Abstract
Exceptional sequences are certain sequences of quiver representations. We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions.
Highlights
Exceptional sequences are certain sequences of quiver representations with strong homological properties
Crawley-Boevey showed that the braid group acts transitively on the set of complete exceptional sequences [CB93]
Exceptional sequences have been connected to many other areas of mathematics since their invention:
Summary
Exceptional sequences are certain sequences of quiver representations with strong homological properties. Crawley-Boevey showed that the braid group acts transitively on the set of complete exceptional sequences (i.e., exceptional sequences of maximal length) [CB93] This result was generalized to hereditary Artin algebras by Ringel [Rin94]. Let us denote such a representation by Xi,j, where is a vector that keeps track of the orientation of the quiver, and i + 1 and j are the positions where the string of 1’s begins and ends, respectively This simple description allows us to view exceptional sequences as combinatorial objects. We show that exceptional collections (i.e., the underlying set of representations in an exceptional sequence) are classified by strand diagrams (see Theorem 12). We give combinatorial proofs that any two reddening sequences produce isomorphic ice quivers (see [Kel12] for a general proof in all types using deep category-theoretic techniques) and that there is a bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions
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