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

Ind_H^Gσ.

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 ${\displaystyle (V,X,(V_{x})_{x\in X})}$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 ${\displaystyle X}$.

References

• "Monomial representation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Karpilovsky, Gregory. "Projective representations of finite groups." New York-Basel (1985).