Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of ‘plactic-like’ monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-n case. For the hypoplactic monoid, this is n−1; for the sylvester and taiga monoids, at least n−1 and at most n; for the stalactic monoid, 3 (except for ranks 1 and 2, when it is respectively 0 and 1); for the plactic monoid, at least n−1 and at most 2n−3. The current state of knowledge, including new and previously-known results, is summarized in a table.authorsversionpublishe

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

The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep connections to several areas of mathematics, in particular, to the theory of symmetric functions. An active research topic is the identities satisfied by the plactic monoids of finite rank. It is known that there is no “global" identity satisfied by the plactic monoid of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic monoid (the monoid of quasi-ribbon tableaux), sylvester monoid (the monoid of binary search trees) and Baxter monoid (the monoid of pairs of twin binary search trees), satisfy global identities, and the shortest identities have been characterized. In this thesis, we present new results on the identities satisfied by the hypoplactic, sylvester, #-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, thus showing that the identity checking problems of these monoids are in the complexity class P, and prove that the varieties generated by these monoids have finite axiomatic rank, by giving a finite basis for them. We also give a subdirect representation ofmultihomogeneous monoids by finite subdirectly irreducible Rees factor monoids, thus showing that they are residually finite.O ubíquo monóide plático, também conhecido como o monóide dos diagramas de Young, tem ligações profundas a várias áreas de Matemática, em particular à teoria das funções simétricas. Um tópico de pesquisa ativo é o das identidades satisfeitas pelos monóides pláticos de característica finita. Sabe-se que não existe nenhuma identidade “global” satisfeita pelos monóides pláticos de cada característica. Em contraste, sabe-se que monóides ligados ao monóide plático, como o monóide hipoplático (o monóide dos diagramas quasifita), o monóide silvestre (o monóide de árvores de busca binárias) e o monóide de Baxter (o monóide de pares de árvores de busca binária gémeas), satisfazem identidades globais, e as identidades mais curtas já foram caracterizadas. Nesta tese, apresentamos novos resultados acerca das identidades satisfeitas pelos monóides hipopláticos, silvestres, silvestres-# e de Baxter. Mostramos como mergulhar estes monóides, de característica estritamente maior que 2, num produto direto de cópias do monóide correspondente de característica 2. Confirmamos assim que todos os monóides da mesma família, de característica maior ou igual a 2, satisfazem exatamente as mesmas identidades. A seguir, damos uma caracterização completa dessas identidades, mostrando assim que os problemas de verificação de identidades destes monóides estão na classe de complexidade P, e provamos que as variedades geradas por estes monóides têm característica axiomática finita, ao apresentar uma base finita para elas. Também damos uma representação subdireta de monóides multihomogéneos por monóides fatores de Rees finitos e subdiretamente irredutíveis, mostrando assim que são residualmente finitos

Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, and related monoids

fellowship (IF/01622/2013/CP1161/CT0001). project PTDC/MHC-FIL/2583/2014.The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid of Young tableaux) and the sylvester monoid (the monoid of binary search trees), connected components consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper discusses new results on the diameters of connected components of the cyclic shift graphs of the finite-rank analogues of these monoids, showing that the maximum diameter of a connected component is dependent only on the rank. The proof techniques are explained in the case of the sylvester monoid.authorsversionpublishe