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Alan J. Cain

Publications and source records attributed to Alan J. Cain.

At least 19 recordsLinked to original sources
arXiv2025

Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions

Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a combinatorial study of these objects. We explain a previously-observed isomorphism of components of the quasi-crystal graph, and provide an explicit description using a new combinatorial structure called a quasi-array. Then two conjectures of Maas-Gariépy on the interaction of fundamental quasi-symmetric functions and Schur functions and on the arrangement of quasi-crystal components within crystal components are answered, the former positively, the latter negatively.

math.CO↗
arXiv2023

Commutative nilpotent transformation semigroups

Cameron, et al. determined the maximum size of a null subsemigroup of the full transformation semigroup $\mathcal{T}(X)$ on a finite set $X$ and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when $X$ is finite, the maximum order of a commutative nilpotent subsemigroup of $\mathcal{T}(X)$ is equal to the maximum order of a null subsemigroup of $\mathcal{T}(X)$ and we prove that the largest commutative nilpotent subsemigroups of $\mathcal{T}(X)$ are the null semigroups previoulsy characterized by Cameron, et al..

math.GR↗
arXiv2023

A local characterization of quasi-crystal graphs

It is provided a local characterization of quasi-crystal graphs, by presenting a set of local axioms, similar to the ones introduced by Stembridge for crystal graphs of simply-laced root systems. It is also shown that quasi-crystal graphs satisfying these axioms are closed under the tensor product recently introduced by Cain, Guilherme and Malheiro. It is deduced that each connected component of such a graph has a unique highest weight element, whose weight is a composition, and it is isomorphic to a quasi-crystal graph of semistandard quasi-ribbon tableaux.

math.CO↗
arXiv2022

Quasi-crystals for arbitrary root systems and associated generalizations of the hypoplactic monoid

The hypoplactic monoid was introduced by Krob and Thibon through a presentation and through quasi-ribbon tableaux and an insertion algorithm. Just as Kashiwara crystals enriched the structure of the plactic monoid and allowed its generalization, the first and third authors of this paper introduced a construction of the hypoplactic monoid by identifying vertices in a quasi-crystal graph derived from the crystal graph associated to the general linear Lie algebra. Although this construction is based on Kashiwara's work, it cannot be extended to other crystal graphs, since the analogous quasi-Kashiwara operators on words do not admit a recursive definition. This paper addresses these issues. A general notion of quasi-crystal is introduced, followed by a study of its properties and relation with crystals. A combinatorial study of quasi-crystals is then made by associating a quasi-crystal graph to each quasi-crystal, which for the class of seminormal quasi-crystals results in a one-to-one correspondence. To model the binary operation of the hypoplactic monoid by quasi-crystals, a notion of quasi-tensor product of quasi-crystals is introduced, along with a combinatorial way of computing it similar to the signature rule for the tensor product of crystals. This framework allows the generalization of the classical hypoplactic monoid to a family of hypoplactic monoids associated to the various simple Lie algebras. The quasi-crystal structure is then used to establish algebraic properties of the hypoplactic monoid associated to the symplectic Lie algebra.

math.CO↗
arXiv2021

Identities and bases in the hypoplactic monoid

This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it.

math.RA↗
arXiv2021

Representations and identities of plactic-like monoids

We exhibit faithful representations of the hypoplactic, stalactic, taiga, sylvester, Baxter and right patience sorting monoids of each finite rank as monoids of upper triangular matrices over any semiring from a large class including the tropical semiring and fields of characteristic $0$. By analysing the image of these representations, we show that the variety generated by a single hypoplactic (respectively, stalactic or taiga) monoid of rank at least $2$ coincides with the variety generated by the natural numbers together with a fixed finite monoid $\mathcal{H}$ (respectively, $\mathcal{F}$) forming a proper subvariety of the variety generated by the plactic monoid of rank $2$.

math.RA↗
arXiv2021

Identities and bases in the sylvester and Baxter monoids

This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.

math.RA↗
arXiv2021

Reconstructing Young Tableaux

This paper completely characterizes the standard Young tableaux that can be reconstructed from their sets or multisets of $1$-minors. In particular, any standard Young tableau with at least $5$ entries can be reconstructed from its set of $1$-minors.

math.CO↗
arXiv2019

Two applications of monoid actions to cross-sections

Using a construction that builds a monoid from a monoid action, this paper exhibits an example of a direct product of monoids that admits a prefix-closed regular cross-section, but one of whose factors does not admit a regular cross-section; this answers negatively an open question from the theory of Markov monoids. The same construction is then used to show that for any full trios $\mathfrak{C}$ and $\mathfrak{D}$ such that $\mathfrak{C}$ is not a subclass of $\mathfrak{D}$, there is a monoid with a cross-section in $\mathfrak{C}$ but no cross-section in $\mathfrak{D}$.

math.GR↗
arXiv2019

Context-free word problem semigroups

This paper studies the classes of semigoups and monoids with context-free and deterministic context-free word problem. First, some examples are exhibited to clarify the relationship between these classes and their connection with the notions of word-hyperbolicity and automaticity. Second, a study is made of whether these classes are closed under applying certain semigroup constructions, including direct products and free products, or under regressing from the results of such constructions to the original semigroup(s) or monoid(s).

math.GR↗
arXiv2019

Crystal monoids \& crystal bases: rewriting systems and biautomatic structures for plactic monoids of types $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, and $G_2$

The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$ (corresponding to the $q$-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time.

math.GR↗
arXiv2018

A Language Hierarchy of Binary Relations

Motivated by the study of word problems of monoids, we explore two ways of viewing binary relations on $A^*$ as languages. We exhibit a hierarchy of classes of binary relations on $A^*$, according to the class of languages the relation belongs to and the chosen viewpoint. We give examples of word problems of monoids distinguishing the various classes.

cs.FL↗
arXiv2018

Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids

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.

math.CO↗
arXiv2018

Conjugacy in Patience Sorting monoids

The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the $\mathrm{rps}$ monoid and the $\mathrm{lps}$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the $\mathrm{rps}$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the $\mathrm{rps}$ monoid. Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the $\mathrm{rps}$ and the $\mathrm{lps}$ monoids.

math.CO↗
arXiv2018

Visual thinking and simplicity in proof

This paper studies how spatial thinking interacts with simplicity in [informal] proof, by analysing a set of example proofs mainly concerned with Ferrers diagrams (visual representations of partitions of integers, and comparing them to proofs that do not use spatial thinking. The analysis shows that using diagrams and spatial thinking can contribute to simplicity by (for example) avoiding technical calculations, division into cases, and induction, and creating a more surveyable and explanatory proof (both of which are connected to simplicity). In response to one part of Hilbert's 24th Problem, the area between two proofs is explored in one example, showing that between a proof that uses spatial reasoning and one that does not, there is a proof that is less simple than either.

math.HO↗
arXiv2018

Crystals and trees: quasi-Kashiwara operators, monoids of binary trees, and Robinson--Schensted-type correspondences

Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the $q$-analogue of the general linear Lie algebra $\mathfrak{gl}_{n}$, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence. The authors previously constructed an analogous `quasi-crystal' structure for the related hypoplactic monoid (the monoid of quasi-ribbon tableaux), which has similarly neat combinatorial properties. This paper constructs an analogous `crystal-type' structure for the sylvester and Baxter monoids (the monoids of binary search trees and pairs of twin binary search trees, respectively). Both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph. The interaction of the structure with the sylvester and Baxter versions of the Robinson-Schensted-Knuth correspondence is studied. The structure is then applied to prove results on the number of factorizations of elements of these monoids, and to prove that both monoids satisfy non-trivial identities.

math.CO↗
arXiv2018

Combinatorics of patience sorting monoids

This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words and pairs of tableaux. We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number $n$, the $n$-th Bell number is given by the number of standard tableaux containing $n$ symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a `hook length formula' that gives the number of standard tableaux of a given shape and deduce some consequences.

math.CO↗