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Good bases for G-modules

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Abstract

We prove a conjecture of I. M. Gelfand, A. V. Zelevinsky and K. Baclawski about the existence of good bases for G-modules. We deduce the result from a previously proved theorem [21] about weak B-modules. There are hopes that good bases can be useful for finding new combinatorial formulas of tensor product multiplicities (cf. [2], [6]).

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References

  1. Baclawski, K., ‘A new rule for computing Clebsh-Gordan series’, Adv. in Appl. Math. 5 (1984), 416–432.

    Google Scholar 

  2. Berenstein, A. D. and Zelevinsky, A. V., ‘Tensor product multiplicities and convex polytopes in partition spaces’ (preprint, 1989).

  3. de Concini, C. and Kazdhan, D., ‘Special bases for S n and GL n ’, Israel J. Math. 40 (1981), 275–290.

    Google Scholar 

  4. Demazure, M., ‘Désingularisation des variétés de Schubert généralisées’, Ann. Sci. E.N.S. 7 (1974), 53–88.

    Google Scholar 

  5. Gelfand, I. M. and Zelevinsky, A. V., ‘Polytopes in the pattern space and canonical basis in irreducible representations of gl(3)’, Funct. Anal. Appl. 19-2 (1985), 72–75.

    Google Scholar 

  6. Gelfand, I. M. and Zelevinsky, A. V., ‘Multiplicities and regular basis for gl(n)’, in Group Theoretical Methods in Physics, Proc. Third Seminar, Yurmala, Nauka, Moscow, Vol. 2, 1985, pp. 22–31.

    Google Scholar 

  7. Gelfand, I. M. and Zelevinsky, A. V., ‘Canonical basis in irreducible representations of gl(3) and its applications’ in Group Theoretical Methods in Physics, Proc. Third Seminar, Yurmala, Nauka, Moscow, Vol. 2, 1985, pp. 31–45.

    Google Scholar 

  8. Howe, R., ‘Highly symmetrical dynamical systems’ (preprint).

  9. Kac, V. G. and Peterson, D. H., ‘Infinite dimensional Lie algebras, theta functions and modular forms’, Adv. in Math. 53 (1984), 125–264.

    Google Scholar 

  10. Kac, V. G. and Wakimoto, M., ‘Modular and conformal invariance constraints in representation theory of affine algebras’, Adv. in Math. 70 (1988), 156–236.

    Google Scholar 

  11. Kostant, B., ‘A formula for the multiplicity of a weight’, Trans. Amer. Math. Soc. 93 (1959), 53–73.

    Google Scholar 

  12. Kempf, G., The Grothendieck-Cousin complex of an induced representation’, Adv. in Math. 29 (1978), 310–396.

    Google Scholar 

  13. Littelmann, P., ‘A generalization of the Littlewood-Richardson rule’ (preprint, Dec. 1987).

  14. Louck, J. D. and Biedenharn, L. C., ‘Some properties of the intertwining numbers of the general linear group’, Adv. in Math. Suppl. Studies 10 (1986), 265–311.

    Google Scholar 

  15. Lakshmibai, V., Musili, C., and Seshadri, C., ‘Geometry of G/P-4’, Proc. Ind. Nat. Sci. Acad. 88 (1979), 279–362.

    Google Scholar 

  16. Lakshmibai, V. and Seshadri, C., ‘Geometry of G/P-5’, J. Algebra 100 (1986), 462–577.

    Google Scholar 

  17. Mathieu, O., ‘Formule de caractère pour les algèbres de Kac-Moody générales’, Astérique 159–160 (1988).

  18. Mathieu, O., ‘Filtrations of B-modules’, Duke Math. J. 59 (1989), 421–442.

    Google Scholar 

  19. Mathieu, O., ‘Frobenius action on the B-cohomology’, Proc. of Infinite Dimensional Lie Algebras and Groups. (ed. V. G. Kac), World Scientific; Adv. Ser. in Math. Physic. 7 (1989), 39–51.

    Google Scholar 

  20. Mathieu, O., ‘Filtrations de G-modules’, C. R. Acad. Sci. Paris 309 (1989), 273–276.

    Google Scholar 

  21. Mathieu, O., ‘Filtrations of G-modules’ (preprint, May 1989) To appear in Ann. Sci. E.N.S.

  22. Mathieu, O., ‘Bonnes bases pour les G-modules’, C.R. Acad. Sci. Paris 310 (1990), 1–4.

    Google Scholar 

  23. Polo, P., ‘Variété de Schubert et excellentes filtrations’, Proc. Orbites Unipotentes et représentations (eds M. Andler, F. Digne, and M. Duflo); Astérique 173–174 (1990), 281–311.

    Google Scholar 

  24. Polo, P., ‘Modules associés aux variétés de Schubert’, C.R. Acad. Sci. Paris 308 (1989), 37–60.

    Google Scholar 

  25. Polo, P., ‘Modules associés aux variétés de Schubert’ (preprint, June 1989).

  26. A. Ramanathan, ‘Schubert varieties are arithmetically Cohen Macaulay’, Invent. Math. 80 (1985), 283–294.

    Google Scholar 

  27. Retakh, V. S. and Zelevinsky, A. V., ‘Base affine space and canonical basis in irreducible representations of Sp(4)’, Dokl. Acad. Nauk USSR 300 (1988), 31–35.

    Google Scholar 

  28. Steinberg, R., ‘A general Clebsch-Gordan theorem’, Bull. Amer. Math. Soc. 67 (1961), 406–407.

    Google Scholar 

  29. van der Kallen, W., ‘Longest weight vectors and excellent filtration’, Math. Z. 201 (1989), 19–31.

    Google Scholar 

  30. Zelevinsky, A. V., ‘Multiplicities in finite dimensional representations of semi-simple Lie algebras’, Uspekhi Math. Nauk (1989).

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Authors and Affiliations

  1. 28 bd de l'Hôpital, 75005, Paris, France

    Olivier Mathieu

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  1. Olivier Mathieu
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Additional information

Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday

Research supported by NSF Grant No. DMS-8610730.

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Mathieu, O. Good bases for G-modules. Geom Dedicata 36, 51–66 (1990). https://doi.org/10.1007/BF00181464

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  • DOI: https://doi.org/10.1007/BF00181464

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