arXiv subjects

Marianne Johnson

Publications and source records attributed to Marianne Johnson.

30 records · Page 2Linked to original sources

arXiv2016

Pure Dimension and Projectivity of Tropical Polytopes

We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a close connection between pure dimension of tropical convex sets, and projectivity (in the sense of ring theory). These results lead to a geometric understanding of idempotency for tropical matrices. As well as their direct interest, our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra (in particular, ring theory) in tropical geometry.

math.RA↗

arXiv2016

NP-completeness in the gossip monoid

Gossip monoids form an algebraic model of networks with exclusive, transient connections in which nodes, when they form a connection, exchange all known information. They also arise naturally in pure mathematics, as the monoids generated by the set of all equivalence relations on a given finite set under relational composition. We prove that a number of important decision problems for these monoids (including the membership problem, and hence the problem of deciding whether a given state of knowledge can arise in a network of the kind under consideration) are NP-complete. As well as being of interest in their own right, these results shed light on the apparent difficulty of establishing the cardinalities of the gossip monoids: a problem which has attracted some attention in the last few years.

math.CO↗

arXiv2016

Face monoid actions and tropical hyperplane arrangements

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropical hyperplane arrangement, in terms of partial bijections which attain permanents of submatrices of a matrix which naturally encodes the arrangement.

math.CO↗

arXiv2014

Convexity of tropical polytopes

We study the relationship between min-plus, max-plus and Euclidean convexity for subsets of $\mathbb{R}^n$. We introduce a construction which associates to any max-plus convex set with compact projectivisation a canonical matrix called its dominator. The dominator is a Kleene star whose max-plus column space is the min-plus convex hull of the original set. We apply this to show that a set which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii) a Euclidean polytope must also be the third. In particular, these results answer a question of Sergeev, Schneider and Butkovic and show that row spaces of tropical Kleene star matrices are exactly the "polytropes" studied by Joswig and Kulas.

math.MG↗

arXiv2013

Exact rings and semirings

We introduce and study an abstract class of semirings, which we call exact semirings, defined by a Hahn-Banach-type separation property on modules. Our motivation comes from the tropical semiring, and in particular a desire to understand the often surprising extent to which it behaves like a field. The definition of exactness abstracts an elementary property of fields and the tropical semiring, which we believe is fundamental to explaining this similarity. The class of exact semirings turns out to include many other important examples of both rings (proper quotients of principal ideal domains, matrix rings and finite group rings over these and over fields), and semirings (the Boolean semiring, generalisations of the tropical semiring, matrix semirings and group semirings over these).

math.RA↗

arXiv2012

Tropical matrix groups

We study the subgroup structure of the semigroup of finitary tropical matrices under multiplication. We show that every maximal subgroup is isomorphic to the full linear automorphism group of a related tropical polytope, and that each of these groups is the direct product of the real numbers with a finite group. We also show that there is a natural and canonical embedding of each full rank maximal subgroup into the group of units of the semigroup of matrices over the tropical semiring with minus infinity. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space, and that every full rank subgroup has a common eigenvector.

math.GR↗

arXiv2012

Idempotent tropical matrices and finite metric spaces

There is a well known correspondence between the triangle inequality for a distance function on a finite set, and idempotency of an associated matrix over the tropical semiring. Recent research has shed new light on the structure (algebraic, combinatorial and geometric) of tropical idempotents, and in this paper we explore the consequences of this for the metric geometry of tropical polytopes. We prove, for example, that every n-point metric space is realised by the Hilbert projective metric on the vertices of a pure n-dimensional tropical polytope in tropical n-space. More generally, every n-point asymmetric distance function is realised by a residuation operator on the vertices of such a polytope. In the symmetric case, we show that the maximal group of tropical matrices containing the idempotent associated to a metric space is a direct product of the real numbers with the isometry group of the space; it follows that every direct product of a finite group with the real numbers arises as a maximal subgroup of a sufficiently large finitary full tropical matrix semigroup. In the process we also prove some new results about tropical idempotent matrices, and note some semigroup-theoretic consequences which may be of independent interest.

math.RA↗

arXiv2012

Green's J-order and the rank of tropical matrices

We study Green's J-order and J-equivalence for the semigroup of all n-by-n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between tropical convex sets. We establish connections between the J-order, isometries of tropical convex sets, and various notions of rank for tropical matrices. We also study the relationship between the relations J and D; Izhakian and Margolis have observed that $D \neq J$ for the semigroup of all 3-by-3 matrices over the tropical semiring with $-\infty$, but in contrast, we show that $D = J$ for all full matrix semigroups over the finitary tropical semiring.

math.RA↗

arXiv2011

Lie powers of the natural module for GL(2,K)

In recent work of R. M. Bryant and the second author a (partial) modular analogue of Klyachko's 1974 result on Lie powers of the natural $\rm{GL}(n,K)$ was presented. There is was shown that nearly all of the indecomposable summands of the $r$th tensor power also occur up to isomorphism as summands of the $r$th Lie power provided that $r\neq p^m$ and $r \neq 2p^m$, where $p$ is the characteristic of $K$. In the current paper we restrict attention to ${\rm GL}(2,K)$ and consider the missing cases where $r = p^m$ and $r = 2p^m$. In particular, we prove that the indecomposable summand of the $r$th tensor power of the natural module with highest weight $(r-1,1)$ is a summand of the $r$th Lie power if and only if $r$ is a not power of $p$.

math.RT↗

arXiv2009

Periodicity of Adams operations on the Green ring of a finite group

The Adams operations $ψ_Λ^n$ and $ψ_S^n$ on the Green ring of a group $G$ over a field $K$ provide a framework for the study of the exterior powers and symmetric powers of $KG$-modules. When $G$ is finite and $K$ has prime characteristic $p$ we show that $ψ_Λ^n$ and $ψ_S^n$ are periodic in $n$ if and only if the Sylow $p$-subgroups of $G$ are cyclic. In the case where $G$ is a cyclic $p$-group we find the minimum periods and use recent work of Symonds to express $ψ_S^n$ in terms of $ψ_Λ^n$.

math.RT↗

arXiv2009

Adams operations on the Green ring of a cyclic group of prime-power order

We consider the Green ring $R_{KC}$ for a cyclic $p$-group $C$ over a field $K$ of prime characteristic $p$ and determine the Adams operations $ψ^n$ in the case where $n$ is not divisible by $p$. This gives information on the decomposition into indecomposables of exterior powers and symmetric powers of $KC$-modules.

math.RT↗

arXiv2009

Multiplicative structure of 2x2 tropical matrices

We study the algebraic structure of the semigroup of all $2 \times 2$ tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups of this semigroup.

math.GR↗