The Multisegment Duality and the Preprojective Algebras of Type A

ABSTRACT. The multisegment (or Zelevinsky) duality ζ plays an important role in the representation theory of the groups GLn over a p-adic field, but also for the quantum groups of type A. Recently, Knight and Zelevinsky have exhibited a formula which allows a direct calculation of ζ; their proof uses the representation theory of a linearly ordered quiver of type A. Some of their considerations may be interpreted homologically. It is well-known that the multisegment duality can easily be defined in terms of the corresponding preprojective algebra. The use of certain modules over the preprojective algebra seems to illuminate the considerations of Knight and Zelevinsky. We are going to outline all the essential steps, only few details are left to the reader. Throughout the paper we fix some natural number n. We denote by S the set of pairs (i, j) where 1 ≤ i ≤ j ≤ n; or also the set of corresponding segments [i, j] = {i, i+1,..., j} of natural numbers. Note that we may identify S with the set of positive roots in the root system of type An; the one-element segment [i, i] corresponds to the simple root ai and [i, j] to ai + ai+1 + · · · + aj. The elements d of NS0 may be considered as forma