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# flip -- isomorphism map of commutativity of tensor product

## Synopsis

• Usage:
flip(F, G)
• Inputs:
• F, ,
• G, ,
• Outputs:
• , the matrix representing the natural isomorphism $F \otimes G \to G \otimes F$

## 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 = | x | | 0 | | 0 | 3 o10 : R i11 : v = y * G_1 o11 = | 0 | | y | | 0 | 3 o11 : R i12 : u ** v o12 = | 0 | | xy | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | 9 o12 : R i13 : v ** u o13 = | 0 | | 0 | | 0 | | xy | | 0 | | 0 | | 0 | | 0 | | 0 | 9 o13 : R i14 : f * (u ** v) o14 = | 0 | | 0 | | 0 | | xy | | 0 | | 0 | | 0 | | 0 | | 0 | 9 o14 : R i15 : f * (u ** v) === v ** u o15 = true

## Ways to use flip :

• flip(Module,Module)

## For the programmer

The object flip is .