Character table

The character table of a finite group lists its irreducible complex characters on its conjugacy classes. The number of rows equals the number of conjugacy classes, and the squared row degrees sum to the group order.

For the nontrivial action of $C_3$ on $C_7$, the conjugacy class sizes are $1,7,7,3,3$. Three linear characters factor through the quotient $C_3$, and the remaining two irreducible characters have degree three. Put $\omega =e^{2\pi i/3}$ and let $\alpha ,\beta =(-1\pm i\sqrt{7})/2$. With suitable class labels the table isOne can derive the last two rows by inducing nontrivial characters of $C_7$ along the two orbits of the order-three automorphism on their exponents. Their degree-three orbit sums vanish off $C_7$. On a generator they give the two sums of seventh roots over quadratic residues and nonresidues. Their sum is $-1$ and their product is $2$: the nine pairwise exponent sums between the two triples contain zero three times and every nonzero residue once. Thus they are the roots of $X^2+X+2$. Alternatively column orthogonality forces the zeros on the centralizers of order three, and orthogonality with the identity column interchanges $\alpha$ and $\beta$ in the last row. This finite table illustrates how centralizer sizes and a small number of given values determine missing character values.

A group of order thirty-six with a normal $C_3^2$ and a cyclic quotient of order four acting freely on its eight nonzero elements has six conjugacy classes of sizes $1,4,4,9,9,9$. Its four linear characters are lifted from the cyclic quotient. The remaining irreducible degrees are four and four, by the sum of squared degrees. Column orthogonality forces the last three values of both degree-four rows to vanish. The square of the order-four action is inversion on $C_3^2$, making the first two nontrivial class values real. Orthogonality then gives sum minus one and product minus two, so the rows are $(4,1,-2,0,0,0)$ and $(4,-2,1,0,0,0)$. This illustrates why reality must be justified when reconstructing a character table from orthogonality.

In the conjugacy class order $1,(123),(12)(34),(12345),(13452)$, the five complex irreducible characters of the alternating group $A_5$ areThe class sizes are $1,20,15,12,12$. Restricting the degree-four and degree-five irreducible characters of $S_5$ gives rows of squared character inner product one. The degree-six restriction has squared norm two and is orthogonal to those rows and the trivial row, so it splits into the two remaining irreducibles. The sum of squared degrees forces their degrees to be three. Orthogonality forces their last two values to have sum one and product minus one, producing the displayed roots. Since a normal subgroup is a union of classes and its order divides sixty, these class sizes also prove the Simplicity of the alternating group A5.

For a finite group, the number of irreducible characters which are real-valued characters equals the number of real conjugacy classes. On the complex vector space of class functions, the linear operator $Tf(g)=f(g^{-1})$ permutes the basis of class indicators by inversion and the basis of irreducible characters by pointwise complex conjugation, using the character inversion identity. The trace of a basis permutation is its number of fixed members. Equality of the trace in the two bases proves the theorem. The linear operator $T$ is complex-linear; pointwise complex conjugation on arbitrary class functions is not.

Column orthogonality and the regular character give quadratic and linear equations for missing character values once their degrees are known. These equations can admit spurious rational solutions. Every character value is an algebraic integer, so a rational candidate must be an ordinary integer. Applying this extra condition can remove ambiguities that orthogonality alone leaves.

Forthere are $2n$ linear characters, given by $a\mapsto {\xi }^j$, $b\mapsto 1$ for a primitive $(2n)$th root of unity $\xi$, and $n$ two-dimensional irreducible representationswhere ${\omega }^3=1$ and $\omega \ne 1$. The two-dimensional representations parametrized by $\epsilon$ and $-\epsilon$ are isomorphic. The sum of squares of irreducible degrees is $2n+4n=6n$, so this list is complete.

The conjugacy classes of $G_{6n}$ arefor $0\le r<n$. With $0\le j<2n$ and $0\le k<n$, the linear character ${\lambda }_j$ and two-dimensional character ${\chi }_k$ have values

The degree of an irreducible character $\chi$ is $\chi (1)$, the dimension of the corresponding irreducible representation.

For a finite group $G$ and prime number $p$, the notationdenotes the irreducible characters whose irreducible character degree is coprime to $p$.

A linear character is the character of a one-dimensional representation and therefore has degree one.

The trivial character takes the value $1$ on every group element and is afforded by the one-dimensional trivial representation.

A nonlinear irreducible character has degree greater than one.