Representation theory of finite groups

In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. Later the modular representation theory of Richard Brauer was developed.

Contents

1 Basic definitions 2 Applying Schur's lemma 3 Character theory 4 Real representations 5 Examples 6 References

Basic definitions

All the linear representations in this article are finite dimensional and assumed to be complex unless otherwise stated. We say that ρ is a representation of G if the mapping

<math>\rho:G\rightarrow GL(n,\mathbf{C})<math>

from G to the general linear group GL(n,C) is a group homomorphism, that is

<math>\rho(g h) = \rho(g) \rho(h)<math>

for all <math>\{g, h\} \subset G<math>.

We say that ρ is a real representation of G if the matrices are real:

<math>\rho:G\rightarrow GL(n,\mathbf{R})<math>

Applying Schur's lemma

According to Schur's lemma, if ρ is a representation of G, then all linear transformations mapping the rep to itself which are invariant under the action of G are multiples of the identity if and only if the rep is irreducible. Invariant here means that the linear transformation commutes with ρ(g) for all g in G. In other language, a G-intertwiner between two irreps is either zero or bijective.

This doesn't work with real reps. Take the cyclic group C₃ for example. It has a 2-dimensional real irrep but yet the linear transformation generated by any nonidentity element of C₃ is an invariant proper rotation!

If we have a subrep of a rep, then the quotient space is another rep.

Lemma: If

<math>f:A\otimes B\rightarrow C<math>

is a G-intertwiner, then the corresponding linear transformation obtained by dualizing B,

<math>f':A\rightarrow C\otimes \bar{B}<math>

is also an intertwiner. Similarly, if

<math>g:A\rightarrow B\otimes C<math>

is a G-intertwiner, dualizing it will give another intertwiner :<math>g':A\otimes \bar{C}\rightarrow B<math>.

The dual rep is just the dual vector space with the group action defined as

<math><\bar{\rho}(g^{-1})[x],y>=<math>

for all x and y which is well defined and a rep.

The group algebra C[G] is a |G|-dimensional algebra over the complex numbers, on which G acts. (See Peter-Weyl for the case of compact groups.) In fact C[G] is a representation for G×G. More specifically, if g₁ and g₂ are elements of G and h is an element of C[G] corresponding to the element h of G,

(g₁,g₂)[h]=g₁h g^-1₂.

C[G] can also be considered as a representation of G in three different ways:

• Conjugation: g[h]=ghg^-1
• As a left action: g[h]=gh (a regular representation)
• As a right action: g[h]=hg^-1 (also);

these are all to be 'found' inside the G×G action.

If ρ is an n-dimensional irrep of G with the underlying vector space V, then we can define a G×G-intertwiner

<math>f:C[G]\otimes (1_G\otimes V)\rightarrow (V\otimes 1_G)<math>

as follows where 1_G is the trivial representation of G.

<math>f(g\otimes x)=\rho(g)[x]<math>

for all g in G and x in V. This defines a G×G intertwiner, as can be explicitly checked.

Do the dualization trick above and obtain the G×G-intertwiner

<math>f':\bar{V}\otimes V\rightarrow \overline{C[G]}<math>.

The dual rep of C[G] as a G×G-rep is equivalent to C[G]. An isomorphism is given if we define the contraction

<g,h>=δ_gh,

as you may check. So, we end up with a G×G-intertwiner

<math>f'':\bar{V}\otimes V\rightarrow C[G]<math>.

It turns out

<math>f''(x\otimes y)=\sum_{g\in G}g<math>

for all x in <math>\bar{V}<math> and y in V.

By Schur's lemma, the image of f′′ is a G×G irrep, which is therefore n×n dimensional, which also happens to be a subrep of C[G] (f′′ most definitely is nonzero).

This, of course would be n direct sum equivalent copies V. Note that if ρ₁ and ρ₂ are equivalent G-irreps, the respective images of the intertwiners would give rise to the same G×G-irrep of C[G].

Here, we use the fact that if f is a function over G, then

<math>\sum_{g\in G}f(g)hgk^{-1}=\sum_{g\in G}f(h^{-1}gk)g<math>

We convert C[G] into a Hilbert space by introducing the norm where <g,h> is 1 if g is h and zero otherwise. This is different from the 'contraction' given a couple of paragraphs back, in that this form is sesquilinear. This makes C[G] a unitary representation of G×G. In particular, we now have the concepts of orthogonal complement and orthogonality of subreps.

In particular, if C[G] contains two inequivalent irreducible G×G subreps, then both subreps are orthogonal to each other. To see this, note that for every subspace of a Hilbert space, there exists a unique linear transformation from the Hilbert space to itself which maps points on the subspace to itself while mapping points on its orthogonal complement to zero. This is called the projection map. The projection map associated with the first irrep is an intertwiner. Restricted to the second irrep, it gives an intertwiner from the second irrep to the first. Using Schur's lemma, this has got to be zero.

Now suppose <math>A\otimes B<math> is a G×G-irrep of C[G]. (The complex irreps of G×H are always a direct product of a complex irrep of G and a complex irrep of H. This is not the case for real irreps.

As an example, there is a 2 dimensional real irrep of C₃×C₃ which transforms nontrivially under both copies of C₃ which can't be expressed as the direct product of two Z₃ irreps.) This rep is also a G-rep (n_A direct sum copies of B where n_A is the dimension of A). If Y is an element of this rep (and hence also of C[G]) and X an element of its dual rep (which is a subrep of the dual rep of C[G]), then

<math>f''(X\otimes Y)=\sum_{g\in G}g=\sum_{g\in G}g=\sum_{g\in G}gY=\sum_{g\in G}(g,e)[Y]<math>

where e is the identity of G. I know the f′′ defined a couple of paragraphs back is only defined for G-irreps and <math>A\otimes B<math> isn't a G-irrep in general. But since <math>A\otimes B<math> is simply the direct sum of copies of B's and we've shown that each copy all maps to the same subG×G-irrep of C[G], we've just showed that <math>\bar{B}\otimes B<math> as an irreducible G×G-subrep of C[G] is contained in <math>A\otimes B<math> as another irreducible G×G-subrep of C[G]. Using Schur's lemma again, this means both irreps are the same.

Putting all of this together,

<math>C[G]\simeq \sum_{inequivalent G-irreps V} \bar{V}\otimes V.<math>

Corollary: If there are p inequivalent G-irreps, V_i, each of dimension n_i, then

|G| = Σ n_i².

Character theory

There is a mapping from G to the complex numbers for each rep called the character given by the trace of the linear transformation upon the rep generated by the element of G in question

χ_ρ(g)=Tr[ρ(g)].

All elements of G belonging to the same conjugacy class have the same character: in other words χ_ρ is a class function on G. This follows from

Tr[ρ(ghg^-1)]=Tr[ρ(g)ρ(h)ρ(g)^-1]=Tr[ρ(h)]

by the cyclic property of the trace of a matrix.

What are the characters of C[G]? Using the property that gh^-1 is only the same as g if h=e, χ_C[G](g) is |G| if g=e and 0 otherwise.

The character of a direct sum of reps is simply the sum of their individual characters.

Putting all of this together,

<math>\sum_{i=1}^p n_i \chi_{V_i}(g)=|G|\delta_{ge}<math>

with the Kronecker delta on the RHS.

Repeat this, working now with G×G characters this time instead of G characters, which I'll call χ′.

Then,

<math>\chi'_{C[G]}((g,h))<math>

is the number of elements in G satisfying

gkh^-1 = k.

This is equal to

<math>\sum_{i=1}^p \chi_{\bar{V}_i}(g)\chi_{V_i}(h)=\sum_{i=1}^p \chi_{V_i}(g^{-1})\chi_{V_i}(h)=\sum_{i=1}^p \chi_{V_i}(g)^*\chi_{V_i}(h)<math>

where * denotes complex conjugation. After all, C[G] is a unitary rep and any subrep of a finite unitary rep is another unitary rep; and all irreps are (equivalent to) a subrep of C[G].

Consider

<math>\sum_{h \in G}\chi'_{C[G]}((g,hkh^{-1}))<math>.

This is |G| times the number of elements which commute with G; which is |G|² divided by the size of the conjugacy class of g, if g and k belong to the same conjugacy class, but zero otherwise. Therefore, for each conjugacy class C_i of size m_i, the characters are the same for each element of the conjugacy class and so we can just call

χ_ρ(C_i)

by an "abuse" of notation). Then,

<math>\frac{|G|}{|C_i|}\delta_{ij}=\sum_{k=1}^p\chi_{V_k}(C_i)^*\chi_{V_k}(C_j)<math>.

Note that

<math>\sum_{g\in G}\rho(g)<math>

is a self-intertwiner (or invariant). This linear transformation, when applied to C[G] (as a rep of the second copy of G×G), would give as its image the 1-dimensional subrepresentation generated by

<math>\sum_{g\in G}g<math>;

which is obviously the trivial representation.

Since we know C[G] contains all irreps up to equivalence and using Schur's lemma, we conclude that

<math>\sum_{g\in G}\rho(g)<math>

for irreps is zero if it's not the trivial irrep; and it's of course |G|1 if the irrep is trivial.

Given two irreps V_i and V_j, we can construct a G-rep :<math>\bar{V}_i\otimes V_j<math>,

this time not as a G×G rep but an ordinary G-rep. See direct product of representations. Then,

<math>\chi_{\bar{V}_i\otimes V_j}(g)=\chi_{\bar{V}_i}(g)\chi_{V_j}(g)=\chi_{V_i}(g)^*\chi_{V_k}(g)<math>.

It can be shown that any irrep can be turned into a unitary irrep. So, the direct product of two irreps can also be turned into a unitary reps and now, we have the neat orthogonality property allowing us to decompose the direct product into a direct sum of irreps (we're also using the property that for finite dimensional reps, if you keep taking proper subreps, you'll hit an irrep eventually. There's no infinite strictly decreasing sequence of positive integers). See Maschke's theorem.

If i&neq;j, then this decomposition does not contain the trivial rep (Otherwise, we'd have a nonzero intertwiner from V_j to V_i contradicting Schur's lemma). If i=j, then it contains exactly one copy of the trivial rep (Schur's lemma states that if A and B are two intertwiners from V_i to itself, since they're both multiples of the identity, A and B are linearly dependent).

Therefore,

<math>\sum_{g\in G}\chi_{V_i}(g)^*\chi_{V_j}(g)=\sum_{k}|C_k|\chi_{V_i}(C_k)^*\chi_{V_j}(C_k)=|G|\delta_{ij}<math>

Applying a result of linear algebra to both orthogonality relations (|C_i| is always positive), we find that the number of conjugacy classes is greater than or equal to the number of inequivalent irreps; and also at the same time less than or equal to. The conclusion, then, is that the number of conjugacy classes of G is the same as the number of inequivalent irreps of G.

Corollary: If two reps have the same characters, then they are equivalent.

Proof: Characters can be thought of as elements of a q-dimensional vector space where q is the number of conjugacy classes. Using the orthogonality relations derived above, we find that the q characters for the q inequivalent irreps forms a basis set. Also, according to Maschke's theorem, both reps can be expressed as the direct sum of irreps. Since the character of the direct sum of reps is the sum of their characters, from linear algebra, we see they are equivalent.

We know that any irrep can be turned into a unitary rep. It turns out the Hilbert space norm is unique up to multiplication by a positive number. To see this, note that the conjugate rep of the irrep is equivalent to the dual irrep with the Hilbert space norm acting as the intertwiner. Using Schur's lemma, all possible Hilbert space norms can only be a multiple of each other.

Real representations

How about real irreducible representations? It turns out if we have a complete classification of complex irreps, it's easy.

Real reps can be complexified to get a complex rep of the same dimension and complex reps can be converted into a real rep of twice the dimension by treating the real and imaginary components separately. Also, since all finite dimensional complex reps can be turned into a unitary rep. for unitary reps, the dual rep is also a (complex) conjugate rep because the Hilbert space norm gives an antilinear bijective map from the rep to its dual rep. Self dual complex irreps correspond to either real irreps of the same dimension or real irreps of twice the dimension called pseudoreal representations (but not both) and non self dual complex irreps correspond to a real irrep of twice the dimension. Note for the latter case, both the complex irrep and its dual give rise to the same real irrep. An example of a pseudoreal rep would be the four dimensional real irrep of the quaternion group Q₈.

Just as for any complex rep ρ,

<math>\frac{1}{|G|}\sum_{g\in G}\rho(g)<math>

is a self-intertwiner, for any integer n,

<math>\frac{1}{|G|}\sum_{g\in G}\rho(g)^n<math>

is also a self-intertwiner. By Schur's lemma, this will be a multiple of the identity for irreps. The trace of this self-intertwiner is called the n^th Frobenius-Schur indicator.

The original case of the Frobenius-Schur indicator is that for n = 2. The zeroth indicator is the dimension of the irrep, the first indicator would be 1 for the trivial representation and zero for the other irreps.

It resembles the Casimir invariants for Lie algebra irreps. In fact, since any rep of G can be thought of as a module for C[G] and vice versa, we can look at the center of C[G]. This is analogous to looking at the center of the universal enveloping algebra of a Lie algebra. It is simple to check that

<math>\sum_{g\in G}g^n<math>

belongs to the center of C[G], which is simply the subspace of class functions on G .

If V is the underlying vector space of a rep, then

<math>V\otimes V<math>

can be decomposed as the direct sum of two subreps, the symmetric tensor product

<math>V\otimes_S V<math>

and the antisymmetric tensor product

<math>V\otimes_A V<math>.

It's easy to show that

<math>\chi_{V\otimes_S V}(g)=\frac{1}{2}[\chi_V(g)^2+\chi_V(g^2)]<math>

and

<math>\chi_{V\otimes_A V}(g)=\frac{1}{2}[\chi_V(g)^2-\chi_V(g^2)]<math>

using a basis set.

<math>\frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_S V}(g)<math>

and

<math>\frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_A V}(g)<math>

are the number of copies of the trivial rep in

<math>V\otimes_S V<math>

and

<math>V\otimes_A V<math>,

respectively. As observed above, if V is an irrep,

<math>V\otimes V<math>

contains exactly one copy of the trivial rep if V is equivalent to its dual rep and no copies otherwise. For the former case, the trivial rep could either lie in the symmetric product, or the antisymmetric product.

Putting all of this together, for an irrep, the second Frobenius-Schur indicator is zero if the irrep isn't self-dual, 1 if it's self-dual and there's a nonzero symmetric intertwiner from <math>V\otimes V<math> to the trivial rep and -1 if it's self-dual and there's a nonzero antisymmetric intertwiner from <math>V\otimes V<math> to the trivial rep; and there are no other possibilities.

Examples

See representations of S_N.

References

• William Fulton, Joe Harris, Representation Theory, A First Course, (1991) Springer, New York. ISBN 0-387-97495-4