Monomial representation

In the mathematical fields of representation theory and group theory, a linear representation $$\rho$$ (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 $$\rho$$ is equivalent to the induced representation $$\mathrm{Ind}_H^{G_\sigma}$$.

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 $$(V,X,(V_x)_{x\in X})$$where $$V$$is a finite-dimensional complex vector space, $$X$$ is a finite set and $$(V_x)_{x\in X}$$ is a family of one-dimensional subspaces of $$V$$ such that $$V=\oplus_{x\in X}V_x$$.

Now Let $$G$$ be a group, the monomial representation of $$G$$ on $$V$$ is a group homomorphism $$\rho:G\to \mathrm{GL}(V)$$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 $$X$$.