Intersection statistics for antichains in minuscule posets

James Propp

Intersection statistics for antichains in minuscule posets

James Propp

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.

Intersection statistics for antichains in minuscule posets

Abstract

For a finite poset

, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of

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.

Paper Structure (2 theorems, 11 equations)

This paper contains 2 theorems, 11 equations.

Key Result

Theorem 1

Let $P$ be one of the following posets. Then is given by:

Theorems & Definitions (4)

Theorem
Lemma
proof
proof