Absolutely indecomposable representations and Kac-Moody Lie algebras

By W. Crawley‐Boevey, Michel van den Bergh. 2001

127 citations recorded by the source corpus.

A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/82251116bfa397ef10268f0895ddc1b0b257830c

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The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Tamás Hausel
2. O. Schiffmann
3. E. Letellier
4. Markus Reineke
5. Fernando Rodriguez-Villegas
6. Ben Davison
7. T. Bridgeland
8. B. Keller

In Geometry and Topology

1. Tamás Hausel
2. Fernando Rodriguez-Villegas
3. O. Schiffmann
4. E. Letellier
5. Markus Reineke
6. T. Bridgeland
7. Ben Davison
8. B. Keller

In Mathematical Physics

1. Tamás Hausel
2. Fernando Rodriguez-Villegas
3. E. Letellier
4. Markus Reineke
5. O. Schiffmann
6. T. Bridgeland
7. Anton Mellit
8. Ben Davison

Citations in the corpus

• Arithmetic harmonic analysis on character and quiver varieties II (2008, background): “3 Another formula for Kac polynomials When the dimension vector vμ is indivisible, it is known by Crawley-Boevey and van den Bergh [1] that the polynomial Aμ(q) equals (up to some power of q) to the polynomial which counts the number of points of some quiver variety over Fq.”
• Kac’s conjecture from Nakajima quiver varieties (2008, background): “1After this result and Theorem 1 were announced in [4] Mozgovoy in [12] found an alternative proof of Theorem 1 by combining [1] and [7].”
• The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli (2002, methodology): “d . A standard argument (see ([Kir], 15.), ([G¨o], 1.2.)), together with the generic compatibility of formation of invariants and base change (see ([ CBV ], Lemma B.4.)) shows that the number of rational points (viewed as a function of v) is precisely the Poincare polynomial of the cohomology of Mss”
• Moduli of representations of quivers (2008, background): “The first conjecture is proved in the case of indivisible d in [ 14 ]; a proof for arbitrary d is announced in [30].”
• Mixed Hodge polynomials of character varieties (2006, methodology): “Since Sg with this dimension vector is divisible (for n > 1) the conjecture is still open (the indivisible case was proved in [5] ).”