We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of all general linear groups over principal ideal local rings of arbitrary length in the function field case.
Let o be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over o and let g denote its Lie algebra. For every positive integer ℓ, let K ℓ be the ℓ-th principal congruence subgroup of G(o). A continuous irreducible representation of G(o) is called regular of level ℓ if it is trivial on K ℓ+1 and its restriction to K ℓ /K ℓ+1 ≃ g(k) consists of characters with G(k)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(o), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.
Let I be a monomial ideal in a polynomial ring A = K[x 1 , . . . , x n ]. We call a monomial ideal J to be a minimal monomial reduction ideal of I if there exists no proper monomial ideal L ⊂ J such that L is a reduction ideal of I. We prove that there exists a unique minimal monomial reduction ideal J of I and we show that the maximum degree of a monomial generator of J determines the slope p of the linear function reg(I t ) = pt + c for t ≫ 0. We determine the structure of the reduced fiber ring F (J) red of J and show that F (J) red is isomorphic to the inverse limit of an inverse system of semigroup rings determined by convex geometric properties of J.
We study the complex irreducible representations of special linear, symplectic, orthogonal and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. The case for general linear groups has already been proved by author.2000 Mathematics Subject Classification. Primary 20G05; Secondary 20C15.
An irreducible character of a finite group [Formula: see text] is called quasi [Formula: see text]-Steinberg character for a prime [Formula: see text] if it takes a nonzero value on every [Formula: see text]-regular element of [Formula: see text]. In this paper, we classify the quasi [Formula: see text]-Steinberg characters of Symmetric ([Formula: see text]) and Alternating ([Formula: see text]) groups and their double covers. In particular, an existence of a nonlinear quasi [Formula: see text]-Steinberg character of [Formula: see text] implies [Formula: see text] and of [Formula: see text] implies [Formula: see text].
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