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REPRESENTATIONS ON A COMPLEX FINITE-DIMENSIONAL VECTOR SPACE Let us first discuss representations acting on finite-dimensional complex vector spaces. A of ''V''. For ''n''-dimensional ''V'', the automorphism group of ''V'' is identified with a subset of complex square-matrices of order ''n''. The automorphism group of ''V'' is given the structure of a smooth manifold using this identification. The condition that Ψ is smooth, in the definition above, means that Ψ is a smooth map from the smooth manifold ''G'' to the smooth manifold ''Aut(V)''. If a basis for the complex vector space ''V'' is chosen, the representation can be expressed as a homomorphism into GL(''n'',C) . This is known as a ''matrix representation''. REPRESENTATIONS ON A FINITE-DIMENSIONAL VECTOR SPACE OVER AN ARBITRARY FIELD A Representation of a Lie Group ''G'' on a Vector Space ''V'' (over a Field ''K'') is a Smooth (i.e. respecting the differential structure) Group Homomorphism ''G''→Aut(''V'') from ''G'' to the Automorphism Group of ''V''. If a basis for the vector space ''V'' is chosen, the representation can be expressed as a homomorphism into GL(''n'',''K'') . This is known as a ''matrix representation''. Two representations of ''G'' on vector spaces ''V'', ''W'' are ''equivalent'' if they have the same matrix representations with respect to some choices of bases for ''V'' and ''W''. On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to End(''V'') preserving the Lie Bracket {Link without Title} . See Representation Of Lie Algebras for the Lie algebra theory. If the homomorphism is in fact an Monomorphism , the representation is said to be ''faithful''. A Unitary Representation is defined in the same way, except that G maps to Unitary Matrices ; the Lie algebra will then map to Skew-hermitian matrices. If ''G'' is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one. REPRESENTATIONS ON HILBERT SPACES A Representation of a Lie Group ''G'' on a complex Hilbert Space ''V'' is a Group Homomorphism Ψ:''G'' → B(''V'') from ''G'' to B(''V''), the group of bounded linear operators of ''V'' which have a bounded inverse, such that the map ''G''×''V'' → ''V'' given by (''g'',''v'') → Ψ(''g'')''v'' is continuous. This definition can handle representations on infinite-dimensional Hilbert spaces. Such representations can be found in e.g. quantum mechanics, but also in Fourier analysis as shown in the following example. Let ''G''=R, and let the complex Hilbert space ''V'' be ''L''2(R). We define the representation Ψ:R → B(''L''2(R)) by Ψ(''r''){''f''(''x'')} → ''f''(''r''-1''x''). See also Wigner's Classification for representations of the PoincarĂ© Group . CLASSIFICATION If G is a Semisimple group, its finite-dimensional representations can be decomposed as Direct Sum s of Irreducible Representation s. The irreducibles are indexed by highest Weight ; the allowable (''dominant'') highest weights satisfy a suitable positivity condition. In particular, there exists a set of ''fundamental weights'', indexed by the vertices of the Dynkin Diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights. The characters of the irreducible representations are given by the Weyl Character Formula . If G is a commutative for this case. A quotient representation is a Quotient Module of the Group Ring . FORMULAIC EXAMPLES Let F''q'' be a finite field of order ''q'' and characteristic ''p''. Let ''G'' be a finite group of Lie type, that is, ''G'' is the F''q''-rational points of a connected reductive group ''G'' defined over F''q''. For example, if ''n'' is a positive integer GL(''n'', F''q'') and SL(n, F''q'') are finite groups of Lie type. Let , where ''I''n is the ''n''×''n'' identity matrix. Let | ||
| <center><math>M | \left \{\left\begin{pmatrix}A_1 & 0 & \cdots & 0 \ 0 & A_2 & \cdots & 0 \ dots & \ddots & \ddots & dots \ 0 & \cdots & 0 & A_r\end{pmatrix}
ightA_j \in GL_{n_j}(\mathbb{F}_q), 1 \le j \le r
ight \}</math>,</center> |
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