Highest weight modules and polarized embeddings of shadow spaces
Rieuwert J. Blok
DOI: 10.1007/s10801-010-0263-3
Abstract
The present paper was inspired by the work on polarized embeddings by Cardinali et al. (J. Algebr. Comb. 25(1):7-23, 2007) although some of our results in it date back to 1999. They study polarized embeddings of certain dual polar spaces, and identify the minimal polarized embeddings for several such geometries. We extend some of their results to arbitrary shadow spaces of spherical buildings, and make a connection to work of Burgoyne, Wong, Verma, and Humphreys on highest weight representations for Chevalley groups.
Pages: 67–113
Keywords: keywords building; shadow space; Grassmannian; polarized embedding; Chevalley group; highest weight module; representation theory
Full Text: PDF
References
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1993. London Math. Soc. Lecture Note Ser., vol. 207, pp. 215-232. Cambridge University Press, Cambridge (1995)
41. Shult, E.E.: Embeddings and hyperplanes of the lie incidence geometry of type e7,1. J. Geom. 59, 152-172 (1997)
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1641. Springer, Berlin (1996)
2. Adamovich, A.M.: The submodule lattices of Weyl modules of symplectic groups with fundamental highest weights. Vestn. Mosk. Univ. Ser. I Mat. Mekh. (2), 7-10, 112 (1986)
3. Blok, R.J.: On geometries related to buildings. Ph.D. thesis, Delft University of Technology (1999). Supervisor: Prof. Dr. A.E. Brouwer
4. Blok, R.J., Brouwer, A.E.: The geometry far from a residue. In: Groups and Geometries, Siena,
1996. Trends Math., pp. 29-38. Birkhäuser, Basel (1998)
5. Blok, R.J., Brouwer, A.E.: Spanning point-line geometries in buildings of spherical type. J. Geom. 62, 26-35 (1998)
6. Blok, R.J., Pasini, A.: On absolutely universal embeddings. Discrete Math. 267(1-3), 45-62 (2003). Combinatorics 2000 (Gaeta)
7. Bourbaki, N.: Groupes et Algèbres de Lie. Diffusion C.C.L.S., Paris (1975). Chapitres 7 et 8
8. Bourbaki, N.: Groupes et Algèbres de Lie. Masson, Paris (1981). Chapitres 4, 5 et 6
9. Brouwer, A.E.: The complement of a geometric hyperplane in a generalized polygon is usually connected. In: Finite Geometry and Combinatorics, pp. 53-57. Cambridge University Press, Cambridge (1993)
10. Buekenhout, F.: Foundations of incidence geometry. In: Handbook of Incidence Geometry, pp. 63-
105. North-Holland, Amsterdam (1995)
11. Buekenhout, F., Cohen, A.M.: Diagram Geometry. Book manuscript
12. Buekenhout, F., Lefèvre, C.: Generalized quadrangles in projective spaces. Arch. Math. 25, 540-552 (1974)
13. Cardinali, I., De Bruyn, B., Pasini, A.: Minimal full polarized embeddings of dual polar spaces. J. Al- gebr. Comb. 25(1), 7-23 (2007)
14. Carter, R.W.: Simple Groups of Lie Type. Pure and Applied Math., vol.
28. Wiley, New York (1972) J Algebr Comb (2011) 34: 67-113
15. Carter, R.W.: Lie Algebras of Finite and Affine Type. Cambridge Studies in Advanced Mathematics, vol.
96. Cambridge University Press, Cambridge (2005)
16. Cohen, A.M.: Point-line spaces related to buildings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry. North-Holland, Amsterdam (1995). Chapter 12
17. Cooperstein, B.N.: Geometry of groups of Lie type. In: Proceedings of the Conference on Finite Groups, Univ. Utah, Park City, Utah, 1975, pp. 503-512. Academic Press, San Diego (1976)
18. Cooperstein, B.N.: Some geometries associated with parabolic representations of groups of Lie type. Can. J. Math. 28(5), 1021-1031 (1976)
19. Cooperstein, B.N., Shult, E.E.: Geometric hyperplanes of embeddable Lie incidence geometries. In: Hirschfeld, J.W.P., Hughes, D.R., Thas, J.A. (eds.) Advances in Finite Geometries and Designs. Ox- ford University Press, London (1991). Proceedings of the 3rd Isle of Thorns Conference
20. Cooperstein, B.N., Shult, E.E.: Frames and bases of Lie incidence geometries. J. Geom. 60(1-2), 17-46 (1997)
21. Cooperstein, B.N., Shult, E.E.: Geometric hyperplanes of Lie incidence geometries. Geom. Dedic. 64(1), 17-40 (1997)
22. Dienst, K.J.: Verallgemeinerte Vierecke in pappusschen projektiven Räumen. Geom. Dedic. 9(2), 199-206 (1980)
23. Fulton, W.: Young Tableaux. London Math. Soc. Student Texts, vol.
35. Cambridge University Press, Cambridge (1997)
24. Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol.
129. Springer, New York (1991). A first course, Readings in Mathematics
25. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Math, vol.
9. Springer, New York (1972). Second printing, first edition,
26. Humphreys, J.E.: Finite and infinite dimensional modules for semisimple Lie algebras. In: Lie Theories and Their Applications, Proc. Ann. Sem. Canad. Math. Congr., Queen's Univ., Kingston, Ont.,
1977. Queen's Papers in Pure and Appl. Math., vol. 48, pp. 1-64. Queen's Univ., Kingston (1978)
27. Humphreys, J.E.: Modular Representations of Finite Groups of Lie Type. London Mathematical So- ciety Lecture Note Series, vol.
326. Cambridge University Press, Cambridge (2006)
28. Jacobson, N.: Lie Algebras. Interscience Tracts in Pure and Applied Mathematics, vol.
10. Interscience Publishers (a division of John Wiley & Sons), New York-London (1962)
29. Kasikova, A.: Characterization of some subgraphs of point-collinearity graphs of building geometries. Eur. J. Combin. 28(5), 1493-1529 (2007)
30. Kasikova, A.: Characterization of some subgraphs of point-collinearity graphs of building geometries. II. Adv. Geom. 9(1), 45-84 (2009)
31. Kasikova, A., Shult, E.E.: Absolute embeddings of point-line geometries. J. Algebra 238(1), 265-291 (2001)
32. Lefèvre-Percsy, C.: Polar spaces embedded in a projective space. In: Finite Geometries and Designs. London Math. Soc. Lecture Note Series, vol. 49, pp. 216-220. Cambridge University Press, Cambridge (1981). Proceedings of the 2nd Isle of Thorns Conference
33. Pasini, A.: Diagram Geometries. Oxford Science Publications. Clarendon, Oxford University Press, London (1994)
34. Premet, A.A., Suprunenko, I.D.: The Weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights. Commun. Algebra 11(12), 1309-1342 (1983)
35. Ree, R.: On some simple groups defined by C. Chevalley. Trans. Am. Math. Soc. 84, 392-400 (1957)
36. Ronan, M.A.: Embeddings and hyperplanes of discrete geometries. Eur. J. Combin. 8, 179-185 (1987)
37. Ronan, M.A.: Lectures on Buildings. Perspectives in Mathematics, vol.
7. Academic Press, San Diego (1989)
38. Shult, E.E.: Geometric hyperplanes of embeddable Grassmannians. J. Algebra 145, 55-82 (1992)
39. Shult, E.E.: Geometric hyperplanes of the half-spin geometries arise from embeddings. Bull. Bel. Math. Soc. 1(3), 439-453 (1994). Dedicated to J.A. Thas on his fiftieth birthday
40. Shult, E.E.: Embeddings and hyperplanes of Lie incidence geometries. In: Groups of Lie Type and Their Geometries, Como,
1993. London Math. Soc. Lecture Note Ser., vol. 207, pp. 215-232. Cambridge University Press, Cambridge (1995)
41. Shult, E.E.: Embeddings and hyperplanes of the lie incidence geometry of type e7,1. J. Geom. 59, 152-172 (1997)
42. Shult, E.E.: Points and Lines. Book manuscript
43. Shult, E.E., Thas, J.A.: Hyperplanes of dual polar spaces and the spin module. Arch. Math. 59, 610-23 (1992)
44. Steinberg, R.: Representations of algebraic groups. Nagoya Math. J. 22, 33-56 (1963)
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