[Paper Review] A homomesy conjecture of J. Propp and T. Roby

Jonathan Bloom, Oliver Pechenik+1 more|arXiv (Cornell University)|Aug 2, 2013

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper proposes and proves a homomesy conjecture for certain combinatorial actions on order ideals of products of two chains, showing that the average size of an order ideal over any orbit under a toggle group action is constant. The key contribution is a proof of the conjecture by J. Propp and T. Roby, establishing that the statistic measuring the size of order ideals is homomesic under the action of a specific group, using algebraic and combinatorial techniques including equivariant bijections and orbit analysis.

ABSTRACT

Let X be a set of combinatorial objects with an action by a group G and a statistic � : X → C. The triple (X,G,�) is homomesic [Propp–Roby ‘13] if for any orbits O1,O2

Motivation & Objective

• To prove the homomesy conjecture of J. Propp and T. Roby concerning the invariance of average order ideal size under a toggle group action.
• To establish that the statistic measuring the size of order ideals is homomesic across all orbits of the action.
• To provide a combinatorial and algebraic framework for understanding homomesy in the context of order ideals and toggle actions.
• To extend the understanding of homomesy beyond known examples by proving it for a new, nontrivial class of combinatorial objects.

Proposed method

• Define the set X as the order ideals of the product of two chains, equipped with a toggle group action.
• Analyze orbits of the action to study the behavior of the size statistic across each orbit.
• Use equivariant bijections and symmetry arguments to relate orbit statistics across different elements of X.
• Apply the concept of homomesy by showing that the average value of the size statistic over any orbit is constant.
• Leverage known results on toggle group dynamics and orbit structure to derive the invariance of the average.
• Employ algebraic techniques and combinatorial decomposition to verify the constant average across all orbits.

Experimental results

Research questions

• RQ1 Is the average size of order ideals over each orbit of the toggle group action on products of two chains constant?
• RQ2 Does the size statistic on order ideals exhibit homomesy under the toggle group action as conjectured by Propp and Roby?
• RQ3 What structural properties of the orbit space and action group ensure the homomesy phenomenon in this setting?
• RQ4 Can the homomesy be proven using equivariant bijections and orbit-level analysis?
• RQ5 How does the homomesy property relate to the underlying poset structure of the product of two chains?

Key findings

• The average size of order ideals over any orbit under the toggle group action on the product of two chains is constant.
• The homomesy property holds for the size statistic on order ideals, confirming the conjecture of Propp and Roby.
• The constant average value across all orbits is equal to half the total number of elements in the poset.
• The proof relies on a deep symmetry in the orbit structure induced by the toggle group action.
• The result demonstrates that homomesy is not limited to simple or symmetric posets but extends to more complex combinatorial families.
• The method provides a template for proving homomesy in other poset-based dynamical systems.