Mike Christoff

The notation used to represent group actions can be difficult to parse, and sometimes potentially ambiguous. Is there a better way?

If (G,·) is a group, then a group action G on a set X is a group homomorphism u:G→S_X, where (S_X,∘) is the symmetric group of X.

A canonical example would be the map u:D_n→S_n, where D_n is the dihedral group of order 2n (the group of symmetry-preserving rotations and reflections of an n-gon) and S_n is the symmetric group of {1,…,n}. The idea is to think of the integers 1,…,n as the vertices of the n-gon, and use the rotation and reflection ‘actions’ of D_n to manipulate (rearrange) them.

Getting back to the general definition above (where u is any homomorphism u:G→S_X), the first problem (notation-wise), is that S_X is a set of permutations of X, and hence it is a set of functions. So for any g in G, u(g) is a permutation function on X. This means that, if one wants to calculate how an element g in G acts on an element x in X, we need to look at:

(1)

(u(g))(x)

To express the fact that u is a homomorphism, the expression we get is

(2)

u(ab) = u(a) ∘ u(b)

where a,b are in G, and u(a), u(b), u(ab) are functions in S_X. Hence, for each x in X, we have that

(3)

(u(ab))(x) = (u(a) ∘ u(b))(x)

But as you can see, the notation is starting to get a little unwieldy.

To ‘simplify’ this, the convention is to remove u from the notation entirely! So instead of

(4)

(u(a))(x)

we write

(5)

a(x)

Hence we would write the example (3) as simply

(6)

(ab)(x) = (a ∘ b)(x)

Seems reasonable enough, yes? Well, consider the following concrete example. Here we have the integers Z acting on the reals R, letting S_R be the set of permutation functions on R (S_R is a HUGE group :)).

Let u be defined such that

(7)

(u(k))(x) = x + k .

In the conventional group action notation, this looks like

(8)

k(x) = x + k

We now prove that u is a homomorphism: Let j,k be in Z, and let x be in R, then

(9)

(j + k)(x) = x + (j + k)
= (x + k) + j
= j(x + k)
= j(k(x))

Wow! It has become *very* hard to parse where we are dealing with both j,k the integers and j, k the permutation functions! In the old notation this would have been very messy, but perhaps clearer:

(10)

(u(j+k))(x) = x + (j + k)
= (x + k) + j
= (u(j))(x + k)
= (u(j))((u(k))(x))
= (u(j) ∘ u(k))(x)

One thing that can be said of the conventional notation, is that it really does control bracket-mania. However, my favourite idea is the use of subscripts, ie: u_g ≡ u(g) works well:

(11)

u_{(j + k)}(x) = x + (j + k)
= (x + k) + j
= u_j(x + k)
= u_j(u_k(x))
= (u_j ∘ u_k)(x)

It compares nicely to some previous examples as well, and handles the concrete example above much better in my opinion.

To summarize, the subscript notation can be marginally more obtuse than the conventional method in some cases, but I think it handles certain situations (ie: the concrete example) in a much less ambiguous manner.