Hochschild cohomology of truncated quiver algebras
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We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras Rof type A when e = 0 (the linear quiver) or e ≥ n. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When e = 0 we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category Opreviously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when e = 0 our graded Schur algebras are Koszul over field of characteristic zero. Finally, we give an LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when e = 0.
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Schur algebra
Zero (linguistics)
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We prove a formula relating the fermionic forms and the Poincare polynomials of quiver varieties associated to a finite quiver. Applied to quivers of type ADE, our result implies a version of the fermionic conjecture of Lusztig.
Quiver
Poincaré conjecture
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We prove $\textsf{NP-hardness}$ results for determining whether quivers are mutation equivalent to quivers with given properties. Specifically, determining whether a quiver is mutation-equivalent to a quiver with exactly $k$ arrows between any two of its vertices is $\textsf{NP-hard}$. Also, determining whether a quiver is mutation equivalent to a quiver with no edges between frozen vertices is $\textsf{strongly NP-hard}$. Finally, we present a characterization of mutation classes of quivers with two mutable vertices.
Quiver
Characterization
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Recently, Ringel introduced the resolution quiver for a connected Nakayama algebra. It is known that each connected component of the resolution quiver has a unique cycle. We prove that all cycles in the resolution quiver are of the same size. We introduce the notion of weight for a cycle in the resolution quiver. It turns out that all cycles have the same weight.
Quiver
Low resolution
Component (thermodynamics)
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We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to skew-symmetrizable matrices and obtain other related results. In particular, we show that any quiver is a full subquiver of a quiver mutation equivalent to a quiver of a plabic graph.
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We develop the theory of 2-quivers and quiver 2-categories to run in parallel with the classical theory of quiver algebras. A quiver 2-category is always finitary, and, conversely, every finitary 2-category will be bi-equivalent with a quiver 2-category for a unique underlying reduced 2-quiver. As an application, we produce an example of a fiat 2-category, one of whose Duflo involutions is not self-adjoint.
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Finitary
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Consider the quiver ~D_n and its finite dimensional representations over the field k. We know due to Ringel in that indecomposable representations without self extensions (called exceptional representations) can be exhibited using matrices involving as coefficients only 0 and 1, such that the number of nonzero coefficients is precisely d-1, where d is the global dimension of the representation. This means that the corresponding ''coefficient quiver'' is a tree, so we will call such a presentation a ''tree presentation''.
In this paper we describe explicit tree presentations for the indecomposable preprojective and preinjective representations of the quiver ~D_n. In this way we generalize results obtained by Mr\' oz for the quiver ~D_4 and by Lorinczi and Szanto in for the quiver ~D_5.
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Indecomposable module
Tree (set theory)
Representation
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We give a description of the Namikawa-Weyl group of affinizations of smooth Nakajima quiver varieties using combinatorial data of the underlying quiver, and compute some explicit examples. This extends a result of McGerty and Nevins for quiver varieties coming from Dynkin quivers.
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