On the global dimension of Nakayama algebras

TL;DR

This work links the global dimension of Nakayama algebras to Dyck-path combinatorics. It establishes an equidistribution between $\mathrm{gldim}$ for connected linear Nakayama algebras with $n$ simples and the height statistic of Dyck paths of semilength $n-1$, via a tree-based bijection and a decomposition into $(m,L,R,M)$. It also shows that sincere Nakayama algebras correspond to Dyck paths with $\mathrm{gldim}=2\,b_D$, enabling Catalan-enumeration for magnitude-1 cases and a precise area-sequence/Dyck-path mapping for the finite-global-dimension sincere class. Overall, the paper forges a deep connection between representation-theoretic invariants and classical Dyck-path combinatorics, yielding explicit calculation rules and exact enumerative results.

Abstract

We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic Nakayama algebras an explicit classification of finite global dimension is not known. However, we show that in certain special cases cyclic Nakayama algebras with finite global dimension can again be interpreted as Dyck paths. In particular, we show that there is a natural bijection between sincere Nakayama algebras and Dyck paths. In this case, we find that the global dimension is in fact twice the bounce count of the corresponding Dyck path.

Figures (2)

• Figure 1: The tree corresponding to the Kupisch series $[3,4,4,3,2,1]$.
• Figure 2: The naturally labelled trees corresponding to the Kupisch series $[5,6,5,4,4,3,3,3,2,3,2,1]$ and to $[2,1,5,5,5,6,5,4,3,2,2,1]$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• proof
• Proposition 2.6
• proof
• Example 2.7
• Example 2.8