Finite characters: conventions and exact methods

The finite-character workbench decomposes representations of $C_3$, $S_3$, $D_8$, $Q_8$, $A_4$, $S_4$, and $A_5$. Its category is finite-dimensional ordinary complex representations in characteristic zero. Every irreducible character is included for each of these seven groups. The declared scope concerns these groups and operations; the broader Atlas contains many additional reference records.

For a representation $V$ with character $\chi_V$, the multiplicity of an irrep $R$ is

$$m_R(V)=\frac{1}{|G|}\sum_C |C|\,\chi_V(C)\,\overline{\chi_R(C)}.$$

The sum runs over conjugacy classes. Tensor products multiply their characters, so

$$m_R(V\otimes W) =\frac{1}{|G|}\sum_C |C|\,\chi_V(C)\chi_W(C)\overline{\chi_R(C)}.$$

These are the character orthogonality and multiplicity formulas discussed in Etingof et al., §§4.4–4.5 and 4.9 and implemented for ordinary characters by GAP’s scalar-product operations, §72.8. The website evaluates them locally using exact algebraic arithmetic.

Scope and notation

Group	Order	Irreducible labels in this workbench	Character values
$C_3$	3	$\mathbf1,\chi,\chi^2$	$\mathbb Z[\omega]$
$S_3$	6	$\mathbf1,\mathrm{sgn},\mathbf2$	Integers
$D_8$	8	$\mathbf1,a,b,c,\mathbf2$	Integers
$Q_8$	8	$\mathbf1,a,b,c,\mathbf2$	Integers
$A_4$	12	$\mathbf1,\mathbf1',\mathbf1'',\mathbf3$	$\mathbb Z[\omega]$
$S_4$	24	$\mathbf1,\mathrm{sgn},\mathbf2,\mathbf3,\mathbf3'$	Integers
$A_5$	60	$\mathbf1,\mathbf3,\mathbf3',\mathbf4,\mathbf5$	$\mathbb Z[\varphi]$

Here

$$\omega=e^{2\pi i/3},\qquad \omega^2+\omega+1=0, \qquad \varphi=\frac{1+\sqrt5}{2},\qquad\varphi^2-\varphi-1=0.$$

The character tables can be checked against the GroupNames character-table collection; each workbench table links to its individual source record. Conjugacy-class representatives are displayed because row and column labels can vary across references.

Three conventions deserve particular attention:

• $D_8$ has order eight. It is the square-symmetry group with $r^4=s^2=1$ and $srs=r^{-1}$. GroupNames denotes this group by $D_4$.
• The two $S_4$ triplets are inequivalent. Here $\mathbf3$ is the four-letter permutation representation with its invariant line removed, and $\mathbf3'=\mathrm{sgn}\otimes\mathbf3$ is the orientation-preserving cubic-rotation representation. This convention fixes which triplet a dimension label denotes. See Etingof et al., §4.8.
• The two $A_5$ triplets are separately of real type. The prime does not mean complex conjugation. With $c=(12345)$, our $\mathbf3$ takes character value $\varphi$ on $c$, while $\mathbf3'$ takes $1-\varphi$. The two rows are displayed in GroupNames: $A_5$.

The interface selects an irrep and allows 1–12 copies of it as a direct sum. It tensors that sum with a second selected irrep, or takes its tensor, symmetric, or exterior power with degree $0\leq n\leq12$; the input dimension must be at most 60. The underlying exact engine additionally accepts mixed finite direct sums, with each input multiplicity at most 12 and each input dimension at most 60. These bounds control released calculations; they do not bound the underlying mathematical representation theory.

Why power maps belong to the data

A power map records the class containing $g^k$ for each conjugacy class of $g$. This information is needed for symmetric and exterior powers. It cannot in general be recovered uniquely from the matrix of irreducible character values. The GAP manual, §73.1, makes this distinction explicit.

The pair $D_8,Q_8$ provides an immediate example. With the columns ordered as

$$\begin{array}{c|ccccc} D_8&1&r^2&r&s&rs\\ Q_8&1&-1&i&j&k \end{array}$$

the two character matrices agree. Their squaring maps differ: reflections in $D_8$ square to the identity, whereas $i,j,k$ in $Q_8$ square to $-1$. Consequently, identical tensor-product rules coexist with different symmetric squares, exterior squares, and reality types. The group presentations and table data appear in GroupNames: $D_4$ and GroupNames: $Q_8$.

The current dataset stores a complete power cycle for each class representative. Powers from 1 through 12 are exposed with the table; periodicity determines every positive power.

Symmetric and exterior powers

If the eigenvalues of $\rho_V(g)$ are $\alpha_1,\ldots,\alpha_d$, define the formal generating functions

$$H_g(t)=\prod_{j=1}^d\frac{1}{1-t\alpha_j}, \qquad E_g(t)=\prod_{j=1}^d(1+t\alpha_j).$$

Their coefficients are the characters of $\operatorname{Sym}^n V$ and $\Lambda^nV$. Logarithmic differentiation gives

$$\begin{aligned} n h_n(g)&=\sum_{k=1}^n\chi_V(g^k)h_{n-k}(g),\\ n e_n(g)&=\sum_{k=1}^n(-1)^{k-1}\chi_V(g^k)e_{n-k}(g), \end{aligned} \qquad h_0=e_0=1.$$

These recurrences give the class characters directly. A subsequent character inner product gives the full irrep decomposition. At degree two they reduce to

$$\chi_{\operatorname{Sym}^2V}(g) =\frac{\chi_V(g)^2+\chi_V(g^2)}{2}, \qquad \chi_{\Lambda^2V}(g) =\frac{\chi_V(g)^2-\chi_V(g^2)}{2}.$$

The result is checked against

$$\dim(V^{\otimes n})=d^n,\qquad \dim\operatorname{Sym}^nV=\binom{d+n-1}{n},\qquad \dim\Lambda^nV=\binom dn.$$

The zeroth power is the trivial representation. An exterior power with $n>d$ is the zero representation. These conventions are handled explicitly. The equivalent symmetrization operations and independent $A_5$ cubic examples are documented in GAP, §72.11.

Reality and the Frobenius–Schur indicator

For an irreducible ordinary complex representation, the second indicator is

$$\nu(R)=\frac{1}{|G|}\sum_C |C|\,\chi_R(C^2) =\begin{cases} +1,&R\text{ has real type},\\ 0,&R\text{ has complex type},\\ -1,&R\text{ has quaternionic type}. \end{cases}$$

An irrep of real type admits a nondegenerate invariant symmetric complex bilinear form. An irrep of quaternionic type admits an alternating one. Both have real-valued characters. The indicator therefore contains information beyond whether displayed character values are real. See Etingof et al., §5.1 and GAP, §71.12-5.

For an irrep, this can also be read as

$$\nu(R)=\dim(\operatorname{Sym}^2 R)^G-\dim(\Lambda^2R)^G.$$

For a reducible representation the same character average is additive, but its value does not supply a single irrep reality classification. The table’s reality column always refers to irreducible constituents.

Exactness and verification

The algebraic values are stored as integer coefficient pairs $a+b\omega$ or $a+b\varphi$. Products use the quadratic relations above. In the cubic-root field, complex conjugation acts as

$$\overline{a+b\omega}=(a-b)-b\omega.$$

Integer coefficients use arbitrary-precision arithmetic. Every division in the power recurrence and multiplicity formula is checked to be exact. Multiplicities are then required to be nonnegative integers. This avoids selecting an integer by rounding an approximate algebraic sum.

Validation checks class sizes, the sum of squared dimensions, row and column orthogonality, conjugate pairs, and Frobenius–Schur indicators. Independent finite-group enumeration checks every released class power map. Each calculated decomposition must reproduce the original character on every class and conserve total dimension. The test fixtures also include the $A_5$ symmetric/exterior cubes printed in the GAP manual and the contrasting $D_8,Q_8$ squares.

Preserve the calculation

The workbench records the convention profile finite-ordinary-complex-v1, version 1.0.0, the character dataset version, algorithm version, inputs, and source locators. Retain the exported result when using a calculation in research; an ordinary share link uses the currently published engine. See reproducibility and citation.

The output gives multiplicities and invariant counts. Explicit contraction tensors require a chosen basis and representation model. Modular, projective, and real representation categories require their own data and operations. Independent human expert review remains recorded separately from these source and algorithm checks.

Continue with From characters to physical couplings for worked examples, or open the finite-character workbench.