A theory of pictures for quasi-posets

Abstract

Abstract. The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, combinatorial (e.g. since semi-standard tableaux can be viewed as double quasi-posets) and topological (quasi-posets identify with finite topologies) lead to extend the theory to quasi-posets. This is the object of the present article. IntroductionThe theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as preorder diagrams and tableaux. The theory captures for example the Robinson-Schensted (RS) correspondence or the Littlewood-Richardson formula, as already shown by Zelevinsky in the seminal article [20]. Recently, the theory was extended to double posets (pairs of orders coexisting on a given finite set -hereafter, "order" means "partial order"; an order on X defines a poset structure on X) and developed from the point of view of combinatorial Hopf algebras which led to new advances in the field [16, 8, 9, 10].In applications, a fundamental property that has not been featured enough, is that often pictures carry themselves implicitly a double poset structure. A typical example is given by standard Young tableaux, which can be put in bijection with certain pictures (this is one of the nicest way in which their appearance in the RS correspondence can be explained [20]) and carry simultaneously a poset structure (induced by their embeddings into N × N equipped with the coordinate-wise partial order) and a total order (the one induced by the integer labelling of the entries of the tableaux).However, objects such as tableaux with repeated entries, such as semistandard tableaux, although essential, do not fit into this framework. They should actually be thought of instead as double quasi-posets (pairs of preorders on a given finite set): the first preorder is the same than for standard tableaux (it is an order), but the labelling by (possibly repeated) integers is naturally captured by a preorder on the entries of the tableau (the one for which two entries are equivalent if they have the same label and else are ordered according to their labels).Besides the fact that these ideas lead naturally to new results and structures on preorders, other observations and motivations have led us to develop on systematic bases in the present article a theory of pictures for quasi-posets. Let us point out in particular recent developments (motivated by applications to multiple zeta values, Rota-Baxter algebras, stochastic integrals... [4,2,3]) that extend to surjections [18,17,14,13] the theory of combinatorial Hopf algebra structures on permutations [15,7]. New results on surjections will be obtained in the last section of the article.Lastly, let us mention our previous works on finite topologies (equivalent to quasi-posets) [11,12] (see also [5, 6] for recent developments) which featured the two products defined on finite topologies by disjoint unio...

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

“…The definition of quasi-monotone partitions below is also inspired independently by similar constructions on quasi-posets and finite topologies related to quasi-symmetric functions [21].…”

Surjections and double posets

“…We denote by H W P P the vector space generated by isomorphism classes of weak plane posets and show below how definitions and results in [19,13] apply in this context (definitions and results relative to double posets are taken from [19]).…”

“…The Hopf algebra of double poset H DP is self-dual for this pairing. By Proposition 24 of [13]: Proposition 7. The Hopf algebra H W P P is a self-dual Hopf subalgebra of the Hopf algebra of double poset H DP .…”