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2015•Glasgow Mathematical JournalOpen access

ON IMAGES OF REAL REPRESENTATIONS OF SPECIAL LINEAR GROUPS OVER COMPLETE DISCRETE VALUATION RINGS

Talia Fernós, Pooja Singla

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Abstract

Abstract In this paper, we investigate the abstract homomorphisms of the special linear group SLn( $\mathfrak{O}$ ) over complete discrete valuation rings with finite residue field into the general linear group GLm( $\mathbb{R}$ ) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn( $\mathfrak{O}$ ). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$ .

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Abstract In this paper, we investigate the abstract homomorphisms of the special linear group SLn( $\mathfrak{O}$ ) over complete discrete valuation rings with finite residue field into the general linear group GLm( $\mathbb{R}$ ) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn( $\mathfrak{O}$ ). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$ .

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Available abstract

Abstract In this paper, we investigate the abstract homomorphisms of the special linear group SLn( $\mathfrak{O}$ ) over complete discrete valuation rings with finite residue field into the general linear group GLm( $\mathbb{R}$ ) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn( $\mathfrak{O}$ ). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$ .

Key concepts: Mathematics, Discrete valuation, Homomorphism, Residue field, Discrete valuation ring, Valuation (finance), Finite field, Combinatorics

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