Representation Theory

Let \(F\) be a field of characteristic \(p\). Let \(S_n\) be the group of permutations of \(X = \{1,2,\ldots ,n\}\). Let \(V = F[X]\) be the space of \(F\)-valued functions. It is a linear representation of \(S_n\), where \(\sigma \in S_n\) acts on \(f \in F[X]\cong F^n\) by \((\pi (\sigma )f)(x)=f(\sigma ^{-1}x)\). We have an intertwining map \(P:F[X]\to F\), where on \(F\) we have the trivial action of \(S_n\), given by \(f\mapsto f(1)+f(2)+\cdots +f(n)\). Prove that this map splits, that is, there exists an intertwining map \(Q\) in the opposite direction such that \(P\circ Q=1\), if and only if \(p\) does not divide \(n\).

Let \(G\) be a finite group. Let \(V_1\) and \(V_2\) be two non-isomorphic irreducible complex representations. Let \(V = V_1 \oplus V_2\). Compute the algebra of linear transformations of \(V\) intertwining the action of \(G\): \[ \operatorname {Hom}_G(V,V). \]

Find a complete list of irreducible representations of the alternating group \(A_4\).

Recall that the standard representation of \(S_n\) is the subspace

with \(S_n\) acting by permutation of the coordinates. Show that \(\wedge ^2 V_5\) is an irreducible representation of \(S_5\).

Suppose \(\sigma \) and \(\tau \) are two elements of \(A_5\) such that, for every finite dimensional complex representation \(\rho \) of \(S_5\), \(\chi _\rho (\sigma )=\chi _\rho (\tau )\) (here \(\chi _\rho \) denotes the character of \(\rho \)). Are \(\sigma \) and \(\tau \) conjugate in \(A_5\)?

Let \(G\) be the group of symmetries of an equilateral triangle. Show that the 2 dimensional representation (as symmetries of the triangle) is irreducible by calculating the character table.

Let \(D_8\) be the group of symmetries of a square. Show that the 2 dimensional representation (as symmetries of the square) is irreducible by calculating the character table.

Let \(G\) be a finite group. Prove the following version of Schur’s Lemma: if \(\rho \colon G\to M_n(\mathbb {C})\) is a finite dimensional irreducible representation of \(G\) over the complex numbers, then every element of \(M_n(\mathbb {C})\) that commutes with \(\rho (g)\) for all \(g\in G\) is a scalar matrix.

Let \(G\) be a finite group. Prove that every short exact sequence of \(\mathbb {C}[G]\)-modules splits, where \(\mathbb {C}[G]\) is the group algebra of \(G\) over the complex numbers.