Exercise 9, § 4

Tag 03FD, permanent. Cite it as [Bourbaki, Tag 03FD].

Spectral Theories, V, p. 507

Let $k$ be a finite field with 5 elements and $G =\mathbf{S}\mathbf{L}(2, k)$ .

a) The conjugacy classes of the group G are represented by $e,-e$ and the seven elements

(1 1)(1 2)(-1 1)(-1 2)

g_3=,g_4=,g_5=,g_6=

0 1 0 1 0 $-1$ 0 $-1$

(2 0)(0 1)(01)

g_7=,g_8=,g_9=

0-2-1 1-1-1

Determine the cardinal of each conjugacy class.

\surd

b) Let $\alpha =\frac{1}{2}(1 +$ 5). There exists a faithful irreducible unitary representation $\pi$ of dimension 2 of G whose character $\chi$ satisfies $\chi (-e) =-2$ and

\chi (g_4) =\alpha ,\chi (g_6) =-\alpha ,\chi (g_3) =-\alpha -1,\chi (g_5) =\alpha + 1

\chi (g_7) = 0,\chi (g_8) = 1,\chi (g_9) =-1

(One may consider the restriction to G of the irreducible representations of dimension 4 of $\mathbf{G}\mathbf{L}(2, k)$ , cf. A, VIII, p. 422, exercise 27.)

c) One has $M_4(\pi ) = 2$ .

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