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

Synopsis

• Usage:

  sumDecompositionString(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, a symmetric bilinear form defined over a field $k$
• Outputs:
  • a string, the decomposition as a sum of hyperbolic and rank one forms

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 = gwClass(M);

i3 : sumDecompositionString(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 = gwClass(M);

i6 : sumDecompositionString(beta)

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

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

Citations:

• [L05] T.Y. Lam, Introduction to quadratic forms over fields, American Mathematical Society, 2005.

See also

• sumDecomposition -- produces a simplified diagonal representative of a Grothendieck Witt class