Orthogonal representations of twisted forms of $\operatorname {SL}_2$
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- by Skip Garibaldi
- Represent. Theory 12 (2008), 435-446
- DOI: https://doi.org/10.1090/S1088-4165-08-00335-X
- Published electronically: December 8, 2008
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Abstract:
For every absolutely irreducible orthogonal representation of a twisted form of $\operatorname {SL}_2$ over a field of characteristic zero, we compute the âuniqueâ symmetric bilinear form that is invariant under the group action. We also prove the analogous result for Weyl modules in prime characteristic (including characteristic 2) and an isomorphism between two symmetric bilinear forms given by binomial coefficients.References
- Ricardo Baeza, Quadratic forms over semilocal rings, Lecture Notes in Mathematics, Vol. 655, Springer-Verlag, Berlin-New York, 1978. MR 0491773, DOI 10.1007/BFb0070341
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7â9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
- Armand Borel and Jacques Tits, Groupes rĂ©ductifs, Inst. Hautes Ătudes Sci. Publ. Math. 27 (1965), 55â150 (French). MR 207712, DOI 10.1007/BF02684375
- N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589â592. MR 23257, DOI 10.2307/2304500
- Andrew Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253â276. MR 1483922
- Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
- Detlev W. Hoffmann and Ahmed Laghribi, Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4019â4053. MR 2058517, DOI 10.1090/S0002-9947-04-03461-0
- J.E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, 1980, Third printing, revised.
- N. Jacobson, Exceptional Lie algebras, Lecture notes in pure and applied mathematics, vol. 1, Marcel-Dekker, New York, 1971.
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929, DOI 10.1090/gsm/067
- J-P. Serre, Galois cohomology, Springer-Verlag, 2002, originally published as Cohomologie galoisienne (1965).
- Jean-Pierre Serre, Cohomological invariants, Witt invariants, and trace forms, Cohomological invariants in Galois cohomology, Univ. Lecture Ser., vol. 28, Amer. Math. Soc., Providence, RI, 2003, pp. 1â100. Notes by Skip Garibaldi. MR 1999384
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- Peter Sin and Wolfgang Willems, $G$-invariant quadratic forms, J. Reine Angew. Math. 420 (1991), 45â59. MR 1124565, DOI 10.1515/crll.1991.420.45
- J. Tits, Formes quadratiques, groupes orthogonaux et algĂšbres de Clifford, Invent. Math. 5 (1968), 19â41 (French). MR 230747, DOI 10.1007/BF01404536
- J. Tits, ReprĂ©sentations linĂ©aires irrĂ©ductibles dâun groupe rĂ©ductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196â220 (French). MR 277536, DOI 10.1515/crll.1971.247.196
- P. W. Winter, On the modular representation theory of the two-dimensional special linear group over an algebraically closed field, J. London Math. Soc. (2) 16 (1977), no. 2, 237â252. MR 486169, DOI 10.1112/jlms/s2-16.2.237
Bibliographic Information
- Skip Garibaldi
- Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- MR Author ID: 622970
- ORCID: 0000-0001-8924-5933
- Email: skip@member.ams.org
- Received by editor(s): November 6, 2007
- Received by editor(s) in revised form: August 4, 2008
- Published electronically: December 8, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 12 (2008), 435-446
- MSC (2000): Primary 20G05; Secondary 11E04, 11E76, 20G15
- DOI: https://doi.org/10.1090/S1088-4165-08-00335-X
- MathSciNet review: 2465801