A Universe of Sorts

§ The similarity between labellings and representations

One way to think about labellings is that we track the "entire history" of the object.
it's hard to count unlabelled objects. it's easier to count labelled objects.
for example, suppose we have graphs $g = (v, e)$ and $h = (v', e')$ . an isomorphism of these as unlabelled graphs is a bijection function $f: v \rightarrow v'$ such that $s e t$ if and only if $f(s) e f(t)$ .
there could be many such $f$ , or no such $f$ . it's hard to find out!
Now let's suppose the graphs have labellings, so we have labels $l: V \rightarrow [|V|]$ and $l': V' \rightarrow [|V'|]$ where $[n] \equiv \{1, 2, \dots, n\}$ .
An isomorphism of labelled graphs is an unlabelled isomorphism along with the constraint that $l'(f(v)) = l(v)$ . That is, we must preseve labels. So, for example, the graph:

a:1 -- b:2
c:2 -- d:1

are isomorphic since I can send a -> d and b -> c.

a:1-b:2-c:3
d:1-e:3-f:2

is not isomorphic (though they would be if we forget the numbering), since the center vertices b and e have different labels.

Let's think of the equation $l'(f(v)) = l(v)$ . Since $f$ is a bijection, we have $|V| = |V'|$ , so $l$ and $l'$ are both bijections to the same set $[|V|] = [|V'|]$ . So we can invert the equation to write $f(v) = l'^{-1}(l(v))$ . This tells us that $f$ is determined by the labellings!
The point of having a labelling is that it forces upon us a unique isomorphism (if it exists), given by the equation $f(v) \equiv l^{-1}(l(v))$ .
This collapses hom sets to either empty, or a unique isomorphism, which is far tamer than having many possible graph isomorphisms that we must search for / enumerate !
In analogy to representation theory, if we consider two irreducible representations of a group $G$ , say $\alpha: G \rightarrow GL(V)$ and $\beta: G \rightarrow GL(W)$ , Schur's lemma tells us that the Hom-set between the two representations (an intertwining map) is either the zero map (which is like having no isos) or a scaling of the identity map (which is like having a uniquely determined iso).
In this sense, we can think of an irrep as a "labelling" of group elements in a particularly nice way, since it constrains the potential isomorphisms of the "labelled objects"!