In the mathematical fields of representation theory and group theory, a linear representation [rho](https://bliptext.com/articles/rho) (rho) of a group G is a monomial representation if there is a finite-index subgroup H and a one-dimensional linear representation \sigma of H, such that [rho](https://bliptext.com/articles/rho) 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 \sigma 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 V is a finite-dimensional complex vector space, X is a finite set and is a family of one-dimensional subspaces of V such that. Now Let G be a group, the monomial representation of G on V is a group homomorphism such that for every element g\in G, \rho(g) permutes the V_x's, this means that \rho induces an action by permutation of G on X.