ON CONSTRUCTING REPRESENTATIONS OF THE SYMMETRIC GROUPS

Abstract. Let G be a symmetric group. In this paper we describe a method that for a certain irreducible character χ of G it finds a subgroup H such that the restriction of χ on H has a linear con-stituent with multiplicity one. Then using a well known algorithm we can construct a representation of G affording χ. We recall some facts about characters of the symmetric group Sn. For a general reference see, for example, [3]. A partition λ = (λ1, λ2,..., λl) of a natural number n is a weakly decreasing sequence λ1> λ2> · · ·> λl> 0 of integers with |λ | = ∑l i=1 λi = n, for short we write λ ` n. The numbers λi are the parts of λ. We also write the partition exponentially as λ = (la11,..., l am m), l1> · · ·> lm> 0, when we have ai parts of size li. Since the number of irreducible characters of a group is equal to the number of conjugacy classes, which in the case of Sn is the number of partitions of n, the irreducible characters of Sn are labelled by partitions of n. We denote the irreducible character labelled by the partition λ by [λ], so Irr(Sn) = {[λ] | λ ` n}. If G is a finite group and χ is an irreducible character of G, an efficient and simple method to construct representations of finite groups has been presented in [2]. This method is applicable whenever G has a subgroup H such that χH has a linear constituent with multiplicity 1. We call such a subgroup H a χ-subgroup. In practice this algorithm is quite fast when H has a small order, but can be very slow for a large H. For using this method to construct representations of G, we need to find a χ-subgroup for each irreducible character χ of G. If G is a simple group or a covering group of a simple group, then a χ-subgroup for each nontrivial irreducible character χ of G of degree < 32 has been found in [1]. Also if λ is a partition of n and λ ′ is the conjugat