On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups
Patrick Headley
DOI: 10.1023/A:1008621126402
Abstract
Let k Ĩ Z k \in Z , let H( a, k) H(α,k) be the hyperplane { v Ĩ V: á a, v ñ = k} \{ v \in V:\left\langle {α,v} \right\rangle = k\} . We define a set of hyperplanes H = { H( d,1): d Ĩ F + } È{ H( d,0): d Ĩ F + } \mathcal{H} = \{ H(δ,1):δ\in Φ^ + \} \cup \{ H(δ,0):δ\in Φ^ + \} . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of H \mathcal{H} is ( 1 + ht ) n \left( {1 + ht} \right)^n , where n is the rank of 
and h is the Coxeter number of the finiteCoxeter group corresponding to .
and h is the Coxeter number of the finiteCoxeter group corresponding to .Pages: 331–338
Keywords: hyperplane arrangement; Weyl group; Poincaré polynomial
Full Text: PDF
References
1. C.A. Athanasiadis, “Characteristic polynomials of subspace arrangements and finite fields,” Advances in Mathematics (to appear).
2. L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1974.
3. P. Headley, “Reduced Expressions in Infinite Coxeter Groups,” Ph.D. thesis, University of Michigan, 1994.
4. J.W. Moon, “Counting labelled trees,” Canadian Mathematical Monographs, No. 1, 1970.
5. P. Orlik and L. Solomon, “Coxeter arrangements,” Singularities, Part 2, Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 40 (1983), 269-291.
6. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.
7. J.-Y. Shi, “The Kazhdan-Lusztig cells in certain affine Weyl groups,” Lecture Notes in Mathematics, Springer- Verlag, Berlin, Vol. 1179, 1986.
8. J.-Y. Shi, “Sign types corresponding to an affine Weyl group,” Journal London Mathematical Society, 35 (1987), 56-74.
9. T. Zaslavsky, “Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes,” Mem. Amer. Math. Soc. No. 154, 1975.
2. L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1974.
3. P. Headley, “Reduced Expressions in Infinite Coxeter Groups,” Ph.D. thesis, University of Michigan, 1994.
4. J.W. Moon, “Counting labelled trees,” Canadian Mathematical Monographs, No. 1, 1970.
5. P. Orlik and L. Solomon, “Coxeter arrangements,” Singularities, Part 2, Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 40 (1983), 269-291.
6. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.
7. J.-Y. Shi, “The Kazhdan-Lusztig cells in certain affine Weyl groups,” Lecture Notes in Mathematics, Springer- Verlag, Berlin, Vol. 1179, 1986.
8. J.-Y. Shi, “Sign types corresponding to an affine Weyl group,” Journal London Mathematical Society, 35 (1987), 56-74.
9. T. Zaslavsky, “Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes,” Mem. Amer. Math. Soc. No. 154, 1975.