Exercise set 5

The character of a finite dimensional (complex) representation V is denoted by  _V .

(1)

(Difficulty level 2) Let G be a finite group and V a finite dimensional complex linear representation of G. Observe the following:

(a)

Each element g of G is diagonalizable as an operator on V .

(b)

The value at g of the character of V is the sum of some n^th roots of unity where n is the order of g.

(c)

Any character value that is rational is actually integral.

(2)

(Difficulty level 2, 3) Compute the primitive central idempotents in the group ring ℂG, for G a cyclic group; for G the symmetric group 𝔖_n, n = 3, n = 4, n = 5, …

(3)

(Difficulty level 1) True or false?: every finite dimensional representation over the real numbers of a finite group is orthogonal.

(4)

• (Difficulty level 1) True or false?: If two group elements of a finite group G act the same way on all irreducible representations over the rational numbers of the finite group, then they are equal.
• (Difficulty level 2) True or false?: If two elements of the complex group ring ℂG of a finite group G act the same way on all irreducible representations over the complex numbers of the finite group, then they are equal.
• (Difficulty level 2) True or false?: If two elements of a ring act the same way on all simple modules, then they are equal.

(5)

(Difficulty level 2) An element g in a finite group G belongs to the centre of G if and only if | _V (g)| = dimV for every complex irreducible representation of G.

(6)

(Difficulty level 3) If a finite group G admits a faithful irreducible complex linear representation, then its centre is cyclic. (A representation V is called faithful if the group homomorphism G → GL(V ) defining it is injective.)

(7)

(Difficulty level 3) Count the number of k-dimensional vector subspaces in a fixed n-dimensional vector space over the finite field F_q of q elements.

(8)

(Difficulty level 3) The cycle type of a permutation of 25 elements is 4,4,4,3,3,2,2,2,1. Compute the order of its centralizer.

Additional problems (Optional)

(1)

(Difficulty level 2) Check that the isomorphism End_R(RR) ≃ R^opp in natural (where R denotes a ring with identity).

(2)

(Difficulty level 3) Construct a ring (with identity) which is not isomorphic to its opposite.