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Signature






Mathematica Notation

Traditional Notation









Integer Functions > Signature[{n1,n2,...,nd}] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/04.21.16.0001.01









  


  










Input Form





Signature[{Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, d]}] Signature[{Subscript[m, 1], Subscript[m, 2], \[Ellipsis], Subscript[m, d]}] == -Sum[Signature[{Subscript[m, 1], Subscript[m, 2], \[Ellipsis], Subscript[m, d]}] Product[KroneckerDelta[Subscript[n, k], Subscript[m, k]], {k, 1, d}], Permutations[Subscript[m, 1], Subscript[m, 2], \[Ellipsis], Subscript[m, d]]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "d"]]], "}"]], "]"]], RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["m", "1"], ",", SubscriptBox["m", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["m", "d"]]], "}"]], "]"]]]], "\[Equal]", RowBox[List["-", RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["m", "1"], ",", SubscriptBox["m", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["m", "d"]]], "}"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "d"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["n", "k"], ",", SubscriptBox["m", "k"]]], "]"]]]]]], ",", RowBox[List["Permutations", "[", RowBox[List[SubscriptBox["m", "1"], ",", SubscriptBox["m", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["m", "d"]]], "]"]]]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> &#949; </mi> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mtext> </mtext> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mtext> </mtext> <mo> , </mo> <mtext> </mtext> <msub> <mi> n </mi> <mi> d </mi> </msub> </mrow> </msub> <mo> &#8290; </mo> <msub> <mi> &#949; </mi> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mtext> </mtext> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mtext> </mtext> <mo> , </mo> <mtext> </mtext> <msub> <mi> m </mi> <mi> d </mi> </msub> </mrow> </msub> </mrow> <mtext> </mtext> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <munder> <mo> &#8721; </mo> <mrow> <mi> permutations </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mi> d </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </munder> <mrow> <msub> <mi> &#949; </mi> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mtext> </mtext> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mtext> </mtext> <mo> , </mo> <mtext> </mtext> <msub> <mi> m </mi> <mi> d </mi> </msub> </mrow> </msub> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> d </mi> </munderover> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> , </mo> <msub> <mi> m </mi> <mi> k </mi> </msub> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#949; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> d </ms> </apply> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> &#949; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> d </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8721; </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> permutations </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> d </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#949; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> d </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> d </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> InterpretationBox </ci> <ms> &#948; </ms> <ci> KroneckerDelta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> k </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> k </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "d_"]]], "}"]], "]"]], " ", RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["m_", "1"], ",", SubscriptBox["m_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["m_", "d_"]]], "}"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["mm", "1"], ",", SubscriptBox["mm", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["mm", "d"]]], "}"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "d"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["nn", "k"], ",", SubscriptBox["mm", "k"]]], "]"]]]]]], ",", RowBox[List["Permutations", "[", RowBox[List[SubscriptBox["mm", "1"], ",", SubscriptBox["mm", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["mm", "d"]]], "]"]]]], "]"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





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