Singular Manual: exteriorPower

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