On the Dowling and Rhodes lattices and wreath products

Stuart Margolis; John Rhodes; Pedro V. Silva

doi:10.1016/j.jalgebra.2021.02.018

On the Dowling and Rhodes lattices and wreath products

Stuart Margolis, John Rhodes, Pedro V. Silva

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of boolean representable simplicial complexes. This turns out to be the direct sum of a uniform matroid with the lift matroid of the complete gain link graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar gain graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups.

Original language	English
Pages (from-to)	55-91
Number of pages	37
Journal	Journal of Algebra
Volume	578
DOIs	https://doi.org/10.1016/j.jalgebra.2021.02.018
State	Published - 15 Jul 2021

Bibliographical note

Publisher Copyright:
© 2021 Elsevier Inc.

Funding

The first author acknowledges support from the Binational Science Foundation (BSF) of the United States and Israel, grant number 2012080. The second author acknowledges support from the Simons Foundation (Simons Travel Grant Number 313548). The third author was partially supported by CMUP, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P. under the project with reference UIDB/00144/2020.

Funders	Funder number
United States and Israel	2012080
Simons Foundation	313548
United States-Israel Binational Science Foundation
Fundação para a Ciência e a Tecnologia	UIDB/00144/2020
Centro de Matemática Universidade do Porto

Keywords

Boolean representable simplicial complex
Brandt groupoid
Dowling lattice
Lift matroid
Matroid
Rhodes lattice
Wreath product

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10.1016/j.jalgebra.2021.02.018

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abstract = "Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of boolean representable simplicial complexes. This turns out to be the direct sum of a uniform matroid with the lift matroid of the complete gain link graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar gain graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups.",

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AB - Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of boolean representable simplicial complexes. This turns out to be the direct sum of a uniform matroid with the lift matroid of the complete gain link graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar gain graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups.

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On the Dowling and Rhodes lattices and wreath products

Abstract

Bibliographical note

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Keywords

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