@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:-MTRGFQ82-F
  skos:prefLabel "permutation group"@en, "groupe de permutations"@fr ;
  a skos:Concept ;
  skos:narrower psr:-M9WLXDDG-X .

psr:-M9WLXDDG-X
  dc:modified "2023-08-24"^^xsd:date ;
  skos:broader psr:-MTRGFQ82-F, psr:-W9LN9ZRK-5, psr:-LP057SP3-B ;
  dc:created "2023-08-18"^^xsd:date ;
  skos:prefLabel "élément de Jucys-Murphy"@fr, "Jucys-Murphy element"@en ;
  skos:definition """In mathematics, the  <b>Jucys–Murphy elements</b> in the group algebra <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 \\\\mathbb {C} [S_{n}]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">C</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">[</mo>
<br/>        <msub>
<br/>          <mi>S</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {C} [S_{n}]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03192415c4c777069325cf912b2d2e8ef259d00b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.615ex; height:2.843ex;" alt="{\\\\mathbb  {C}}[S_{n}]"></span> of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:
<br/>
<br/><dl><dd><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}=0,~~~X_{k}=(1\\\\;k)+(2\\\\;k)+\\\\cdots +(k-1\\\\;k),~~~k=2,\\\\dots ,n.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>X</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <msub>
<br/>          <mi>X</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mspace width="thickmathspace"></mspace>
<br/>        <mi>k</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>2</mn>
<br/>        <mspace width="thickmathspace"></mspace>
<br/>        <mi>k</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>+</mo>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>k</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <mspace width="thickmathspace"></mspace>
<br/>        <mi>k</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>,</mo>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mi>k</mi>
<br/>        <mo>=</mo>
<br/>        <mn>2</mn>
<br/>        <mo>,</mo>
<br/>        <mo>…<!-- … --></mo>
<br/>        <mo>,</mo>
<br/>        <mi>n</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle X_{1}=0,~~~X_{k}=(1\\\\;k)+(2\\\\;k)+\\\\cdots +(k-1\\\\;k),~~~k=2,\\\\dots ,n.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d14126a2809b0e802e3dfa3bc8a188d67cd98e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:61.375ex; height:2.843ex;" alt="{\\\\displaystyle X_{1}=0,~~~X_{k}=(1\\\\;k)+(2\\\\;k)+\\\\cdots +(k-1\\\\;k),~~~k=2,\\\\dots ,n.}"></span></dd></dl>
<br/>They play an important role in the representation theory of the symmetric group. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element">https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element</a>)"""@en ;
  a skos:Concept ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element> ;
  skos:inScheme psr: .

psr:-W9LN9ZRK-5
  skos:prefLabel "group representation"@en, "représentation de groupe"@fr ;
  a skos:Concept ;
  skos:narrower psr:-M9WLXDDG-X .

psr: a skos:ConceptScheme .
psr:-LP057SP3-B
  skos:prefLabel "fonction symétrique"@fr, "symmetric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-M9WLXDDG-X .