Abstracts of Recent Papers

##
Title and Abstract for 'JDT':

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

###
(Preprint.)

`
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.
`