Representations of Symmetric Groups

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 = (λ ₁, λ ₂, …, λ _k), ∑ λ _i = n, are the row subgroup R _T ⊂ S _n and the column subgroup C _T ⊂ S _n 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 $C_{T} \simeq S_{\lambda _{1}^{t}} \times S_{\lambda _{2}^{t}} \times \,\cdots \, \times S_{\lambda _{m}^{t}}$, whereλ ^t = (λ ₁ ^t, λ ₂ ^t, …, λ _m ^t) is the transposed Young diagram. For example, the standard filling