Finite groups · research workbench

Finite characters & tensor products

Decompose tensor, symmetric, and exterior powers using exact ordinary complex characters.

Scope: C₃, S₃, D₈, Q₈, A₄, S₄, A₅; characteristic zero. Seven curated groups; base representations of complex dimension ≤60; multiplicities ≤12; powers of degree 0–12.

Exact resultExact checks passed

\left(\mathbf{3}\right)\otimes \left(\mathbf{3}\right)=\mathbf{1}\oplus \mathbf{1}'\oplus \mathbf{1}''\oplus 2\,\mathbf{3}

Total complex dimension: $9$
Trivial-irrep multiplicity: $1$
Distinct summands: 4
Character field: $\mathbb Q(\omega)$

Irreducible decomposition
Irrep	Multiplicity	Dimension	Reality type	Conjugate
$\mathbf{1}$	$1$	$1$	Real	$\mathbf{1}$
$\mathbf{1}'$	$1$	$1$	Complex	$\mathbf{1}''$
$\mathbf{1}''$	$1$	$1$	Complex	$\mathbf{1}'$
$\mathbf{3}$	$2$	$3$	Real	$\mathbf{3}$

This counts internal representation multiplicities. A Lorentz-invariant operator basis additionally needs the full field content, contractions, derivatives, and statistics. Copies in the first input form a direct sum.

Full character table and class data

Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123). ω = exp(2πi/3), ω² + ω + 1 = 0

Ordinary complex irreducible characters
Irrep	1	(12)(34)	(123)	(132)	Indicator
$\mathbf{1}$	$1$	$1$	$1$	$1$	1
$\mathbf{1}'$	$1$	$1$	$\omega$	$\omega^2$	0
$\mathbf{1}''$	$1$	$1$	$\omega^2$	$\omega$	0
$\mathbf{3}$	$3$	$-1$	$0$	$0$	1

Conjugacy-class data
Class	Size	Element order	Representative
$1$	1	1	()
$(12)(34)$	3	2	(12)(34)
$(123)$	4	3	(123)
$(132)$	4	3	(132)

The JSON export retains exact algebraic coefficients and the calculation. Download this table with its power maps.

✓ integral multiplicities✓ nonnegative multiplicities✓ dimension✓ character reconstruction

Try a worked example

Method & interpretation

Exact character arithmetic

Class-size-weighted character inner products give multiplicities. Symmetric and exterior powers use exact power maps and Newton identities. Orthogonality, every class value, and total dimension are checked.

Read the worked guide · Mathematical method

Download script for an independent environment; not executed by this website. GAP checks the serialized character data and exact decomposition; it does not import an independently identified upstream table.

Sources, conventions, and version

Computed locally with exact integer or algebraic arithmetic. Independent human expert review is pending. The export records the conventions and implementation version.

Tim Dokchitser, GroupNames: A4Character table, class sizes, group presentation, and representation type. Our explicit representatives and row labels fix any reordering.
Etingof et al., Introduction to Representation TheorySections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators
GAP Reference Manual, Chapter 72: Class FunctionsSections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers
GAP Reference Manual, Chapter 71: Character TablesSection 71.12-5: second Frobenius–Schur indicator; class data and power maps
GAP Reference Manual, Chapter 73: Maps Concerning Character TablesSection 73.1: power maps are extra class data and need not follow from the character matrix alone

Convention: finite-ordinary-complex-v1
Algorithm: finite-characters-1.0.0

Convention profiles · Reproduce and cite

How calculation links, versions, and exports work