Transformation representations of diagram monoids

Reinis Cirpons; James East; James D. Mitchell

Transformation representations of diagram monoids

Reinis Cirpons, James East, James D. Mitchell

Research output: Working paper › Preprint

Abstract

We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid 𝒫_n has degree 1 + [B(n+2)-B(n+1)+B(n)]/2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

Original language	English
Publisher	arXiv
Number of pages	35
Publication status	Published - 22 Nov 2024

Keywords

Diagram monoids
Transformation representations
Transformation degrees

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https://arxiv.org/abs/2411.14693v1Licence: CC BY

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Minimum transformation representations of diagram monoids
Cirpons, R., East, J. & Mitchell, J. D., 12 Mar 2026, In: International Mathematics Research Notices. 2026, 6, 36 p., rnag041.
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title = "Transformation representations of diagram monoids",

abstract = "We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid 𝒫n has degree 1 + [B(n+2)-B(n+1)+B(n)]/2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.",

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author = "Reinis Cirpons and James East and Mitchell, \{James D.\}",

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T1 - Transformation representations of diagram monoids

AU - Cirpons, Reinis

AU - East, James

AU - Mitchell, James D.

PY - 2024/11/22

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N2 - We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid 𝒫n has degree 1 + [B(n+2)-B(n+1)+B(n)]/2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

AB - We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid 𝒫n has degree 1 + [B(n+2)-B(n+1)+B(n)]/2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

KW - Diagram monoids

KW - Transformation representations

KW - Transformation degrees

M3 - Preprint

BT - Transformation representations of diagram monoids

PB - arXiv

ER -

Transformation representations of diagram monoids

Abstract

Keywords

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Research output

Minimum transformation representations of diagram monoids

Cite this