12  Character Theory

The character table of a finite group is a square matrix that encodes every irreducible representation as a single row of complex numbers. It is the Fourier-analytic backbone of the group: just as the DFT matrix decomposes signals into frequencies, the character table decomposes functions on the group into irreducible components.

For a programmer, the character table is a precomputed lookup table. Once you have it, Fourier transforms, inner products, and decompositions reduce to matrix-vector operations.

(ns harmonica-book.character-theory
  (:require
   [scicloj.harmonica :as hm]
   [scicloj.harmonica.linalg.complex :as cx]
   [harmonica-book.book-helpers :refer [allclose?]]
   [tech.v3.datatype.functional :as dfn]
   [scicloj.kindly.v4.kind :as kind]))

What is a character?

A representation \(\rho : G \to GL(V)\) realizes each group element as an invertible matrix, with group multiplication becoming matrix multiplication. The character is the trace of this matrix:

\[\chi_\rho(g) = \text{tr}(\rho(g))\]

Why is the trace — a single number — enough to capture a matrix? Three reasons:

  1. Class function: The trace is invariant under conjugation (\(\text{tr}(ABA^{-1}) = \text{tr}(B)\)), so conjugate group elements have the same character value. The character depends only on the conjugacy class, not the individual element.

  2. Determines the representation: Two irreducible representations with the same character are equivalent. The trace loses no information that matters.

  3. Orthogonality: The irreducible characters form an orthonormal basis for the space of class functions — exactly what you need for Fourier analysis.

The character table collects all irreducible characters into one matrix. Rows are irreducible representations, columns are conjugacy classes.

Cyclic groups — the DFT matrix

For \(\mathbb{Z}/n\mathbb{Z}\), the character table is the DFT matrix: \(\chi_k(g) = \omega^{kg}\) where \(\omega = e^{2\pi i/n}\). All entries lie on the unit circle. We saw this in detail in The DFT as Group Fourier Transform.

(let [ct (hm/character-table (hm/cyclic-group 6))]
  (count (:table ct)))
6
(hm/show-character-table (hm/character-table (hm/cyclic-group 6)))
0 1 2 3 4 5
0 1 1 1 1 1 1
1 1 0.500+0.866i -0.500+0.866i -1 -0.500-0.866i 0.500-0.866i
2 1 -0.500+0.866i -0.500-0.866i 1 -0.500+0.866i -0.500-0.866i
3 1 -1 1 -1 1 -1
4 1 -0.500-0.866i -0.500+0.866i 1 -0.500-0.866i -0.500+0.866i
5 1 0.500-0.866i -0.500-0.866i -1 -0.500+0.866i 0.500+0.866i

Every entry has magnitude 1:

(allclose? (cx/cabs (:table (hm/character-table (hm/cyclic-group 8)))) 1.0)
true

Symmetric groups — integer entries

For \(S_n\), the character table is computed by the Murnaghan-Nakayama rule — a combinatorial algorithm based on border-strip tableaux. A remarkable fact: all entries are real integers.

(let [table (:table (hm/character-table (hm/symmetric-group 4)))
      re-vals (cx/re table)]
  (and (allclose? (cx/im table) 0.0)
       (allclose? re-vals (dfn/rint re-vals))))
true

Known character tables

\(S_3\)

The character table of \(S_3\) has three rows (one per partition of 3): the trivial representation \([3]\), the standard representation \([2,1]\), and the sign representation \([1,1,1]\).

(hm/show-character-table (hm/character-table (hm/symmetric-group 3)))
[1 1 1] [2 1] [3]
[3] 1 1 1
[2 1] 2 0 -1
[1 1 1] 1 -1 1 
(let [ct (hm/character-table (hm/symmetric-group 3))
      re-table (mapv (fn [row] (mapv #(Math/round (cx/re %)) row))
                     (:table ct))]
  re-table)
[[1 1 1] [2 0 -1] [1 -1 1]]

The first column is the identity class — these entries are the dimensions of each irrep (\(1, 2, 1\)). The dimension-squared sum equals \(|S_3| = 6\): \(1^2 + 2^2 + 1^2 = 6\).

\(S_4\)

(hm/show-character-table (hm/character-table (hm/symmetric-group 4)))
[1 1 1 1] [2 1 1] [2 2] [3 1] [4]
[4] 1 1 1 1 1
[3 1] 3 1 -1 0 -1
[2 2] 2 0 2 -1 0
[2 1 1] 3 -1 -1 0 1
[1 1 1 1] 1 -1 1 1 -1 
(let [ct (hm/character-table (hm/symmetric-group 4))
      re-table (mapv (fn [row] (mapv #(Math/round (cx/re %)) row))
                     (:table ct))]
  re-table)
[[1 1 1 1 1] [3 1 -1 0 -1] [2 0 2 -1 0] [3 -1 -1 0 1] [1 -1 1 1 -1]]

Row orthogonality

The central identity of character theory (Schur orthogonality): different irreps are orthogonal as class functions.

\[\frac{1}{|G|}\sum_C |C| \, \chi_i(C) \, \overline{\chi_j(C)} = \delta_{ij}\]

This is the Fourier inversion formula for class functions. It guarantees that the character table is invertible — that we can decompose any class function into irreducible components.

Let’s verify for \(S_4\) — a \(5 \times 5\) character table.

(let [ct (hm/character-table (hm/symmetric-group 4))
      {:keys [table class-sizes]} ct
      order (hm/order (:group ct))
      n (count table)
      inner (fn [i j]
              (reduce + (map-indexed
                         (fn [k sz]
                           (let [ci ((table i) k) cj ((table j) k)]
                             (* (double sz)
                                (+ (* (cx/re ci) (cx/re cj))
                                   (* (cx/im ci) (cx/im cj))))))
                         class-sizes)))
      max-err (apply max
                     (for [i (range n) j (range n)]
                       (Math/abs (- (inner i j)
                                    (if (= i j) (double order) 0.0)))))]
  (< max-err 1e-8))
true

The inner product matrix is \(|G| \cdot I\): row \(i\) dotted with row \(j\) (weighted by class sizes) gives \(|G|\) on the diagonal and 0 off it.

Column orthogonality

The dual identity: different conjugacy classes are orthogonal across irreps.

\[\sum_\rho \chi_\rho(C_i) \, \overline{\chi_\rho(C_j)} = \frac{|G|}{|C_i|} \, \delta_{ij}\]

(let [ct (hm/character-table (hm/symmetric-group 4))
      {:keys [table class-sizes]} ct
      order (hm/order (:group ct))
      n (count class-sizes)
      max-err (apply max
                     (for [i (range n) j (range n)]
                       (let [ip (reduce + (map (fn [row]
                                                 (let [ci (row i) cj (row j)]
                                                   (+ (* (cx/re ci) (cx/re cj))
                                                      (* (cx/im ci) (cx/im cj)))))
                                               table))
                             expected (if (= i j)
                                        (/ (double order) (double (nth class-sizes i)))
                                        0.0)]
                         (Math/abs (- ip expected)))))]
  (< max-err 1e-8))
true

Dihedral groups

For \(D_n\), the character table has:

  • Odd \(n\): 2 one-dimensional irreps + \((n-1)/2\) two-dimensional
  • Even \(n\): 4 one-dimensional irreps + \((n/2 - 1)\) two-dimensional

All entries are real (dihedral characters can be realized over \(\mathbb{R}\)).

(hm/show-character-table (hm/character-table (hm/dihedral-group 4)))
[:r 0] [:r 2] [:r 1] [:s 0] [:s 1]
:trivial 1 1 1 1 1
:sign 1 1 1 -1 -1
:sign-rot 1 1 -1 1 -1
:sign-both 1 1 -1 -1 1
[:dim2 1] 2 -2 0 0 0

The \(D_3 \cong S_3\) isomorphism means their character dimensions match:

(let [ct-d3 (hm/character-table (hm/dihedral-group 3))
      dims (sort (mapv #(Math/round (cx/re (% 0))) (:table ct-d3)))
      ct-s3 (hm/character-table (hm/symmetric-group 3))
      dims-s3 (sort (mapv #(Math/round (cx/re (% 0))) (:table ct-s3)))]
  (= dims dims-s3))
true

For the structure of dihedral character tables across many values of \(n\), see Algebraic Identities.

Murnaghan-Nakayama spot checks

The trivial character \(\chi_{[n]}\) has value 1 for every conjugacy class.

(let [trivial-row (first (:table (hm/character-table (hm/symmetric-group 5))))]
  (allclose? (cx/re trivial-row) 1.0))
true

The sign character \(\chi_{[1^n]}(\mu) = (-1)^{n-k}\) where \(k\) is the number of parts of \(\mu\).

(let [ct (hm/character-table (hm/symmetric-group 5))
      sign-label [1 1 1 1 1]
      labels (:irrep-labels ct)
      row-idx (.indexOf labels sign-label)
      sign-row (nth (:table ct) row-idx)
      classes (:classes ct)
      expected (mapv (fn [mu] (Math/pow -1 (- 5 (count mu)))) classes)]
  (allclose? (cx/re sign-row) expected))
true

The character table of \(S_5\)

(hm/show-character-table (hm/character-table (hm/symmetric-group 5)))
[1 1 1 1 1] [2 1 1 1] [2 2 1] [3 1 1] [3 2] [4 1] [5]
[5] 1 1 1 1 1 1 1
[4 1] 4 2 0 1 -1 0 -1
[3 2] 5 1 1 -1 1 -1 0
[3 1 1] 6 0 -2 0 0 0 1
[2 2 1] 5 -1 1 -1 -1 1 0
[2 1 1 1] 4 -2 0 1 1 0 -1
[1 1 1 1 1] 1 -1 1 1 -1 -1 1

What comes next

Characters are the “scalar” side of representation theory. The full picture includes the actual matrices:

source: notebooks/harmonica_book/character_theory.clj