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Minor changes to get BettiCharacters and HighestWeights to work with …
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@mikestillman
mikestillman committed Oct 16, 2024
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38 changes: 19 additions & 19 deletions M2/Macaulay2/packages/BettiCharacters.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -1083,7 +1083,7 @@ Node
I1=ideal apply({4,5,6,7}, i -> (x_1-x_2)*(x_3-x_i));
I2=ideal apply(subsets({3,4,5,6,7},2), s -> (x_1-x_(s#0))*(x_2-x_(s#1)));
I=I1+I2
RI=res I
RI=freeResolution I
betti RI
Text
Next we set up the group action on the resolution.
Expand Down Expand Up @@ -1152,7 +1152,7 @@ Node
Example
R=QQ[x_1..x_6]
I=intersect(apply(subsets(gens R,4),x->(ideal x)^3))
RI=res I
RI=freeResolution I
betti RI
Text
Next, we set up the group action on the resolution.
Expand Down Expand Up @@ -1215,10 +1215,10 @@ Node
H = jacobian transpose jacobian f4
f6 = -1/54*det(H)
I = minors(2,jacobian matrix{{f4,f6}})
RI = res I
RI = freeResolution I
betti RI
I2 = I^2;
RI2 = res I2
RI2 = freeResolution I2
betti RI2
Text
The unique simple group of order 168 acts as described
Expand Down Expand Up @@ -1330,7 +1330,7 @@ Node
Example
R = QQ[x_1,x_2,y_1,y_2,y_3,Degrees=>{2:{1,0},3:{0,1}}]
I = intersect(ideal(x_1,x_2),ideal(y_1,y_2,y_3))
RI = res I
RI = freeResolution I
G = {
matrix{{x_1,x_2,y_2,y_3,y_1}},
matrix{{x_1,x_2,y_2,y_1,y_3}},
Expand Down Expand Up @@ -1424,7 +1424,7 @@ Node
Example
R = QQ[x,y,z]
I = ideal(x*y,x*z,y*z)
RI = res I
RI = freeResolution I
S3 = symmetricGroupActors R
A = action(RI,S3)
a = character A
Expand Down Expand Up @@ -1521,7 +1521,7 @@ Node
Example
R = QQ[x_1..x_4]
I = ideal apply(subsets(gens R,2),product)
RI = res I
RI = freeResolution I
G = {matrix{{x_2,x_3,x_4,x_1}},
matrix{{x_2,x_3,x_1,x_4}},
matrix{{x_2,x_1,x_4,x_3}},
Expand Down Expand Up @@ -1566,7 +1566,7 @@ Node
these matrices by permuting columns of the identity.
Example
M = module I
RM = res M
RM = freeResolution M
G' = { (id_(R^6))_{2,4,5,0,1,3},
(id_(R^6))_{2,0,1,4,5,3},
(id_(R^6))_{0,4,3,2,1,5},
Expand All @@ -1586,7 +1586,7 @@ Node
module). This can be achieved as follows.
Example
E = Ext^3(R^1/I,R^{-4})
RE = res E
RE = freeResolution E
G'' = toList(5:id_(R^1))
action(RE,G,G'',3)
Caveat
Expand Down Expand Up @@ -1694,7 +1694,7 @@ Node
resolution.
Example
E = Ext^3(R^1/I,R^{-4})
RE = res E
RE = freeResolution E
G'' = toList(5:id_(R^1))
B = action(RE,G,G'',3)
G' = actors(B,0)
Expand Down Expand Up @@ -1729,7 +1729,7 @@ Node
Example
R = QQ[x_1..x_4]
I = ideal apply(subsets(gens R, 2), product)
RI = res I
RI = freeResolution I
S4 = symmetricGroupActors(R)
A = action(RI,S4)
G = {map(RI_3, RI_3, {{0, -1, 1}, {1, 1, 0}, {0, 1, 0}}),
Expand Down Expand Up @@ -1798,7 +1798,7 @@ Node
Example
R = QQ[x_1..x_4]
I = ideal apply(subsets(gens R,2),product)
RI = res I
RI = freeResolution I
G = {matrix{{x_2,x_3,x_4,x_1}},
matrix{{x_2,x_3,x_1,x_4}},
matrix{{x_2,x_1,x_4,x_3}},
Expand Down Expand Up @@ -1933,7 +1933,7 @@ Node
Example
R = QQ[x_1..x_4]
J = intersect(apply(subsets(gens R,3),x->(ideal x)^2))
RJ = res J
RJ = freeResolution J
G = { matrix{{x_2,x_3,x_4,x_1}},
matrix{{x_2,x_3,x_1,x_4}},
matrix{{x_2,x_1,x_4,x_3}},
Expand Down Expand Up @@ -1991,7 +1991,7 @@ Node
Example
R = QQ[x_1..x_4]
J = intersect(apply(subsets(gens R,3),x->(ideal x)^2))
RJ = res J
RJ = freeResolution J
G = { matrix{{x_2,x_3,x_4,x_1}},
matrix{{x_2,x_3,x_1,x_4}},
matrix{{x_2,x_1,x_4,x_3}},
Expand Down Expand Up @@ -2708,7 +2708,7 @@ Node
Example
R = QQ[x_1..x_4]
I = ideal apply(subsets(gens R,2),product)
RI = res I
RI = freeResolution I
G = { (id_(R^4))_{1,2,3,0},
(id_(R^4))_{1,2,0,3},
(id_(R^4))_{1,0,3,2},
Expand Down Expand Up @@ -2856,7 +2856,7 @@ TEST ///
clearAll
R = QQ[x,y,z]
I = ideal(x*y,x*z,y*z)
RI = res I
RI = freeResolution I
S3 = {matrix{{y,z,x}},matrix{{y,x,z}},matrix{{x,y,z}}}
assert(S3 == symmetricGroupActors(R))
A = action(RI,S3)
Expand Down Expand Up @@ -2918,7 +2918,7 @@ I = ideal(
(x_1-x_2)*(x_3-x_5),
(x_1-x_2)*(x_3-x_4)
)
RI = res I
RI = freeResolution I
S5 = for p in partitions(5) list (
L := gens R;
g := for u in p list (
Expand Down Expand Up @@ -3030,7 +3030,7 @@ TEST ///
clearAll
R = QQ[x_1,x_2,y_1,y_2,Degrees=>{2:{1,0},2:{0,1}}]
I = intersect(ideal(x_1,x_2),ideal(y_1,y_2))
RI = res I
RI = freeResolution I
G = {
matrix{{x_2,x_1,y_2,y_1}},
matrix{{x_2,x_1,y_1,y_2}},
Expand Down Expand Up @@ -3060,7 +3060,7 @@ assert(character(C,{0,2}) ++ character(C,{2,0}) == b)
TEST ///
clearAll
R = QQ[x_1..x_4]
K = res ideal vars R
K = freeResolution ideal vars R
S4 = symmetricGroupActors(R)
A = action(K,S4)
c = character A
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2 changes: 1 addition & 1 deletion M2/Macaulay2/packages/HighestWeights/doc.m2
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Expand Up @@ -538,7 +538,7 @@ doc ///
R=QQ[x_(1,1)..x_(3,4)];
G=genericMatrix(R,4,3)
I=minors(2,G);
resI=res I
resI=freeResolution I
betti resI
D=dynkinType{{"A",2},{"A",3}};
U={{1,0,1,0,0},{1,0,-1,1,0},{1,0,0,-1,1},{1,0,0,0,-1},{-1,1,1,0,0},{-1,1,-1,1,0},{-1,1,0,-1,1},{-1,1,0,0,-1},{0,-1,1,0,0},{0,-1,-1,1,0},{0,-1,0,-1,1},{0,-1,0,0,-1}};
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12 changes: 6 additions & 6 deletions M2/Macaulay2/packages/HighestWeights/examples.m2
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Expand Up @@ -28,7 +28,7 @@ doc ///

Example
I=Grassmannian(1,5,CoefficientRing=>QQ); R=ring I;
RI=res I; betti RI
RI=freeResolution I; betti RI
Text
Now we assign weights to the variables of $R$. First we input the weights of $e_0,\ldots,e_5$ in a list @TT "L"@.

Expand Down Expand Up @@ -91,7 +91,7 @@ doc ///
Example
R=QQ[x_(1,1)..x_(6,3)];
G=genericMatrix(R,3,6)
M=coker G; BR=res M; betti BR
M=coker G; BR=freeResolution M; betti BR
Text
The ring $R$ carries a degree compatible action of $SL_6 ({\mathbb C}) \times SL_3 ({\mathbb C})$. Define the map of graded free $R$-modules
$$\phi : E \otimes R(-1) \rightarrow F^* \otimes R, e_i \otimes 1 \mapsto \sum_{j=1}^3 f_j^* \otimes x_{i,j}$$
Expand Down Expand Up @@ -151,7 +151,7 @@ doc ///
R=QQ[x_(1,1)..x_(3,2),y_(1,1)..y_(4,2),Degrees=>{6:{1,0},8:{0,1}}]
G=genericMatrix(R,2,3)|genericMatrix(R,y_(1,1),2,4)
I=minors(2,G);
EN=res I; betti EN
EN=freeResolution I; betti EN
Text
Notice how the first three columns of $G$ involve only the $x_{i,j}$ variables while the other columns involve only the $y_{i,j}$ variables. In fact, $G$ is the matrix of the map
$$\phi : E\otimes R(-1,0) \oplus F\otimes R(0,-1) \rightarrow H^* \otimes R, e_i \otimes 1 \mapsto \sum_{k=1}^2 h_k^* \otimes x_{i,k}, f_j \otimes 1 \mapsto \sum_{k=1}^2 h_k^* \otimes y_{j,k}$$
Expand Down Expand Up @@ -229,7 +229,7 @@ doc ///
Next we generate the resolution and obtain its decomposition.

Example
EFW=res coker f; betti EFW
EFW=freeResolution coker f; betti EFW
highestWeightsDecomposition(EFW,0,W)
Text
We conclude that with the action of $SL(E)$ the complex has the
Expand Down Expand Up @@ -300,7 +300,7 @@ doc ///

Example
I=ideal jacobian ideal Delta;
RI=res I; betti RI
RI=freeResolution I; betti RI
Text
The root system of type $C_3$ is contained in $\RR^3$. It is easy
to express the weight of each variable of the ring $R$ with respect
Expand Down Expand Up @@ -399,7 +399,7 @@ doc ///
x_{0,2}*x_{1,2,3,4}-x_{1,2}*x_{0,2,3,4}+x_{2,3}*x_{0,1,2,4}-x_{2,4}*x_{0,1,2,3},
-x_{0,3}*x_{1,2,3,4}+x_{1,3}*x_{0,2,3,4}-x_{2,3}*x_{0,1,3,4}+x_{3,4}*x_{0,1,2,3},
x_{0,4}*x_{1,2,3,4}-x_{1,4}*x_{0,2,3,4}+x_{2,4}*x_{0,1,3,4}-x_{3,4}*x_{0,1,2,4});
RI=res I; betti RI
RI=freeResolution I; betti RI
Text
The root system of type $D_5$ is contained in $\RR^5$. It is easy
to express the weight of each variable of the ring $R$ with respect
Expand Down

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