7: Representations of Finite Groups
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- Feb 2, 2025
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( \newcommand{\kernel}{\mathrm{null}\,}\)
Throughout this chapter, G is a finite group and F is a field. All vector spaces are finite-dimensional.
An F-algebra is a ring A containing F in its centre and finite dimensional as an F-vector space. We do not assume A to be commutative; for example, A could be the matrix algebra Mn(F). Let {e1,…,en} be a basis for A as an F-vector space; then eiej=∑kakijekfor some akij∈F, called the structure constants of A relative to the basis; once a basis has been chosen, the algebra A is uniquely determined by its structure constants.
All A-modules are finite-dimensional when regarded as F-vector spaces. For an A-module V, mV denotes the direct sum of m copies of V.
The opposite Aopp of an F-algebra A is the same F-algebra as A but with the multiplication reversed, i.e., Aopp=(A,+,⋅′) with a⋅′b=ba. In other words, there is a one-to-one correspondence a↔a′:A↔Aopp which is an isomorphism of F-vector spaces and has the property that a′b′=(ba)′.
An A-module M is simple if it is nonzero and contains no submodules except 0 and M, and it is semisimple if it is isomorphic to a direct sum of simple modules.
