35 Irreducible Characters Form a Basis; Characters of S₅ and A₅
· Denis Auroux
Last time
Characters are class functions .
Let
a Hermitian inner product on the space of class functions . Then .
Characters of irreducible representations are orthonormal:
Every , with , and . Applying this to the regular representation , we got .
We used this to find the character tables of and .
Characters of irreducibles form a basis of class functions
Last time we said but didn’t prove: the characters of irreducible representations are actually an orthonormal basis (for ) of the space of class functions .
The proof uses a more general averaging/projection formula. Last time we saw that
is the projection onto the invariant subspace ( the trivial summands in ).
Proposition 35.1. Given any class function and any representation of , let
Then is -linear (equivariant).
Proof.∎
Theorem 35.2. The characters of the irreducible representations of form an orthonormal basis (for ) of the space of class functions , and
Proof. To show that the characters of the irreducible representations span all class functions, it suffices to show: for all implies .
Given any class function and an irreducible representation , consider as above. Then by Schur’s lemma, , where ,. So
So if for all irreducible , then for all ; hence, by considering direct sums, for all representations of . In particular this holds for the regular representation of (the permutation representation for left multiplication on ). So for the regular representation,
Since the are linearly independent, this implies for all , i.e. .∎
Along the way, we found the following. For irreducible, look at for . Then , where
Proposition 35.3. If is any representation of and is its decomposition into irreducibles, then
is the projection onto the summand (i.e. the identity on that summand, on the others).
(The case of the trivial representation is our previous projection formula for .)
The representation ring of
(Or: what are general class functions “the characters of”?)
Fix a group and consider the set of (finite-dimensional, over ) representations of up to isomorphism. There are two operations, and , which are commutative, associative, and distributive: . So this is a ring?… Almost! We’re missing additive inverses. We’ll just add them!
Let
the formal linear combinations with integer coefficients of representations of , and consider the additive subgroup generated by all . Let be the quotient of by this subgroup. (So in ,, but we can subtract representations!)
Definition 35.4. is now a ring: the representation ring of . (We extend the operations , to formal sums/differences of representations by linearity!)
As a set,
where the are the irreducible representations of (by complete reducibility and uniqueness of the decomposition into irreducibles). That is, is a free abelian group (, irreducibles). General elements () are called virtual representations; actual representations, i.e. elements such that for all , form a cone inside it (i.e. a subset closed under addition).
Next: the character, , can be extended by linearity to a map
This is a ring homomorphism! (,.)
The image of this map is the set of virtual characters().
Passing to complex linear combinations instead of integer ones, our results about irreducible characters forming a basis say:
Remark. There are theorems of Artin and Brauer that describe the lattice of virtual characters inside .
Representations of
Next we’ll look at representations of and , for extra practice with characters, and to motivate the discussion of restriction and induction of representations between and its subgroups.
One can start building the character table of the usual way: start with known representations. First we have (trivial) and (alternating), and (the standard representation, dimension 4). Use: the permutation representation , so
(class size)
1
10
20
30
24
15
20
1
1
1
1
1
1
1
1
1
1
1
4
2
1
0
0
4
1
0
0
1
Then we need more. Since , we’re still missing 3 irreducibles with . The most effective way to find them is to keep building tensor products: namely, look at (dimension 16), or rather its two pieces (dimension 10) and (dimension 6).
Note (Observe). If has eigenvalues (,), then the corresponding map on
has eigenvalues , (recall: basis of basis of );
has eigenvalues , (recall: basis of ).
Now
so
(This is true for any representation.)
This formula lets us calculate and for the standard representation of .
(class size)
1
10
20
30
24
15
20
4
2
1
0
0
6
0
0
0
1
0
10
4
1
0
0
2
1
Observe:
so is irreducible! Whereas
so splits into 3 irreducible summands. And
Similar calculations show that also contains with multiplicity 1, and does not contain or . Hence
for some irreducible 5-dimensional representation . Subtracting, we find , and one more, , which completes the list (see the table below).
Remark. The standard representation and its exterior powers ,, and are all irreducible! This is in fact a general property: for all , the exterior powers of the standard representation of are all irreducible (see Fulton–Harris, Representation Theory: A First Course, §3.2).
Character table of :
(class size)
1
10
20
30
24
15
20
1
1
1
1
1
1
1
1
1
1
1
4
2
1
0
0
4
1
0
0
1
6
0
0
0
1
0
()
(10)
(4)
(1)
(0)
(0)
(2)
(1)
5
1
0
1
1
5
1
0
1
Notes not reached in this lecture, covered at the start of {{LINK:L36}}
Representations of
Next, move on to . The starting point is to restrict the irreducible representations of to and see which ones remain irreducible or decompose. Of course, different irreducible representations of can become isomorphic after restriction: namely, elements of act by on , so becomes trivial, and the restrictions of and become isomorphic; similarly for . The character table of gives, after restriction:
(class size)
1
20
12
12
15
1
1
1
1
1
4
1
0
5
0
0
1
6
0
1
1
Calculating , we find that ,, are irreducible, while , so breaks into the direct sum of 2 distinct irreducibles. Also doesn’t contain , or , so , the last two irreducible representations of .
From we find . How do we find and ? Using orthogonality: , so
Hence , where
Remark. In fact, either
and ; or
and ,.
The existence of a real 3-dimensional representation (see below) implies it’s .