2.1 Permutations in RepLAB

In RepLAB, a permutation \(g\) is described using the row vector of permutation images \((g(1), g(2), \ldots, g(n))\). This corresponds to the behavior of the randperm and perms functions, and enables the use of such vectors of indices as matrix/vector subscripts.

Let us define two permutations. Note that in RepLAB permutations act on a set of given size: [1 2 3] and [1 2 3 4] are not equivalent, even though they both represent the identity of a permutation group.

g = [2 1 3];
h = [3 2 1];

This behavior is different from other computer algebra systems. For example, GAP System represents permutations as a product of cycles. SymPy has a specific type of object to describe permutations. In RepLAB we prefer using existing conventions of the MATLAB/Octave ecosystem every time it is possible.

To compute the action of a permutation on a point, we write:

g(2)

ans = 1

Let \(k = g h\) be the composition of \(g\) and \(h\). It is well defined if we ask that \(k(i) = g(h(i))\), then:

gh2 = g(h(2))
k = g(h)
k2 = k(2)

gh2 = 1
k =

   3   1   2

k2 = 1

In some textbooks, for example [HEOBrien05], the exponent notation is used for permutations: one then writes \(2^g\) for the image of 2 under the permutation \(g\). The composition \(g h\) is then defined as \((2^g)^h\), and \(g\) is applied first! This differs from the RepLAB convention, which rather matches the standard definition of function composition. One should then be careful when mixing different textbooks or computer algebra systems.

But how can one compute the inverse of a permutation? Luckily, we can construct the group of all permutations of a given size in RepLAB, and call its methods to perform group operations. The function replab.S(n) creates the symmetric group acting on \(\{1,2,\ldots,n\}\). The returned object is of type replab.PermutationGroup and it proposes notably the identity, compose and inverse methods:

G = replab.S(3);
gh = G.compose(g, h)
gInv = G.inverse(g)
g_gInv = G.compose(g, gInv)

gh =

   3   1   2

gInv =

   2   1   3

g_gInv =

   1   2   3