×

On characters of Chevalley groups vanishing at the non-semisimple elements. (English) Zbl 1383.20014

Summary: Let \(G\) be a finite simple group of Lie type. In this paper, we study characters of \(G\) that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of \(G\). Such characters can be viewed as a natural generalization of the Steinberg character. For groups \(G\) of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

MSC:

20C33 Representations of finite groups of Lie type
20C40 Computational methods (representations of groups) (MSC2010)
20C15 Ordinary representations and characters

Software:

CHEVIE
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 1. F. M. Bleher, Finite groups of Lie type of small rank, Pacific J. Math.187(2) (1999) 215-239. genRefLink(16, ’S0218196716500351BIB001’, ’10.2140
[2] 2. R. W. Carter, Finite Groups of Lie type: Conjugacy Classes and Complex Characters (John Wiley & Sons, 1985). · Zbl 0567.20023
[3] 3. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups (Oxford University Press, 1985). · Zbl 0568.20001
[4] 4. Ch. W. Curtis and I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, Vol. 1 (John Wiley & Sons, 1981).
[5] 5. Ch. W. Curtis and I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, Vol. 2 (John Wiley & Sons, 1987).
[6] 6. F. Digne and J. Michel, Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts, Vol. 21 (Cambridge University Press, 1991). genRefLink(16, ’S0218196716500351BIB006’, ’10.1017 · Zbl 0815.20014
[7] 7. W. Feit, The Representation Theory of Finite Groups (North-Holland Mathematical Library, 1982). · Zbl 0493.20007
[8] 8. M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE - A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Eng. Comm. Comput.7 (1996) 175-210. genRefLink(16, ’S0218196716500351BIB008’, ’10.1007
[9] 9. I. M. Gelfand and M. I. Graev, Construction of irreducible representations of simple algebraic groups over finite fields, Dokl. Acad. Nauk SSSR147 (1962) 529-532 (Soviet Math. Doklady3 (1962) 1646-1648).
[10] 10. D. Gorenstein, R. Lyons and R. Solomon, The Classification of Finite Simple Groups, Vol. 3 (American Mathematical Society, 1998). · Zbl 0890.20012
[11] 11. F. Himstedt and S.-C. Huang, Dade’s invariant conjecture for the Ree groups 2F4(q) in defining characteristic, Comm. Algebra40 (2012) 452-496. genRefLink(16, ’S0218196716500351BIB011’, ’10.1080
[12] 12. J. Humphreys, Modular Representations of Finite Groups of Lie Type (Cambridge University Press, 2006). · Zbl 1113.20016
[13] 13. D. Kotlar, On the irreducible constituents of degenerate Gelfand-Graev characters, J. Algebra173 (1995) 348-360. genRefLink(16, ’S0218196716500351BIB013’, ’10.1006
[14] 14. G. Lusztig, On the representations of reductive groups with disconnected centre. Orbites unipotentes et représentations, I, Astérisque168(10) (1988) 157-166.
[15] 15. G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type (Cambridge University Press, 2011). genRefLink(16, ’S0218196716500351BIB015’, ’10.1017 · Zbl 1256.20045
[16] 16. G. Malle and T. Weigel, Finite groups with minimal 1-PIM, Manuscripta Math.126 (2008) 315-332. genRefLink(16, ’S0218196716500351BIB016’, ’10.1007
[17] 17. M. A. Pellegrini, A description of the Steinberg character using Gelfand-Graev characters, Results Math.67 (2015) 71-85. genRefLink(16, ’S0218196716500351BIB017’, ’10.1007
[18] 18. A. Przygocki, Schur indices of symplectic groups, Comm. Algebra10(3) (1982) 279-310. genRefLink(16, ’S0218196716500351BIB018’, ’10.1080
[19] 19. K. Shinoda, The characters of the finite conformal symplectic group CSp(4,q), Comm. Algebra10(13) (1982) 1369-1419. genRefLink(16, ’S0218196716500351BIB019’, ’10.1080
[20] 20. B. Srinivasan, The characters of the finite symplectic group Sp(4,q), Trans. Amer. Math. Soc.131 (1968) 488-525. genRefLink(128, ’S0218196716500351BIB020’, ’A1968B141600016’); · Zbl 0213.30401
[21] 21. R. Steinberg, Lectures on Chevalley Groups (Yale University, 1968). · Zbl 1196.22001
[22] 22. F. D. Veldkamp, Regular elements in anisotropic tori, in Contributions to Algebra, eds. H. Bass, P. J. Cassidy and J. Kovacic (Academic Press, 1977), pp. 389-424. genRefLink(16, ’S0218196716500351BIB022’, ’10.1016 · Zbl 0372.20034
[23] 23. W. Willems and A. Zalesski, Quasi-projective and quasi-liftable characters, J. Algebra442 (2015) 548-559. genRefLink(16, ’S0218196716500351BIB023’, ’10.1016
[24] 24. A. Zalesski, Low dimensional projective indecomposable modules for Chevalley groups in defining characteristic, J. Algebra377 (2013) 125-156. genRefLink(16, ’S0218196716500351BIB024’, ’10.1016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.