Key Concepts of Character Tables

Why This Matters

Character tables are the Rosetta Stone of representation theory—they compress everything you need to know about a finite group's irreducible representations into a single, elegant matrix. When you're working with symmetry groups in quantum mechanics, classifying finite groups, or decomposing representations, the character table is your primary computational tool. You're being tested on your ability to read, construct, and apply these tables, which means understanding the deep connections between conjugacy classes, orthogonality relations, and representation dimensions.

The concepts here tie directly to fundamental questions: How many irreducible representations does a group have? How do we verify that a representation is irreducible? How can we decompose an arbitrary representation into irreducibles? Don't just memorize the orthogonality formulas—know why characters are class functions, how the Great Orthogonality Theorem constrains the table's structure, and what each entry actually represents.

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The Foundation: What Character Tables Encode

Character tables work because characters capture essential information about representations while discarding unnecessary detail. The trace of a matrix is invariant under conjugation, which is why characters are naturally class functions.

Definition of Character Tables

• Square matrix structure—rows correspond to irreducible representations, columns to conjugacy classes, with the number of each being equal
• Entries are traces of representation matrices $\chi_\rho(g) = \text{Tr}(\rho(g))$, evaluated at a representative element of each conjugacy class
• Complex number values that encode representation behavior without requiring explicit matrix forms

Irreducible Representations

• Cannot be decomposed into direct sums of smaller representations—these are the "atoms" of representation theory
• Count equals conjugacy classes—a finite group has exactly as many distinct irreducible representations as it has conjugacy classes
• Dimension constraints follow from $\sum_i d_i^2 = |G|$, where $d_i$ are the dimensions and $|G|$ is the group order

Class Functions

• Constant on conjugacy classes—if $g$ and $hgh^{-1}$ are conjugate, then $f(g) = f(hgh^{-1})$ for any class function $f$
• Characters are prime examples, forming a distinguished basis for the space of class functions
• Inner product structure $\langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}$ makes this space a Hilbert space

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The Orthogonality Engine

Orthogonality relations are what make character tables computationally powerful. They transform abstract representation theory into linear algebra, letting you extract information through inner products.

Orthogonality Relations

• Row orthogonality—distinct irreducible characters satisfy $\langle \chi_i, \chi_j \rangle = \delta_{ij}$, where the inner product sums over group elements
• Column orthogonality—summing over representations gives $\sum_i \chi_i(g)\overline{\chi_i(h)} = \frac{|G|}{|C_g|}\delta_{C_g, C_h}$, where $|C_g|$ is the conjugacy class size
• Decomposition tool—to find how many times irreducible $\chi_i$ appears in representation $\chi$, compute $\langle \chi, \chi_i \rangle$

Great Orthogonality Theorem

• Master theorem stating $\sum_{g \in G} \rho_{ij}(g)\overline{\sigma_{kl}(g)} = \frac{|G|}{d_\rho}\delta_{\rho\sigma}\delta_{ik}\delta_{jl}$ for matrix entries of irreducible representations
• Character orthogonality follows by taking traces—this is the deeper result from which row/column relations derive
• Structural implications include the dimension formula and the fact that irreducible characters form an orthonormal basis

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Building and Reading the Table

Construction requires systematic identification of conjugacy classes and clever use of constraints. The table is overdetermined—orthogonality relations often force entries once you know a few.

Character Table Construction

• Step 1: Find conjugacy classes by identifying which elements are related by $g \sim hgh^{-1}$; the number of classes determines table size
• Step 2: Determine dimensions using $\sum d_i^2 = |G|$ and the fact that $d_i$ divides $|G|$ for each irreducible representation
• Step 3: Compute characters starting with the trivial representation (all 1s), then using orthogonality to constrain remaining entries

Character Table Symmetries

• First column always contains the dimensions $d_i = \chi_i(e)$ since the identity maps to the identity matrix
• Complex conjugate pairing—if $\chi$ is a character, so is $\overline{\chi}$, and they're equal iff the representation is real
• Abelian group simplification—all irreducible representations are 1-dimensional, so entries are roots of unity

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Applications and Connections

Character tables aren't just classification tools—they connect representation theory to counting problems, physics, and group structure. The table encodes more than it appears to.

Character Table Applications

• Group classification—non-isomorphic groups can have identical character tables, but the table still constrains structure significantly
• Quantum mechanics applications—selection rules, molecular orbital symmetries, and spectroscopic transitions all use character table analysis
• Representation decomposition—given any representation, inner products with irreducible characters reveal its decomposition

Burnside's Theorem

• Orbit counting formula $|X/G| = \frac{1}{|G|}\sum_{g \in G}|X^g|$ counts orbits as the average number of fixed points
• Character connection—the permutation character $\chi(g) = |X^g|$ links Burnside's lemma to representation theory
• Practical applications include counting distinct colorings, necklaces, and other combinatorial objects under symmetry

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Quick Reference Table

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Self-Check Questions

1. Why must the number of irreducible representations equal the number of conjugacy classes? Which orthogonality relation demonstrates this?

2. Given a character table, how would you verify that a particular row corresponds to an irreducible (rather than reducible) representation?

3. Compare the character tables of $\mathbb{Z}_6$ and $S_3$: both groups have order 6, but how do their tables differ, and why?

4. If you're given a representation $\rho$ and want to decompose it into irreducibles, explain the computational procedure using inner products.

5. How does Burnside's theorem connect to character theory? If an FRQ asks you to count orbits, when would you use character methods versus direct counting?