On the arithmetic of weight spaces in the root lattice
HTML articles powered by AMS MathViewer
- by Robert W. Deckhart PDF
- Proc. Amer. Math. Soc. 106 (1989), 81-86 Request permission
Abstract:
Using the notion of $i$-blocks in a weight space of the positive cone of the root lattice we obtain, after suitable reduction to nontrivial cases, a surprising strongly inductive formula for Kostant’s partition function in terms of exactly 3 lower weight spaces. The proof is elementary, using arithmetic properties of $i$-blocks from [2], and avoiding any character formulas.References
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Robert W. Deckhart, The Kostant cone and the combinatorial structure of the root lattice, J. Algebra 122 (1989), no. 1, 81–97. MR 994936, DOI 10.1016/0021-8693(89)90238-X —, Modular maximal vectors and the Kostant cone, J. Alg., to appear. —, ${A_n}$ modular maximal vectors, I: Bases, rank 2 families, and an introduction to products, Comm. Alg., (to appear).
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- James E. Humphreys, Ordinary and modular representations of Chevalley groups, Lecture Notes in Mathematics, Vol. 528, Springer-Verlag, Berlin-New York, 1976. MR 0453884
- R. V. Moody and J. Patera, Fast recursion formula for weight multiplicities, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 237–242. MR 656202, DOI 10.1090/S0273-0979-1982-15021-2
- R. V. Moody and J. Patera, Characters of elements of finite order in Lie groups, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 359–383. MR 752042, DOI 10.1137/0605037
- A. Pianzola, On the regularity and rationality of certain elements of finite order in Lie groups, J. Reine Angew. Math. 377 (1987), 40–48. MR 887398, DOI 10.1515/crll.1987.377.40
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 106 (1989), 81-86
- MSC: Primary 17B35; Secondary 17B20
- DOI: https://doi.org/10.1090/S0002-9939-1989-0967484-7
- MathSciNet review: 967484