Monomial representation

In the mathematical fields of representation theory and group theory, a linear representation \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 \mathrm{Ind}_H^{G_\sigma} . 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 .

In the mathematical fields of representation theory and group theory, a linear representation \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 \mathrm{Ind}_H^{G_\sigma} . 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 . 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 X}) where V is a finite-dimensional complex vector space, X is a finite set and (V_x)_{x\in X} is a family of one-dimensional subspaces of V such that V=\oplus_{x\in X}V_x . Now Let G be a group, the monomial representation of G on V is a group homomorphism \rho:G\to \mathrm{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==

Text derived from https://en.wikipedia.org/w/index.php?oldid=1249369131 under CC BY-SA 4.0; modifications have been made.