A Combinatorial Approach to the Fusion Process for the Symmetric Group

In this survey paper we give a detailed account of a construction strictly related to the so-called fusion process for the symmetric group Sn . This procedure gives a way of building up the primitive idempotents in C[Sn]—indeed the Young idempotents are obtained—affording the irreducible representations of Sn as a product of factors 1+ ci, j (i, j) ∈ C[Sn]; these factors should be regarded as values of special rational functions on the hyperplane ∑n i=1 zi = 0 in Cn . This was first proved by Jucys [4], then by Cherednik [1, 2] and finally generalized by Nazarov [7] to describe the projective representations of Sn . The fusion process associates to each partition λ ` n, more precisely to the standard column tableau 3 of shape λ, the following data: