Monomial representation
In the mathematical fields of representation theory and group theory, a linear representation ρ (rho) 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 IndHGσ.
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 [math]\displaystyle{ (V,X,(V_x)_{x\in X}) }[/math]where [math]\displaystyle{ V }[/math]is a finite-dimensional complex vector space, [math]\displaystyle{ X }[/math]is a finite set and [math]\displaystyle{ (V_x)_{x\in X} }[/math]is a family of one-dimensional subspaces of [math]\displaystyle{ V }[/math]such that [math]\displaystyle{ V=\oplus_{x\in X}V_x }[/math].
Now Let [math]\displaystyle{ G }[/math] be a group, the monomial representation of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ V }[/math] is a group homomorphism [math]\displaystyle{ \rho:G\to \mathrm{GL}(V) }[/math]such that for every element [math]\displaystyle{ g\in G }[/math], [math]\displaystyle{ \rho(g) }[/math] permutes the [math]\displaystyle{ V_x }[/math]'s, this means that [math]\displaystyle{ \rho }[/math] induces an action by permutation of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ X }[/math].
References
- Hazewinkel, Michiel, ed. (2001), "Monomial representation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Monomial_representation
- Karpilovsky, Gregory (1985). Projective Representations of Finite Groups. M. Dekker. ISBN 978-0-8247-7313-7. https://books.google.com/books?id=K-nuAAAAMAAJ.
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