Monomial representation

In the mathematical fields of representation theory and group theory, a linear representation of a group is a monomial representation if there is a finite-index subgroup and a one-dimensional linear representation of, such that is equivalent to the induced representation.
Alternatively, one may define it as a representation whose image is in the monomial matrices.
Here for example and 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 on the cosets of. It is necessary only to keep track of scalars coming from applied to elements of.

Definition

To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple where is a finite-dimensional complex vector space, is a finite set and is a family of one-dimensional subspaces of such that.
Now Let be a group, the monomial representation of on is a group homomorphism such that for every element, permutes the 's, this means that induces an action by permutation of on.