Dale R. Worley's Math Page

To combinatorialists, the bottom lines are the Yang-Baxter equation, the representation theory of the classical groups, and the Schensted algorithm. — Gian-Carlo Rota, “Bottom Lines”, Indiscrete Thoughts

Research Program

Developing a general theory of Schensted-type tableau insertion algorithms and their relationship to the representations of the classical groups.

Research Works

Note: The e-prints below are draft versions which may differ from the published papers. Only the published versions are definitive.

Constructing/analyzing differential distributed lattices [Wor2026e]
arXiv:2603.23741 (math.CO)

We restate a process presented by Stanley as a technique to prove that there exists exactly one d-differential distributive lattice for any positive integer d. This process can be trivially extended to apply to distributive finitary lattices that have a variety of differential poset structures. It can be viewed as an algorithm for constructing such lattices. Alternatively, it can be viewed as an algorithm for analyzing and characterizing such lattices. We show that the process can be used to prove properties of all weighted-differential lattices with positive weights. We present this with the hope that this approach can be used as the basis for a complete characterization of distributive lattices with a weighted-differential structure with positive weights.

An extension of Birkhoff's representation theorem to locally-finite distributive lattices [Wor2026c]
arXiv:2603.05841 (math.CO)
Presentation [Wor2026d]

Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. We present a simplified version of Stone's extension of this theorem to general distributive lattices. We then apply this formulation to locally finite distributive lattices to produce a novel representation theorem: The lattice is isomorphic to the order ideals of the poset of prime filters of the lattice whose symmetric difference from a particular ideal is finite.

On the combinatorics of tableaux — Classification of lattices underlying Schensted correspondences [Wor2026b]
Publication draft
Extended preprint arXiv:2511.07611 (math.CO)

The celebrated Robinson–Schensted algorithm and each of its variants that have attracted substantial attention can be constructed using Fomin's "growth diagram" construction from a modular lattice that is also a weighted-differential poset. We classify all such "Fomin" lattices that meet certain criteria; the main criterion is that the lattice is distributive. Intuitively, these criteria seem excessively strict, but all known Fomin lattices satisfy all of these criteria, with the sole exception of one family that is not even distributive, the Young–Fibonacci lattices and cartesian products involving them. We discover a new class of Fomin lattices, but unfortunately they cannot be used to construct Robinson--Schensted algorithms.

On the structure of modular lattices — Axioms for gluing [Wor2025b]
arXiv:2504.05507 (math.CO)

This paper explores alternative statements of the axioms for lattice gluing, focusing on lattices that are modular, locally finite, and have finite covers, but may have infinite height. We give a set of "maximal" axioms that maximize what can be immediately adduced about the structure of a valid gluing. We also give a set of "minimal" axioms that minimize what needs to be adduced to prove that a system of blocks is a valid gluing. This system appears to be novel in the literature. A distinctive feature of the minimal axioms is that they involve only relationships between elements of the skeleton which are within an interval [x∧y,x∨y] where either x and y cover x∧y or they are covered by x∨y. That is, they have a decidedly local scope, despite that the resulting sum lattice, being modular, has global structure, such as the diamond isomorphism theorem.

On the structure of modular lattices — Unique gluing and dissection [Wor2025a]
arXiv:2502.08934 (math.CO)

This work proves that the process of gluing finite lattices to form a larger lattice is bijective, that is each lattice is the glued sum of a unique system of finite lattices, provided the class of lattices is constrained to modular, locally-finite lattices with finite covers. The results of this work are not surprising given the prior literature, but this seems to be the first proof that the processes of gluing and dissection can be made inverses, and hence that gluing is bijective.

Extending Birkhoff's representation theorem to modular lattices [Wor2024c]
PDF

Goal: Construct a nice way to represent/classify modular lattices
Applications:

• Construct "tableaux" that nicely represent saturated chains in general modular lattices.
• Find/classify (weighted) differential structures on general modular lattices for use in RSK algorithms.

S-glued Sums of Lattices by Christian Herrmann [Herr1973-en]
arXiv:2409.10738 (math.CO)

My translation from the German of Christian Herrmann's S-verklebte Summen von Verbänden.

A survey of lattice properties: modular, Arguesian, linear, and distributive [Wor2024b]
arXiv:2403.19677 (math.HO)

This is a survey of characterizations and relationships between some properties of lattices, particularly the modular, Arguesian, linear, and distributive properties, but also some other related properties. The survey emphasizes finite and finitary lattices and deemphasizes complemented lattices.

A final section is a restatement of the open questions, which may prove to be a source of thesis problems.

On the combinatorics of tableaux — Graphical representation of insertion algorithms [Wor2023c]
arXiv:2306.11140 (math.CO)

Introduces "insertion diagrams" for representing the R-correspondences which are the add-in to growth diagrams that define different insertion algorithms. Includes a listing of many of the described insertion algorithms.

An extension of Birkhoff's Representation Theorem to infinite distributive lattices [Wor2023a]
arXiv:2303.04267 (math.CO)

Proof of an extension of Birkhoff's Representation Theorem to a class of infinite distributive lattices. Includes an alternative antecedent that is conditions on infinite ascending and descending chains.

A theory of shifted Young tableaux [Wor1984a]
PDF

This thesis develops a combinatorial theory of shifted Young tableaux that parallels the existing theory of (unshifted) Young tableaux. A shifted Young tableau is similar to an unshifted Young tableau, except that each successive row is indented one position rightward from the preceding row. The structure of the theory is modeled after that given by Schützenberger in the Strasbourg conference in 1976 (LNM 579), and contains the following elements: (1) a version of the Robinson-Schensted-Knuth correspondence which transforms sequences of numbers into pairs of shifted tableaux, (2) the Knuth relations, a set of elementary transformations on sequences of numbers which characterize the sets of sequences which produce the same first tableau under the insertion algorithm, (3) a combinatorial function similar to the one discovered by Greene that characterizes these sets of sequences, and (4) a version of Schützenberger's operation "jeu de taquin" under which a diagonal line of numbers is reduced to the first tableau of its corresponding pair. Several applications of the theory to symmetric functions and several open problems are discussed.

On the combinatorics of tableaux — A notebook of open problems

[Wor2026a]
Latest arXiv version: arXiv:2509.25446 (math.CO)
Current dynamic version: PDF

On the combinatorics of tableaux — A survey

Current draft: PDF

Cumulative Bibliography

Brief PDF

With abstracts PDF