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• W1507260030 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group generated by unitary reflections in a Hermitian space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=zeta> <mml:semantics> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:annotation encoding=application/x-tex>zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a root of unity. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a subspace of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, maximal with respect to the property of being a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=zeta> <mml:semantics> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:annotation encoding=application/x-tex>zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-eigenspace of an element of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding=application/x-tex>C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the parabolic subgroup of elements fixing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> pointwise. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi> <mml:semantics> <mml:mi>χ<!-- χ --></mml:mi> <mml:annotation encoding=application/x-tex>chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any linear character of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we give a condition for the restriction of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi> <mml:semantics> <mml:mi>χ<!-- χ --></mml:mi> <mml:annotation encoding=application/x-tex>chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding=application/x-tex>C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be trivial in terms of the invariant theory of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and give a formula for the polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation Underscript x element-of upper G Endscripts chi left-parenthesis x right-parenthesis upper T Superscript d left-parenthesis x comma zeta right-parenthesis> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:munder> <mml:mi>χ<!-- χ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>sum _{xin G}chi (x)T^{d(x,zeta )}</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=d left-parenthesis x comma zeta right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>d(x,zeta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the dimension of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=zeta> <mml:semantics> <mml:mi>ζ<!-- ζ --></mml:mi> <mml:annotation encoding=application/x-tex>zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-eigenspace of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Applications include criteria for regularity, and new connections between the invariant theory and the structure of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1507260030 created "2016-06-24" @default.
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• W1507260030 date "2005-05-02" @default.
• W1507260030 modified "2023-09-23" @default.
• W1507260030 title "Remarks concerning linear characters of reflection groups" @default.
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