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

## Synopsis

• Usage:
gwMultiply(beta, gamma)
• Inputs:
• beta, , the isomorphism class of a non-degenerate symmetric bilinear form represented by a matrix M
• gamma, , the isomorphism class of a non-degenerate symmetric bilinear form represented by a matrix N
• Outputs:
• , 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

• 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

## Ways to use gwMultiply :

• gwMultiply(GrothendieckWittClass,GrothendieckWittClass)

## For the programmer

The object gwMultiply is .