Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.13445

Involutions of (twisted) diagram monoids

James East·P. A. Azeef Muhammed
View original source ↗Read source document ↗Export metadata ↓

Abstract

We classify the involutions of all of the most well-studied diagram monoids -- namely the partition, planar partition, partial Brauer, Motzkin, Brauer and Temperley--Lieb monoids -- and characterise those that give rise to star-regular or regular star-monoid structures. We then complete the same program for the associated twisted diagram monoids, with respect to both the canonical float-counting twisting, and the recently-discovered rank-based twisting. This necessitates developing a general theory of involutions of twisted products. Some of our results were quite unexpected. For example, a Brauer monoid is star-regular with respect to many of its involutions, but only a regular star-monoid for one of them. We will also see that twisted diagram monoids over the integers are always star-regular, thereby providing new and very natural examples of star-regular monoids. Along the way, we also obtain (by necessity) a number of results of independent interest; specifically, we classify the automorphisms of the Motzkin and Temperley--Lieb monoids (and hence also of the planar partition monoids), and we show that all of our diagram monoids generically have trivial centre.

Explore related subjects

math.RAmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

James East, P. A. Azeef Muhammed. 2026-07-15. Involutions of (twisted) diagram monoids. https://arxiv.org/abs/2607.13445

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Normal Quaternionic Matrices and Finitely Generated Witt Rings

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix satisfying a small additional condition that particularly holds for Witt rings of fields. Via computational search, we find all these matrices for $n$ up to $7$. This verifies that all Witt rings of fields with up to $128$ square classes are of elementary type.

math.RA↗
arXiv2026

Graded differential polynomial rings

We study differential polynomial rings $R[t;δ]$ over $Γ$-graded rings, where $Γ$ is an arbitrary group. We show that $R[t;δ]$ admits a $Γ$-grading compatible with that of $R$ if and only if $δ$ is a $γ$-derivation for some $γ\in C_Γ(Γ_R)$, and that this grading is unique once $°(t)=γ$ is fixed; if $δ\neq0$, then $γ$ is itself uniquely determined by $δ$. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;δ])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to $δ$-gr-simplicity of $R$ and $γ$-outerness of $δ$, extending Jordan's simplicity criterion to the graded setting. We further show that $R[t;δ]$ is gr-prime if and only if $R$ is $δ$-gr-prime, and that gr-Noetherianity of $R$ passes to $R[t;δ]$, recovering a graded Hilbert basis theorem as a special case. When $Γ$ is abelian, gr-simplicity and gr-primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to $R[t;δ]$ via a compatible idempotent is again a graded differential polynomial ring.

math.RA↗
arXiv2026

Affinization of algebraic structures: Poisson algebras

An affinization of the notion of a Poisson algebra is presented. This is termed a Poisson affgebra and consists of an affine space together with an associative bi-affine multiplication and a bi-affine Lie bracket that acts as an affine derivation for the associative product. The constructive relation between Poisson affgebras and Poisson algebras is described and several low-dimensional examples are studied in detail.

math.RA↗