In the mathematical fields of representation theory and group theory, a linear representation
is a monomial representation if there is a finite-index subgroup
and a one-dimensional linear representation
is equivalent to the induced representation
Alternatively, one may define it as a representation whose image is in the monomial matrices.
may be finite groups, so that induced representation has a classical sense.
The monomial representation is only a little more complicated than the permutation representation of
It is necessary only to keep track of scalars coming from
applied to elements of
To define the monomial representation, we first need to introduce the notion of monomial space.
A monomial space is a triple
is a finite-dimensional complex vector space,
is a finite set and
is a family of one-dimensional subspaces of
be a group, the monomial representation of
is a group homomorphism
induces an action by permutation of
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