Appendix F — Homework 5

Exercise F.1 Recall that \(\mathcal{S}_{4}\) encodes the orientation-preserving isometries of the cube; there is one such rotation for each permutation of the \(4\) diagonals of the cube. In particular, \(\mathcal{S}_{4}\) acts (a) on the 8 vertices of the cube and (b) on the 12 edges of the cube.

  1. Decompose the permutation representation of \(\mathcal{S}_{4}\) on the vertices of the cube into irreducibles.

  2. Decompose the permutation representation of \(\mathcal{S}_{4}\) on the edges of the cube into irreducibles.

Exercise F.2 Compute the character table of \(\mathcal{D}_{10}\).

Exercise F.3 In this problem we compute the character table of \(\mathcal{S}_{5}\). We start with only knowledge of the conjugacy classes and their sizes: \[ \begin{array}{c|c|c|c|c|c|c} 1 & 10 & 20 & 15 & 30 & 20 & 24 \\ \mathbb{I}& (1\ 2) & (1\ 2\ 3) & (1\ 2)(3\ 4) & (1\ 2\ 3\ 4) & (1\ 2\ 3)(4\ 5) & (1\ 2\ 3\ 4\ 5) \end{array} \]

  1. Let \(U,\) \(U',\) and \(V\) be the trivial, alternating, and standard representations of \(\mathcal{S}_{5}\), respectively. Find the characters of these representations and confirm that they are irreducible.

  2. Set \(V' = V \otimes U'\), find its character, and confirm that it is a distinct irreducible representation from \(U\), \(U'\), and \(V\).

  3. If \(Y\) is a representation of \(G\), then we can compute the characters of its symmetric powers \(\operatorname{Sym}^{k}({Y})\) (c.f. Exercise E.3) in terms of \(\chi_Y\). In particular, you may use the following formula without proof: \[ \chi_{\operatorname{Sym}^{2}({Y})}(g) = \frac{1}{2} \left( \chi_Y(g)^2 + \chi_Y(g^2) \right). \] Compute the character for \(\operatorname{Sym}^{2}({V})\) (with \(V\) the standard representation) and use this to show that \(\operatorname{Sym}^{2}({V})\) is a direct sum of three distinct irreducible representations.

  4. Show that \(\operatorname{Sym}^{2}({V})\) is the direct sum of \(U\), \(V\), and one previously unknown irreducible—call it \(W\).

  5. Complete the character table for \(\mathcal{S}_{5}\).

Exercise F.4 Consider the class function \(f: \mathcal{S}_{5} \to \mathbb{C}\), where \(f(\sigma)\) counts the number of \(2\)-cycles in the cycle notation for \(\sigma\), e.g.: \[ \begin{aligned} f((2\ 3\ 4)) & = 0, \\ f((1\ 5)) & = 1, \\ f((1\ 2)(3\ 4)) & = 2. \end{aligned} \] Write \(f\) in the basis of irreducible characters for \(\mathscr{C}({\mathcal{S}_{5}})\).

Exercise F.5 Let \(V\) be an irreducible representation of \(G\).

  1. Show that \(V^*\) is also an irreducible representation of \(G\).

  2. If \(W\) is a \(1\)-dimensional representation of \(G\), show that \(V \otimes W\) is also an irreducible representation of \(G\).

Exercise F.6 (Bonus) If \(\rho: G \to \operatorname{GL}(V)\) is faithful,1 show that every irreducible representation of \(G\) is contained in some tensor power \(V^{\otimes k}\).

Hint: If W is an irreducible representation, consider the generating function \[ \sum_{k=0}^\infty \langle \chi_W, \chi_{V^{\otimes k}} \rangle t^k \] If you can show this series converges to a nontrivial rational function, it must contain (infinitely many) non-zero coefficients!

  1. Recall that a representation \(\rho\) is faithful if \(\ker \rho = \{e\}\).↩︎