# tensorProduct -- tensor product of sheaves on the elliptic curve or sheaf times CliffordModule

## Synopsis

• Usage:
eta = tensorProduct(phi,psi)
G = tensorProduct(M,V)
L = tensorProduct(L1,L2)
• Inputs:
• Outputs:

## Description

Sheaves on the hyperelliptic curve y^2 -(-1)^{g}* f(s,t) are represented as sheaves on PP^1 together with the action of y. Clifford modules are represented as the action of maps eOdd_i: M_1 \to M_0 and eEv_i:M_0 \to M_1 between the even and odd parts of the module. The result are the corresponding data for the tensor product.

 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 : (uOdd,uEv)=cliffordOperators(Mu1,Mu2,R); i5 : symMatrix(uOdd,uEv) o5 = | -5t -50s 6t -6t | | -50s 0 -9t 5t | | 6t -9t -s-30t 3t | | -6t 5t 3t -48t | 4 4 o5 : Matrix R <--- R i6 : f=det symMatrix(uOdd,uEv); i7 : M = cliffordModule(uOdd, uEv); i8 : L = randomLineBundle(0,f); i9 : L.yAction o9 = {-1} | -18s2-13st-43t2 13s2+45st-16t2 | {-1} | 45s2-14st-t2 18s2+13st+43t2 | 2 2 o9 : Matrix R <--- R i10 : L2 = tensorProduct(L,L) o10 = VectorBundleOnE{...1...} o10 : VectorBundleOnE i11 : L2.yAction o11 = {-1} | -18s2+6st+19t2 13s2-3st-16t2 | {-1} | 45s2-5st+7t2 18s2-6st-19t2 | 2 2 o11 : Matrix R <--- R i12 : M' = tensorProduct(M,L) o12 = CliffordModule{...6...} o12 : CliffordModule i13 : M.evenCenter o13 = {-4} | -49st-24t2 -24s2t+47st2-42t3 -10t3 -25st2-43t3 | {-3} | -50s 49st+24t2 -25t2 -45t2 | {-3} | -9t 5st+49t2 49st-17t2 50s2-15st+7t2 | {-3} | 5t -2t2 24st+41t2 -49st+17t2 | 4 4 o13 : Matrix R <--- R i14 : M'.evenCenter o14 = {-4} | 47st+18t2 45s2t-33st2-27t3 24st2-11t3 -6st2+11t3 | {-3} | -7s+37t -30st+7t2 34st+5t2 14st-5t2 | {-3} | 35s-34t -3s2-5st-7t2 -6s2+19st-t2 -50s2-49st+43t2 | {-3} | -26s+50t -36s2-37st+36t2 29s2+44st-35t2 6s2-36st-24t2 | 4 4 o14 : Matrix R <--- R