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converted canned example to regular example
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@galettof
galettof committed Jul 26, 2021
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244 changes: 36 additions & 208 deletions BettiCharacters.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -661,114 +661,18 @@ Node
characters, we work over the cyclotomic field obtained by
adjoining a primitive 7th root of unity to $\mathbb{Q}$.
(This example was precompiled by the package author.)
CannedExample
i1 : kk=toField(QQ[a]/ideal(sum apply(7,i->a^i)))

o1 = kk

o1 : PolynomialRing
i2 : R=kk[x,y,z]

o2 = R

o2 : PolynomialRing
i3 : f4=x^3*y+y^3*z+z^3*x

3 3 3
o3 = x y + y z + x*z

o3 : R
i4 : H=jacobian transpose jacobian f4

o4 = {-3} | 6xy 3x2 3z2 |
{-3} | 3x2 6yz 3y2 |
{-3} | 3z2 3y2 6xz |

3 3
o4 : Matrix R <--- R
i5 : f6=-1/54*det(H)

5 5 2 2 2 5
o5 = x*y + x z - 5x y z + y*z

o5 : R
i6 : I=minors(2,jacobian matrix{{f4,f6}})

3 5 7 7 4 2 2 4 3 2 5 8 7
o6 = ideal (14x y - 5x z - 3y z - 35x y z + 35x*y z - 7x y*z + z , 3x y -
------------------------------------------------------------------------
8 4 3 5 2 5 3 2 2 4 7 8 7 5 2
y - 35x y z + 7x*y z - 14x z + 35x y z + 5y*z , x - 5x*y - 7x y z
------------------------------------------------------------------------
2 4 2 3 4 3 5 7
- 35x y z + 35x y*z + 14y z - 3x*z )

o6 : Ideal of R
i7 : RI=res I

1 3 2
o7 = R <-- R <-- R <-- 0

0 1 2 3

o7 : ChainComplex
i8 : betti RI

0 1 2
o8 = total: 1 3 2
0: 1 . .
1: . . .
2: . . .
3: . . .
4: . . .
5: . . .
6: . . .
7: . 3 .
8: . . .
9: . . 1
10: . . .
11: . . 1

o8 : BettiTally
i9 : I2=I^2;

o9 : Ideal of R
i10 : RI2=res I2

1 6 6 1
o10 = R <-- R <-- R <-- R <-- 0

0 1 2 3 4

o10 : ChainComplex
i11 : betti RI2

0 1 2 3
o11 = total: 1 6 6 1
0: 1 . . .
1: . . . .
2: . . . .
3: . . . .
4: . . . .
5: . . . .
6: . . . .
7: . . . .
8: . . . .
9: . . . .
10: . . . .
11: . . . .
12: . . . .
13: . . . .
14: . . . .
15: . 6 . .
16: . . . .
17: . . 3 .
18: . . . .
19: . . 3 .
20: . . . .
21: . . . 1

o11 : BettiTally
Example
kk=toField(QQ[a]/ideal(sum apply(7,i->a^i)))
R=kk[x,y,z]
f4=x^3*y+y^3*z+z^3*x
H=jacobian transpose jacobian f4
f6=-1/54*det(H)
I=minors(2,jacobian matrix{{f4,f6}})
RI=res I
betti RI
I2=I^2;
RI2=res I2
betti RI2
Text
The unique simple group of order 168 acts as described
in §2.2 of @HREF("https://doi.org/10.1093/imrn/rnx329",
Expand All @@ -782,86 +686,24 @@ Node
The action of the group on the resolution of
both ideals is described in the second proof of
Proposition 8.1.
CannedExample
i12 : g=matrix{{a^4,0,0},{0,a^2,0},{0,0,a}}

o12 = | a4 0 0 |
| 0 a2 0 |
| 0 0 a |

3 3
o12 : Matrix kk <--- kk
i13 : h=matrix{{0,1,0},{0,0,1},{1,0,0}}

o13 = | 0 1 0 |
| 0 0 1 |
| 1 0 0 |

3 3
o13 : Matrix ZZ <--- ZZ
i14 : i=(2*a^4+2*a^2+2*a+1)/7 * matrix{
{a-a^6,a^2-a^5,a^4-a^3},
{a^2-a^5,a^4-a^3,a-a^6},
{a^4-a^3,a-a^6,a^2-a^5}
}

o14 = | 3/7a5+1/7a4+1/7a3+3/7a2-1/7 -1/7a5+2/7a4+2/7a3-1/7a2-2/7
| -1/7a5+2/7a4+2/7a3-1/7a2-2/7 -2/7a5-3/7a4-3/7a3-2/7a2-4/7
| -2/7a5-3/7a4-3/7a3-2/7a2-4/7 3/7a5+1/7a4+1/7a3+3/7a2-1/7
-----------------------------------------------------------------------
-2/7a5-3/7a4-3/7a3-2/7a2-4/7 |
3/7a5+1/7a4+1/7a3+3/7a2-1/7 |
-1/7a5+2/7a4+2/7a3-1/7a2-2/7 |

3 3
o14 : Matrix kk <--- kk
i15 : j=-1/(2*a^4+2*a^2+2*a+1) * matrix{
{a^5-a^4,1-a^5,1-a^3},
{1-a^5,a^6-a^2,1-a^6},
{1-a^3,1-a^6,a^3-a}
}

o15 = | -1/7a5-1/7a4+2/7a2-2/7a+2/7 1/7a5+4/7a4+2/7a3+2/7a2+4/7a+1/7
| 1/7a5+4/7a4+2/7a3+2/7a2+4/7a+1/7 1/7a5-1/7a4+1/7a3+3/7a+3/7
| -2/7a5-1/7a3+2/7a2+2/7a-1/7 1/7a5+3/7a4-1/7a3+3/7a2+1/7a
-----------------------------------------------------------------------
-2/7a5-1/7a3+2/7a2+2/7a-1/7 |
1/7a5+3/7a4-1/7a3+3/7a2+1/7a |
2/7a4-1/7a3-2/7a2-1/7a+2/7 |

3 3
o15 : Matrix kk <--- kk
i16 : G={id_(R^3),i,h,j,g,inverse g};
i17 : A1=action(RI,G,Sub=>false)

o17 = ChainComplex with 6 actors

o17 : ActionOnComplex
i18 : A2=action(RI2,G,Sub=>false)

o18 = ChainComplex with 6 actors

o18 : ActionOnComplex
i19 : character A1

o19 = HashTable{0 => GradedCharacter{{0} => Character{1, 1, 1, 1, 1, 1}} }
4 2 4 2
1 => GradedCharacter{{8} => Character{3, -1, 0, 1, a + a + a, - a - a - a - 1}}
2 => GradedCharacter{{11} => Character{1, 1, 1, 1, 1, 1}}
{13} => Character{1, 1, 1, 1, 1, 1}

o19 : HashTable
i20 : character A2

o20 = HashTable{0 => GradedCharacter{{0} => Character{1, 1, 1, 1, 1, 1}} }
1 => GradedCharacter{{16} => Character{6, 2, 0, 0, -1, -1}}
4 2 4 2
2 => GradedCharacter{{19} => Character{3, -1, 0, 1, a + a + a, - a - a - a - 1}}
4 2 4 2
{21} => Character{3, -1, 0, 1, a + a + a, - a - a - a - 1}
3 => GradedCharacter{{24} => Character{1, 1, 1, 1, 1, 1}}

o20 : HashTable
Example
g=matrix{{a^4,0,0},{0,a^2,0},{0,0,a}}
h=matrix{{0,1,0},{0,0,1},{1,0,0}}
i=(2*a^4+2*a^2+2*a+1)/7 * matrix{
{a-a^6,a^2-a^5,a^4-a^3},
{a^2-a^5,a^4-a^3,a-a^6},
{a^4-a^3,a-a^6,a^2-a^5}
}
j=-1/(2*a^4+2*a^2+2*a+1) * matrix{
{a^5-a^4,1-a^5,1-a^3},
{1-a^5,a^6-a^2,1-a^6},
{1-a^3,1-a^6,a^3-a}
}
G={id_(R^3),i,h,j,g,inverse g};
A1=action(RI,G,Sub=>false)
A2=action(RI2,G,Sub=>false)
elapsedTime character A1
elapsedTime character A2
Text
The character of the resolution of the ideal shows the
free module in homological degree two is a direct sum
Expand All @@ -879,26 +721,12 @@ Node
24 (by local duality); in
particular, it is a trivial representation. We can verify
this directly.
CannedExample
i21 : needsPackage "SymbolicPowers"

o21 = SymbolicPowers

o21 : Package
i22 : Is2 = symbolicPower(I,2);

o22 : Ideal of R
i23 : M = Is2 / I2;
i24 : B = action(M,G,Sub=>false)

o24 = Module with 6 actors

o24 : ActionOnGradedModule
i25 : character(B,21)

o25 = GradedCharacter{{21} => Character{1, 1, 1, 1, 1, 1}}

o25 : GradedCharacter
Example
needsPackage "SymbolicPowers"
Is2 = symbolicPower(I,2);
M = Is2 / I2;
B = action(M,G,Sub=>false)
elapsedTime character(B,21)


Node
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