Pairs of eventually constant maps and nilpotent pairs

TL;DR

The paper extends Leinster’s nilpotent-pair correspondence from a single vector space to a two-vertex quiver setting, deriving an explicit count for nilpotent pairs and a finite-field nilpotency probability. It develops a linear-algebraic decomposition (a Fitting-type framework) and introduces balanced vectors to obtain a constructive classification of nilpotent pairs. On the set-theoretic side, it analyzes eventually constant maps between finite sets, obtaining a Cayley-type enumeration via a bijection with spanning trees in a complete bipartite graph. Together, these results connect representation-theoretic nilpotent cones to combinatorial tree counts and bipartite graph structures, yielding precise enumerations and probabilistic insights.

Abstract

Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space $V$ and the set of pairs consisting of a nilpotent operator and a vector in $V$. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in $V$. We generalize this correspondence to pairs of operators between pairs of vector spaces and determine the probability that a random pair of operators be nilpotent. We also determine the set-theoretical counterpart of this construction and compute the number of eventually constant pairs of maps between two finite sets, closely related to the number of spanning trees in a complete bipartite graph.

Figures (2)

• Figure 1: Left: a graph with a vertex and a loop. Right: graph $\Gamma$ with 2 vertices and 2 arrows forming an oriented 2-cycle.
• Figure 2: Example of a balanced vector $v$ of length $\ell$ and its images under compositions of $f$ and $g$.

Theorems & Definitions (18)

• Lemma 2.1: Cayley's formula
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• Corollary 2.5
• proof
• Lemma 3.1