Binary Search Tree insertion, the Hypoplactic insertion, and Dual Graded Graphs

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-Kash…

Dual graded graphs for Kac-Moody algebras

Motivated by affine Schubert calculus, we construct a family of dual graded graphs $(Γ_s,Γ_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. The graded graphs have the Weyl group $W$ of $\g(A)$ as vertex set and are labeled versions of the strong and weak orders of $W$ respectively. Using a construction of Lusztig for quivers with an admissible automorphism, we define folded insertion for a Kac-Moody algebra and obtain Sagan-Worley shifted insertion from Robinson-Schensted insertion as a special case. Drawing on work of Stembridge, we analyze the induced subgraphs of $(Γ_s,Γ_w)$ which are distri…

Dual Filtered Graphs

We define a K-theoretic analogue of Fomin's dual graded graphs, which we call dual filtered graphs. The key formula in the definition is DU-UD= D + I. Our major examples are K-theoretic analogues of Young's lattice, of shifted Young's lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Mobius construction. The Pieri construc…

On the Young-Fibonacci insertion algorithm

This work is concerned with some properties of the Young-Fibonacci insertion algorithm and its relation with Fomin's growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebra associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are \emph{conveniently} interpreted as chains in the Young-Fibonacci lattice. A property of Killpatrick's evacuation is given a simpler proof, but this evacuat…

非平面二叉树上的霍普夫代数

A Hopf algebra on nonplanar binary forests

本文研究非平面二叉树生成的多项式代数，通过定义禁止同时切断任意顶点两个子节点的余乘积，构建霍普夫代数结构，并揭示其与布伦德霍普夫代数及预李运算的联系。