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• W1986263670 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, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f colon upper G right-arrow bold upper C> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>C</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>f:G to {mathbf {C}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a function, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho> <mml:semantics> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:annotation encoding=application/x-tex>rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an irreducible representation 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>. The Fourier transform is defined as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove f With caret left-parenthesis rho right-parenthesis equals normal upper Sigma Subscript s element-of upper G Baseline f left-parenthesis s right-parenthesis rho left-parenthesis s right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>f</mml:mi> <mml:mo stretchy=false>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>s</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>hat f(rho ) = {Sigma _{s in G}}f(s)rho (s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Direct computation for all irreducible representations involves order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue upper G EndAbsoluteValue squared> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{left | G right |^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> operations. We derive fast algorithms and develop them for the symmetric group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{S_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. There, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis n factorial right-parenthesis squared> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{(n!)^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is reduced to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n left-parenthesis n factorial right-parenthesis Superscript a slash 2> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>n{(n!)^{a/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=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the constant for matrix multiplication (2.38 as of this writing). Variations of the algorithm allow efficient computation for “small” representations. A practical version of the algorithm is given on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{S_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Numerical evidence is presented to show a speedup by a factor of 100 for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n equals 9> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n = 9</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W1986263670 date "1990-01-01" @default.
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• W1986263670 title "Efficient computation of the Fourier transform on finite groups" @default.
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