Polyvector representations of $\operatorname{GL}_n$

N. A. Vavilov   ; E. Ya. Perelman

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 69-97

Résumé

In the present paper we characterise $\bigwedge^n(\operatorname{GL}(n,R))$ over any commutative ring $R$ as the connected component of the stabiliser of Plücker ideal. This folk theorem is classically known for algebraically closed fields and should be also well-known in general. However, we are not aware of any obvious reference, so we produce a detailed proof which follows a general scheme developed by W. C. Waterhouse. The present paper is a technical preliminary for a subsequent paper, where we construct decomposition of transvections in polyvector representations of $\operatorname{GL}_n$.

Bibliographie

[1] E. Artin, Geometricheskaya algebra, Nauka, M., 1969 | MR | Zbl

[2] N. Burbaki, Algebra, Glavy I–III, M., 1962

[3] N. Burbaki, Gruppy i algebry Li, Glavy VII, VIII, M., 1978 | MR

[4] N. A. Vavilov, Podgruppy rasschepimykh klassicheskikh grupp, Dokt. diss., Leningr. gos. un-t, 1987 | Zbl

[5] N. A. Vavilov, “Kak uvidet znaki strukturnykh konstant?”, Algebra i analiz, 19:4 (2007), 34–68 | MR

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

[7] N. A. Vavilov, E. Ya. Perelman, “Transvektsii v polivektornykh predstavleniyakh $\mathrm{GL}_n$” (to appear)

[8] N. A. Vavilov, E. B. Plotkin, A. V. Stepanov, “Vychisleniya v gruppakh Shevalle nad kommutativnymi koltsami”, Dokl. AN SSSR, 40:1 (1990), 145–147 | MR

[9] E. B. Vinberg, A. G. Elashvili, “Klassifikatsiya trivektorov v devyatimernom prostranstve”, Trudy Seminara po Vekt. i Tenz. Analizu, 18, 1978, 197–233 | MR | Zbl

[10] F. Griffits, Dzh. Kharris, Printsipy algebraicheskoi geometrii, I, Mir, M., 1982 | MR

[11] T. tom Dik, Gruppy preobrazovanii i teoriya predstavlenii, Mir, M., 1982 | MR | Zbl

[12] Zh. Dedonne, Geometriya klassicheskikh grupp, Mir, M., 1974 | MR

[13] E. B. Dynkin, “Maksimalnye podgruppy klassicheskikh grupp”, Trudy Mosk. Mat. ob-va, 1, 1952, 39–166 | MR | Zbl

[14] P. Kon, Svobodnye koltsa i ikh svyazi, Mir, M., 1975 | MR

[15] A. V. Stepanov, Usloviya stabilnosti v teorii lineinykh grupp nad koltsami, Kand. diss., Leningr. gos. un-t, 1987, 1–112

[16] V. Khodzh, D. Pido, Metody algebraicheskoi geometrii, I, IL, M., 1954

[17] I. R. Shafarevich, Osnovy algebraicheskoi geometrii, I, Nauka, M., 1988 | MR

[18] A. Bak, N. Vavilov, “Structure of hyperbolic unitary groups. I: Elementary subgroups”, Algebra Colloquium, 7:2 (2000), 159–196 | DOI | MR | Zbl

[19] M. Barnabei, A. Brini, G.-C. Rota, “On the exterior calculus of invariant theory”, J. Algebra, 96 (1985), 120–160 | DOI | MR | Zbl

[20] H. Bass, “Clifford algebras and spinor norms over a commutative ring”, Amer. J. Math., 96 (1974), 156–206 | DOI | MR | Zbl

[21] J. B. Bertin, M. J. Bertin, Algèbre linèaire et geométrie classique, Masson, Paris, 1981 | MR | Zbl

[22] B. Capdevielle, “Classification des formes trilinéaires alternées en dimension 6”, Enseignement Math., 18 (1972), 225–243 | MR

[23] C. Chevalley, Séminaire sur la classification des groupes de Lie algèbriques, I, II, Ecole Norm. Sup., Paris, 1956–1958

[24] W. L. Chow, “On the geometry of algebraic homogeneous spaces”, Ann. Math., 50 (1949), 32–67 | DOI | MR | Zbl

[25] A. M. Cohen, A. G. Helminck, “Trilinear alternating forms on a vector space of dimension 7”, Commun. Algebra, 16:1 (1988), 1–25 | DOI | MR | Zbl

[26] B. N. Cooperstein, “Nearly maximal representations for the special linear group”, Michigan Math. J., 27 (1980), 3–19 | DOI | MR | Zbl

[27] M. Demazure, P. Gabriel, Groupes algèbriques, I, North Holland, Amsterdam et al., 1970 | MR

[28] M. Demazure, P. Gabriel, Introduction to algebraic geometry and algebraic groups, North Holland, Amsterdam et al., 1980, 357 | MR | Zbl

[29] M. Demazure, A. Grothendieck, Schémas en groupes, I, II, III, Lecture Notes Math., 151, 1971, 564; 152, 654 ; 153, 529 | Zbl | Zbl

[30] J. D. Dixon, “Rigid embeddings of simple groups in the general linear group”, Canad. J. Math., 29:2 (1977), 384–391 | DOI | MR | Zbl

[31] D. Ž. -Djoković, “Classification of trivectors of an eight-dimensional real vector space”, Linear and Multilinear Algebra, 13 (1983), 3–39 | DOI | MR

[32] A. J. Hahn, O. T. O'Meara, The classical groups and K-theory, Springer, Berlin et al., 1989 | MR

[33] N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloquium Public., Providence, RI, 1968 | MR | Zbl

[34] J. C. Jantzen, Representations of algebraic groups, Academic Press, NY, 1987 | MR | Zbl

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

[36] E. B. Plotkin, A. A. Semenov, N. A. Vavilov, “Visual basic representations: an atlas”, Int. J. Algebra and Computations, 8:1 (1998), 61–97 | DOI | MR

[37] Ph. Revoy, “Trivecteurs de rang 6”, Bull. Soc. Math. France, 59 (1979), 141–155 | MR | Zbl

[38] Ph. Revoy, “Formes trilinéaires alternées de rang inférieur ou égal á 7”, Actes du $110^e$ Congrès National des Sociétés savantes (Montpellier, 1985), no. 3, Paris, 1985, 189–194 | MR

[39] Ph. Revoy, “Formes trilinéaires alternées en dimension 6 et 7”, Bulletin Sci. Math., 112 (1988), 357–368 | MR | Zbl

[40] G. M. Seitz, The maximal subgroups of classical algebraic groups, Memoirs Amer. Math. Soc., 67, 1987, 286 | MR

[41] T. A. Springer, F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Berlin et al., Springer, 2000 | MR | Zbl

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

[43] D. M. Testerman, “Irreducible subgroups of exceptional algebraic groups”, Memoirs Amer. Math. Soc., 75, 1988, 1–190 | MR

[44] N. A. Vavilov, “Structure of Chevalley groups over commutative rings”, Proc. Conf. Non-associative algebras and related topics (Hiroshima, 1990), World Sci., London et al., 1991, 219–335 | MR | Zbl

[45] N. A. Vavilov, “A third look at weight diagrams”, Rendiconti del Seminario Matem. dell'Univ. di Padova, 204 (2000), 1–45

[46] N. A. Vavilov, E. B. Plotkin, “Chevalley groups over commutative rings. I: Elementary calculations”, Acta Applicandae Math., 45 (1996), 73–115 | DOI | MR

[47] W. C. Waterhouse, Introduction to affine group schemes, Springer-Verlag, New York et al., 1979 | MR

[48] W. C. Waterhouse, “Automorphisms of quotients of $\prod\mathrm{GL}(n_i)$”, Pacif. J. Math., 102 (1982), 221–233 | MR | Zbl

[49] W. C. Waterhouse, “Automorphisms of $\det(X_{ij})$: the group scheme approach”, Adv. Math., 65:2 (1987), 171–203 | DOI | MR | Zbl

[50] R. Westwick, “Real trivectors of rank seven”, Linear and Multilinear Algebra, 10 (1981), 183–204 | DOI | MR | Zbl