An inductive approach to Coxeter arrangements and Solomon's descent algebra
J.Matthew Douglass
, Götz Pfeiffer
and Gerhard Röhrle
DOI: 10.1007/s10801-011-0301-9
Abstract
In our recent paper (Douglass et al. arXiv:1101.2075 ( 2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.
Pages: 215–235
Keywords: keywords Coxeter groups; reflection arrangements; descent algebra; dihedral groups
Full Text: PDF
References
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2. Bergeron, F., Bergeron, N., Howlett, R.B., Taylor, D.E.: A decomposition of the descent algebra of a finite Coxeter group. J. Algebr. Comb. 1(1), 23-44 (1992)
3. Brieskorn, E.: Sur les groupes de tresses [d'après V. I. Arnol'd]. In: Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., vol. 317, pp. 21-44. Springer, Berlin (1973)
4. Douglass, J.M., Pfeiffer, G., Röhrle, G.: Coxeter arrangements and Solomon's descent algebra. (2011)
5. Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras. London Mathematical Society Monographs. New Series, vol.
21. Clarendon Press, New York (2000)
6. Hanlon, P.: The action of Sn on the components of the Hodge decomposition of Hochschild homology. Mich. Math. J. 37(1), 105-124 (1990)
7. Howlett, R.B.: Normalizers of parabolic subgroups of reflection groups. J. Lond. Math. Soc. 21, 62-80 (1980)
8. Konvalinka, M., Pfeiffer, G., Röver, C.: A note on element centralizers in finite Coxeter groups. J. Group Theory (2011). doi:
9. Lehrer, G.I., Solomon, L.: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes. J. Algebra 104(2), 410-424 (1986)
10. Orlik, P., Solomon, L.: Coxeter arrangements. In: Singularities, Part 2, Arcata, Calif.,
1981. Proc. Sympos. Pure Math., vol. 40, pp. 269-291. Amer. Math. Soc, Providence (1983)
11. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol.
300. Springer, Berlin, (1992)
12. Pfeiffer, G.: A quiver presentation for Solomon's descent algebra. Adv. Math. 220(5), 1428-1465 (2009)
13. Pfeiffer, G., Röhrle, G.: Special involutions and bulky parabolic subgroups in finite Coxeter groups. J. Aust. Math. Soc. 79(1), 141-147 (2005)
14. Schocker, M.: Über die höheren Lie-Darstellungen der symmetrischen Gruppen. Bayreuth. Math.
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