Algebra II pp 151-171

# Representations of Symmetric Groups

• Alexey L. Gorodentsev
Chapter

## Abstract

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

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