, 3) has:
l(w) = 1
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 1
w = s2 = (1, 3, 2) has:
l(w) = 1
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 1
w = s1*s2*s1 = (3, 2, 1) has:
l(w) = 3
d(w) = deg P(1,w) = deg( 1 ) = 0
a(w) = l(w) - 2d(w) = 3
(All K-L polynomials are trivial.) To do the same for the group $S_4$, using a less verbose output...
W = WeylGroup('A3', prefix='s')
print(f"W = {W}")
print(f"W has order |W| = {W.order()}\n")
R.<q> = LaurentPolynomialRing(QQ)
KL = KazhdanLusztigPolynomial(W, q)
involutions = [w for w in W if w.order() in (1, 2)]
for w in involutions:
lw = w.length()
pw = KL.P( W(1), w )
dw = pw.degree()
aw = lw - 2*dw
print(f"w = {w} = {w.to_permutation()} has:\n"
f"a(w) = l(w) - 2d(w) = {lw} - 2deg({pw}) = {aw}\n")
Results:
W = Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)
W has order |W| = 24
w = 1 = (1, 2, 3, 4) has:
a(w) = l(w) - 2d(w) = 0 - 2deg(1) = 0
w = s3 = (1, 2, 4, 3) has:
a(w) = l(w) - 2d(w) = 1 - 2deg(1) = 1
w = s2 = (1, 3, 2, 4) has:
a(w) = l(w) - 2d(w) = 1 - 2deg(1) = 1
w = s2*s3*s2 = (1, 4, 3, 2) has:
a(w) = l(w) - 2d(w) = 3 - 2deg(1) = 3
w = s2*s3*s1*s2 = (3, 4, 1, 2) has:
a(w) = l(w) - 2d(w) = 4 - 2deg(1 + q) = 2
w = s1 = (2, 1, 3, 4) has:
a(w) = l(w) - 2d(w) = 1 - 2deg(1) = 1
w = s3*s1 = (2, 1, 4, 3) has:
a(w) = l(w) - 2d(w) = 2 - 2deg(1) = 2
w = s1*s2*s3*s2*s1 = (4, 2, 3, 1) has:
a(w) = l(w) - 2d(w) = 5 - 2deg(1 + q) = 3
w = s1*s2*s1 = (3, 2, 1, 4) has:
a(w) = l(w) - 2d(w) = 3 - 2deg(1) = 3
w = s1*s2*s3*s1*s2*s1 = (4, 3, 2, 1) has:
a(w) = l(w) - 2d(w) = 6 - 2deg(1) = 6
I would like to have the KL-cells inside sage, but i do not know how to get them now.
Personal note: There was some years ago the chance to use gap3+chevie, then make sage know about them, then get the corresponding information from chevie. Since gap4 i no longer use gap. After repeated switches between linux distros and linux reinstallations, i decided that it is simpler to rewrite all pieces of code inside chevie to be inside sage. But i need time and some support... maybe this could be a project for this year.
Let's try to get the same numbers using the RSK correspondence. The code results are also formatted to be printed in an array...
W = WeylGroup('A3', prefix='s')
print(f"W = {W}")
print(f"W has order |W| = {W.order()}\n")
R.<q> = LaurentPolynomialRing(QQ)
KL = KazhdanLusztigPolynomial(W, q)
def a_involution(w):
lw = w.length()
pw = KL.P( W(1), w )
dw = pw.degree()
aw = lw - 2*dw
return aw
def a_page198(shape):
return sum([binomial(entry, 2) for entry in shape])
def transposed_shape(shape):
"""this should be a basic method, but i could not find it...
For the shape [3,2,2,1] we return [4,3,1] and conversely...
[3,2,2,1] is [4,3,1] is
XXX XXXX
XX XXX
XX X
X
"""
n = shape[0]
return [len([rowlength for rowlength in shape if rowlength > k]) for k in range(n)]
for w in W:
wp = w.to_permutation()
Q = RSK( list(wp) )[1]
sh = list(Q.shape())
sht = transposed_shape(sh)
if w.order() in (1, 2):
a_inv = a_involution(w)
else:
a_inv = ''
aw = a_page198(sht)
print(r"{} & {} & {} & {} & {} & {} & {}\\\hline"
.format(latex(w), wp, Q, sh, sht, a_inv, aw))
Results:
W = Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
W has order |W| = 6
1 & (1, 2, 3) & [[1, 2, 3]] & [3] & [1, 1, 1] & 0 & 0\\hline
0\\\hline
s_{1}s_{2} & (2, 3, 1) & [[1, 2], [3]] & [2, 1] & [2, 1] & & 1\\hline
1\\\hline
s_{2}s_{1} & (3, 1, 2) & [[1, 3], [2]] & [2, 1] & [2, 1] & & 1\\hline
1\\\hline
s_{1} & (2, 1, 3) & [[1, 3], [2]] & [2, 1] & [2, 1] & 1 & 1\\hline
1\\\hline
s_{2} & (1, 3, 2) & [[1, 2], [3]] & [2, 1] & [2, 1] & 1 & 1\\hline
1\\\hline
s_{1}s_{2}s_{1} & (3, 2, 1) & [[1], [2], [3]] & [1, 1, 1] & [3] & 3 & 3\\hline
In latex:
$$
\begin{array}[|l|c|l|l|l|r|r|]
\hline
w\in W & w\in S_3 & Q &\text{shape}(Q) &\text{shape}(Q)^t & a\text{ if inv.} &a\\hline\hline
&a\\\hline\hline
1 & (1, 2, 3) & [[1, 2, 3]] & [3] & [1, 1, 1] & 0 & 0\\hline
0\\\hline
s_{1}s_{2} & (2, 3, 1) & [[1, 2], [3]] & [2, 1] & [2, 1] & & 1\\hline
1\\\hline
s_{2}s_{1} & (3, 1, 2) & [[1, 3], [2]] & [2, 1] & [2, 1] & & 1\\hline
1\\\hline
s_{1} & (2, 1, 3) & [[1, 3], [2]] & [2, 1] & [2, 1] & 1 & 1\\hline
1\\\hline
s_{2} & (1, 3, 2) & [[1, 2], [3]] & [2, 1] & [2, 1] & 1 & 1\\hline
1\\\hline
s_{1}s_{2}s_{1} & (3, 2, 1) & [[1], [2], [3]] & [1, 1, 1] & [3] & 3 & 3\\hline
3\\\hline
\end{array}
$$
Let us do the same also for the $A_3$ case. The code remains, we only change the type in the WeylGroup constructor to A3, the result is in a table...
Later edit: the hline's are not parsed here, so i remove them... (Same computing code, the one print format is changed.)
$$
\begin{array}[|l|c|l|l|l|r|r|]
\hline
\\
w\in W & w\in S_3 & Q &\text{shape}(Q) &\text{shape}(Q)^t & a\text{ if inv.} &a\\\\
1 & (1, 2, 3, 4) & [[1, 2, 3, 4]] & [4] & [1, 1, 1, 1] & 0 & 0\\hline
0\\
s_{3} & (1, 2, 4, 3) & [[1, 2, 3], [4]] & [3, 1] & [2, 1, 1] & 1 & 1\\hline
1\\
s_{3}s_{2} & (1, 4, 2, 3) & [[1, 2, 4], [3]] & [3, 1] & [2, 1, 1] & & 1\\hline
1\\
s_{3}s_{2}s_{1} & (4, 1, 2, 3) & [[1, 3, 4], [2]] & [3, 1] & [2, 1, 1] & & 1\\hline
1\\
s_{2} & (1, 3, 2, 4) & [[1, 2, 4], [3]] & [3, 1] & [2, 1, 1] & 1 & 1\\hline
1\\
s_{2}s_{3} & (1, 3, 4, 2) & [[1, 2, 3], [4]] & [3, 1] & [2, 1, 1] & & 1\\hline
1\\
s_{2}s_{3}s_{2} & (1, 4, 3, 2) & [[1, 2], [3], [4]] & [2, 1, 1] & [3, 1] & 3 & 3\\hline
3\\
s_{2}s_{3}s_{2}s_{1} & (4, 1, 3, 2) & [[1, 3], [2], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline
3\\
s_{2}s_{1} & (3, 1, 2, 4) & [[1, 3, 4], [2]] & [3, 1] & [2, 1, 1] & & 1\\hline
1\\
s_{2}s_{3}s_{1} & (3, 1, 4, 2) & [[1, 3], [2, 4]] & [2, 2] & [2, 2] & & 2\\hline
2\\
s_{2}s_{3}s_{1}s_{2} & (3, 4, 1, 2) & [[1, 2], [3, 4]] & [2, 2] & [2, 2] & 2 & 2\\hline
2\\
s_{2}s_{3}s_{1}s_{2}s_{1} & (4, 3, 1, 2) & [[1, 4], [2], [3]] & [2, 1, 1] & [3, 1] & & 3\\hline
3\\
s_{1} & (2, 1, 3, 4) & [[1, 3, 4], [2]] & [3, 1] & [2, 1, 1] & 1 & 1\\hline
1\\
s_{3}s_{1} & (2, 1, 4, 3) & [[1, 3], [2, 4]] & [2, 2] & [2, 2] & 2 & 2\\hline
2\\
s_{3}s_{1}s_{2} & (2, 4, 1, 3) & [[1, 2], [3, 4]] & [2, 2] & [2, 2] & & 2\\hline
2\\
s_{3}s_{1}s_{2}s_{1} & (4, 2, 1, 3) & [[1, 4], [2], [3]] & [2, 1, 1] & [3, 1] & & 3\\hline
3\\
s_{1}s_{2} & (2, 3, 1, 4) & [[1, 2, 4], [3]] & [3, 1] & [2, 1, 1] & & 1\\hline
1\\
s_{1}s_{2}s_{3} & (2, 3, 4, 1) & [[1, 2, 3], [4]] & [3, 1] & [2, 1, 1] & & 1\\hline
1\\
s_{1}s_{2}s_{3}s_{2} & (2, 4, 3, 1) & [[1, 2], [3], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline
3\\
s_{1}s_{2}s_{3}s_{2}s_{1} & (4, 2, 3, 1) & [[1, 3], [2], [4]] & [2, 1, 1] & [3, 1] & 3 & 3\\hline
3\\
s_{1}s_{2}s_{1} & (3, 2, 1, 4) & [[1, 4], [2], [3]] & [2, 1, 1] & [3, 1] & 3 & 3\\hline
3\\
s_{1}s_{2}s_{3}s_{1} & (3, 2, 4, 1) & [[1, 3], [2], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline
3\\
s_{1}s_{2}s_{3}s_{1}s_{2} & (3, 4, 2, 1) & [[1, 2], [3], [4]] & [2, 1, 1] & [3, 1] & & 3\\hline
3\\
s_{1}s_{2}s_{3}s_{1}s_{2}s_{1} & (4, 3, 2, 1) & [[1], [2], [3], [4]] & [1, 1, 1, 1] & [4] & 6 & 6\\hline
6\\
\end{array}
$$
