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Cones, crystals, and patterns. (English) Zbl 0908.17010
This work describes an interpretation of the Gelfand-Tsetlin patterns in terms of crystal graphs and generalizes such patterns to arbitrary complex semisimple algebraic groups. For each element of the crystal basis, the author gets a sequence of integers. He then considers the cone \(C\) spanned by all such sequences. In some cases \(C\) has a simple description. He finds a description in terms of the root ordering for all rank two Kac-Moody algebras.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
11B83 Special sequences and polynomials
Full Text: DOI
[1] N. Bourbaki,Algèbre de Lie VI?VII, Chap. 4-6, Hermann, Paris, 1968. Russian translation: ?. ???????,?????? ? ??????? ??. ????? IV?VI. ??????, ???, 1972.
[2] A. D. Berenstein and A. V. Zelevinsky,Tensor product multiplicities and convex polytopes in partition space. J. Geom. and Phys.5 (1989), 453-472. · Zbl 0712.17006 · doi:10.1016/0393-0440(88)90033-2
[3] A. D. Berenstein and A. V. Zelevinsky,String bases for quantum groups of type A r, Advances in Soviet Math.16 (1993), 51-89. · Zbl 0794.17007
[4] A. D. Berenstein and A. V. Zelevinsky,Canonical bases for the quantum group of type A r and piecewise linear combinatorics, Duke Math. J.82 (1996), 473-502. · Zbl 0898.17006 · doi:10.1215/S0012-7094-96-08221-6
[5] S. R. Hansen,A q-analogue of Kempf’s vanishing theorem, PhD thesis (1994). · Zbl 1062.17013
[6] A. Joseph,Quantum Groups and their Primitive Ideals, Springer Verlag, Berlin, 1995. · Zbl 0808.17004
[7] M. Kashiwara,The crystal base and Littelmann’s refined Demazure character formula Duke Math J.71 (1993), 839-858. · Zbl 0794.17008 · doi:10.1215/S0012-7094-93-07131-1
[8] M. Kashiwara,Crystal bases of modified quantized enveloping algebra, Duke Math. J.73 (1994), 383-414. · Zbl 0794.17009 · doi:10.1215/S0012-7094-94-07317-1
[9] M. Kashiwara,Similarities of crystal bases, Lie Algebras and their Representations (Seoul 1995). Contemp. Mat.194 (1996), 177-186.
[10] M. Kashiwara and T. Nakashima,Crystal graphs for the representations of the q-analogue of classical Lie algebras, J. of Algebra165 (1994), 295-345. · Zbl 0808.17005 · doi:10.1006/jabr.1994.1114
[11] V. Lakshmibai,Bases for quantum Demazure modules II, Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods. Proc. Sympos. Pure Math.56 (1994), 149-168. · Zbl 0848.17020
[12] V. Lakshmibai and C. S. Seshadri,Standard monomial theory, Proceedings of the Hyderabad Conference on Algebraic Groups, Manoj Prakashan, 1991. · Zbl 0785.14028
[13] P. Littelmann,Paths and root operators in representation theory, Annals of Math.142 (1995), 499-525. · Zbl 0858.17023 · doi:10.2307/2118553
[14] P. Littelmann,Crystal graphs and Young tableaux, J. of Algebra175 (1995), 65-87. · Zbl 0831.17004 · doi:10.1006/jabr.1995.1175
[15] P. Littelmann,A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math.116 (1994), 329-346. · Zbl 0805.17019 · doi:10.1007/BF01231564
[16] P. Littelmann,A plactic algebra for semisimple Lie algebras, Adv. Math.124 (1996), 312-331. · Zbl 0892.17009 · doi:10.1006/aima.1996.0085
[17] P. Littelmann,An algorithm to compute bases and representation matrices for SL n+1-representations, Proceedings of the MEGA conference (Eindhoven 1995). J. of Pure and Appl. Algebra117 & 118 (1997), 447-468. · Zbl 0973.17008 · doi:10.1016/S0022-4049(97)00022-4
[18] G. Lusztig,Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc.3 (1990), 447-498. · Zbl 0703.17008 · doi:10.1090/S0894-0347-1990-1035415-6
[19] G. Lusztig,Canonical bases arising from quantized enveloping algebras II. Prog. Theor. Phys.102 (1990), 175-201. · Zbl 0776.17012 · doi:10.1143/PTPS.102.175
[20] G. Lusztig,Introduction to Quantum Groups, Birkhäuser Verlag, Boston, 1993. · Zbl 0788.17010
[21] T. Nakashima and A. V. Zelevinsky,Polyhedral realizations of crystal bases for quantized Kac-Moody Algebras, preprint (1997). · Zbl 0897.17014
[22] M. Reineke,On the coloured graph structure of Lusztig’s Canonical Basis, Math. Ann.307 (1997), 705-723. · Zbl 0881.17010 · doi:10.1007/s002080050058
[23] J. Sheats,A symplectic Jeu de Taquin bijection between the tableaux of King and of De Concini, prepint (1995). · Zbl 0940.05069
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