14 research outputs found
A Combinatorial Model for Exceptional Sequences in Type A
Exceptional sequences are certain ordered sequences of quiver
representations. We use noncrossing edge-labeled trees in a disk with boundary
vertices (expanding on T. Araya's work) to classify exceptional sequences of
representations of Q, the linearly-ordered quiver with n vertices. We also show
how to use variations of this model to classify c-matrices of Q, to interpret
exceptional sequences as linear extensions, and to give a simple bijection
between exceptional sequences and certain chains in the lattice of noncrossing
partitions. In the case of c-matrices, we also give an interpretation of
c-matrix mutation in terms of our noncrossing trees with directed edges.Comment: 18 page
Equivariant log concavity and representation stability
We expand upon the notion of equivariant log concavity, and make equivariant
log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil
algebras of oriented matroids, and Orlik--Terao algebras of hyperplane
arrangements. In the case of the Coxeter arrangement for the Lie algebra
, we exploit the theory of representation stability to give
computer assisted proofs of these conjectures in low degree
A combinatorial model for exceptional sequences in type A
Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of , the linearly ordered quiver with vertices. We also show how to use variations of this model to classify -matrices of , to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of -matrices, we also give an interpretation of -matrix mutation in terms of our noncrossing trees with directed edges
Singular Hodge theory for combinatorial geometries
We introduce the intersection cohomology module of a matroid and prove that
it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the
Hodge-Riemann relations. As applications, we obtain proofs of Dowling and
Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of
Kazhdan-Lusztig polynomials for all matroids.Comment: 101 pages; v3: major improvements to the exposition, particularly in
Sections 8, 9, 10. Fixed a gap in the proof of the main result of Section 10.
Added one new result (Theorem 1.4), giving the monotonicity of coefficients
of Kazhdan-Lusztig polynomials of matroids under contraction of flat
The Deligne-Simpson problem for connections on with a maximally ramified singularity
The classical additive Deligne-Simpson problem is the existence problem for
Fuchsian connections with residues at the singular points in specified adjoint
orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem
in terms of quiver varieties. A more general version of this problem, solved by
Hiroe, allows additional unramified irregular singularities. We apply the
theory of fundamental and regular strata due to Bremer and Sage to formulate a
version of the Deligne-Simpson problem in which certain ramified singularities
are allowed. These allowed singular points are called toral singularities; they
are singularities whose leading term with respect to a lattice chain filtration
is regular semisimple. We solve this problem in the important special case of
connections on with a maximally ramified singularity at and
possibly an additional regular singular point at infinity. We also give a
complete characterization of all such connections which are rigid, under the
additional hypothesis of unipotent monodromy at infinity.Comment: 27 pages. Minor correction
A combinatorial model for exceptional sequences in type A
Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of , the linearly ordered quiver with vertices. We also show how to use variations of this model to classify -matrices of , to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of -matrices, we also give an interpretation of -matrix mutation in terms of our noncrossing trees with directed edges.Les suites exceptionnelles sont certaines suites ordonnées de représentations de carquois. Nous utilisons des arbres aux arêtes étiquetés et aux sommets dans le bord d’un disque (expansion sur le travail de T. Araya) pour classifier les suites exceptionnelles de représentations du carquois linéairement ordonné à sommets. Nous exploitons des variations de ce modèle pour classifier les -matrices dudit carquois, pour interpréter les suites exceptionnelles comme des extensions linéaires, et pour donner une bijection élémentaire entre les suites exceptionnelles et certaines chaînes dans le réseau des partitions sans croisement. Dans le cas des -matrices, nous donnons également une interprétation de la mutation des -matrices en termes des arbres sans croisement aux arêtes orientés