Weak Exceptional Sequences

TL;DR

This paper introduces weak exceptional sequences as a new module sequence concept, extending the standard case, with unique properties for algebras of higher global dimension, and provides combinatorial formulas for their enumeration.

Contribution

It defines weak exceptional sequences, explores their properties for algebras with higher global dimension, and derives formulas for counting these sequences in specific algebra classes.

Findings

Weak exceptional sequences generalize standard exceptional sequences.

For algebras with global dimension > 1, sequence size can exceed algebra rank.

Closed-form formulas for counting sequences in Nakayama algebras.

Abstract

We introduce weak exceptional sequence of modules which can be viewed as another modification of the standard case, different than the works of Igusa-Todorov \cite{Igusa-Todorov} and Buan-Marsh \cite{Buan-Marsh}. For hereditary algebras it is equivalent to standard exceptional sequences. One important new feature is: if global dimension of algebra is greater than one, then the size of the full sequence can exceed the rank of the algebra. We use both cyclic and linear Nakayama algebras to test combinatorial aspects of this new sequence. For some particular classes we give closed form formulas which returns the number of the full weak exceptional sequences, and compare them with the number of exceptional sequences of types $\mathbb{A}$ and linear radical square zero Nakayama algebras \cite{sen19}.