Can one see the signs of structure constants?

N. A. Vavilov

Algebra i analiz, Tome 19 (2007) no. 4, pp. 34-68

Résumé

It is described how one can see the signs of action structure constants directly in the weight diagram of microweight and adjoint representations for groups of types $\mathrm{E}_6$, $\mathrm{E}_7$ and $\mathrm{E}_8$. This generalizes the results of the preceding paper, “A third look at weight diagrams”, where a similar algorithm was discussed for microweight representations of $\mathrm{E}_6$ and $\mathrm{E}_7$. The proofs are purely combinatorial and can be viewed as an elementary construction of Lie algebras and Chevalley groups of types $\mathrm{E}_l$ .

Détail

Keywords: Microweight representation, adjoint representation, weight diagram
Mots-clés : structure constants.

Bibliographie

[1] Borel A., Svoistva i lineinye predstavleniya grupp Shevalle, Seminar po algebraicheskim gruppam, Mir, M., 1973, 9–59 | MR

[2] Burbaki N., Gruppy i algebry Li, Gl. IV–VI, Mir, M., 1972 ; Группы и алгебры Ли, Гл. VII, VIII, 1978 | MR | Zbl

[3] Vavilov N. A., “Razlozhenie unipotentov v prisoedinennom predstavlenii gruppy Shevalle tipa $\mathrm{E}_6$”, Algebra i analiz (to appear)

[4] Vavilov N. A., Gavrilovich M. R., “$\mathrm{A}_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm{E}_6$ i $\mathrm{E}_7$”, Algebra i analiz, 16:4 (2004), 54–87 | MR | Zbl

[5] Vavilov N. A., Gavrilovich M. R., Nikolenko S. I., “Stroenie grupp Shevalle: Dokazatelstvo iz knigi”, Zap. nauchn. sem. POMI, 330, 2006, 36–76 | MR | Zbl

[6] Vavilov N. A., Luzgarev A. Yu., Pevzner I. M., “Gruppa Shevalle tipa $\mathrm{E}_6$ v 27-mernom predstavlenii”, Zap. nauchn. sem. POMI, 338, 2006, 5–68 | Zbl

[7] Vavilov N. A., Nikolenko S. I., “$\mathrm{A}_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipa $\mathrm{F}_4$”, Algebra i analiz, 20:4 (2008), 27–63 | MR

[8] Vavilov N. A., Plotkin E. B., “Setevye podgruppy grupp Shevalle”, Zap. nauchn. sem. LOMI, 94, 1979, 40–49 | MR | Zbl

[9] Vavilov N. A., Plotkin E. B., Stepanov A. V., “Vychisleniya v gruppakh Shevalle nad kommutativnymi koltsami”, Dokl. AN SSSR, 307:4 (1989), 788–791 | MR | Zbl

[10] Vinberg E. B., Gorbatsevich V. V., Onischik A. L., “Stroenie grupp i algebr Li”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 41, VINITI, M., 1990, 5–253 | MR

[11] Vinberg E. B., Elashvili A. G., “Klassifikatsiya trivektorov devyatimernogo prostranstva”, Tr. Sem. po vekt. i tenzor. anal., 18, 1978, 197–233 | MR | Zbl

[12] Semenov N. S., “Diagrammy Khasse i motivy odnorodnykh proektivnykh mnogoobrazii”, Zap. nauchn. sem. POMI, 330, 2006, 236–246 | Zbl

[13] Springer T. A., “Lineinye algebraicheskie gruppy”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 55, VINITI, M., 1989, 5–136 | MR

[14] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[15] Khamfri Dzh., Vvedenie v teoriyu algebr Li i ikh predstavlenii, MTsNMO, M., 2003

[16] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[17] Azad H., Barry M., Seitz G. M., “On the structure of parabolic subgroups”, Comm. Algebra, 18 (1990), 551–562 | DOI | MR | Zbl

[18] Baez J. C., “The octonions”, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 145–205 | DOI | MR | Zbl

[19] Baston R. J., Eastwood M. G., The Penrose transform. Its interactions with representation theory, Claredon Press, New York, 1989 | MR | Zbl

[20] Brouwer A. E., Cohen A. M., Neumaier A., Distance-regular graphs, Ergeb. Math. Grenzgeb. (3), 18, Springer-Verlag, Berlin, 1989 | MR | Zbl

[21] Brylinski R., Kostant B., “Minimal representations of $\mathrm{E}_6$, $\mathrm{E}_7$ and $\mathrm{E}_8$ and the generalized Capelli identity”, Proc. Nat. Acad. Sci. U.S.A., 91 (1994), 2469–2472 | DOI | MR | Zbl

[22] Burgoyne N., Williamson C., Some computations involving simple Lie algebras, Proc. 2nd Sympos. Symbolic and Algebraic Manipulation, Ass. Comput. Mach., New York, 1971

[23] Carter R. W., Simple groups of Lie type, John Wiley and Sons, New York, 1989 | MR | Zbl

[24] Chaput P.-E., Manivel L., Perrin N., Quantum cohomology of minuscule homogeneous spaces, , 2006 arXiv: /math.AG/0607492 | MR

[25] Cohen A. M., “Point-line spaces related to buildings”, Handbook of Incidence Geometry, North-Holland, Amsterdam, 1995, 647–737 | MR | Zbl

[26] Cohen A. M., Cushman R. H., “Gröbner bases and standard monomial theory”, Computational Algebraic Geometry (Nice, 1992), Progr. Math., 109, Birkhäuser, Boston, MA, 1993, 41–60 | MR | Zbl

[27] Cohen A. M., Griess R. L., Lisser B., “The group $L(2,61)$ embeds in the Lie group of type $\mathrm{E}_8$”, Comm. Algebra, 21:6 (1993), 1889–1907 | DOI | MR | Zbl

[28] Curtis C. W., Iwahori N., Kilmoyer R., “Hecke algebras and characters of parabolic type of finite groups with $(B,N)$-pairs”, Inst. Hautes Études Sci. Publ. Math., 40, 1971, 81–116 | MR | Zbl

[29] Donnelly R. G., The numbers game, geometric representations of Coxeter groups and Dynkin diagram classification results, , 2006 arXiv: /math.CO/0610702 | MR

[30] Everitt B., “Coxeter groups and hyperbolic manifolds”, Math. Ann., 330:1 (2004), 127–150 | DOI | MR | Zbl

[31] Frenkel I. B., Kac V., “Basic representations of affine Lie algebras and dual resonance models”, Invent. Math., 62:1 (1980), 23–66 | DOI | MR | Zbl

[32] Frenkel I. B., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure Appl. Math., 134, Acad. Press, Boston, MA, 1988 | MR | Zbl

[33] Gilkey P., Seitz G. M., “Some representations of exceptional Lie algebras”, Geometries and Groups (Noordwijkerhout, 1986), Geom. Dedicata, 25, no. 1–3, 1988, 407–416 | MR | Zbl

[34] Grélaud G., Quitté Cl., Tauvel P., Bases de Chevalley et $\mathfrak{sl}(2)$-triplets des algèbres de Lie simples exceptionnelles, Prepubl. No 52, Univ. Poitiers, 1990

[35] Hiller H., Geometry of Coxeter groups, Res. Notes in Math., 54, Pitman, Boston, MA-London, 1982 | MR | Zbl

[36] Howe R., “Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond”, The Schur Lectures, Israel Math. Conf. Proc. (Tel Aviv, 1992), 8, Bar-Ilan Univ., Ramat Gan, 1995, 1–182 | MR | Zbl

[37] Iliev A., Manivel L., “The Chow ring of the Cayley plane”, Compositio Math., 141 (2005), 146–160 | DOI | MR | Zbl

[38] Joseph A., Quantum groups and their primitive ideals, Ergeb. Math. Grenzgeb. (3), 29, Springer-Verlag, Berlin, 1995 | MR

[39] Kac V., Infinite-dimensional Lie algebras, 2nd ed., Cambridge Univ. Press, Cambridge, 1985 | MR | Zbl

[40] Kashiwara M., “Crystalizing the $q$-analogue of universal enveloping algebras”, Comm. Math. Phys., 133 (1990), 249–260 | DOI | MR | Zbl

[41] Kashiwara M., “On crystal bases of $Q$-analogue of universal enveloping algebras”, Duke Math. J., 63 (1991), 465–516 | DOI | MR | Zbl

[42] Kashiwara M., Nakashima T., “Crystal graphs for representations of the $q$-analogue of classical Lie algebras”, J. Algebra, 165 (1994), 295–345 | DOI | MR | Zbl

[43] Lakshmibai V., Seshadri C. S., “Geometry of $G/P$, V”, J. Algebra, 100 (1986), 462–557 | DOI | MR | Zbl

[44] Lakshmibai V., Seshadri C. S., “Standard monomial theory”, Hyderabad Conf. on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, 1991, 279–322 | MR | Zbl

[45] Lichtenstein W., “A system of quadrics describing the orbit of the highest weight vector”, Proc. Amer. Math. Soc., 84:4 (1982), 605–608 | DOI | MR | Zbl

[46] Littelmann P., “Crystal graphs and Young tableaux”, J. Algebra, 175:1 (1995), 65–87 | DOI | MR | Zbl

[47] Littelmann P., “Paths and root operators in representation theory”, Ann. of Math. (2), 142 (1995), 499–525 | DOI | MR | Zbl

[48] Littelmann P., “The path model for representations of symmetrizable Kac-Moody algebras”, Proc. of Internat. Congr. Math., V. 1 (Zürich, 1994), Birkhäuser, Basel, 1995, 298–308 | MR | Zbl

[49] Lusztig G., “Canonical bases arising from quantized enveloping algebras”, J. Amer. Math. Soc., 3 (1990), 447–498 | DOI | MR | Zbl

[50] Lusztig G., “Canonical bases arising from quantized enveloping algebras, II”, Common trends in mathematics and quantum field theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl., 102 (1990), 1991, 175–201 | MR

[51] Lusztig G., Introduction to quantum groups, Progr. Math., 110, Birkhäuser, Boston, MA, 1993 | MR | Zbl

[52] Marsh R. J., On the adjoint module of a quantum group, Preprint No 79, Univ. Warwick, 1994 | MR

[53] Mathieu O., “Bases des représentations des groupes simples complexes”, Séminaire Bourbaki, (1990–1991), no. 201–203, eds. Kashiwara, Lusztig, Ringel et al., 1991; Astérisque, 743, 1992, 421–442

[54] Mathieu O., “Le modéle des chemins”, Séminaire Bourbaki, (1994–1995), no. 237, eds. P. Littelmann; Astérisque, 798, no. 4, 1996, 209–224 | MR | Zbl

[55] Matsumoto H., “Sur les sous-groupes arithmétiques des groupes semi-simples déployés”, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–62 | MR | Zbl

[56] Michel L., Patera J., Sharp R., “The Demazure-Tits subgroup of a simple Lie group”, J. Math. Phys., 29:4 (1988), 777–796 | DOI | MR | Zbl

[57] Mizuno K., “The conjugate classes of Chevalley groups of type $\mathrm{E}_6$”, J. Fac. Sci. Univ. Tokyo. Sect. LA Math., 24:3 (1977), 525–563 | MR | Zbl

[58] Mizuno K., “The conjugate classes of unipotent elements of the Chevalley groups $\mathrm{E}_6$ and $\mathrm{E}_8$”, Tokyo J. Math., 3:2 (1980), 391–461 | MR

[59] Nikolenko S., Semenov N., Chow ring structure made simple, , 2006 arXiv: /math.AG/0606335

[60] Nikolenko S., Semenov N., Zainoulline K., Motivic decomposition of anisotropic varieties of type $\mathrm{E}_8$ and generalized Rost motives, arXiv: /math.AG/0502382

[61] Parker Ch., Röhrle G. E., Minuscule representations, Preprint No 72, Univ. Bielefeld, 1993

[62] Parker Ch., Röhrle G. E., “The restriction of minuscule representations to parabolic subgroups”, Math. Proc. Cambridge Philos. Soc., 135:1 (2003), 59–79 | DOI | MR | Zbl

[63] Petrov V., Semenov N., Zainoulline K., Zero cycles on a twisted Cayley plane, arXiv: /math.AG/0508200v2 | MR

[64] Plotkin E. B., “Stability theorems for $k_1$-functor for Chevalley groups”, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, 1991, 203–217 | MR | Zbl

[65] Plotkin E. B., “On the stability of the $K_1$-functor for Chevalley groups of type $\mathrm{E}_7$”, J. Algebra, 210 (1998), 67–85 | DOI | MR | Zbl

[66] Plotkin E. B., Semenov A. A., Vavilov N. A., “Visual basic representations: an atlas”, Internat. J. Algebra Comput., 8:1 (1998), 61–95 | DOI | MR | Zbl

[67] Proctor R. A., “Bruhat lattices, plane partition generating functions, and minuscule representations”, European J. Combin., 5 (1984), 331–350 | MR | Zbl

[68] Proctor R. A., “A Dynkin diagram classification theorem arising from a combinatorial problem”, Adv. Math., 62:2 (1986), 103–117 | DOI | MR | Zbl

[69] Proctor R. A., “Minuscule elements of Weyl groups, the numbers game, and $d$-complete posets”, J. Algebra, 213 (1999), 272–303 | DOI | MR | Zbl

[70] Proctor R. A., “Dynkin diagram classification of $\lambda$-minuscule Bruhat lattices and of $d$-complete posets”, J. Algebraic Combin., 9:1 (1999), 61–94 | DOI | MR | Zbl

[71] Ringel C. M., “Hall polynomials for the representation-finite hereditary algebras”, Adv. Math., 84 (1990), 137–178 | DOI | MR | Zbl

[72] Röhrle G. E., “On the structure of parabolic subgroups in algebraic groups”, J. Algebra, 157:1 (1993), 80–115 | DOI | MR | Zbl

[73] Scharlau R., Buildings, Handbook of Incidence Geometry, North Holland, Amsterdam, 1995, 477–645 | MR | Zbl

[74] Segal G., “Unitary representations of some infinite-dimensional groups”, Comm. Math. Phys., 80 (1981), 301–342 | DOI | MR | Zbl

[75] Seshadri C. S., Geometry of G/P. I., “Theory of standard monomials for minuscule representations”, C. P. Ramanujam – a Tribute, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978, 207–239 | MR

[76] Springer T. A., Linear algebraic groups, 2nd ed., Progr. Math., 9, Birkhäuser, Boston, MA, 1981 | MR | Zbl

[77] Stein M. R., “Generators, relations and coverings of Chevalley groups over commutative rings”, Amer. J. Math., 93:4 (1971), 965–1004 | DOI | MR | Zbl

[78] Stein M. R., “Stability theorems for $\mathrm{K}_1$, $\mathrm{K}_1$K2 and related functors modeled on Chevalley groups”, Japan. J. Math. (N.S.), 4:1 (1978), 77–108 | MR | Zbl

[79] Stembridge J. R., “On minuscule representations, plane partitions and involutions in complex Lie groups”, Duke Math. J., 73:2 (1994), 469–490 | DOI | MR | Zbl

[80] Stembridge J. R., “Quasi-minuscule quotients and reduced words for reflections”, J. Algebraic Combin., 13:3 (2001), 275–293 | DOI | MR | Zbl

[81] Stepanov A. V., Vavilov N. A., “Decomposition of transvections: a theme with variations”, $K$-Theory, 19 (2000), 109–153 | DOI | MR | Zbl

[82] Thomas H., Yong A., A combinatorial rule for (co)minuscule Schubert calculus, Preprint UCLA, 2006

[83] Tits J., “Sur les constantes de structure et le théorème d'existence des algébres de Lie semi-simples”, Inst. Hautes Études Sci. Publ. Math., 31, 1966, 21–58 | MR

[84] Vavilov N. A., “Structure of Chevalley groups over commutative rings”, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, NJ, 1991, 219–335 | MR | Zbl

[85] Vavilov N. A., “Intermediate subgroups in Chevalley groups”, Groups of Lie Type and their Geometries (Como, 1993), London Math. Soc. Lecture Note Ser., 207, Cambridge Univ. Press, Cambridge, 1995, 233–280 | MR | Zbl

[86] Vavilov N. A., “A third look at weight diagrams”, Rend. Sem. Mat. Univ. Padova, 104:1 (2000), 201–250 | MR | Zbl

[87] Vavilov N. A., “Do it yourself structure constants for Lie algebras of type $\mathrm E_l$”, Zap. nauchn. sem. POMI, 281, 2001, 60–104 | MR | Zbl

[88] Vavilov N. A., “An $\mathrm{A}_3$-proof of structure theorems for Chevalley groups of types $\mathrm{E}_7$ and $\mathrm{E}_7$”, J. Pure Appl. Algebra (to appear)

[89] Vavilov N. A., Plotkin E. B., “Chevalley groups over commutative rings. I. Elementary calculations”, Acta Appl. Math., 45:1 (1996), 73–113 | DOI | MR | Zbl

[90] Wildberger N. J., A combinatorial construction for simply-laced Lie algebras, Preprint, Univ. New South Wales, 2000, 11 pp. | MR

[91] Wildberger N. J., Minuscule posets from neighbourly graph sequences, Preprint, Univ. New South Wales, 2001, 19 pp. | MR | Zbl

[92] Wildberger N. J., An easy construction of $\mathrm{G}_2$, Preprint, Univ. New South Wales, 2001, 9 pp. | MR | Zbl