[Paper Review] Plane partitions in the work of Richard Stanley and his school

C. Krattenthaler|arXiv (Cornell University)|Mar 19, 2015

Architecture and Art History Studies4 citations

TL;DR

This paper surveys the theory of plane partitions through the lens of Richard Stanley's work and his school, emphasizing connections to symmetric functions, representation theory, and combinatorial enumeration. It presents key results such as MacMahon's generating function for plane partitions in an a×b×c box, derived via non-intersecting lattice paths and Schur functions, and establishes a bijection between plane partitions and rhombus tilings of hexagons, enabling elegant combinatorial proofs and enumeration via determinantal formulas.

ABSTRACT

These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.

Motivation & Objective

• To survey the development of plane partition theory as advanced by Richard Stanley and his collaborators.
• To clarify the deep connections between plane partitions, symmetric functions, and representation theory of classical groups.
• To present the combinatorial bijection between plane partitions and rhombus tilings of hexagons with side lengths a, b, c.
• To explain how non-intersecting lattice path methods and determinant/Pfaffian formulas enable enumeration of symmetry classes of plane partitions.
• To demonstrate the equivalence of MacMahon's generating function for plane partitions to a specialized Schur function.

Proposed method

• Utilizes the 3D cube pile representation of plane partitions to establish a bijection with rhombus tilings of a hexagon with side lengths a, b, c, a, b, c.
• Applies the Lindström–Gessel–Viennot lemma to count non-intersecting lattice paths, leading to determinantal formulas for enumeration.
• Employs the minor summation theorem of Ishikawa and Wakayama to generalize path counting when starting or ending points vary in sets.
• Transforms plane partitions into semistandard Young tableaux via rotation and addition of row indices, linking them to Schur functions.
• Uses the Weyl character formula and Vandermonde determinants to evaluate the specialized Schur function, recovering MacMahon's generating function.
• Applies the principal specialization x_i = q^i to Schur functions to express the generating function for plane partitions in terms of q-series.

Experimental results

Research questions

• RQ1 How can plane partitions be enumerated using non-intersecting lattice paths and determinantal formulas?
• RQ2 What is the connection between plane partitions and rhombus tilings of hexagons with side lengths a, b, c?
• RQ3 How do symmetric functions and representation theory of SL_n(C) relate to the enumeration of plane partitions?
• RQ4 What is the role of the minor summation theorem in extending lattice path enumeration to symmetry classes of plane partitions?
• RQ5 How does the principal specialization of Schur functions recover MacMahon’s generating function for plane partitions in an a×b×c box?

Key findings

• The generating function for plane partitions in an a×b×c box is given by the product formula ∏_{i=1}^a ∏_{j=1}^b ∏_{k=1}^c (i+j+k-1)/(i+j+k-2), which is the q→1 limit of MacMahon’s formula.
• There is a bijection between plane partitions of shape (b,b,…,b) with entries ≤c and rhombus tilings of a hexagon with side lengths a, b, c, a, b, c.
• The number of plane partitions in an a×b×c box equals the number of semistandard Young tableaux of shape (b,b,…,b) with entries in {1,2,…,a+c}, via a rotation and shift transformation.
• The generating function ∑_π q^|π| for plane partitions in an a×b×c box equals q^{-b binom(a+1,2)} times the Schur function s_{(b,b,…,b)}(q, q^2, …, q^{a+c}).
• The Weyl character formula, when specialized with x_i = q^i, yields the Vandermonde determinant evaluation that confirms MacMahon’s formula.
• The use of Pfaffian formulas via the minor summation theorem allows for the enumeration of symmetry classes of plane partitions where starting or ending points of paths are not fixed.