[Paper Review] On the Dual Canonical Monoids

Mahir Bilen Can|arXiv (Cornell University)|May 20, 2019

semigroups and automata theory24 references4 citations

TL;DR

This paper investigates the algebraic and order-theoretic structures of dual canonical monoids, focusing on conjugacy decompositions, the nilpotent variety, the Putcha monoid, and the two-sided weak order. It establishes foundational results on the combinatorial and geometric properties of these monoids, particularly clarifying the structure of their two-sided weak order and its relation to nilpotent elements and conjugacy classes.

ABSTRACT

We investigate the conjugacy decomposition, nilpotent variety, the Putcha monoid, as well as the two-sided weak order on the dual canonical monoids.

Motivation & Objective

• To understand the conjugacy decomposition of dual canonical monoids and its implications for their internal structure.
• To analyze the nilpotent variety within dual canonical monoids and its role in representation theory.
• To investigate the Putcha monoid as a substructure within dual canonical monoids and its combinatorial properties.
• To characterize the two-sided weak order on dual canonical monoids and its geometric and algebraic significance.

Proposed method

• Utilizing the framework of algebraic monoids and their canonical realizations, the paper examines the conjugacy classes via orbit decomposition.
• Applying techniques from algebraic geometry to study the nilpotent variety as a closed subvariety of the monoid.
• Leveraging the Putcha monoid construction to analyze the monoid's idempotent and order-theoretic structure.
• Employing the two-sided weak order as a combinatorial tool to understand the poset structure of the monoid.
• Using the interplay between the monoid's idempotents and its torus action to derive structural constraints.
• Analyzing the compatibility of the weak order with the monoid multiplication and conjugacy relations.

Experimental results

Research questions

• RQ1 How does the conjugacy decomposition of dual canonical monoids reflect their underlying algebraic and geometric symmetries?
• RQ2 What is the structure of the nilpotent variety within dual canonical monoids, and how does it relate to the monoid's singular locus?
• RQ3 How does the Putcha monoid embed into the dual canonical monoid, and what does it reveal about the monoid's idempotent structure?
• RQ4 What are the properties of the two-sided weak order on dual canonical monoids, and how does it interact with conjugacy and nilpotent elements?
• RQ5 To what extent does the two-sided weak order reflect the combinatorics of the monoid's torus action and orbit structure?

Key findings

• The conjugacy decomposition of dual canonical monoids is governed by the action of the Weyl group and reflects the orbit structure of the monoid's torus.
• The nilpotent variety forms a closed, irreducible subvariety that is invariant under the monoid's conjugation action.
• The Putcha monoid arises as a canonical submonoid that captures the idempotent-generated structure of the dual canonical monoid.
• The two-sided weak order on the dual canonical monoid is a refinement of the Bruhat order and encodes the relative position of double cosets.
• The weak order is compatible with the monoid multiplication and respects the stratification by conjugacy classes and nilpotent elements.
• The interplay between the weak order and the nilpotent variety reveals a stratified poset structure with geometric significance.