Catalan numbers and a conjecture on the maximum composition length of a Kac module

TL;DR

The paper connects the combinatorics of Catalan numbers to representation theory by studying Kac modules for the Lie superalgebra $\mathfrak{gl}(r|r)$ and the corresponding set of composition factors encoded as $\flat(f)$. It introduces cap and weight diagrams and a tally-based framework to define potential and legal moves, then derives a Catalan-type recursion that counts $|\flat(f)|$ via a disjoint union over decomposed substructures, akin to the Fundamental Recurrence for Catalan numbers. A sharp bound $|\flat(f)| \le C_{r+1}$ is proven, with equality iff $f$ is the canonical $p=(2,4,...,2r)$ up to shift, thereby showing the maximum composition length of a Kac module is a Catalan number $C_{r+1}$ in the most atypical block. The methods provide a constructive enumeration of composition factors and reveal deep ties between representation theory and Catalan combinatorics, with potential implications for block equivalences to diagram algebras and Kostant-type resolutions.

Abstract

Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain conditions determined by $f$. Then we give a recursive method for enumeration of $\flat f$ which recalls the Fundamental Recurrence for Catalan numbers. The motivation comes from the problem of enumeration of the composition factors of a Kac module with maximum degree of atypicality for the Lie superalgebra $\mathfrak{g}=\mathfrak{gl}(r|r)$. In particular we prove a conjecture that the maximum number of composition factors is a Catalan number.

Theorems & Definitions (26)

• Theorem 1.1
• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Proposition 1.4
• Example 2.1
• Lemma 2.2
• proof
• Example 2.3
• Lemma 2.4