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arXiv · 1307.7573

The number of complete exceptional sequences for a Dynkin algebra

Mustafa A. A. Obaid·S. Khalid Nauman·Wafa S. Al Shammakh·Wafaa M. Fakieh·Claus Michael Ringel
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Abstract

We consider Dynkin algebras, these are the hereditary artin algebras of finite representation type. The indecomposable modules for a Dynkin algebra correspond bijectively to the positive roots of a Dynkin diagram. Given a Dynkin algebra with n simple modules, a complete exceptional sequence is a sequence M_1,..., M_n of indecomposable modules such that Hom(M_i,M_j) = 0 = Ext(M_i,M_j) for i > j. The aim of this paper is to determine the number of complete exceptional sequences for any Dynkin algebra. There are direct connections between the representation theory of a Dynkin algebra A and the lattice L of non-crossing partitions of the same Dynkin type: As Ingalls and Thomas have shown, the lattice of the thick subcategories of mod A can be identified with L. Hubery and Krause have pointed out that this identification provides a bijection between the complete exceptional sequences for A and the maximal chains in L. Thus, our calculations may also be considered as a categorification of results concerning non-crossing partitions.

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Mustafa A. A. Obaid, S. Khalid Nauman, Wafa S. Al Shammakh, Wafaa M. Fakieh, Claus Michael Ringel. 2013-11-23. The number of complete exceptional sequences for a Dynkin algebra. https://arxiv.org/abs/1307.7573

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