Computational Approaches to Finding Irreducible Representations
نویسندگان
چکیده
Among the various branches of algebra, linear algebra has the distinctions of being well understood and applicable to a wide variety of scientific and mathematical problems. In particular, it can be used to convert mathematical problems into a format where computers can easily be applied to perform calculations that would be impractical for humans to attempt by hand. This project deals with one such application of linear algebra, in a branch of group theory called representation theory. Given a group G and a field F, a representation φ is a homomorphism from G to GLn(F), the invertible n× n matrices with entries in F. A representation φ is irreducible if the only subspaces invariant under the matrices in the image φ(G) are {0} and F. In much the same way that one can factor an integer into primes, a representation can be factored into irreducible representations. This factorization is unique, up to a change of basis, and can reveal important information about the group. However, as with factoring primes, expressing a representation as a product of irreducible representations requires considerable computation.
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