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On the spanning vectors of Lusztig cones

Author: Robert Bédard
Journal: Represent. Theory 4 (2000), 306-329
MSC (2000): Primary 16G20, 16G70, 17B37
Published electronically: July 31, 2000
MathSciNet review: 1773864
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Abstract: For each reduced expression ${\mathbf i}$ of the longest element $w_0$ of the Weyl group $W$ of a Dynkin diagram $\Delta$ of type $A$, $D$ or $E$, Lusztig defined a cone ${\mathcal C}_{\mathbf i}$ such that there corresponds a monomial in the quantized enveloping algebra ${\mathbf U}$ of $\Delta$ to each element of ${\mathcal C}_{\mathbf i}$ and he asked under what circumstances these monomials belong to the canonical basis of ${\mathbf U}$. In this paper, we consider the case where ${\mathbf i}$ is a reduced expression adapted to a quiver $\Omega$ whose graph is $\Delta$ and we describe ${\mathcal C}_{\mathbf i}$ as the set of non-negative integral combination of spanning vectors. These spanning vectors are themselves described by using the Auslander-Reiten quiver of $\Omega$and homological algebra.

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  • 1. M. Auslander, Relations for Grothendieck groups of artin algebras., Proc. Amer. Math. Soc. 91 (1984), 336-340. MR 85e:16038
  • 2. M. Auslander, I. Reiten and S.O. Smalø, Representation theory of artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge Univ. Press, Cambridge, 1995. MR 98e:16011; MR 96c:16015
  • 3. R. Bédard, On commutation classes of reduced words in Weyl groups, European J. Combin. 20 (1999), 483-505. CMP 99:16
  • 4. I.N. Bernstein, I.M. Gelfand and V.A. Ponomarev, Coxeter functors and Gabriel's theorem, Russian Math. Surveys 28 (1973), 17-32.
  • 5. R.W. Carter and R.J. Marsh, Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type $A_4$, forthcoming paper 1998.
  • 6. P. Gabriel, Unzerlegbare darstellungen. I, Manuscripta Math. 6 (1972), 71-103. MR 48:11212
  • 7. P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, Lecture Notes in Math. 831, Springer-Verlag, Berlin-Heidelberg-New York, 1980, pp. 1-71. MR 82i:16030
  • 8. M. Kashiwara, On crystal bases of the $Q$-analogue of the universal enveloping algebras, Duke Math. J. 63 (1991), 465-516. MR 93b:17045
  • 9. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498. MR 90m:17023
  • 10. G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Common trends in mathematics and quantum field theories (T. Eguchi et. als., eds), Progr. Theoret. Phys. Suppl., 102, 1990, 175-201. MR 93g:17019
  • 11. G. Lusztig, Tight monomials in quantized enveloping algebras, Quantum deformations of algebras and their representations (A. Joseph and S. Shnider, eds.), Israel Math. Conf. Proc 7 1993, pp. 117-132. MR 95i:17016
  • 12. R.J. Marsh, More tight monomials in quantized enveloping algebras, J. Algebra 204 (1998), 711-732. MR 99j:17023
  • 13. R.J. Marsh, The Lusztig cones of a quantized enveloping algebra of type $A$, preprint, 1998.
  • 14. R.J. Marsh, Rectangle diagrams for the Lusztig cones of quantized enveloping algebras of type $A$, preprint, 1998.
  • 15. C.M. Ringel, Tame algebras (On algorithms for solving vector space problems. II), Lecture Notes in Math, 831, Springer-Verlag, Berlin-Heidelberg-New York, 1980, pp. 137-287. MR 82j:16056
  • 16. C.M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 87f:16027
  • 17. J. Tits, Le problème des mots dans les groupes de Coxeter, Symposia Mathematica (INDAM, Rome 1967/68), Academic Press, London 1969, vol. 1, pp. 175-185. MR 40:7339

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Additional Information

Robert Bédard
Affiliation: Département de Mathématiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada

Received by editor(s): December 2, 1999
Received by editor(s) in revised form: May 27, 2000
Published electronically: July 31, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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