@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-JM0F4GVV-0
  skos:prefLabel "Young tableau"@en, "tableau de Young"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-S0L8K1MV-2
  skos:prefLabel "opération d'Adams"@fr, "Adams operation"@en ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-CCWWKDXS-K
  skos:prefLabel "symmetrization"@en, "symétrisation"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-LTZS5RBS-J
  skos:prefLabel "quasisymmetric function"@en, "fonction quasi-symétrique"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-M6N11QFV-P
  skos:prefLabel "treillis de Young"@fr, "Young's lattice"@en ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-MQ44ZV7K-7
  skos:prefLabel "correspondance de Robinson-Schensted-Knuth"@fr, "Robinson-Schensted-Knuth correspondence"@en ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-BMFQ647R-D
  skos:prefLabel "alternating polynomial"@en, "polynôme alterné"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-XGCPJH3S-J
  skos:prefLabel "Giambelli's formula"@en, "formule de Giambelli"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-M9WLXDDG-X
  skos:prefLabel "Jucys-Murphy element"@en, "élément de Jucys-Murphy"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr: a skos:ConceptScheme .
psr:-G4DL5W27-W
  skos:prefLabel "Pieri's formula"@en, "formule de Pieri"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-MSWK90XH-7
  skos:prefLabel "Monk's formula"@en, "formule de Monk"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-QRR42L58-B
  skos:prefLabel "Stanley symmetric function"@en, "fonction symétrique de Stanley"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-F279704J-T
  skos:prefLabel "Kronecker coefficient"@en, "coefficient de Kronecker"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-L2BN0W1T-P
  skos:prefLabel "fonction"@fr, "function"@en ;
  a skos:Concept ;
  skos:narrower psr:-LP057SP3-B .

psr:-HS1X95S1-9
  skos:prefLabel "symmetric polynomial"@en, "polynôme symétrique"@fr ;
  a skos:Concept ;
  skos:broader psr:-LP057SP3-B .

psr:-LP057SP3-B
  skos:narrower psr:-JM0F4GVV-0, psr:-MQ44ZV7K-7, psr:-HS1X95S1-9, psr:-G4DL5W27-W, psr:-XGCPJH3S-J, psr:-BMFQ647R-D, psr:-CCWWKDXS-K, psr:-M9WLXDDG-X, psr:-F279704J-T, psr:-LTZS5RBS-J, psr:-M6N11QFV-P, psr:-S0L8K1MV-2, psr:-MSWK90XH-7, psr:-QRR42L58-B ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:broader psr:-L2BN0W1T-P ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """En mathématiques, une fonction symétrique est une fonction invariante par permutation de ses variables. Le cas le plus fréquent est celui d'une fonction polynomiale symétrique, donnée par un polynôme symétrique. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_sym%C3%A9trique">https://fr.wikipedia.org/wiki/Fonction_sym%C3%A9trique</a>)"""@fr, """In mathematics, a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\\\\displaystyle n}"></span> variables is <b>symmetric</b> if its value is the same no matter the order of its arguments. For example, a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f\\\\left(x_{1},x_{2}\\
ight)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mrow>           <mo>(</mo>           <mrow>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>             <mo>,</mo>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\left(x_{1},x_{2}\\
ight)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462cb2a17f3647e32711978c6d602319a2844810" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.277ex; height:2.843ex;" alt="{\\\\displaystyle f\\\\left(x_{1},x_{2}\\
ight)}"></span> of two arguments is a symmetric function if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f\\\\left(x_{1},x_{2}\\
ight)=f\\\\left(x_{2},x_{1}\\
ight)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mrow>           <mo>(</mo>           <mrow>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>             <mo>,</mo>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>         <mo>=</mo>         <mi>f</mi>         <mrow>           <mo>(</mo>           <mrow>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>             <mo>,</mo>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\left(x_{1},x_{2}\\
ight)=f\\\\left(x_{2},x_{1}\\
ight)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0723e094e952c164f40cf98317c0a121fdf29d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.652ex; height:2.843ex;" alt="{\\\\displaystyle f\\\\left(x_{1},x_{2}\\
ight)=f\\\\left(x_{2},x_{1}\\
ight)}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x_{1}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x_{1}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\\\\displaystyle x_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x_{2}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x_{2}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\\\\displaystyle x_{2}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\left(x_{1},x_{2}\\
ight)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow>           <mo>(</mo>           <mrow>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>             <mo>,</mo>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left(x_{1},x_{2}\\
ight)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f9102ed82f9fc85fcd78e827ca38cd5fb0bf34" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.611ex; height:2.843ex;" alt="{\\\\displaystyle \\\\left(x_{1},x_{2}\\
ight)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\left(x_{2},x_{1}\\
ight)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow>           <mo>(</mo>           <mrow>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msub>             <mo>,</mo>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>1</mn>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left(x_{2},x_{1}\\
ight)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d125b04c5063ffb30920a4625a442ad066dbbf46" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.611ex; height:2.843ex;" alt="{\\\\displaystyle \\\\left(x_{2},x_{1}\\
ight)}"></span> are in the domain of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\\\\displaystyle f.}"></span> The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle k}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>k</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\\\\displaystyle k}"></span>-tensors on a vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle V}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>V</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle V}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\\\\displaystyle V}"></span> is isomorphic to the space of homogeneous polynomials of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle k}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>k</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\\\\displaystyle k}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle V.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>V</mi>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle V.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\\\\displaystyle V.}"></span> Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Symmetric_function">https://en.wikipedia.org/wiki/Symmetric_function</a>)"""@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_sym%C3%A9trique>, <https://en.wikipedia.org/wiki/Symmetric_function> ;
  skos:prefLabel "fonction symétrique"@fr, "symmetric function"@en ;
  dc:created "2023-08-18"^^xsd:date .