Monomial representation

In the mathematical fields of representation theory and group theory, a linear representation ${\displaystyle \rho }$ (rho) of a group ${\displaystyle G}$ is a monomial representation if there is a finite-index subgroup ${\displaystyle H}$ and a one-dimensional linear representation ${\displaystyle \sigma }$ of ${\displaystyle H}$, such that ${\displaystyle \rho }$ is equivalent to the induced representation ${\displaystyle \mathrm {Ind} _{H}^{G_{\sigma }}}$.

Alternatively, one may define it as a representation whose image is in the monomial matrices.

Here for example ${\displaystyle G}$ and ${\displaystyle H}$ 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 ${\displaystyle G}$ on the cosets of ${\displaystyle H}$. It is necessary only to keep track of scalars coming from ${\displaystyle \sigma }$ applied to elements of ${\displaystyle H}$.

To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple ${\displaystyle (V,X,(V_{x})_{x\in X})}$ where ${\displaystyle V}$ is a finite-dimensional complex vector space, ${\displaystyle X}$ is a finite set and ${\displaystyle (V_{x})_{x\in X}}$ is a family of one-dimensional subspaces of ${\displaystyle V}$ such that ${\displaystyle V=\oplus _{x\in X}V_{x}}$.

Now Let ${\displaystyle G}$ be a group, the monomial representation of ${\displaystyle G}$ on ${\displaystyle V}$ is a group homomorphism ${\displaystyle \rho :G\to \mathrm {GL} (V)}$ such that for every element ${\displaystyle g\in G}$, ${\displaystyle \rho (g)}$ permutes the ${\displaystyle V_{x}}$'s, this means that ${\displaystyle \rho }$ induces an action by permutation of ${\displaystyle G}$ on ${\displaystyle X}$.