Monomial representations, monomial characters, and M-groups

Throughout this page $G$ is a finite group, "representation" means a finite-dimensional complex representation (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree), and $H\le G$ means that $H$ is a subgroup of $G$ (Subgroup).

A linear character of a finite group $H$ is a group homomorphism $\lambda :H\to {\mathbb{C}}^{\times }$. Equivalently it is a one-dimensional complex character of $H$, namely the character of the one-dimensional representation on which $h$ acts as multiplication by $\lambda (h)$ (The character ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ of a finite-dimensional complex representation). Its kernel is $\ker \lambda =\{h\in H:\lambda (h)=1\}$, the kernel of that representation.

Monomial representation. A nonzero finite-dimensional complex $G$-representation $V$ is monomial if there are a subgroup $H\le G$ and a one-dimensional $H$-representation $L$ with

$V\cong {\mathrm{Ind}}_H^{G}L$

as complex $G$-representations (The induced $R$-linear $G$-module ${\mathrm{Ind}}_H^{G}W$ as $H$-covariant functions on $G$). Such an $L$ is determined by the linear character $\lambda :H\to {\mathbb{C}}^{\times }$ that is its representing map, and one writes ${\mathrm{Ind}}_H^G\lambda$ for ${\mathrm{Ind}}_H^{G}L$.

Monomial character. A complex character $\chi$ of $G$ is monomial if there are a subgroup $H\le G$ and a linear character $\lambda$ of $H$ with

$\chi ={\mathrm{Ind}}_H^G\lambda$

(The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character).

$M$-group. A finite group $G$ is an $M$-group if every irreducible complex character of $G$ is monomial (An irreducible complex character). Since a nonzero representation is monomial exactly when its character is, by Finite-dimensional complex representations of a finite group are determined up to isomorphism by their characters together with the defining equation ${\chi }_{{\mathrm{Ind}}_H^{G}L}={\mathrm{Ind}}_H^G{\chi }_L$ of the induced character, this is equivalent to asking that every irreducible complex representation of $G$ is monomial.

• If $\chi ={\mathrm{Ind}}_H^G\lambda$ is monomial with $\lambda$ linear, then $\chi (1)=[G:H]\dim L=[G:H]$, so an induced character is linear exactly when $H=G$ (The dimension of an induced finite-dimensional representation is $[G:H]\dim W$). Consequently a nonlinear monomial irreducible of $G$ is induced from a proper subgroup.

• The definition quantifies over honest irreducible characters. An integral combination of monomial characters need not be an irreducible character and need not be monomial itself, so exhibiting a virtual character as a $\mathbb{Z}$-combination of induced linear characters says nothing about whether a given irreducible is one of the summands. The page keeps the Brauer-type statement for virtual characters and the $M$-group property strictly apart.

• All conventions above are the ones used later on this page: induction is the covariant-function model of the linked definition, characters are complex, and "linear" always means one-dimensional.