#### 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 considered as a field
    ==> // 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 considered as a field
    ==> // 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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