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<p>For the ring of square matrices Mat<sub><i>n</i></sub>(𝕜) of order n over a field 𝕜, one can construct the orthogonality graph O(Mat<sub><i>n</i></sub>(𝕜)), whose vertices are the zero divisors of the ring Mat<sub><i>n</i></sub>(𝕜). Two vertices <i>A</i> and <i>B</i> are connected by an edge if <i>AB</i> = <i>BA</i> = 0. The notion of the distance between two elements of the ring naturally implies that one can consider the set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7709_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({\text{O}}_{n}^{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mtext>O</mtext> <mrow> <mi>n</mi> </mrow> <mi>d</mi> </msubsup> </math></EquationSource> </InlineEquation> of pairs of elements lying within distance at most <i>d.</i></p><p>It is proved that such sets form affine algebraic varieties; a decomposition of these varieties into irreducible components is provided, and their dimensions are calculated. Also the sets that are similarly defined for the ring of upper triangular matrices are described, and generalizations of these results to arbitrary finite-dimensional algebras are suggested.</p>

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Geometry of Varieties of Mutually Orthogonal Matrices

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • A. E. Guterman,
  • S. A. Zhilina,
  • K. D. Mukhanov

摘要

For the ring of square matrices Matn(𝕜) of order n over a field 𝕜, one can construct the orthogonality graph O(Matn(𝕜)), whose vertices are the zero divisors of the ring Matn(𝕜). Two vertices A and B are connected by an edge if AB = BA = 0. The notion of the distance between two elements of the ring naturally implies that one can consider the set \({\text{O}}_{n}^{d}\) O n d of pairs of elements lying within distance at most d.

It is proved that such sets form affine algebraic varieties; a decomposition of these varieties into irreducible components is provided, and their dimensions are calculated. Also the sets that are similarly defined for the ring of upper triangular matrices are described, and generalizations of these results to arbitrary finite-dimensional algebras are suggested.