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<p>The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and Hermitian forms over fields of good characteristic. Explicit results are given for the partitions (<i>n</i>), <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1399_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\((1^n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mn>1</mn> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1399_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\((2,1^{n-2})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1399_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\((3,1^{n-3})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <msup> <mn>1</mn> <mrow> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as well as for all partitions of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1399_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\le 7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>. For orthogonal groups, these symmetrizations are not irreducible, and we continue to find the determinants of their irreducible constituents, the refined symmetrizations, over fields of characteristic 0.</p>

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Symmetrizations of quadratic and Hermitian forms

  • 期刊论文

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  • 发表日期：

    2025年3月20日

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  • Gabriele Nebe

摘要

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and Hermitian forms over fields of good characteristic. Explicit results are given for the partitions (n), \((1^n)\) ( 1 n ) , \((2,1^{n-2})\) ( 2 , 1 n - 2 ) and \((3,1^{n-3})\) ( 3 , 1 n - 3 ) as well as for all partitions of \(n\le 7\) n ≤ 7 . For orthogonal groups, these symmetrizations are not irreducible, and we continue to find the determinants of their irreducible constituents, the refined symmetrizations, over fields of characteristic 0.