2018/10/06 by Soham Swadhin Pradhan, Pradhan, Soham Swadhin
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1810.04015
PhD thesis, Indian Institute of Technology, Bombay (2018)
arxiv created 2018/10/06 · arxiv updated 2018/10/10
The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group G of order N = p1p2… pn, where pi's are primes, there always exists a subnormal series: ⟨ e ⟩ = Go < G1 < … < Gn = G such that Gi/Gi-1 is isomorphic to a cyclic group of order pi, i = 1,2,…,n. Associated with this series, there exists a system of generators consisting n elements x1, x2, …, xn (say), such that Gi = ⟨ x1, x2, …, xi ⟩, i = 1,2,…,n, which is called a "long system of generators". In terms of this system of generators and conjugacy class sum of xi in Gi, i = 1,2, …, n, we present an algorithm for constructing the irreducible matrix representations of G over ℂ within the group algebra ℂ[G]. This algorithmic construction needs the knowledge of primitive central idempotents, a well defined set of primitive (not necessarily central) idempotents and the "diagonal subalgebra" of ℂ[G]. In terms of this system of generators, we give simple expressions for the primitive central idempotents, a well defined system of primitive (not necessarily central) idempotents and a convenient set of generators of the "diagonal subalgebra" of ℂ[G]. For a finite abelian group, we present an algorithm for constructing the inequivalent irreducible matrix representations over a field of characteristic 0 or prime to the order of the group and a systematic way of computing the primitive central idempotents of the group algebra. Besides that, we give simple expressions of the primitive central idempotents of the rational group algebra of a finite abelian group using a "long presentation" and it's Wedderburn decomposition.