Coset Representations
Next up is a family of interesting representations that are also applicable to any group 
. The main ingredient is a subgroup 
— a subset of the elements of so that the inverse of any element in is also in , and the product of any two elements of is also in .
Our next step is to use to break up into cosets. We consider 
and 
to be equivalent if 
. It’s easy to check that this is actually and equivalence relation (reflexive, symmetric, and transitive), and so it breaks up into equivalence classes. We write the coset of all 
that are equivalent to 
as 
, and we write the collection of all cosets of as 
.
We should note that we don’t need to worry about being a normal subgroup of , since we only care about the set of cosets. We aren’t trying to make this set into a group — the quotient group — here.
Okay, now multiplication on the left by shuffles around the cosets. That is, we have a group action of on the quotient set , and this gives us a permutation representation of !
Let’s work out an example to see this a bit more explicitly. For our group, take the symmetric group 
, and for our subgroup let 
. Indeed, is closed under both inversion and multiplication. And we can break up into cosets:
where we have picked a “transversal” — one representative of each coset so that we can write them down more simply. It doesn’t matter whether we write 
or 
, since both are really the same set. Now we can write down the multiplication table for the group action. It takes 
and 
, and tells us which coset 
falls in:
This is our group action. Since there are three elements in the set , the permutation representation we get will be three-dimensional. We can write down all the matrices just by working them out from this multiplication table:
It turns out that these matrices are the same as we saw when writing down the defining representation of . There’s a reason for this, which we will examine later.
As special cases, if 
, then there is one coset for each element of , and the coset representation is the same as the left regular representation. At the other extreme, if 
, then there is only one coset and we get the trivial representation.
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September 20, 2010 -
Posted by John Armstrong |
Algebra, Group theory, Representation Theory
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Hello. Thanks for the great series on representation theory! Just to let you know, I think there is a slight typo in this post. In decomposing your set into cosets, in the indented equation a few lines before the multiplication table, I believe (12) should be paired with (132) and (13) with (123).
It’s a matter of convention which order the multiplication in a symmetric group is written. When I introduced the symmetric group, I chose the composition to reflect functional notation. That is, just like we write
to say “do 
first, then 
“, I write 
to say “do 
first, then 
“.