( q, t)-analogues and GL n(\mathbb F q) GL_{n}({\mathbb{F}}_{q})
Victor Reiner
and Dennis Stanton
DOI: 10.1007/s10801-009-0194-z
Abstract
We start with a ( q, t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These ( q, t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald's “7 th variation” of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL n(\mathbb F q) GL_{n}({\mathbb{F}}_{q}) .
Pages: 411–454
Keywords: keywords $q$-binomial; $q$-multinomial; finite field; Gaussian coefficient; invariant theory; Coxeter complex; Tits building; Steinberg character; principal specialization
Full Text: PDF
References
1. Benson, D.J.: Polynomial Invariants of Finite Groups. London Mathematical Society Lecture Note Series, vol.
190. Cambridge University Press, Cambridge (1993)
2. Björner, A.: Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings. Adv. Math. 52(3), 173-212 (1984)
3. Foata, D., Schützenberger, M.-P.: Major index and inversion number of permutations. Math. Nachr. 83, 143-159 (1978)
4. Hewett, T.J.: Modular invariant theory of parabolic subgroups of GLn(Fq ) and the associated Steenrod modules. Duke Math. J. 82(1), 91-102 (1996)
5. James, G.D.: Representations of General Linear Groups. London Mathematical Society Lecture Note Series, vol.
94. Cambridge University Press, Cambridge (1984)
6. James, G.D., Peel, M.H.: Specht series for skew representations of symmetric groups. J. Algebra 56, 343-364 (1979)
7. Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer, New York (2002)
8. Kuhn, N., Mitchell, S.: The multiplicity of the Steinberg representation of GLnFq in the symmetric algebra. Proc. Amer. Math. Soc. 96(1), 1-6 (1986)
9. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. With Contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1995)
10. Macdonald, I.G.: Schur functions: theme and variations. In: Séminaire Lotharingien de Combinatoire, Saint-Nabor,
1992. Publ. Inst. Rech. Math. Av., vol. 498, pp. 5-39. Univ. Louis Pasteur, Strasbourg (1992)
11. Mathas, A.: A q-analogue of the Coxeter complex. J. Algebra 164(3), 831-848 (1994)
12. Mui, H.: Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22(3), 319-369 (1975)
13. Olsson, J.B.: On the blocks of GL(n, q). I. Trans. Amer. Math. Soc. 222, 143-156 (1976)
14. Reiner, V., Stanton, D., White, D.: The cyclic sieving phenomenon. J. Combin. Theory Ser. A 108, 17-50 (2004)
15. Roichman, Y.: On permutation statistics and Hecke algebra characters. In: Combinatorial Methods in Representation Theory, Kyoto,
1998. Adv. Stud. Pure Math., vol. 28, pp. 287-304. Kinokuniya, Tokyo (2000)
16. Smith, S.D.: On decomposition of modular representations from Cohen-Macaulay geometries. J. Al- gebra 131(2), 598-625 (1990)
17. Solomon, L.: A decomposition of the group algebra of a finite Coxeter group. J. Algebra 9, 220-239 (1968)
18. Stanley, R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1, 475-511 (1979)
19. Stanley, R.P.: Enumerative Combinatorics, Vol.
1. Cambridge Studies in Advanced Mathematics, vol.
49. Cambridge University Press, Cambridge (1997)
20. Stanley, R.P.: Enumerative Combinatorics, Vol.
2. Cambridge Studies in Advanced Mathematics, vol.
62. Cambridge University Press, Cambridge (1999)
21. Steinberg, R.: A geometric approach to the representations of the full linear group over a Galois field.
190. Cambridge University Press, Cambridge (1993)
2. Björner, A.: Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings. Adv. Math. 52(3), 173-212 (1984)
3. Foata, D., Schützenberger, M.-P.: Major index and inversion number of permutations. Math. Nachr. 83, 143-159 (1978)
4. Hewett, T.J.: Modular invariant theory of parabolic subgroups of GLn(Fq ) and the associated Steenrod modules. Duke Math. J. 82(1), 91-102 (1996)
5. James, G.D.: Representations of General Linear Groups. London Mathematical Society Lecture Note Series, vol.
94. Cambridge University Press, Cambridge (1984)
6. James, G.D., Peel, M.H.: Specht series for skew representations of symmetric groups. J. Algebra 56, 343-364 (1979)
7. Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer, New York (2002)
8. Kuhn, N., Mitchell, S.: The multiplicity of the Steinberg representation of GLnFq in the symmetric algebra. Proc. Amer. Math. Soc. 96(1), 1-6 (1986)
9. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. With Contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1995)
10. Macdonald, I.G.: Schur functions: theme and variations. In: Séminaire Lotharingien de Combinatoire, Saint-Nabor,
1992. Publ. Inst. Rech. Math. Av., vol. 498, pp. 5-39. Univ. Louis Pasteur, Strasbourg (1992)
11. Mathas, A.: A q-analogue of the Coxeter complex. J. Algebra 164(3), 831-848 (1994)
12. Mui, H.: Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22(3), 319-369 (1975)
13. Olsson, J.B.: On the blocks of GL(n, q). I. Trans. Amer. Math. Soc. 222, 143-156 (1976)
14. Reiner, V., Stanton, D., White, D.: The cyclic sieving phenomenon. J. Combin. Theory Ser. A 108, 17-50 (2004)
15. Roichman, Y.: On permutation statistics and Hecke algebra characters. In: Combinatorial Methods in Representation Theory, Kyoto,
1998. Adv. Stud. Pure Math., vol. 28, pp. 287-304. Kinokuniya, Tokyo (2000)
16. Smith, S.D.: On decomposition of modular representations from Cohen-Macaulay geometries. J. Al- gebra 131(2), 598-625 (1990)
17. Solomon, L.: A decomposition of the group algebra of a finite Coxeter group. J. Algebra 9, 220-239 (1968)
18. Stanley, R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1, 475-511 (1979)
19. Stanley, R.P.: Enumerative Combinatorics, Vol.
1. Cambridge Studies in Advanced Mathematics, vol.
49. Cambridge University Press, Cambridge (1997)
20. Stanley, R.P.: Enumerative Combinatorics, Vol.
2. Cambridge Studies in Advanced Mathematics, vol.
62. Cambridge University Press, Cambridge (1999)
21. Steinberg, R.: A geometric approach to the representations of the full linear group over a Galois field.
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