Granger, Michel, Mond, D. (David) and Schulze, Mathias. (2012) Partial normalizations of coxeter arrangements and discriminants. Moscow Mathematical Journal, Vol.12 (No.2). pp. 335-367. ISSN 1609-4514

Preview

Text WRAP_Mond_rccox.pdf - Accepted Version Download (537Kb) | Preview

Abstract

We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin’s Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also describe an independent approach to these structures via duality of maximal Cohen–Macaulay fractional ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter group. Finally, we show that our partial normalizations give rise to new free divisors.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Coxeter groups, Representations of groups
Journal or Publication Title:	Moscow Mathematical Journal
Publisher:	American Mathematical Society
ISSN:	1609-4514
Date:	2012
Volume:	Vol.12
Number:	No.2
Page Range:	pp. 335-367
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
References:	[Arn79] V. I. Arnol′d, Indexes of singular points of 1-forms on manifolds with boundary, convolutions of invariants of groups generated by reflections, and singular projections of smooth surfaces, Uspekhi Mat. Nauk 34 (1979), no. 2(206), 3–38. MR 535708 (81e:58041) (document), 6.2 [BEGvB09] Ragnar-Olaf Buchweitz, Wolfgang Ebeling, and Hans-Christian Graf von Bothmer, Low-dimensional singular- ities with free divisors as discriminants, J. Algebraic Geom. 18 (2009), no. 2, 371–406. MR 2475818 3.9 [Bri71] E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congr`es International des Math´ematiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 279–284. MR 0437798 (55 #10720) 4.4 [Che55] Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 0072877 (17,345d) 5 [dJvS90] T. de Jong and D. van Straten, Deformations of the normalization of hypersurfaces, Math. Ann. 288 (1990), no. 3, 527–547. MR 1079877 (92d:32050) 3.1, 3.1, 3.4, 3.2 [Dub98] Boris Dubrovin, Differential geometry of the space of orbits of a Coxeter group, Surveys in differential geometry: integral systems [integrable systems], Surv. Differ. Geom., IV, Int. Press, Boston, MA, 1998, pp. 181–211. MR 1726929 (2001f:53182) (document), 2 [Eis95] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, With a view toward algebraic geometry. MR 1322960 (97a:13001) 3.1 [GR71] H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer-Verlag, Berlin, 1971, Unter Mitarbeit von O. Riemenschneider, Die Grundlehren der mathematischen Wissenschaften, Band 176. MR 0316742 (47 #5290) 3.2 [GR84] Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR 755331 (86a:32001) 3.2 [GS] D. Grayson and M. Stillman, Macaulay2 — a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2. 5 [Her02] Claus Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, vol. 151, Cambridge University Press, Cambridge, 2002. MR 1924259 (2004a:32043) (document), 2, 2, 2 [Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002) 1, 1, 1, 1, 4.3, 4.3, 5 [Meh88] M. L. Mehta, Basic sets of invariant polynomials for finite reflection groups, Comm. Algebra 16 (1988), no. 5, 1083–1098. MR 926338 (88m:20104) 5 [MP89] David Mond and Ruud Pellikaan, Fitting ideals and multiple points of analytic mappings, Algebraic geometry and complex analysis (P´atzcuaro, 1987), Lecture Notes in Math., vol. 1414, Springer, Berlin, 1989, pp. 107–161. MR 1042359 (91e:32035) 2, 3.2, 3.1, 4.2, 6 [MS10] David Mond and Mathias Schulze, Adjoint divisors and free divisors, arXiv.org math.AG (2010), no. 1001.1095. (document), 6, 6, 6, 6 [Orl89] Peter Orlik, Stratification of the discriminant in reflection groups, Manuscripta Math. 64 (1989), no. 3, 377– 388. MR 1003095 (90d:32026) 4.3 [OS88] Peter Orlik and Louis Solomon, The Hessian map in the invariant theory of reflection groups, Nagoya Math. J. 109 (1988), 1–21. MR 931948 (89i:32024) 5 [Sai80] Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. MR 586450 (83h:32023) 1, 4.3, 4.14, 4.4 [Sai93] , On a linear structure of the quotient variety by a finite reflexion group, Publ. Res. Inst. Math. Sci. 29 (1993), no. 4, 535–579. MR 1245441 (94k:32059) 1 [Slo80] Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445 (82g:14037) 4.4 [Sol63] Louis Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57–64. MR 0154929 (27 #4872) 1, 5 [Sol64] , Invariants of Euclidean reflection groups, Trans. Amer. Math. Soc. 113 (1964), 274–286. MR 0165038 (29 #2329) 4.1, 5, 5 [Ter83] Hiroaki Terao, Discriminant of a holomorphic map and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 2, 379–391. MR 722502 (85d:32027) 1 [Vas91] Wolmer V. Vasconcelos, Jacobian matrices and constructions in algebra, Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991), Lecture Notes in Comput. Sci., vol. 539, Springer, Berlin, 1991, pp. 48–64. MR 1229308 3.2 [Vas98] , Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998, With chapters by David Eisenbud, Daniel R. Grayson, J¨urgen Herzog and Michael Stillman. MR 1484973 (99c:13048) 3.2, 3.2
URI:	http://wrap.warwick.ac.uk/id/eprint/52024

Request changes to a record

Actions (login required)

View Item