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1. Definition

A Nakayama algebra is a finite-dimensional algebra over a field $\mathbb{F}$ where every indecomposable projective or injective module is uniserial, see [ARS97], [AF92] and [SY11].

Let $Q$ be a finite quiver with path algebra $\mathbb{F} Q$, and let $I$ be a two-sided ideal in $\mathbb{F} Q$. Let $J$ denote the ideal generated by all arrows in $\mathbb{F} Q$. Then $I$ is called admissible in case $J^m \subseteq I \subseteq J^2$ for some $m \geq 2$. We will restrict in this survey on Nakayama algebras given by quiver and admissible relations (note that over algebraically closed fields this is no loss of generality when doing homological algebra, as any finite dimensional algebra is Morita equivalent to a quiver algebra). In this language, Nakayama algebras can be characterised as algebras $\mathbb{F}Q/I$ for admissible ideals $I$ and a finite quiver $Q$ that is either an oriented linear quiver or an oriented cyclic quiver.

Let $A$ be an Nakayama algebra with $n$ simple modules and let $e_i$ denote the idempotent corresponding to the vertex $i$ in the quiver. The Kupisch series of $A$ is the sequence $[c_0,c_1,...,c_{n-1}]$, where $c_i \geq 1$ denotes the vector space dimension of the indecomposable projective module $e_iA$.

2. Nakayama algebras with linear quiver and Dyck paths

The area sequence $[a_1,\dots,a_n]$ of a Dyck path of semilength $n$ is given by setting $a_i$ to be the number of full boxes between the path and the main diagonal, see St000012.

Sending a Nakayama algebra with $n+1$ simple modules and Kupisch sequence $[c_0,\dots,c_n]$ to unique Dyck path with area sequence $[c_{n-1}-2,\dots,c_1-2,c_0-2]$ is a bijection between Nakayama algebras on $n+1$ simple modules and with linear quiver and Dyck paths of semilength $n$.

All statistics on Dyck paths in the database that describe properties of Nakayama algebras with linear quiver use this bijective identification.

3. Properties

Let $A$ be a Nakayama algebra over the field $\mathbb{F}$ with $n$ simple modules and Kupisch series $[c_0,\dots,c_{n-1}]$ and let $M$ and $N$ be $A$-modules and let $S$ be a simple $A$-module. Denote by $D:=Hom_{\mathbb{F}}(-,\mathbb{F})$ the natural duality.

4. References

[AF92]   F. W. Anderson and R. Kent, Rings and Categories of Modules, Graduate Texts in Mathematics Vol. 13, Springer (1992).

[ARS97]   M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36. Cambridge University Press (1997), xiv+425pp.

[SY11]   A. Skowronski and K. Yamagata, Frobenius Algebras I: Basic Representation Theory, EMS Textbooks in Mathematics (2011).