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

Description

Given a symmetric bilinear form beta over a field $k$, we return a simplified diagonal form of beta.

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 : getSumDecompositionString beta

o3 = 1H + <1>

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 : getSumDecompositionString beta

o6 = 1H + <1> + <-5>

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 getSumDecompositionString:

  • getSumDecompositionString(GrothendieckWittClass)

For the programmer

The object getSumDecompositionString is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/DecompositionDoc.m2:62:0.