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 characterof G it flnds a subgroup H such that the restriction ofon H has a linear con- stituent with multiplicity one. Then using a well known algorithm we can construct a representation of G afiording ´. 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 ‚ by (‚), so Irr(Sn) = f(‚) j ‚ ' ng. If G is a flnite group andis an irreducible character of G, an e-cient and simple method to construct representations of flnite groups has been presented in (2). This method is applicable whenever G has a subgroup H such thatH has a linear constituent with multiplicity 1. We call such a subgroup H a ´-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 flnd a ´-subgroup for each irreducible characterof G. If G is a simple group or a covering group of a simple group, then a ´- subgroup for each nontrivial irreducible characterof G of degree < 32 has been found in (1). Also if ‚ is a partition of n and ‚ 0 is the conjugate