Irreducible characters of GL[subscript]2 (Z/p[superscript]2 Z)

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Representation theory studies the structure of a finite group G by looking at the set of homomorphisms p from G to the group of automorphisms of a complex vector space V. The character x of a representation (p, V) of G is the complex valued function x( g ) = tr(p(g)) for g E G. In general, it is less complicated to work with the characters of a group G than with the representations themselves. Fortunately, a representation is uniquely determined by its character. This thesis focusses on characters of the group G = GL[subscript]2(Z/p[superscript]2 Z), the general linear group of 2 x 2 invertible matrices over the local ring R = Z/p[superscript] 2 Z. In particular, we study GL[subscript]2(Z/p[superscript]2 Z) directly without resorting to SL[subscript]2 (Z/p[superscript]2 Z), the subgroup of G of elements with determinant 1. Let R[superscript]X denote the group of units of R, and let Mu and v be irreducible characters of R[superscript]X. We construct the character Ind[G over B]w[subscript]Muv of G, where B is the Borel (or upper triangular) subgroup in G and w[subscript]Muv ([a b over 0 d]) = Mu(a)v(d). In this thesis, we determine the decomposition of Ind[G over B)w[subscript]Muv, for all pairs of characters {Mu, v} of R[superscript]X, into a direct sum of irreducible characters. Since all representations of a finite group G are composed as the direct sum of irreducible representations, this information can be used to find further characters of G.

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