Multiplicity of an irreducible representation

Definition

Let $V$ be a finite-dimensional completely reducible representation over an algebraically closed field, and let $S$ be an irreducible representation of the same group. If

V\cong\bigoplus_{[T]}m_TT,

then the multiplicity of $S$ in $V$ is the nonnegative integer $m_S$ . Equivalently,

m_S=\dim\operatorname{Hom}_G(S,V).

The same definition applies to completely reducible Lie-algebra representations, with $\operatorname{Hom}_{\mathfrak g}$ in place of $\operatorname{Hom}_G$ .

Isotypic component

The sum of all subrepresentations of $V$ isomorphic to $S$ is its $S$ -isotypic component. It is canonically the image of the evaluation map

S\otimes\operatorname{Hom}_G(S,V)\longrightarrow V, \qquad s\otimes f\longmapsto f(s),

and is isomorphic to $S^{\oplus m_S}$ . The choice of individual copies of $S$ inside that component is generally not canonical.

Computing multiplicity

For compact groups, irreducible characters are orthonormal, so

m_S=\langle\chi_V,\chi_S\rangle =\int_G\chi_V(g)\overline{\chi_S(g)}\,dg.

For finite groups the normalized integral becomes $|G|^{-1}\sum_{g\in G}$ . In highest-weight representations, character formulas and weight data provide another route to the same irreducible multiplicities.

The multiplicity of the trivial representation is $\dim V^G$ , the dimension of the fixed-vector subspace. Multiplicities appearing after restriction are the coefficients in a branching rule.

Cautions

An irreducible multiplicity is not the same as a weight multiplicity $\dim V_\lambda$ : the former counts irreducible summands, while the latter counts vectors having a specified weight inside a representation.

Without complete reducibility, $V$ need not be a direct sum of irreducibles. One may still count composition factors using a finite-length filtration, but those numbers describe Jordan–Hölder multiplicities and need not equal $\dim\operatorname{Hom}(S,V)$ . Over a non-algebraically-closed field, $\operatorname{End}_G(S)$ may be larger than the scalar field, so the displayed dimension formula also requires adjustment.

References

Jean-Pierre Serre, Linear Representations of Finite Groups, Springer, 1977, Chapters 2 and 6. Publisher record.
Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Chapters 4 and 12. Publisher record.