An inductive approach to Coxeter arrangements and Solomon’s descent algebra
- 105 Downloads
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.
KeywordsCoxeter groups Reflection arrangements Descent algebra Dihedral groups
- 1.Bergeron, F., Bergeron, N., Garsia, A.M.: Idempotents for the free Lie algebra and q-enumeration. In: Invariant Theory and Tableaux, Minneapolis, MN, 1988, IMA Vol. Math. Appl., vol. 19, pp. 166–190. Springer, New York (1990) Google Scholar
- 4.Douglass, J.M., Pfeiffer, G., Röhrle, G.: Coxeter arrangements and Solomon’s descent algebra. arXiv:1101.2075 (2011)
- 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) Google Scholar