We study irreducible representations of some nilpotent groups of finite abelian total rank. The main result of the paper states that if a torsion-free minimax group G of nilpotency class 2 admits a faithful irreducible representation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> over a finitely generated field k such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{char}\, k \notin \textrm{Sp}\hspace{0.55542pt}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>char</mtext> <mspace width="0.166667em" /> <mi>k</mi> <mo>∉</mo> <mtext>Sp</mtext> <mspace width="0.55542pt" /> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> then there exist a subgroup N and an irreducible representation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> of the subgroup N over k such that the representation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is induced from <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> and the quotient group <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$N/\textrm{Ker}\,\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">/</mo> <mtext>Ker</mtext> <mspace width="0.166667em" /> <mi>ψ</mi> </mrow> </math></EquationSource> </InlineEquation> is finitely generated.

Irreducible representations of certain nilpotent groups of finite rank

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2025年7月1日

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Anatolii Tushev

摘要

We study irreducible representations of some nilpotent groups of finite abelian total rank. The main result of the paper states that if a torsion-free minimax group G of nilpotency class 2 admits a faithful irreducible representation $\varphi $ $φ$ over a finitely generated field k such that $\textrm{char}\, k \notin \textrm{Sp}\hspace{0.55542pt}(G)$ $char k \notin Sp (G)$ then there exist a subgroup N and an irreducible representation $\psi $ $ψ$ of the subgroup N over k such that the representation $\varphi $ $φ$ is induced from $\psi $ $ψ$ and the quotient group $N/\textrm{Ker}\,\psi $ $N / Ker ψ$ is finitely generated.