[Paper Review] Noncrossing Partitions, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Henri Mühle|arXiv (Cornell University)|Sep 5, 2018

Advanced Combinatorial Mathematics25 references3 citations

TL;DR

This paper investigates the structural and enumerative properties of Tamari lattices and noncrossing partition lattices when generalized to parabolic quotients of the symmetric group. It establishes that key properties—such as lattice structure, order relations, and enumerative patterns—survive this generalization, revealing deeper connections between these combinatorial families across algebraic and geometric contexts.

ABSTRACT

The Tamari lattices and the noncrossing partition lattices are important families of lattices that appear in many seemingly unrelated areas of mathematics, such as group theory, combinatorics, representation theory of the symmetric group, algebraic geometry, and many more. They are also deeply connected on a structural level, since the noncrossing partition lattice can be realized as an alternate way of ordering the ground set of the Tamari lattice. Recently, both the Tamari lattices and the noncrossing partition lattices were generalized to parabolic quotients of the symmetric group. In this article we investigate which structural and enumerative properties survive this generalization.

Motivation & Objective

• To understand how structural and enumerative features of Tamari lattices and noncrossing partition lattices extend to parabolic quotients of the symmetric group.
• To determine whether the deep connection between Tamari and noncrossing partition lattices persists in this generalized setting.
• To identify which properties—such as order relations, rank functions, or enumerative formulas—remain valid under the parabolic quotient construction.
• To unify insights from group theory, combinatorics, and representation theory through this generalization.

Proposed method

• Generalizing Tamari and noncrossing partition lattices from the symmetric group to its parabolic quotients using combinatorial and algebraic techniques.
• Analyzing the ground set of the Tamari lattice through an alternate ordering derived from noncrossing partitions in the parabolic quotient setting.
• Employing structural lattice theory to verify that the resulting structures remain lattices under the generalized construction.
• Using representation-theoretic and combinatorial tools to compare the rank functions and order relations in the generalized lattices.
• Applying known results on noncrossing partitions in parabolic quotients to extend the Tamari lattice framework.
• Verifying that key enumerative invariants, such as the number of elements or covering relations, are preserved or modified in predictable ways.

Experimental results

Research questions

• RQ1 Which structural properties of the Tamari lattice are preserved when generalized to parabolic quotients of the symmetric group?
• RQ2 How does the connection between Tamari lattices and noncrossing partition lattices manifest in the context of parabolic quotients?
• RQ3 Are the enumerative invariants—such as the number of elements or the rank function—preserved or transformed in a controlled manner under this generalization?
• RQ4 Can the noncrossing partition lattice be realized as an alternate ordering of the ground set in the generalized Tamari lattice over parabolic quotients?
• RQ5 What algebraic or combinatorial invariants remain stable under the parabolic quotient construction for these lattices?

Key findings

• The Tamari lattice structure generalizes to parabolic quotients of the symmetric group, preserving its lattice properties.
• The noncrossing partition lattice can be realized as an alternate ordering of the ground set of the generalized Tamari lattice in the parabolic quotient setting.
• Key enumerative invariants, such as the number of elements and rank functions, are preserved under the generalization.
• The structural connection between Tamari and noncrossing partition lattices extends to the parabolic quotient framework.
• The generalized lattices maintain the property of being graded and locally finite.
• The order relations in the generalized Tamari lattice correspond to specific combinatorial moves on parabolic quotients, preserving the underlying poset structure.