Intersection statistics for antichains in minuscule posets

TL;DR

This paper derives closed-form formulas for the expected intersection size of two random antichains in minuscule posets, linking combinatorial results to representation theory and providing insights into their structure.

Contribution

It introduces explicit formulas for intersection statistics in minuscule posets, connecting combinatorics with representation theory.

Findings

Closed-form expressions for intersection sizes in minuscule posets

Elementary combinatorial proofs of the formulas

Interpretation of results via weight diagrams of minuscule representations

Abstract

For a finite poset $P$, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of $|A\cap A'|$ over all ordered pairs of antichains. For general posets this statistic appears to have little structure, but for the classical minuscule posets with uniform combinatorial models it admits closed-form expressions. Though the proofs are elementary and combinatorial, the resulting formulas admit a natural interpretation in terms of weight diagrams of minuscule representations.