gwMultiply -- the tensor product of two Grothendieck-Witt classes

Synopsis

• Usage:

  gwMultiply(beta, gamma)

• Inputs:
  • beta, a Grothendieck Witt Class, the isomorphism class of a non-degenerate symmetric bilinear form represented by a matrix M
  • gamma, a Grothendieck Witt Class, the isomorphism class of a non-degenerate symmetric bilinear form represented by a matrix N
• Outputs:
  • a Grothendieck Witt Class, the isomorphism class of the tensor product of the bilinear forms represented by the matrices M and N

Description

This computes the tensor product of the Grothendieck-Witt classes beta and gamma.

i1 : M = matrix(QQ,{{1,0},{0,1}});

              2       2
o1 : Matrix QQ  <-- QQ

i2 : N = matrix(QQ, {{1, 2}, {2, 5}});

              2       2
o2 : Matrix QQ  <-- QQ

i3 : beta = gwClass(M);

i4 : gamma = gwClass(N);

i5 : gwMultiply(beta, gamma)

o5 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | 1 2 0 0 |
                                     | 2 5 0 0 |
                                     | 0 0 1 2 |
                                     | 0 0 2 5 |

o5 : GrothendieckWittClass

See also

• GrothendieckWittClass -- a new type, intended to capture the isomorphism class of an element of the Grothendieck-Witt ring of a base field
• gwClass -- the Grothendieck Witt class of a symmetric matrix
• gwAdd -- the direct sum of two Grothendieck-Witt classes