Unipotent Brauer Character Values of GL( n, \mathbb F q ) GL(n, \mathbb{F}_q ) and the Forgotten Basis of the Hall Algebra
Jonathan Brundan
DOI: 10.1023/A:1011293414487
Abstract
We give a formula for the values of irreducible unipotent p-modular Brauer characters of GL( n, \mathbb F q ) GL(n, \mathbb{F}_q ) at unipotent elements, where p is a prime not dividing q, in terms of (unknown!) weight multiplicities of quantum GL n and certain generic polynomials S 
, 
( q). These polynomials arise as entries of the transition matrix between the renormalized Hall-Littlewood symmetric functions and the forgotten symmetric functions. We also provide an alternative combinatorial algorithm working in the Hall algebra for computing S , ( q).
, ( q). These polynomials arise as entries of the transition matrix between the renormalized Hall-Littlewood symmetric functions and the forgotten symmetric functions. We also provide an alternative combinatorial algorithm working in the Hall algebra for computing S , ( q).Pages: 137–149
Keywords: symmetric function; general linear group; unipotent representation; Brauer character
Full Text: PDF
References
1. J. Brundan, R. Dipper, and A. Kleshchev, “Quantum linear groups and representations of GLn(Fq ),” Mem. Amer. Math. Soc. 706 (2001), 112.
2. J. Brundan and A. Kleshchev, “Lower bounds for degrees of irreducible Brauer characters of finite general linear groups,” J. Algebra 223 (2000), 615-629.
3. R. Dipper and G.D. James, “The q-Schur algebra,” Proc. London Math. Soc. 59 (1989), 23-50.
4. J. Du, “A note on quantized Weyl reciprocity at roots of unity,” Alg. Colloq. 2 (1995), 363-372.
5. I.M. Gelfand and M.I. Graev, “The construction of irreducible representations of simple algebraic groups over a finite field,” Dokl. Akad. Nauk. USSR 147 (1962), 529-532.
6. G.D. James, “Unipotent representations of the finite general linear groups,” J. Algebra 74 (1982), 443-465.
7. N. Kawanaka, “Generalized Gelfand-Graev representations and Ennola duality,” Advanced Studies in Pure Math. 6 (1985), 175-206.
8. G. Lusztig, “Finite dimensional Hopf algebras arising from quantized universal enveloping algebras,” J. Am. Math. Soc. 3 (1990), 257-297.
9. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, 2nd edn., OUP, 1995.
10. Y. Nakada and K.-I. Shinoda, “The characters of a maximal parabolic subgroup of GLn(Fq ),” Tokyo J. Math. 13(2) (1990), 289-300.
11. T.A. Springer, “Characters of special groups,” in Seminar on Algebraic Groups and Related Finite Groups, A. Borel et al. (Eds.), Springer, 1970, pp. 121-166. Lecture Notes in Math. Vol. 131.
12. R. Steinberg, “A geometric approach to the representations of the full linear group over a Galois field,” Trans. Amer. Math. Soc. 71 (1951), 274-282.
13. A. Zelevinsky, Representations of Finite Classical Groups, Springer-Verlag, Berlin,
1981. Lecture Notes in Math. Vol. 869.
2. J. Brundan and A. Kleshchev, “Lower bounds for degrees of irreducible Brauer characters of finite general linear groups,” J. Algebra 223 (2000), 615-629.
3. R. Dipper and G.D. James, “The q-Schur algebra,” Proc. London Math. Soc. 59 (1989), 23-50.
4. J. Du, “A note on quantized Weyl reciprocity at roots of unity,” Alg. Colloq. 2 (1995), 363-372.
5. I.M. Gelfand and M.I. Graev, “The construction of irreducible representations of simple algebraic groups over a finite field,” Dokl. Akad. Nauk. USSR 147 (1962), 529-532.
6. G.D. James, “Unipotent representations of the finite general linear groups,” J. Algebra 74 (1982), 443-465.
7. N. Kawanaka, “Generalized Gelfand-Graev representations and Ennola duality,” Advanced Studies in Pure Math. 6 (1985), 175-206.
8. G. Lusztig, “Finite dimensional Hopf algebras arising from quantized universal enveloping algebras,” J. Am. Math. Soc. 3 (1990), 257-297.
9. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, 2nd edn., OUP, 1995.
10. Y. Nakada and K.-I. Shinoda, “The characters of a maximal parabolic subgroup of GLn(Fq ),” Tokyo J. Math. 13(2) (1990), 289-300.
11. T.A. Springer, “Characters of special groups,” in Seminar on Algebraic Groups and Related Finite Groups, A. Borel et al. (Eds.), Springer, 1970, pp. 121-166. Lecture Notes in Math. Vol. 131.
12. R. Steinberg, “A geometric approach to the representations of the full linear group over a Galois field,” Trans. Amer. Math. Soc. 71 (1951), 274-282.
13. A. Zelevinsky, Representations of Finite Classical Groups, Springer-Verlag, Berlin,
1981. Lecture Notes in Math. Vol. 869.