View source for 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|group algebra]] of  $  G $
determined by  $  \pi $ (see [[Character of a representation of an associative algebra|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  J. Dixmier,   "$C^\star$ algebras" , North-Holland  (1977)  (Translated from French)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  G.F. Frobenius,   J.-P. Serre (ed.) , ''Gesammelte Abhandlungen'' , Springer  (1968)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Naimark,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.E. Littlewood,   "The theory of group characters and matrix representations of groups" , Clarendon Press  (1950)</TD></TR></table>