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• W4210347000 abstract "Given a root system <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the corresponding finite reflection group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding=application/x-tex>W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H o m left-parenthesis upper W comma ModifyingAbove double-struck upper Z With caret Subscript 2 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Hom</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>W</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {Hom}(W,,widehat {mathbb Z}_2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the group of homomorphisms from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding=application/x-tex>W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove double-struck upper Z With caret Subscript 2> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>widehat {mathbb Z}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove double-struck upper Z With caret Subscript 2 Baseline equals StartSet 1 comma negative 1 EndSet> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>widehat {mathbb Z}_2={1,-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with multiplication. We propose a procedure of constructing subroot systems of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by using homomorphisms <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=eta element-of upper H o m left-parenthesis upper W comma ModifyingAbove double-struck upper Z With caret Subscript 2 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>η<!-- η --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>Hom</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>W</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>eta in operatorname {Hom}(W,,widehat {mathbb Z}_2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This construction is next used for establishing a relation between concepts of total positivity and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=eta> <mml:semantics> <mml:mi>η<!-- η --></mml:mi> <mml:annotation encoding=application/x-tex>eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-total positivity." @default.
• W4210347000 created "2022-02-08" @default.
• W4210347000 creator A5000126125 @default.
• W4210347000 date "2022-06-16" @default.
• W4210347000 modified "2023-09-25" @default.
• W4210347000 title "Subroot systems and total positivity in finite reflection groups" @default.
• W4210347000 cites W1581055187 @default.
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• W4210347000 doi "https://doi.org/10.1090/proc/15979" @default.
• W4210347000 hasPublicationYear "2022" @default.
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