36 Restriction, Induction and Frobenius Reciprocity

Before break

We determined the character table of using known representations, , , and orthogonality. The irreducible representations of are:

• trivial, alternating, standard, (dimension 4);
• (dimension 6);
• (dimension 5) such that , and .

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

Thus:

(class size)	1	20	12	12	15
	3	0
	3	0

What are and ?? Recall: is the group of rotational symmetries of an icosahedron in . So ( and differ by an outer automorphism of : conjugation by a transposition inside .)

Restriction and induction

A more systematic approach: if is a finite group and a subgroup, then we have a restriction operation This is actually a functor [objects representations of , resp. of ; morphisms homomorphisms of representations].

How about the opposite direction?

Suppose is a representation of , and is invariant under (but not all of ). Now for , the subspace depends only on the coset , and each is a representation of , with

The simplest possible scenario is (which implies ). [In general there is no reason for this to hold.]

If this happens, then the representation of is completely determined by that of . Indeed, choose representatives of the cosets of (each coset contains one ). Given , for some , so there exists such that . Then acts by mapping to , with .

That is, fixing one element in each coset, , we can write each uniquely as for .

(Indeed: , so where is invariant under and as an -representation.)

In particular:

The character of an induced representation

We can actually calculate the character of an induced representation! Choose representatives of the cosets of as usual; maps to , where . If , then this doesn’t contribute to . If , then and maps to itself by , so . Summing over the ’s:

Frobenius reciprocity

A key property for understanding induced representations is Frobenius reciprocity.

Notes not reached in this lecture, covered at the start of Lecture 37

Proposition 36.5. If is a representation of , and a representation of , then every -equivariant map extends uniquely to a -equivariant map :

Proof. Choose representatives of the cosets of , and let . Given -equivariant, if is -equivariant and , then necessarily we have a commutative diagram i.e. is given by (the action of on ). This determines uniquely.

To check that is -equivariant, recall that acts on by mapping to such that , via . Given , where in , is acting on . So on for all , hence on .

So does have a unique -equivariant extension .

Conversely, given , is -equivariant, and hence its restriction to is -equivariant. ∎

Comparing dimensions, , which gives:

Corollary 36.6 (Frobenius reciprocity).

(This can also be proved by direct calculation using the above formula for .)

Thus: if is an irreducible representation of and an irreducible representation of , then the number of times appears in is equal to the number of times appears in .

Example 36.7. . The restrictions of the irreducible representations of are:

• trivial: ;
• alternating: ;
• standard: (since the permutation representation restricts to permutation trivial: );
• : (using );
• (factors through ): .

(Or, instead of arguing explicitly, one can just use character tables!)

So by Frobenius reciprocity, (this has dimension ). Similarly, and .

Theorems of Artin and Brauer

Some of the key motivation for studying induced representations comes from two deep theorems of Artin and Brauer.

Theorem 36.8 (Artin). Every character of a representation of is a linear combination with rational coefficients of characters of representations induced from cyclic subgroups of .

Theorem 36.9 (Brauer). Every character of a representation of is a linear combination with integer coefficients of characters of representations induced from 1-dimensional representations of ”elementary” subgroups of ,

where elementary means isomorphic to a product , with a -group () and cyclic, , .

(We won’t prove these. See e.g. Serre’s Linear Representations of Finite Groups.)