Cambrian combinatorics on quiver representations (type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="double-struck">A</mml:mi></mml:math>)

The Combinatorics of Quiver Representations

Contents 1. Introduction 1 1.1. Main results 1 1.2. Horn's conjecture and related problems 2 1.3. The quiver method 4 1.4. Organization 5 2. Preliminaries 6 2.1. Basic notions for quivers 6 2.2. Semi-invariants for quiver representations 7 2.3. Representations in general position 8 2.4. The canonical decomposition 9 2.5. The combinatorics of dimension vectors 11 2.6. Perpendicular categories 12...

A ug 2 00 6 THE COMBINATORICS OF QUIVER REPRESENTATIONS

On Combinatorics of Quiver Component Formulas

Buch and Fulton [7] conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono [21] proved this conjecture as an immediate consequence of their “component formula”. We present an alternative proof of the component formula by substituting combinatorics for Gröbner degeneration [21, 20]. We relate the component formula t...

Quiver Representations of Maximal Rank Type and an Application to Representations of a Quiver with Three Vertices

We introduce the notion of “maximal rank type” for representations of quivers, which requires certain collections of maps involved in the representation to be of maximal rank. We show that real root representations of quivers are of maximal rank type. By using the maximal rank type property and universal extension functors we construct all real root representations of a particular wild quiver w...

Cluster Algebras and Quiver Representations

These notes contain an extended abstract and references for a minicourse given at the Summer School ‘Microlocal and Geometric Methods in Representation Theory’ (Schloss Reisensburg, July 2006) of the International Research Training Group Metz–Paderborn.