Monomial representation


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

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.