Abstract
We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a linear character) using the theory of semi-invariant differential forms. Springer’s theory of regular numbers gives a formula when the group is generated by dim V reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.
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References
S. Ariki, T. Terasoma, and H.-F. Yamada, “Higher Specht polynomials,” Hiroshima Math. J. 27(1) (1997), 177–188.
D. Bessis, “Zariski theorems and diagrams for braid groups,” Invent. Math., Invent. Math. 145(3) (2001), 487–507.
M. Broué, G. Malle, and J. Michel, “Towards Spetses I,” Transform. Groups 4(2–3) (1999), 157–218.
C. Chevalley, ‘Invariants of finite groups generated by reflections,” Amer. J. Math. 77 (1955), 778–782.
R. Kane, Reflection Groups and Invariant Theory, vol. 5 of CMS Books in Mathematics, Springer-Verlag, 2001.
F. Klein, Gesammelte Mathematische Abhandlungen. vol. 2, Springer, 1922.
G. Lehrer and J. Michel, “Invariant theory and eigenspaces for unitary reflection groups,” submitted.
G. Lehrer and T. Springer, “Reflection subquotients of unitary reflection groups,” Canad. J. Math. 51(6) (1999), 1175–1193.
H. Morita and H.-F. Yamada, “Higher Specht polynomials for the complex reflection groups (r,p,n),” Hokkaido Math. J. 27(3) (1998), 505–515.
P. Orlik, V. Reiner, and A. Shepler, “The sign representation for Shephard groups,” Mathematische Annalen 322 (2002), 477–492.
P. Orlik, and L. Solomon, “Unitary reflection groups and cohomology,” Invent. Math. 59 (1980), 77–94.
P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.
G. Shephard and J. Todd, “Finite unitary reflection groups,” Canad. J. Math. 6 (1954), 274–304.
A. Shepler, “Semi-invariants of finite reflection groups,” J. Algebra 220 (1999), 314–326.
L. Smith, Polynomial Invariants of Finite Groups, vol. 6 of Research Notes in Mathematics. A K Peters, 1995.
L. Solomon, “Invariants of finite reflection groups,” Nagoya Math. J. 22 (1963), 57–64.
L. Solomon, “Invariants of Euclidean reflection groups,” Trans. Amer. Math. Soc. 113 (1964), 274–286.
T. Springer, “Regular elements of finite reflection groups,” Invent. Math. 25 (1974), 159–198.
R. Stanley, “Relative invariants of finite groups,” Journal of Algebra 49(1) (1977), 134–148.
R. Steinberg, “Invariants of finite reflection groups,” Canad. J. Math. 12 (1960), 616–618.
R. Steinberg, “Differential equations invariant under finite reflection groups,” Trans. Amer. Math. Soc. 112 (1964), 392–400.
J. Stembridge, “On the eigenvalues of representations of reflection groups and wreath products,” Pacific J. of Math. 140(2) (1989), 353–396.
H. Terao, “The Jacobians and the discriminants of finite reflection groups,” Tohoku Math. J. 41 (1989), 237–247.
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Shepler, A.V. Generalized Exponents and Forms. J Algebr Comb 21, 185–202 (2005). https://doi.org/10.1007/s10801-005-6908-y
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DOI: https://doi.org/10.1007/s10801-005-6908-y