The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a linear representation of a Coxeter group.
32.
The interest for algebra is in the process in the other direction : given a " projective representation ", try to'lift'it to a conventional " linear representation ".
33.
The complete reducibility of finite-dimensional linear representations of compact groups, or connected semisimple Lie groups and complex semisimple Lie algebras goes sometimes under the name of " Weyl's theorem ".
34.
Here, by " tensor product of representations ", the usual meaning is intended : the result should be a linear representation of the same algebra on the product vector space.
35.
However, since we want to construct examples for linear representations, where groups act on vector spaces instead of on arbitrary finite sets, we have to proceed in a different way.
36.
To be precise, the generalized permutation matrices are a ( faithful ) linear representation of this abstract wreath product : a realization of the abstract group as a subgroup of matrices.
37.
If it does, we may take that as origin and regard A as a vector space : in that case, we actually have a linear representation in dimension " n ".
38.
The Debye model and the Lorentz model use a 1st-order and 2nd-order ( respectively ) lumped system parameter linear representation ( such as an RC and an LRC resonant circuit ).
39.
It is known that the monodromy representation of KZ equation with a general semi-simple Lie algebra agrees with the linear representation of braid group given by R-matrix of the corresponding quantum group.
40.
This means that they must be tensor products of complex linear representations of, as can be seen by restriction to the subgroup, a compact group to which the Peter Weyl theorem applies.
