# flip(Module,Module) -- matrix of commutativity of tensor product

## Synopsis

• Usage:
flip(F,G)
• Function: flip
• Inputs:
• F,
• G,
• Outputs:
• , the matrix representing the natural isomorphism G ** F <-- F ** G

## Description

 `i1 : R = QQ[x,y];` ```i2 : F = R^{1,2,3} 3 o2 = R o2 : R-module, free, degrees {-1, -2, -3}``` ```i3 : G = R^{10,20,30} 3 o3 = R o3 : R-module, free, degrees {-10, -20, -30}``` ```i4 : f = flip(F,G) o4 = {-11} | 1 0 0 0 0 0 0 0 0 | {-12} | 0 0 0 1 0 0 0 0 0 | {-13} | 0 0 0 0 0 0 1 0 0 | {-21} | 0 1 0 0 0 0 0 0 0 | {-22} | 0 0 0 0 1 0 0 0 0 | {-23} | 0 0 0 0 0 0 0 1 0 | {-31} | 0 0 1 0 0 0 0 0 0 | {-32} | 0 0 0 0 0 1 0 0 0 | {-33} | 0 0 0 0 0 0 0 0 1 | 9 9 o4 : Matrix R <--- R``` ```i5 : isHomogeneous f o5 = true``` ```i6 : target f 9 o6 = R o6 : R-module, free, degrees {-11, -12, -13, -21, -22, -23, -31, -32, -33}``` ```i7 : source f 9 o7 = R o7 : R-module, free, degrees {-11, -21, -31, -12, -22, -32, -13, -23, -33}``` ```i8 : target f === G**F o8 = true``` ```i9 : source f === F**G o9 = true``` ```i10 : u = x * F_0 o10 = {-1} | x | {-2} | 0 | {-3} | 0 | 3 o10 : R``` ```i11 : v = y * G_1 o11 = {-10} | 0 | {-20} | y | {-30} | 0 | 3 o11 : R``` ```i12 : u ** v o12 = {-11} | 0 | {-21} | xy | {-31} | 0 | {-12} | 0 | {-22} | 0 | {-32} | 0 | {-13} | 0 | {-23} | 0 | {-33} | 0 | 9 o12 : R``` ```i13 : v ** u o13 = {-11} | 0 | {-12} | 0 | {-13} | 0 | {-21} | xy | {-22} | 0 | {-23} | 0 | {-31} | 0 | {-32} | 0 | {-33} | 0 | 9 o13 : R``` ```i14 : f * (u ** v) o14 = {-11} | 0 | {-12} | 0 | {-13} | 0 | {-21} | xy | {-22} | 0 | {-23} | 0 | {-31} | 0 | {-32} | 0 | {-33} | 0 | 9 o14 : R``` ```i15 : f * (u ** v) === v ** u o15 = true```