The multisegment duality and the preprojective algebras of type A
Claus Michael Ringel
Algebra Montpellier Announcements 1 (1999)

Abstract:
The multisegment (or Zelevinsky) duality $\zeta$ plays an important role in 
the representation theory of the gruops ${\rm GL}_n$ 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 $\zeta$; 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.