gwClass -- the Grothendieck Witt class of a symmetric matrix

Synopsis

• Usage:

  gwClass(M)

• Inputs:
  • M, a matrix, a symmetric matrix defined over an arbitrary field
• Outputs:
  • a Grothendieck Witt Class, the isomorphism class of a symmetric bilinear form represented by $M$

Description

Given a symmetric matrix, $M$, this command outputs an object of type GrothendieckWittClass. This output has the representing matrix, $M$, and the base field of the matrix stored in its CacheTable.

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

              3       3
o1 : Matrix QQ  <-- QQ

i2 : beta = gwClass(M)

o2 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | 0 0 1 |
                                     | 0 1 0 |
                                     | 1 0 0 |

o2 : GrothendieckWittClass

The matrix representing a GrothendieckWittClass element can be recovered using the matrix command:

i3 : beta.matrix

o3 = | 0 0 1 |
     | 0 1 0 |
     | 1 0 0 |

              3       3
o3 : Matrix QQ  <-- QQ

The base field which the form $\beta$ is implicitly defined over can be recovered with the baseField method.

i4 : baseField beta

o4 = QQ

o4 : Ring

See also

• baseField -- the base field of a Grothendieck Witt class
• GrothendieckWittClass -- a new type, intended to capture the isomorphism class of an element of the Grothendieck-Witt ring of a base field