On dominance and minuscule Weyl group elements
Qëndrim R. Gashi
and Travis Schedler
DOI: 10.1007/s10801-010-0248-2
Abstract
Fix a Dynkin graph and let λ be a coweight. When does there exist an element w of the corresponding Weyl group such that w is λ -minuscule and w( λ ) is dominant? We answer this question for general Coxeter groups. We express and prove these results using a variant of Mozes' game of numbers.
Pages: 383–399
Keywords: keywords dominant weights; minuscule Weyl group elements; numbers game with a cutoff
Full Text: PDF
References
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2. Bourbaki, N.: Elements of Mathematics, Lie Groups and Lie Algebras, Chap. 4-6. Springer, Berlin Heidelberg (2002)
3. Broer, A.: Line bundles on the cotangent bundle of the flag variety. Invent. Math. 113, 1-20 (1993)
4. Donnelly, R.G., Eriksson, K.: The numbers game and Dynkin diagram classification results (2008).
5. Eriksson, K.: Convergence of Mozes' game of numbers. Linear Algebra Appl. 166, 151-165 (1992)
6. Eriksson, K.: Strongly convergent games and Coxeter groups, Ph.D. thesis, KTH, Stockholm (1993)
7. Eriksson, K.: Node firing games on graphs. In: Jerusalem Combinatorics '93: An International Conference in Combinatorics (May 9-17, 1993, Jerusalem, Israel), vol. 178, pp. 117-128. Am. Math. Soc., Providence (1994)
8. Eriksson, K.: Reachability is decidable in the numbers game. Theor. Comput. Sci. 131, 431-439 (1994)
9. Eriksson, K.: The numbers game and Coxeter groups. Discrete Math. 139, 155-166 (1995)
10. Eriksson, K.: Strong convergence and a game of numbers. Eur. J. Comb. 17(4), 379-390) (1996)
11. Gashi, Q.R.: The conjecture of Kottwitz and Rapoport in the case of split groups, Ph.D. thesis, The University of Chicago (2008)
12. Gashi, Q.R.: On a Conjecture of Kottwitz and Rapoport, Ann. Sci. d'E.N.S. (2010, to appear).
13. Gashi, Q.R., Schedler, T., Speyer, D.: Looping of the numbers game and the alcoved hypercube (2009).
14. Haines, T.J.: Test functions for Shimura varieties: the Drinfeld case. Duke Math. J. 106(1), 19-40 (2001)
15. Kottwitz, R.E.: On the Hodge-Newton decomposition for split groups. Int. Math. Res. Not. 26, 1433- 1447 (2003)
16. Kottwitz, R.E., Rapoport, M.: On the existence of F -isocrystals. Comment. Math. Helv. 78, 153-184 (2003)
17. Mazur, B.: Frobenius and the Hodge filtration. Bull. Am. Math. Soc. 78, 653-667 (1972)
18. Mazur, B.: Frobenius and the Hodge filtration (estimates). Ann. Math. (2) 98, 58-95 (1973)
19. Mozes, S.: Reflection processes on graphs and Weyl groups. J. Comb. Theory, Ser. A 53(1), 128-142 (1990)
20. Proctor, R.A.: Bruhat lattices, plane partition generating functions, and minuscule representations.
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