q, t-Fuß-Catalan numbers for finite reflection groups
Christian Stump
DOI: 10.1007/s10801-009-0205-0
Abstract
In type A, the q, t-Fuß-Catalan numbers can be defined as the bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with nonnegative integer coefficients. We prove the conjectures for the dihedral groups and for the cyclic groups. Finally, we present several ideas on how the q, t-Fuß-Catalan numbers could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras and thereby generalize known connections.
Pages: 67–97
Keywords: keywords Catalan number; fuß-Catalan number; $q; t$-Catalan number; nonnesting partition; Dyck path; shi arrangement; Cherednik algebra
Full Text: PDF
References
1. Alfano, J.: A basis for the Y 2 subspace of diagonal harmonics. PhD thesis, University of California, San Diego, USA (1994)
2. Armstrong, D.: Braid groups, clusters and free probability: an outline from the AIM workshop. Available at (2004)
3. Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Am. Math. Soc. 202, 949 (2009)
4. Athanasiadis, C.A.: Deformations of Coxeter hyperplane arrangements and their characteristic polynomials. In: Arrangements, Tokyo
1998. Adv. Stud. Pure Math., vol. 27, pp. 1-26 (2000)
5. Athanasiadis, C.A.: Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. Lond. Math. Soc. 36, 294-302 (2004)
6. Athanasiadis, C.A.: On a refinement of the generalized Catalan numbers for Weyl groups. Trans. Am. Math. Soc. 357, 179-196 (2005)
7. Berest, Y., Etingof, P., Ginzburg, V.: Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118(2), 279-337 (2003)
8. Berest, Y., Etingof, P., Ginzburg, V.: Finite dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 19, 1053-1088 (2003)
9. Bessis, D.: Finite complex reflection arrangements are k(π, 1). Preprint, available at (2007)
10. Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127-190 (1998)
11. Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77(4), 778-782 (1955)
12. Dunkl, C.F., Opdam, E.: Dunkl operators for complex reflection groups. Proc. Lond. Math. Soc. 86, 70-108 (2003)
13. Edelman, P.H., Reiner, V.: Free arrangements and rhombic tilings. Discrete Comput. Geom. 15, 307- 340 (1996)
14. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002)
15. Fulton, W., Harris, J.: Representation Theory: a First Course. Springer, New York (1991)
16. Fürlinger, J., Hofbauer, J.: q-Catalan numbers. J. Comb. Theory Ser. A 40(2), 248-264 (1985)
17. Garsia, A., Haglund, J.: A positivity result in the theory of MacDonald polynomials. Proc. Natl. Acad. Sci. USA 98(8), 4313-4316 (2001)
18. Garsia, A., Haiman, M.: A remarkable q, t -Catalan sequence and q-Lagrange inversion. J. Algebraic Comb. 5, 191-244 (1996)
19. Gordon, I.: On the quotient ring by diagonal harmonics. Invent. Math. 153, 503-518 (2003)
20. Gordon, I., Stafford, J.T.: Rational Cherednik algebras and Hilbert schemes. Adv. Math. 198(1), 222- 274 (2005)
21. Griffeth, S.: Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n). Proc. Edinb. Math. Soc. (2008, to appear). Available at
22. Haglund, J.: Conjectured statistics for the q, t -Catalan numbers. Adv. Math. 175(2), 319-334 (2003)
23. Haglund, J.: The q, t -Catalan numbers and the space of diagonal harmonics, Univ. Lect. Ser., Am. Math. Soc. 41 (2008)
24. Haglund, J., Loehr, N.: A conjectured combinatorial formula for the Hilbert series for diagonal harmonics. Discrete Math. 298, 189-204 (2005)
25. Haiman, M.: Conjectures on the quotient ring by diagonal invariants. J. Algebraic Comb. 3, 17-76 (1994)
26. Haiman, M.: t, q-Catalan numbers and the Hilbert scheme. Discrete Math. 193, 201-224 (1998). Selected papers in honor of Adriano Garsia
27. Haiman, M.: Combinatorics, symmetric functions, and Hilbert schemes. CDM, vol. 2002: Current Developments in Mathematics, pp. 39-111 (2002)
28. Haiman, M.: Notes on MacDonald polynomials and the geometry of Hilbert schemes. In: Proceedings of the NATO Advanced Study Institute, Cambridge, pp. 1-64 (2002)
29. Haiman, M.: Vanishing theorems and character formulas for the Hilbert scheme of points in the plane. Invent. Math. 149, 371-407 (2002)
30. Haiman, M.: Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ.
2. Armstrong, D.: Braid groups, clusters and free probability: an outline from the AIM workshop. Available at (2004)
3. Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Am. Math. Soc. 202, 949 (2009)
4. Athanasiadis, C.A.: Deformations of Coxeter hyperplane arrangements and their characteristic polynomials. In: Arrangements, Tokyo
1998. Adv. Stud. Pure Math., vol. 27, pp. 1-26 (2000)
5. Athanasiadis, C.A.: Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. Lond. Math. Soc. 36, 294-302 (2004)
6. Athanasiadis, C.A.: On a refinement of the generalized Catalan numbers for Weyl groups. Trans. Am. Math. Soc. 357, 179-196 (2005)
7. Berest, Y., Etingof, P., Ginzburg, V.: Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118(2), 279-337 (2003)
8. Berest, Y., Etingof, P., Ginzburg, V.: Finite dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 19, 1053-1088 (2003)
9. Bessis, D.: Finite complex reflection arrangements are k(π, 1). Preprint, available at (2007)
10. Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127-190 (1998)
11. Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77(4), 778-782 (1955)
12. Dunkl, C.F., Opdam, E.: Dunkl operators for complex reflection groups. Proc. Lond. Math. Soc. 86, 70-108 (2003)
13. Edelman, P.H., Reiner, V.: Free arrangements and rhombic tilings. Discrete Comput. Geom. 15, 307- 340 (1996)
14. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002)
15. Fulton, W., Harris, J.: Representation Theory: a First Course. Springer, New York (1991)
16. Fürlinger, J., Hofbauer, J.: q-Catalan numbers. J. Comb. Theory Ser. A 40(2), 248-264 (1985)
17. Garsia, A., Haglund, J.: A positivity result in the theory of MacDonald polynomials. Proc. Natl. Acad. Sci. USA 98(8), 4313-4316 (2001)
18. Garsia, A., Haiman, M.: A remarkable q, t -Catalan sequence and q-Lagrange inversion. J. Algebraic Comb. 5, 191-244 (1996)
19. Gordon, I.: On the quotient ring by diagonal harmonics. Invent. Math. 153, 503-518 (2003)
20. Gordon, I., Stafford, J.T.: Rational Cherednik algebras and Hilbert schemes. Adv. Math. 198(1), 222- 274 (2005)
21. Griffeth, S.: Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n). Proc. Edinb. Math. Soc. (2008, to appear). Available at
22. Haglund, J.: Conjectured statistics for the q, t -Catalan numbers. Adv. Math. 175(2), 319-334 (2003)
23. Haglund, J.: The q, t -Catalan numbers and the space of diagonal harmonics, Univ. Lect. Ser., Am. Math. Soc. 41 (2008)
24. Haglund, J., Loehr, N.: A conjectured combinatorial formula for the Hilbert series for diagonal harmonics. Discrete Math. 298, 189-204 (2005)
25. Haiman, M.: Conjectures on the quotient ring by diagonal invariants. J. Algebraic Comb. 3, 17-76 (1994)
26. Haiman, M.: t, q-Catalan numbers and the Hilbert scheme. Discrete Math. 193, 201-224 (1998). Selected papers in honor of Adriano Garsia
27. Haiman, M.: Combinatorics, symmetric functions, and Hilbert schemes. CDM, vol. 2002: Current Developments in Mathematics, pp. 39-111 (2002)
28. Haiman, M.: Notes on MacDonald polynomials and the geometry of Hilbert schemes. In: Proceedings of the NATO Advanced Study Institute, Cambridge, pp. 1-64 (2002)
29. Haiman, M.: Vanishing theorems and character formulas for the Hilbert scheme of points in the plane. Invent. Math. 149, 371-407 (2002)
30. Haiman, M.: Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ.
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