In the mathematical fields of representation theory and group theory, a linear representation ρ {\displaystyle \rho } (rho) of a group G {\displaystyle G} is a monomial representation if there is a finite-index subgroup H {\displaystyle H} and a one-dimensional linear representation σ {\displaystyle \sigma } of H {\displaystyle H} , such that ρ {\displaystyle \rho } is equivalent to the induced representation I n d H G σ {\displaystyle \mathrm {Ind} _{H}^{G_{\sigma }}} . Alternatively, one may define it as a representation whose image is in the monomial matrices. Here for example G {\displaystyle G} and H {\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 G {\displaystyle G} on the cosets of H {\displaystyle H} . It is necessary only to keep track of scalars coming from σ {\displaystyle \sigma } applied to elements of H {\displaystyle 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 ∈ X ) {\displaystyle (V,X,(V_{x})_{x\in X})} where V {\displaystyle V} is a finite-dimensional complex vector space, X {\displaystyle X} is a finite set and ( V x ) x ∈ X {\displaystyle (V_{x})_{x\in X}} is a family of one-dimensional subspaces of V {\displaystyle V} such that V = ⊕ x ∈ X V x {\displaystyle V=\oplus _{x\in X}V_{x}} . Now Let G {\displaystyle G} be a group, the monomial representation of G {\displaystyle G} on V {\displaystyle V} is a group homomorphism ρ : G → G L ( V ) {\displaystyle \rho :G\to \mathrm {GL} (V)} such that for every element g ∈ G {\displaystyle g\in G} , ρ ( g ) {\displaystyle \rho (g)} permutes the V x {\displaystyle V_{x}} 's, this means that ρ {\displaystyle \rho } induces an action by permutation of G {\displaystyle G} on X {\displaystyle X} .
References "Monomial representation", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Karpilovsky, Gregory (1985). Projective Representations of Finite Groups. M. Dekker. ISBN 978-0-8247-7313-7.