Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-08T21:48:18.774Z Has data issue: false hasContentIssue false
  • English
  • Français

The Face Semigroup Algebra of a Hyperplane Arrangement

Published online by Cambridge University Press:  20 November 2018

Franco V. Saliola
Franco V. Saliola*
Affiliation:
Laboratoire de Combinatoire et d’Informatique, Mathématique (LaCIM), Université du Québec à Montréal, Montréal, QC, e-mail:saliola@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Keywords

52C3505E2516S37
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 904 - 929
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Auslander, M., Reiten, I., and Smalø, S. O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, MA, 1995.Google Scholar
[2] Bacławski, K., Whitney numbers of geometric lattices. Advances in Math. 16(1975), 125–138.Google Scholar
[3] Beilinson, A., Ginzburg, V., and Soergel, W., Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9(1996), no. 2, 473–527.Google Scholar
[4] Benson, D. J., Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second edition, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, Cambridge, 1998.Google Scholar
[5] Bidigare, P., Hanlon, P., and Rockmore, D., A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99(1999), no. 1, 135–174.Google Scholar
[6] Bidigare, T. P., Hyperplane Arrangement Face Algebras and Their Associated Markov Chains. Ph. thesis, D., University of Michigan, 1997.Google Scholar
[7] Björner, A., The homology and shellability of matroids and geometric lattices. In: Matroid applications, Encyclopedia Math. Appl. 40, Cambridge University Press, Cambridge, 1992, pp. 226–283.Google Scholar
[8] Björner, A., Las Vergnas, M., Sturmfels, B., White, N., and Ziegler, G. M., Oriented matroids. Encyclopedia of Mathematics and its Applications 46, Cambridge University Press, Cambridge, 1993.Google Scholar
[9] Bongartz, K., Algebras and quadratic forms. J. London Math. Soc. (2) 28(1983), no. 3, 461–469.Google Scholar
[10] Brown, K. S., Semigroups, rings, and Markov chains. J. Theoret. Probab. 13(2000), no. 3, 871–938.Google Scholar
[11] Brown, K. S. and Diaconis, P., Random walks and hyperplane arrangements. Ann. Probab. 26(1998), no. 4, 1813–1854.Google Scholar
[12] Cibils, C., Hochschild homology of an algebra whose quiver has no oriented cycles. In: Representation theory, I, Lecture Notes in Math. 1177, Springer, Berlin, 1986, pp. 55–59.Google Scholar
[13] Cibils, C., Cohomology of incidence algebras and simplicial complexes. J. Pure Appl. Algebra 56(1989), no. 3, 221–232.Google Scholar
[14] Cooke, G. E. and Finney, R. L., Homology of cell complexes. Based on lectures by Norman E. Steenrod. Princeton University Press, Princeton, NJ, 1967.Google Scholar
[15] Folkman, J., The homology groups of a lattice. J. Math. Mech. 15(1966), 631–636.Google Scholar
[16] Gerstenhaber, M. and Schack, S. D., Simplicial cohomology is Hochschild cohomology. J. Pure Appl. Algebra 30(1983), no. 2, 143–156.Google Scholar
[17] Green, E. L. and Martínez-Villa, R., Koszul and Yoneda algebras. II. In: Algebras and modules, II, C MS Conf. Proc. 24, American Mathematical Society, Providence, RI, 1998, pp. 227–244.Google Scholar
[18] Hozo, I., Inclusion of poset homology into Lie algebra homology. J. Pure Appl. Algebra 111(1996), no. 1-3, 169–180.Google Scholar
[19] Keller, B., Derived invariance of higher structures on the Hochschild complex. http://people.math.jussieu.fr/-keller/publ/index.html.Google Scholar
[20] Munkres, J. R., Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.Google Scholar
[21] Orlik, P. and Terao, H., Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 300, Springer-Verlag, Berlin, 1992.Google Scholar
[22] Polo, P., On Cohen- Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties. Bull. London Math. Soc. 27(1995), no. 5, 425–434.Google Scholar
[23] Saliola, F. V., On the quiver of the descent algebra. J. Algebra 320(2008), no. 11, 3866–3894.Google Scholar
[24] Saliola, F. V., The quiver of the semigroup algebra of a left regular band. Internat. J. Algebra Comput. 17(2007), no. 8, 1593–1610.Google Scholar
[25] Schocker, M., The module structure of the Solomon-Tits algebra of the symmetric group. J. Algebra 301(2006), no. 2, 554–586.Google Scholar
[26] Solomon, L., A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(1976), no. 2, 255–264.Google Scholar
[27] Stanley, R. P., Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, Cambridge, MA, 1997.Google Scholar
[28] Wachs, M. L., Whitney homology of semipure shellable posets. J. Algebraic Combin. 9(1999), no. 2, 173–207.Google Scholar
[29] Woodcock, D., Cohen- Macaulay complexes and Koszul rings. J. London Math. Soc. (2) 57, no. 2, 398–410.Google Scholar
[30] Zaslavsky, T., Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem. Amer. Math. Soc. 1(1975), no. 154.Google Scholar