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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 \(\{0, \pm 1, \pm 2\}\).

06A07 Combinatorics of partially ordered sets
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
Full Text: DOI
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