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D.3.1.31 exteriorPower
Procedure from library matrix.lib (see matrix_lib).
- Usage:
- exteriorPower(A, k); A module, k int
- Return:
- module: the k-th exterior power of A
- Note:
- the chosen bases and most of intermediate data will be shown if
printlevel is big enough. Last rows will be invisible if zero.
Example:
| LIB "matrix.lib";
ring r = (0),(a, b, c, d, e, f), dp;
r; "base ring:";
==> // coefficients: QQ
==> // number of vars : 6
==> // block 1 : ordering dp
==> // : names a b c d e f
==> // block 2 : ordering C
==> base ring:
module B = a*gen(1) + c*gen(2) + e*gen(3),
b*gen(1) + d*gen(2) + f*gen(3),
e*gen(1) + f*gen(3);
print(B);
==> a,b,e,
==> c,d,0,
==> e,f,f
print(exteriorPower(B, 2));
==> df, cf, -de+cf,
==> bf-ef,-e2+af,-be+af,
==> -de, -ce, -bc+ad
print(exteriorPower(B, 3));
==> -de2-bcf+adf+cef
def g = superCommutative(); setring g; g;
==> // coefficients: QQ
==> // number of vars : 6
==> // block 1 : ordering dp
==> // : names a b c d e f
==> // block 2 : ordering C
==> // noncommutative relations:
==> // ba=-ab
==> // ca=-ac
==> // da=-ad
==> // ea=-ae
==> // fa=-af
==> // cb=-bc
==> // db=-bd
==> // eb=-be
==> // fb=-bf
==> // dc=-cd
==> // ec=-ce
==> // fc=-cf
==> // ed=-de
==> // fd=-df
==> // fe=-ef
==> // quotient ring from ideal
==> _[1]=f2
==> _[2]=e2
==> _[3]=d2
==> _[4]=c2
==> _[5]=b2
==> _[6]=a2
module A = a*gen(1), b * gen(1), c*gen(2), d * gen(2);
print(A);
==> a,b,0,0,
==> 0,0,c,d
print(exteriorPower(A, 2));
==> 0,bd,bc,ad,ac,0
module B = a*gen(1) + c*gen(2) + e*gen(3),
b*gen(1) + d*gen(2) + f*gen(3),
e*gen(1) + f*gen(3);
print(B);
==> a,b,e,
==> c,d,0,
==> e,f,f
print(exteriorPower(B, 2));
==> df, cf, de+cf,
==> bf+ef,af, be+af,
==> -de, -ce,bc+ad
print(exteriorPower(B, 3));
==> bcf+adf-cef
| See also:
symmetricPower.
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