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 formula_1where formula_2is a finite-dimensional complex vector space, formula_3is a finite set and formula_4is a family of one-dimensional subspaces of formula_2such that formula_6.
Now Let formula_7 be a group, the monomial representation of formula_7 on formula_2 is a group homomorphism formula_10such that for every element formula_11, formula_12 permutes the formula_13's, this means that formula_14 induces an action by permutation of formula_7on formula_3.