Irreducible representations of finite groups

Obtain relation (4) by writing out explicitly the expression $t_{i j}=\left(\Gamma_{i}, \chi_{j}\right)_{G}$ for the coefficients of the expansion $\Gamma_{i}=\sum_{j} t_{i j} \chi_{j}$ of the basis central function $\Gamma_{i}$ (see (7) from § 4) in terms of the irreducible characters.

Verify (and recall the isomorphism between the vector space $V$ and its dual space $V^{*}$ of linear functions) that the map $\tau : A \longrightarrow \hat{\hat{A}}$, defined by the condition

gives an isomorphism of the abelian group $A$ onto $\hat{\hat{A}}$.

This exercise, together with Theorem 4, establishes part of the so-called duality law for finite abelian groups. An analogous, but much deeper, duality law for topological abelian groups, leading to important consequences, was established in the 1930s by L.S. Pontryagin.

Prove that if a finite abelian group $A$ admits a faithful complex irreducible representation, then $A$ is a cyclic group.

Let $A$ be a finite abelian group, $B$ a subgroup of it. Prove that any character of the group $B$ extends to a character of the group $A$, and the number of such extensions equals the index $(A: B)$.

Justify the phrase before the closing parentheses in part V) of item 4.

Recall what this refers to. Let $\Phi$ be the natural representation of the group $S_{4}$ on the space $V=\langle e_{1}, e_{2}, e_{3}, e_{4}\rangle$ with the action $\Phi (g) e_{i}=e_{g(i)}$; then $\Phi =\Phi^{(1)} \dotplus \Psi$, where $\Phi^{(1)}$ is the unit representation and $\Psi$ is an irreducible three-dimensional representation acting on the space $W=\langle e_{1}-e_{4}, e_{2}-e_{4}, e_{3}-e_{4}\rangle$, with character $\chi_{\Psi }(g)=N(g)-1$, where $N(g)$ is the number of points of $\left\{ 1,2,3,4\right\}$ fixed by $g$. In the character table of the group $S_{4}$ the character $\chi_{5}=\chi_{\Psi }$ takes the values $3, -1, 1, 0, -1$ on the classes $e$, $(1\, 2)(3\, 4)$, $(1\, 2)$, $(1\, 2\, 3)$, $(1\, 2\, 3\, 4)$ respectively. Part V) (irreducible representations of the symmetric group $S_{4}$) ends as follows: "The representation $\Phi^{(5)}=\Psi$ with character $\chi_{5}$ is connected with the 2-transitivity of the group $S_{4}$. It is also equivalent to the representation corresponding to all symmetry transformations of the tetrahedron $\Delta_{4}$ (rotations plus reflections; it is exactly these transformations that are important in describing the vibrations of the phosphorus molecule (Problem 2 of § 2, Chapter 1 in Part I of this book))".

What is the average value $\frac{1}{\left|G\right|} \sum_{g} \chi (g)$ of the values of a complex character $\chi$ on the elements of a finite group $G$?

Collect, from various places, the tables relating to the group $A_{5}$ into a consolidated character table

Give a description of the irreducible representations with characters $\chi_{1}, \chi_{2}, \chi_{3}, \chi_{4}$. Fill in the last row of the table, using the second orthogonality relation (4) for characters.

Let $P= \left\langle A^{i} B^{j} C^{k} ; 0 \leqslant i, j, k \leqslant p-1\right\rangle$ be the group of order $p^{3}$ considered in Exercise 3 from $\S 3$ Ch. $1 ; V= \left\langle e_{0}, e_{1}, \ldots , e_{p-1}\right\rangle_{\mathbb {C}}$ a complex vector space of dimension $p ; \varepsilon$ a primitive $p$-th root of unity $; \mathcal{A}, \mathcal{B}, \mathcal{C}$ linear operators on $V$ defined by the relations

(the subscripts of the basis elements are taken modulo $p$).

Show that the map

gives a linear irreducible representation of the group $P$. The representations $\Phi^{(1)}, \ldots , \Phi^{(p-1)}$ are pairwise inequivalent and, together with the $p^{2}$ one-dimensional representations ($p^{2}$ is the index of the commutator subgroup $P^{\prime }=\langle C\rangle$ in $P$), exhaust all the irreducible complex representations of the group $P$.

Fill in with calculations the following argument. Let

be the dihedral group of order $2 n$, whose properties (including the description of the conjugacy classes) are given in Example 1 from item $5 \S 4$ Ch. 1. Since $\langle a\rangle \triangleleft D_{n}$, the maps $a \mapsto 1, b \mapsto 1$ and $a \mapsto 1, b \mapsto -1$ define two one-dimensional representations. Let $\varepsilon$ be a primitive $n$-th root of unity. Then the map

will define a representation of degree 2. The representation $\Phi^{(j)}$ is irreducible for $j=1,2, \ldots ,[(n-1) / 2]$ ($[\alpha ]$ is the integer part of the real number $\alpha )$. For $n=2 m$, the representation $\Phi^{(m)}$ splits into a direct sum of two one-dimensional representations: $a \mapsto -1, b \mapsto 1$ and $a \mapsto -1, b \mapsto -1$. This agrees with the fact that the commutator subgroup $D_{2 m}^{\prime }$ has index 4 in $D_{2 m}$ and $D_{2 m} / D_{2 m}^{\prime } \cong Z_{2} \times Z_{2}$. All the indicated representations are irreducible and constitute the complete set of irreducible complex representations of the dihedral group. Find a real realization of the representations $\Phi^{(j)}$. Give explicitly the isomorphism (equivalence) $\Phi^{(j)} \approx \Phi^{(k)}, k>m, j \leqslant m$.

Crystallographic groups (relating to Problem 2 from § 2 Ch. 1 in [ВА I]). Let $E$ be an $n$-dimensional Euclidean space and $V$ the vector space associated with it, with Euclidean inner product. To every motion $d$ of the space $E$ there corresponds an orthogonal linear transformation $\bar{d} \subset O(n)$, in such a way that $\overline{d_{1} d_{2}}=\bar{d}_{1} \bar{d}_{2}$. A group $D$ of motions of the space is called a crystallographic group if the $D$-orbit of an arbitrary point is discrete (has no limit points) and there exists a compact set $M \subset E$ for which $D(M)=\bigcup_{d \in D} d(M)=E$. The Schoenflies--Bieberbach theorem holds, according to which a crystallographic group $D$ contains $n$ independent affine translations generating a normal subgroup $L$ in $D$, and $\bar{D} \cong D / L$ is a finite group (the point crystallographic group). In all, there are 32 geometrically distinct point crystallographic groups for $n=3$. Among them there must obviously be groups containing reflections (improper motions). From the crystallographic conditions it follows that every proper rotation from $\bar{D}$ is represented by a matrix similar to

$\operatorname {tr} A=1+2 \cos \theta \in \mathbb {Z}$. Relying on Theorem 2 from § 3 and on the noted consideration, show that for $n=3$ the point crystallographic groups without reflections are precisely the cyclic groups $C_{1}, C_{2}, C_{3}, C_{4}, C_{6}$, the dihedral groups $D_{2}, D_{3}, D_{4}, D_{6}$, the tetrahedron group $\mathbf{T}$, and the group of the cube (octahedron) $\mathbf{O}$.