On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two

Abstract

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 which preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the 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 the general linear groups over principal ideal local rings of arbitrary length in the function field case.

21 pages

References in corpus (2)

• On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings
• Similarity classes of 3x3 matrices over a local principal ideal ring

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• Representation Growth of Compact Special Linear Groups of degree two
• On Representations of Classical Groups over Finite Local Rings of Length Two
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• A multiplicity one theorem for groups of type $A_n$ over discrete valuation rings
• Geometric interpretation of Murphy bases and an application
• The stable representations of ${\mathrm{GL}}_N$ over finite local principal ideal rings
• Consensus Networks over Finite Fields
• The $\ell$-modular representation of reductive groups over finite local rings of length two
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