Lesson plan
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 .
Article summary
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 .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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