Character Theory

The character of a representation — the trace of each representing matrix — is a single class function that determines the representation completely. Characters turn representation theory into linear algebra over a small table of complex numbers, and their orthogonality relations make decomposition into irreducibles a matter of inner products. This page builds on reps-basics.md (Maschke, Schur) and is the computational heart of finite-group representation theory.

Characters

The character of a representation $\rho :G\to GL(V)$ is $${\chi }_{\rho }(g)=\mathrm{tr}\rho (g).$$ Because trace is conjugation-invariant, ${\chi }_{\rho }(hg{h}^{-1})={\chi }_{\rho }(g)$: a character is a class function, constant on conjugacy classes. Basic values and properties:

• ${\chi }_{\rho }(e)=\dim V$ (the degree).
• ${\chi }_{\rho \oplus \sigma }={\chi }_{\rho }+{\chi }_{\sigma }$ and ${\chi }_{\rho \otimes \sigma }={\chi }_{\rho }{\chi }_{\sigma }$.
• ${\chi }_{\rho }(g^{-1})=\overline{{\chi }_{\rho }(g)}$ for unitary $\rho$.
• Characters are algebraic integers, and equivalent representations have equal characters — and, remarkably, the converse holds (below).

The orthogonality relations

Equip class functions with the Hermitian inner product $$⟨\chi ,\psi ⟩=\frac{1}{|G|}\sum _{g\in G}\overline{\chi (g)}\psi (g).$$

First orthogonality relation (rows). The irreducible characters $\{{\chi }_1,\dots ,{\chi }_k\}$ are orthonormal: $⟨{\chi }_i,{\chi }_j⟩={\delta }_{ij}$.

Consequences that make the theory computable:

• Decomposition. Any character decomposes as $\chi ={\sum }_i{m}_i{\chi }_i$ with multiplicities $m_i=⟨\chi ,{\chi }_i⟩$ read off by an inner product.
• Irreducibility test. $\rho$ is irreducible iff $⟨{\chi }_{\rho },{\chi }_{\rho }⟩=1$.
• Characters determine representations. $\rho \cong \sigma$ iff ${\chi }_{\rho }={\chi }_{\sigma }$ — the character is a complete invariant.
• Number of irreps = number of conjugacy classes (the irreducible characters form a basis of the space of class functions).

Second orthogonality relation (columns). Summing over irreducibles, ${\sum }_i\overline{{\chi }_i(g)}{\chi }_i(h)=0$ for non-conjugate $g,h$, and $=|C_G(g)|$ when $g=h$.

The regular representation and the sum of squares

Decomposing the regular representation (character ${\chi }_{\text{reg}}(e)=|G|$, ${\chi }_{\text{reg}}(g)=0$ for $g\ne e$) with the first orthogonality relation gives $m_i=\dim V_i$: each irrep appears with multiplicity equal to its dimension, whence $$\boxed{\sum _i(\dim V_i{)}^2=|G|}$$ This finite identity is the exact analogue of the Peter–Weyl decomposition for compact groups. It severely constrains the possible dimensions of irreps and is often enough to pin down a character table by hand.

Character tables

A character table is the $k\times k$ grid of values ${\chi }_i(g_j)$ (irreducibles by conjugacy classes). Two worked examples:

$S_3$ (order $6$; classes $\{e\}$, three transpositions $\{(12)\dots \}$, two $3$-cycles $\{(123),(132)\}$):

Dimensions $1^2+1^2+2^2=6=|S_3|$. ✓

$Q_8$ and $D_4$ (both order $8$) have the same character table — four $1$-dimensional characters and one $2$-dimensional, with $4\cdot 1^2+2^2=8$ — a reminder that the character table does not determine the group (their element orders differ; see axioms.md).

Payoff theorems

Character theory proves results with no elementary route:

• Burnside's $p^a{q}^b$ theorem: every group whose order has only two prime divisors is solvable (see series-solvable.md). The original proof is pure character theory.
• Frobenius's theorem on Frobenius kernels — the existence of a normal subgroup detected entirely through induced characters (see induced-reps.md).
• Degrees of irreps divide $|G|$; groups with all irreps $1$-dimensional are exactly the abelian ones (recovering cyclic-abelian.md).

References

• Serre, Linear Representations of Finite Groups, Part I.
• Isaacs, Character Theory of Finite Groups — the definitive reference.
• Fulton & Harris, Representation Theory, Ch. 2–3.