Algebra II pp 151-171 | Cite as

Representations of Symmetric Groups

  • Alexey L. Gorodentsev


A Young diagram λ of weight | λ | = n filled by nonrepeating numbers 1, 2,  , n is called a standard filling of shape λ. Given a filling T, we write λ(T) for its shape. Associated with every standard filling T of shape λ = (λ1, λ2, …, λk), ∑ λi = n, are the row subgroup RT ⊂ Sn and the column subgroup CT ⊂ Sn permuting the elements 1, 2,  , n only within the rows and within the columns of T respectively. Thus, \(R_{T} \simeq S_{\lambda _{1}} \times S_{\lambda _{2}} \times \,\cdots \, \times S_{\lambda _{k}}\) and \(C_{T} \simeq S_{\lambda _{1}^{t}} \times S_{\lambda _{2}^{t}} \times \,\cdots \, \times S_{\lambda _{m}^{t}}\), whereλt = (λ1t, λ2t, …, λmt) is the transposed Young diagram. For example, the standard filling


  1. [DK]
    Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRefGoogle Scholar
  2. [Fu]
    Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.zbMATHGoogle Scholar
  3. [FH]
    Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.zbMATHGoogle Scholar
  4. [Mo]
    Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  • Alexey L. Gorodentsev
    • 1
  1. 1.Faculty of MathematicsNational Research University “Higher School of Economics”MoscowRussia

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