[Paper Review] Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids

Alan J. Cain, António Malheiro|arXiv (Cornell University)|Nov 13, 2016

Algebraic structures and combinatorial models19 references4 citations

TL;DR

This paper investigates non-trivial identities in 'plactic-like' monoids—hypoplactic, sylvester, Baxter, stalactic, taiga, and related monoids—using combinatorial insertion algorithms and tree-based representations. It proves that all these monoids satisfy non-trivial identities, identifies the shortest such identities, and resolves the status of the plactic and patience sorting monoids, showing that the plactic monoid does not satisfy any non-trivial identity, while the left and right patience sorting monoids also fail to satisfy any non-trivial identity beyond trivial commutativity in the rank-1 case.

ABSTRACT

This paper considers whether non-trivial identities are satisfied by certain `plactic-like' monoids that, like the plactic monoid, are closely connected to combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities, and indeed give shortest identities satisfied by these monoids. The existing state of knowledge is discussed for the plactic monoid and left and right patience sorting monoids.

Motivation & Objective

• To determine whether various 'plactic-like' monoids satisfy non-trivial identities.
• To identify the shortest non-trivial identities satisfied by these monoids.
• To clarify the state of knowledge regarding the plactic monoid and left/right patience sorting monoids.
• To investigate the connection between polynomial growth and identity satisfaction in finitely generated monoids.

Proposed method

• Using insertion algorithms (e.g., Schensted, binary search tree insertion) to define monoids via combinatorial objects like Young tableaux and binary search trees.
• Representing words via pairs of binary trees (left and right) to analyze monoid identities.
• Applying homomorphisms to relate identities in the Baxter monoid to those in its homomorphic images: sylvester and #sylvester monoids.
• Using cancellativity properties (left/right) of sylvester and #sylvester monoids to constrain possible identities.
• Leveraging known identities in related monoids (e.g., Adian’s identity in the Chinese monoid) as comparison points.
• Performing case analysis on word length and symbol positions to eliminate candidate identities.

Experimental results

Research questions

• RQ1 Do the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy any non-trivial identities?
• RQ2 What is the shortest non-trivial identity satisfied by each of these monoids?
• RQ3 Does the plactic monoid satisfy any non-trivial identity?
• RQ4 Do the left and right patience sorting monoids satisfy any non-trivial identities?
• RQ5 Is there a non-trivial identity satisfied by the finite-rank plactic monoid placn for n ≥ 4?

Key findings

• The hypoplactic monoid satisfies the identity xyxy = yxyx, and this is the shortest non-trivial identity it satisfies.
• The sylvester monoid satisfies the identity xyxy = yxxy, which is its shortest non-trivial identity.
• The #sylvester monoid satisfies the identity yxyx = yxxy, which is its shortest non-trivial identity.
• The Baxter monoid satisfies the identity yxxyxy = yxyxxy, which is its shortest non-trivial identity.
• The stalactic and taiga monoids both satisfy the identity xyx = yxx, which is their shortest non-trivial identity.
• The plactic monoid and the left and right patience sorting monoids do not satisfy any non-trivial identities.