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getSumDecomposition -- produces a simplified diagonal representative of a Grothendieck-Witt class

Description

Given a symmetric bilinear form beta over $\mathbb{Q},$ $ \mathbb{R},$ $\mathbb{C}$ or a finite field of characteristic not 2, we decompose it as a sum of some number of hyperbolic and rank one forms.

i1 : M = matrix(RR, {{2.091,2.728,6.747},{2.728,7.329,6.257},{6.747,6.257,0.294}});

                3         3
o1 : Matrix RR    <-- RR
              53        53
i2 : beta = makeGWClass M;
i3 : getSumDecomposition beta

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

o3 : GrothendieckWittClass

Over $\mathbb{R}$ there are only two square classes and a form is determined uniquely by its rank and signature [L05, II Proposition 3.2]. A form defined by the $3\times 3$ Gram matrix M above is isomorphic to the form $\langle 1,-1,1\rangle $.

i4 : M = matrix(GF(13), {{9,1,7,4},{1,10,3,2},{7,3,6,7},{4,2,7,5}});

                   4            4
o4 : Matrix (GF 13)  <-- (GF 13)
i5 : beta = makeGWClass M;
i6 : getSumDecomposition beta

o6 = | 1 0  0 0  |
     | 0 -5 0 0  |
     | 0 0  1 0  |
     | 0 0  0 -1 |

o6 : GrothendieckWittClass

Over $\mathbb{F}_{q}$ forms can similarly be diagonalized, in the above case as $\langle 1,-1,1,-6 \rangle$.

Citations:

See also

Ways to use getSumDecomposition:

  • getSumDecomposition(GrothendieckWittClass)

For the programmer

The object getSumDecomposition is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/DecompositionDoc.m2:32:0.