Overview of Recent Research

Overview of Recent Research Concerning
the Hook Length and Jeu de Taquin
Properties and d-Complete Posets

( Recent Research Table of Contents points to
further explanations and to pictures of d-complete posets.)

Inspired by certain representation theoretic considerations, we have discovered many new partially ordered sets (posets) which possess both the rare jeu de taquin and the rare hook length properties. The hook length results are joint work with Dale Peterson of the University of British Columbia. The statements of Theorems 1 and 2 appear on my home page.

The definition of the hook length property for posets is a bit technical. Nonetheless, Theorem 2 (all d-complete posets have the hook length property) can be viewed in the following light. One of the earliest nineteenth century classical number theory identities was actually found in the late eighteenth century by Euler, viz. his famous infinite sum = infinite product identity for the sequence of partition counts p(n) of non-negative integers. In 1970 Richard Stanley defined a more general poset setting in which to view this kind of identity, and proved that such sum = product identities exist for the posets defined by 'shapes' (i.e. Young or Ferrers diagrams). Theorem 2 extends Stanley's result to a larger class of posets which arises naturally in the study of representations of Kac-Moody algebras. Hopefully this joint work with Peterson can be regarded as another entry in the list of interesting overlaps between contemporary representation theory and classical number theory which includes the results of Macdonald (Dedekind eta function), Lepowsky-Milne (Rogers-Ramanujan identities), and Kac-Wakimoto (extensions of identities of Jacobi and Gauss).

In 1977 G.P. Thomas proved that shape posets also possess Schutzenberger's jeu de taquin label sliding property. This fact and related procedures (Robinson-Schensted, promotion/evacuation, etc.) play many interesting roles in combinatorial representation theory, such as providing nice proofs of the Littlewood-Richardson rule.

These two results were extended to 'shifted shape' posets 14 to 20 years ago by Gansner, Sagan, and Worley. Both shapes and shifted shapes can be viewed as constituting certain classes of posets. The jeu de taquin property can be defined in the context of arbitrary posets. It has been known that 'tree' posets have both the jeu de taquin and the hook length properties.

(Move to Basic Poset Stuff for precise definitions of shape, shifted shape, and tree posets, if needed.)

d-Complete posets are posets defined by the satisfaction of certain local structural conditions. It is easy to see that shapes, shifted shapes, and trees are d-complete posets. Hence Theorems 1 and 2 provide one large unifying class of posets which gives a uniform setting for all currently known infinite families of hook length or jeu de taquin posets.

Many years ago Dale Peterson found a less explicit version of the corollary in a different but closely related setting, viz. the enumeration of reduced decompositions of certain Kac-Moody Weyl group elements. Dale has studied certain upper 'thin' regions of weight diagrams for representations of Kac-Moody algebras which are indexed by Weyl group elements which Dale calls 'lambda-minuscule.' The following result can also be interpreted in the context of a 'numbers game.' It is a key result in the proof of Theorem 2, which also uses algebraic geometric and representation theoretic steps supplied by Dale Peterson.

Proposition 1. ['Minuscule' paper.] There is a certain equivalence between Peterson's lambda-minuscule Kac-Moody Weyl group elements and d-complete posets.

Lie theorists may wish to link to Lie Theoretic Comments at any time for the definition of Peterson's lambda-minuscule concept and its connection to the d-complete property, comments concerning listing and counting reduced decompositions of certain Coxeter group elements, and comments concerning the Lakshmibai-Seshadri basis results origins of the notion of d-complete posets.

Another useful preliminary result concerning d-complete posets is:

Proposition 2. ['Classification' paper.] 'Slant irrducible' d-complete posets can be completely classified and explicitly described with the use of Y-shaped Dynkin diagrams which are almost always of type A, type D, or of "general type E."

The global structure of a connected d-complete poset is that of a tree. Each slant irreducible component of a d-complete poset falls into one of 15 classes. The first two classes consist of shapes and of shifted shapes respectively. This proposition is used in our proof of Theorem 1, which proceeds by strengthening and extending Kimmo Eriksson's proof of the jeu de taquin property for shapes to the other 14 classes of slant irreducible d-complete posets.

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