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

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.

💡 Research Summary

The paper investigates a combinatorial problem that originates from the representation theory of the general linear Lie superalgebra gl(r|r). For a function f: ℤ → {×,·} that is equal to · at all but finitely many integers, the authors define a set ♭ f of non‑intersecting cap (or arc) diagrams that match the weight diagram D₍wt₎(f). Each element g ∈ ♭ f corresponds to a highest‑weight g such that the cap diagram D₍cap₎(g) covers the weight diagram of f; this set is precisely the collection of simple composition factors of the Kac module K(f). The central question is to determine the cardinality |♭ f|, which is the same as the maximal length of a composition series of K(f).

The authors introduce a “tally” function T_f that records the cumulative excess of × over · as one moves from left to right along the integer line. Formally, T_f(b+1)=T_f(b)+1 if f(b+1)=× and T_f(b+1)=T_f(b)−1 if f(b+1)=·, with the normalization T_f(a_r)=1 where a_r is the rightmost ×. Extending T_f piecewise linearly yields a piecewise‑linear graph whose slopes are ±1. The zeros of T_f identify potential moves: intervals (b,a) that are “balanced” (contain the same number of × and ·) and start at a · position. A potential move replaces the leftmost · in such an interval by a × and deletes the rightmost ×; however, a move is only legal if the tally never becomes negative on the interval and returns to zero at the right endpoint. This condition translates into a simple inequality involving the local maxima and minima of T_f.

Potential moves are collected in the set PM_f, legal moves in LM_f, and the authors augment PM_f by the special element ½ to obtain PM*f. Using these notions they decompose ♭ f as a disjoint union
 ♭ f = ♭{1/2} f ∪ ⋃_{i∈PM_f} ♭i f,
where ♭{1/2} f contains diagrams in which the rightmost × is paired with the immediately adjacent ·, and ♭i f contains diagrams where the rightmost × is paired with the · at position a+1−2i. Each ♭i f is shown to be in bijection with a Cartesian product ♭{L,i} f × ♭{R,i} f, where the left factor records caps that start left of the distinguished cap and the right factor records caps that start to its right.

The crucial combinatorial estimate comes from bounding the sizes of these factors. By interpreting the caps that lie under a given distinguished cap as ballot sequences (or Dyck paths) of length 2(i−1), the authors prove
 |♭{L,i} f| ≤ C{r+1−i}, |♭{R,i} f| ≤ C{i−1},
where C_n denotes the n‑th Catalan number. The base case is the “standard” function p = (2,4,…,2r), for which |♭ p| = C_{r+1}.

With these bounds, an induction yields
 |♭ f| ≤ C_0 C_r + ∑{i∈PM_f} C{r+1−i} C_{i−1} ≤ ∑{i=0}^{r} C{r+1−i} C_{i-1} = C_{r+1},
the last equality being the well‑known Catalan recurrence. Equality holds only when PM*_f contains all indices ½,1,…,r, which forces f to be a shift of p. Consequently the conjecture posed in