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<p>For a prime <i>p</i>, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\mathbb {F}}}_q\)</EquationSource> </InlineEquation> be a finite extension of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({{\mathbb {F}}}_p\)</EquationSource> </InlineEquation>. The restriction of an irreducible mod <i>p</i> representation of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\text {GL}_2({{\mathbb {F}}}_q)\)</EquationSource> </InlineEquation> to its subgroup <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\text {GL}_2({{\mathbb {F}}}_p)\)</EquationSource> </InlineEquation> can be seen as a tensor product of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\([{{\mathbb {F}}}_q:{{\mathbb {F}}}_p]\)</EquationSource> </InlineEquation> irreducible representations of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\text {GL}_2({{\mathbb {F}}}_p)\)</EquationSource> </InlineEquation>. In this paper, we study the restriction of some of these representations of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\text {GL}_2({{\mathbb {F}}}_q)\)</EquationSource> </InlineEquation> to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\text {GL}_2({{\mathbb {F}}}_p)\)</EquationSource> </InlineEquation>, for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(q=p^2\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(p^3\)</EquationSource> </InlineEquation> using combinatorial tools and give explicit socle filtration when <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(q=4\)</EquationSource> </InlineEquation>.</p>

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On the restriction of some irreducible mod p representations

  • 期刊论文

  • 发表日期：

    2026年5月23日

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  • Shubhanshi Gupta

摘要

For a prime p, let \({{\mathbb {F}}}_q\) be a finite extension of \({{\mathbb {F}}}_p\) . The restriction of an irreducible mod p representation of \(\text {GL}_2({{\mathbb {F}}}_q)\) to its subgroup \(\text {GL}_2({{\mathbb {F}}}_p)\) can be seen as a tensor product of \([{{\mathbb {F}}}_q:{{\mathbb {F}}}_p]\) irreducible representations of \(\text {GL}_2({{\mathbb {F}}}_p)\) . In this paper, we study the restriction of some of these representations of \(\text {GL}_2({{\mathbb {F}}}_q)\) to \(\text {GL}_2({{\mathbb {F}}}_p)\) , for \(q=p^2\) and \(p^3\) using combinatorial tools and give explicit socle filtration when \(q=4\) .