§ 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 and . an isomorphism of these as unlabelled graphs is a bijection function such that if and only if .
- there could be many such , or no such . it's hard to find out!
- Now let's suppose the graphs have labellings, so we have labels and where .
- An isomorphism of labelled graphs is an unlabelled isomorphism along with the constraint that . That is, we must preseve labels. So, for example, the graph:
are isomorphic since I can send
a:1 -- b:2
c:2 -- d:1
a -> d and
b -> c.
- On the other hand, the graph:
is not isomorphic (though they would be if we forget the numbering), since the center vertices
e have different labels.
- Let's think of the equation . Since is a bijection, we have , so and are both bijections to the same set . So we can invert the equation to write . This tells us that 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 .
- 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 , say and , 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"!