# Character of a representation of a group

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In the case of a finite-dimensional representation $\pi$ this is the function $\chi _ \pi$ on the group $G$ defined by the formula

$$\chi _ \pi ( g) = \ \mathop{\rm tr} \pi ( g),\ \ g \in G.$$

For arbitrary continuous representations of a topological group $G$ over $\mathbf C$ this definition is generalized as follows:

$$\chi _ \pi ( g) = \ \chi ( \pi ( g)) \ \ \textrm{ for } \ g \in G,$$

where $\chi$ is a linear functional defined on some ideal $I$ of the algebra $A$ generated by the family of operators $\pi ( g)$, $g \in G$, that is invariant under inner automorphisms of $A$. In certain cases the character of a representation $\pi$ is defined as that of the representation of a certain group algebra of $G$ determined by $\pi$ (see Character of a representation of an associative algebra).

The character of a direct sum (of a tensor product) of finite-dimensional representations is equal to the sum (the product) of the characters of these representations. The character of a finite-dimensional representation of a group is a function that is constant on classes of conjugate elements; the character of a continuous finite-dimensional unitary representation of a group is a continuous positive-definite function on the group.

In many cases the character of a representation of a group determines the representation uniquely up to equivalence; for example, the character of an irreducible finite-dimensional representation over a field of characteristic 0 determines the representation uniquely up to spatial equivalence; the character of a finite-dimensional continuous unitary representation of a compact group is determining up to unitary equivalence.

The character of a representation of a locally compact group $G$ admitting an extension to a representation of the algebra of continuous functions of compact support on $G$ can be defined by a measure on $G$; in particular, the character of the regular representation of a unimodular group is given by a probability point measure concentrated at the unit element of $G$. The character of a representation $\pi$ of a Lie group $G$ admitting an extension to a representation of the algebra $C _ {0} ^ \infty ( G)$ of infinitely-differentiable functions of compact support on $G$ can be defined as a generalized function on $G$. If $G$ is a nilpotent or a linear semi-simple Lie group, then the characters of irreducible unitary representations $\pi$ of $G$ are defined by locally integrable functions $\psi _ \pi$ according to the formula

$$\chi _ \pi ( f ) = \ \int\limits _ { G } f ( g) \psi _ \pi ( g) dg,\ \ f \in C _ {0} ^ \infty ( G).$$

These characters determine the representation $\pi$ uniquely up to unitary equivalence.

If the group $G$ is compact, every continuous positive-definite function on $G$ that is constant on classes of conjugate elements can be expanded into a series with respect to the characters of the irreducible representations $\pi _ \alpha$ of $G$. The series converges uniformly on $G$ and the characters $\chi _ {\pi _ \alpha }$ form an orthonormal system in the space $L _ {2} ( G)$ that is complete in the subspace of functions in $L _ {2} ( G)$ that are constant on classes of conjugate elements in $G$. If $\chi _ \rho = \sum _ \alpha m _ \alpha \chi _ {\pi _ \alpha }$ is the expansion of the character of a continuous finite-dimensional representation $\rho$ of the group $G$ with respect to the characters $\chi _ {\pi _ \alpha }$, then the $m _ \alpha$ are integers, namely, the multiplicities with which the $\pi _ \alpha$ occur in $\rho$. If $\rho$ is a continuous representation of $G$ in a quasi-complete, barrelled, locally convex topological space $E$, then there exists a maximal subspace $E _ \alpha$ of $E$ such that the restriction of $\rho$ to $E _ \alpha$ is a multiple of $\pi _ \alpha$, and there is a continuous projection $P _ \alpha$ of $E$ onto $E _ \alpha$, defined by

$$P _ \alpha = \ \chi _ {\pi _ \alpha } ( e) \int\limits _ { G } {\chi _ {\pi _ \alpha } ( g) } bar \rho ( g) dg,$$

where $dg$ is the Haar measure on $G$ for which $\int _ {G} dg = 1$.

#### References

 [1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) [2] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) [3] J. Dixmier, "$C^\star$ algebras" , North-Holland (1977) (Translated from French) [4] G.F. Frobenius, J.-P. Serre (ed.) , Gesammelte Abhandlungen , Springer (1968) [5] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) [6] D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
How to Cite This Entry:
Character of a representation of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_representation_of_a_group&oldid=53349
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article