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The partial order of dominant weights. (English) Zbl 0916.06001
The weight lattice of a crystallographic root system is partially ordered by the rule that $$\lambda>\mu$$ if $$\lambda-\mu$$ is a nonnegative integer linear combination of positive roots. The paper under review is devoted to combinatorial properties of the subposet $$\Lambda^+$$ formed by the dominant weights. It is proved that $$\lambda$$ covers $$\mu$$ in $$\Lambda^+$$ only if $$\lambda-\mu$$ belongs to a distinguished subset of the positive roots. Also, if the root system is irreducible, it is proved that the Möbius function of $$\Lambda^+$$ takes on only the values $$\{0, \pm 1, \pm 2\}$$.

##### MSC:
 06A07 Combinatorics of partially ordered sets 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
coxeter
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##### References:
 [1] Bourbaki, N., Groupes et algèbres de Lie, (1981), Masson Paris [2] Bourbaki, N., Groupes et algèbres de Lie, (1975), Hermann Paris [3] Brylawski, T., The lattice of integer partitions, Discrete math., 6, 201-219, (1973) · Zbl 0283.06003 [4] Greene, C., A class of lattices with Möbius function ±1, 0, Europ. J. combin., 9, 225-240, (1988) · Zbl 0665.06007 [5] Humphreys, J.E., Introduction to Lie algebras and representation theory, (1972), Springer-Verlag Berlin/New York · Zbl 0254.17004 [6] Solomon, L., The Burnside algebra of a finite group, J. combin. theory, 2, 603-615, (1967) · Zbl 0183.03601 [7] Stanley, R.P., Enumerative combinatorics, (1986), Wadsworth & Brooks/Cole Monterey · Zbl 0608.05001 [8] Stembridge, J.R., A Maple package for root systems and finite Coxeter groups, manuscript, (1997)
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