Permanent groups

Abstract

A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A ring upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∘<!-- ∘ --></mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A \circ B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the Hadamard product of matrices <italic>A</italic> and <italic>B</italic>. The set of groups <italic>G</italic> of nonsingular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices which contain the diagonal group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and such that for every pair <italic>A, B</italic> of matrices in <italic>G</italic> we have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A ring upper B Superscript upper T element-of script upper D"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∘<!-- ∘ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A \circ {B^T} \in \mathcal {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is denoted by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {A}_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If the underlying field has at least three elements then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {A}_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consists of permanent groups. A partial converse is available: If a permanent group <italic>G</italic> is generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> together with a set <italic>S</italic> of elementary matrices and a set <italic>Q</italic> of permutation matrices then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals upper H upper K"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G = HK</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <italic>H</italic> is the subgroup generated by <italic>Q</italic> and <italic>K</italic> is generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <italic>S</italic>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K element-of script upper A Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K \in {\mathcal {A}_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Proceedings of the American Mathematical Society - View - PDF