Monomial representation

(Redirected from Monomial representations)

In the mathematical fields of representation theory and group theory, a linear representation (rho) of a group is a monomial representation if there is a finite-index subgroup and a one-dimensional linear representation of , such that is equivalent to the induced representation .

Alternatively, one may define it as a representation whose image is in the monomial matrices.

Here for example and 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 on the cosets of . It is necessary only to keep track of scalars coming from applied to elements of .

Definition

edit

To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple   where   is a finite-dimensional complex vector space,   is a finite set and   is a family of one-dimensional subspaces of   such that  .

Now Let   be a group, the monomial representation of   on   is a group homomorphism   such that for every element  ,   permutes the  's, this means that   induces an action by permutation of   on  .

References

edit
  • "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.