Character of a representation

The character of a finite-dimensional representation $\rho$ is the class function ${\chi }_{\rho }(g)=\mathrm{tr}(\rho (g))$. Isomorphic representations have the same character.

For any group, take the group-algebra image on the direct sum of two finite-dimensional complex semisimple representations. Its radical annihilates their simple summands and hence the faithful direct sum, so the image algebra is semisimple. Equality of traces on group elements gives equality on their linear span. Traces on primitive central blocks then recover every simple multiplicity, proving that equal characters imply isomorphism.

At an element $g$ of a finite group, a complex character value is the sum of the roots of unity which are the eigenvalues of its representing matrix. It is therefore an algebraic integer, and every algebraic conjugate has modulus at most the degree.

An irreducible character is the character of an irreducible representation. Over the complex numbers, every finite-group character decomposes uniquely as a nonnegative integer combination of irreducible characters.

For a real-valued irreducible character of a finite odd-order group, pairing $g$ with $g^{-1}$ gives $⟨\chi ,1_G⟩=(\chi (1)+2A)/|G|$, with $A$ an algebraic integer. If this inner product were zero, $\chi (1)$ would be even, since a rational algebraic integer is an integer. But an irreducible character degree divides the group order, so it is odd. Therefore the character is trivial.

An irreducible character $\phi$ is a constituent of a character $\chi$ when its multiplicity $⟨\chi ,\phi ⟩$ is positive.

For a subgroup $H\le G$, the restriction of a $G$-character $\chi$ to $H$ is the $H$-character obtained by evaluating the same function only on elements of $H$.

If $\chi$ is an irreducible character of a finite group $G$ and $H\le G$, thenEquality holds exactly when $\chi$ vanishes on $G\setminus H$.

Let $H◃G$ have index two, and let $\lambda$ be the nontrivial linear character of $G/H$ inflated to $G$. For an irreducible character $\chi$ of $G$, either ${\mathrm{Res}}_H^G\chi$ is irreducible and $\chi \ne \chi \lambda$, or the restriction is a sum of two distinct irreducible characters and $\chi =\chi \lambda$.

A virtual character is an integer linear combination of characters, equivalently an element of the Grothendieck group of finite-dimensional representations.