Monomial representation explained

\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 Ind ^{G _{\sigma}} _{H} .

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

Now Let

G be a group, the monomial representation of G on V is a group homomorphism \rho:G\to 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 .

References

• • Book: Gregory . Karpilovsky . Projective Representations of Finite Groups . 1985 . M. Dekker . 978-0-8247-7313-7.