cliffordModule -- makes a clifford Module

Synopsis

• Usage:
M = cliffordModule(M1,M2,R)
M = cliffordModule(eOdd,eEv)
• Inputs:
• M1, ,
• M2, , M1, M2 a matrix factorization of a quadratic form qq
• R, a ring, base ring of the quadratic form
• eEv, a list,
• eOdd, a list, lists such as the output of cliffordOperators(M1,M2)
• Outputs:

Description

The keys oddOperators evenOperators are the same as the two lists output by cliffordOperators(M1,M2)

The keys evenCenter oddCenter yield the action of the center of the even Clifford algebra of qq on the even, respectively odd parts of the Clifford module.

The key symmetricM yields the matrix of coefficients of the quadratic form qq

the key hyperellipticBranchEquation yields the branch equation in R -- that is, the equation of the set of points over which the quadratic form is singular.

 i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing i2 : g = 1 o2 = 1 i3 : (S, qq, R, u, M1, M2, Mu1, Mu2)=randomNicePencil(kk,g); i4 : M = cliffordModule(M1,M2, R) o4 = CliffordModule{...6...} o4 : CliffordModule i5 : Mu = cliffordModule(Mu1,Mu2, R) o5 = CliffordModule{...6...} o5 : CliffordModule

The symmetric matrices are the same for both:

 i6 : Mu.symmetricM o6 = | -5t -50s 6t -6t | | -50s 0 -9t 5t | | 6t -9t -s-30t 3t | | -6t 5t 3t -48t | 4 4 o6 : Matrix R <--- R i7 : M.symmetricM o7 = | -5t -50s 6t -6t | | -50s 0 -9t 5t | | 6t -9t -s-30t 3t | | -6t 5t 3t -48t | 4 4 o7 : Matrix R <--- R

But the operators are twice the size for M (in both cases the same size as the corresponding matrix factorization Mu.evenCenter numrows(Mu.evenCenter) == numrows(Mu1) M.evenCenter