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Let L be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of L with weight basis B. The supporting graph P of B is defined to be the directed graph whose vertices are the elements of B and whose colored edges describe the supports of the actions of the Chevalley generators on V . Four… (More)

- Robert G. Donnelly
- J. Comb. Theory, Ser. A
- 1999

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Combinatorial reasoning is used to show that those connected graphs… (More)

- Robert G. Donnelly, Scott J. Lewis, Robert Pervine
- Discrete Mathematics
- 2003

- Robert G. Donnelly
- Eur. J. Comb.
- 2008

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie… (More)

- Robert G. Donnelly, Scott J. Lewis, Robert Pervine
- Discrete Mathematics
- 2006

We further develop Eriksson’s theory of geometric representations of a Coxeter group with respect to certain possibly asymmetric bilinear forms, and we show how certain aspects of the geometry, though different from the standard (symmetric) case, can be fairly well behaved. In particular, we relate the finiteness of certain sets of roots to a combinatorial… (More)

The main results of this paper were found while addressing the question: what do the “nice” bases for the irreducible representations of semisimple Lie algebras look like? Using the Gelfand-Zetlin bases for the irreducible representations of gl(n,C) as our model, we take a combinatorial approach to this question by associating a certain kind of directed… (More)

This work-in-progress is intended as an exposition of the background material, results, and open problems of a particular poset theoretic study of Weyl characters and semisimple Lie algebra representations begun in the late 1970’s and early 1980’s in the work of Richard P. Stanley and Robert A. Proctor.

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Here, the edge amplitudes will be negative integers. Combinatorial… (More)