Up- and Down-Operators on Young's Lattice

The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.

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.

Cohomology rings of the plactic monoid algebra via a Gröbner–Shirshov basis

In this paper, we calculate the cohomology ring [Formula: see text] and the Hochschild cohomology ring of the plactic monoid algebra [Formula: see text] via the Anick resolution using a Gröbner–Shirshov basis.

Finite convergent presentation of plactic monoid for type C

We give an explicit presentation for the plactic monoid for type C using admissible column generators. Thanks to the combinatorial properties of symplectic tableaux, we prove that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for type C satisfy homological finiteness properties.

Algebraic and combinatorial structures on Baxter permutations

International audience We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (\emphi.e. Baxter permutations, pairs of twin binary trees, \emphetc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005]. Nous proposons une nouvelle construction d'une sous-algèbre de Hopf de l'algèbre de Hopf des fonctions quasi-symétriques libres dont les bases sont indexées par les objets de la famille combinatoire de Baxter (\emphi.e. permutations de Baxter, couples d'arbres binaires jumeaux, \emphetc.). Cette construction repose sur la définition du mono\"ıde de Baxter, analogue du mono\"ıde plaxique et du mono\"ıde sylvestre, et d'un algorithme d'insertion analogue à l'algorithme de Robinson-Schensted. Les propriétés algébriques de cette algèbre de Hopf sont étudiées. Cette algèbre de Hopf est apparue pour la première fois dans le travail de Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005].