The Combinatorics of Quiver Representations
نویسنده
چکیده
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 2.7. Exceptional Sequences 14 2.8. Stability and GIT-quotients 17 3. Stability 19 3.1. Harder-Narasimhan and Jordan-Hölder filtrations 19 3.2. The σ-stable decomposition 22 4. Schur sequences 27 5. The faces of the cone R + Σ(Q, α) 30 6. More on the σ-stable decompositions 34 6.1. The set of σ-stable dimension vectors 34 6.2. σ-stable decomposition for quivers with oriented cycles 37 7. Littlewood-Richardson coefficients 39 7.1. The Klyachko cone 39 7.2. Walls of the Klyachko cone 42 7.3. Faces of the Klyachko cone of arbitrary codimension 45 7.4. A multiplicative formula for Littlewood-Richardson coefficients 46 Appendix: Belkale's proof of Fulton's conjecture 48 References 51 1. Introduction 1.1. Main results. Let α, β be dimension vectors for a quiver Q without oriented cycles. If α, β Q = 0, where ·, · Q is the Euler form (or Ringel form), then we will define a number α • β. This number 1 2 HARM DERKSEN AND JERZY WEYMAN can be defined as the dimension of a space SI(Q, β) α,· of semi-invariants (see Definition 2.5). It was shown in [16] that α • β can also be defined in terms of Schubert calculus. In the Schubert calculus approach, α • β counts the number of α-dimensional subrepresentations of a general (α + β)-dimensional representation. In the special case of the triple flag quiver, the number α • β turns out to a be Littlewood-Richardson coefficient c ν λ,µ where the partitions λ(α, β), µ = µ(α, β), ν = ν(α, β) depend on α and β. This allows us to prove many new results about LR-coefficients, and to extend results about Littlewood-Richardson coefficients to the more general setting of quiver representations. Knutson and Tao (see [28]) proved the saturation conjecture for LR-coefficients: if c N ν N λ,N µ > 0 for some positive integer N , then c ν λ,µ > 0 (see Theorem 1.4). It was shown in [13] that the saturation theorem for LR-coefficients generalizes to quivers: if N α • M β > 0 for some …
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