Abstracts of Recent Papers

Title and Abstract for 'JDT':

d-Complete Posets Generalize Young Diagrams for the Jeu de Taquin Property


The jeu de taquin process produced a standard Young tableau from a skew standard Young tableau by shifting its entries to the northwest. We generalize this process to posets: certain partial numberings of any poset are shifted upward. A poset is said to have the jeu de taquin property if the numberings resulting from this process do not depend upon certain choices made during the process. Young diagrams are the posets which underlie standard Young tableaux. These posets have the jeu de taquin property. d-Complete posets are posets which satisfy certain local structual conditions. They are mutual generalizations of Young diagrams, shifted Young diagrams, and rooted trees. We prove that all d-complete posets have the jeu de taquin property. The proof shows that each d-complete poset actually has the stronger "simultaneous" property; this may lead to an algebraic understanding of the main result. A partial converse is stated: "Non-overlapping" simultaneous posets are d-complete.

Title and Abstract for 'Classification':

Dynkin Diagram Classification
of Lambda-Minuscule Bruhat Lattices and
of d-Complete Posets

(Journal of Algebraic Combinatorics 9, 61-94.)

d-Complete posets are defined to be posets which satisfy certain local structural conditions. These posets play or conjecturally play several roles in algebraic combinatorics related to the notions of shapes, shifted shapes, plane partitions, and hook length posets. They also play several roles in Lie theory and algebraic geometry related to lambda-minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to lambda-minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams.

Title and Abstract for 'Minuscule':

Minuscule Elements of Weyl Groups, the Numbers Game,
and d-Complete Posets

(Journal of Algebra 213, 272-303.)

Certain posets associated to a restricted version of the numbers game of Mozes are shown to be distributive lattices. The posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties, which form the definition of d-complete poset. In representation theoretic language, the top "minuscule portions" of weight diagrams for integrable representations of simply laced Kac-Moody algebras are shown to be distributive lattices. These lattices form a certain family of intervals of weak Bruhat orders. These Bruhat lattices are useful in studying reduced decompositions of lambda-minuscule elements of Weyl groups and their associated Schubert varieties. The d-complete posets have recently been proved to possess both the hook length and the jeu de taquin properties.

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