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 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

  • "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.


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