Lecture notes for Introduction to Representation Theory

Thus for g G we have
ν V (g 2 ) = ν S 2 V (g) − ν 2 V (g)
Therefore,
⎞
⎟ 1, if V is of real type
| G| −1 ν 2 V ) G
V ( g 2 ) = ν S 2 V (P )−ν √ 2 V (P ) = dim(S 2 V ) G −dim(√ = −1, if V is of quaternionic type
⎠
gG
0, if V is of complex type
Finally, the number of involutions in G equals
1
dim V ν V ( g 2 ) = dim V − dim V.
|G|
V
gG real V quat V
Corollary 4.5. Assume that all representations of a finite group G are defined over real numbers
(i.e., all complex representations of G are obtained by complexifying real representations). Then
the sum of dimensions of irreducible representations of G equals the number of involutions in G.
Exercise. Show that any nontrivial finite group of odd order has an irreducible representation
which is not defined over R (i.e., not realizable by real matrices).
4.2 Frobenius determinant
Enumerate the elements of a finite group G as follows: g 1 , g 2 , . . . , g n . Introduce n variables indexed
with the elements of G :
x g1 , x g2 , . . . , x gn .
Definition 4.6. Consider the matrix X G with entries a ij = x gi g j
. The determinant of X G is some
polynomial of degree n of x g1 , x g2 , . . . , x gn that is called the Frobenius determinant.
The following theorem, discovered by Dedekind and proved by Frobenius, became the starting
point for creation of representation theory (see [Cu]).
Theorem 4.7.
r
P j (x) deg P j
det X G =
j=1
for some pairwise non-proportional irreducible polynomials P j (x), where r is the number of conjugacy
classes of G.
We will need the following simple lemma.
Lemma 4.8. Let Y be an n × n matrix with entries y ij . Then det Y is an irreducible polynomial
of {y ij }.
n
Proof. Let Y = t·
Id+ ⎨ i=1 x iE i,i+1 , where i+1 is computed modulo n, and E i,j are the elementary
matrices. Then det(Y ) = t n − (−1) n x 1 ...x n , which is obviously irreducible. Hence det(Y ) is
irreducible (since factors of a homogeneous polynomial are homogeneous).
Now we are ready to proceed to the proof of Theorem 4.7.
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