Poincaré duality in modular coinvariant rings
HTML articles powered by AMS MathViewer
- by Müfit Sezer and Wenliang Zhang PDF
- Proc. Amer. Math. Soc. 144 (2016), 5113-5120 Request permission
Abstract:
We classify the modular representations of a cyclic group of prime order whose corresponding rings of coinvariants are Poincaré duality algebras. It turns out that these algebras are actually complete intersections. For other representations we demonstrate that the dimension of the top degree of the coinvariants grows at least linearly with respect to the number of summands of dimension at least four in the representation.References
- J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771, DOI 10.1017/CBO9780511623592
- H. E. A. Eddy Campbell and David L. Wehlau, Modular invariant theory, Encyclopaedia of Mathematical Sciences, vol. 139, Springer-Verlag, Berlin, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR 2759466, DOI 10.1007/978-3-642-17404-9
- Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 72877, DOI 10.2307/2372597
- Harm Derksen and Gregor Kemper, Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical Sciences, 130. MR 1918599, DOI 10.1007/978-3-662-04958-7
- W. G. Dwyer and C. W. Wilkerson, Poincaré duality and Steinberg’s theorem on rings of coinvariants, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3769–3775. MR 2661576, DOI 10.1090/S0002-9939-2010-10429-X
- P. Fleischmann, M. Sezer, R. J. Shank, and C. F. Woodcock, The Noether numbers for cyclic groups of prime order, Adv. Math. 207 (2006), no. 1, 149–155. MR 2264069, DOI 10.1016/j.aim.2005.11.009
- Ian Hughes and Gregor Kemper, Symmetric powers of modular representations, Hilbert series and degree bounds, Comm. Algebra 28 (2000), no. 4, 2059–2088. MR 1747371, DOI 10.1080/00927870008826944
- Richard Kane, Poincaré duality and the ring of coinvariants, Canad. Math. Bull. 37 (1994), no. 1, 82–88. MR 1261561, DOI 10.4153/CMB-1994-012-3
- Ernst Kunz, Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111–115. MR 330158, DOI 10.1016/0021-8693(74)90025-8
- Tzu-Chun Lin, Poincaré duality algebras and rings of coinvariants, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1599–1604. MR 2204269, DOI 10.1090/S0002-9939-05-08170-0
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- Jean-Pierre Serre, Groupes finis d’automorphismes d’anneaux locaux réguliers, Colloque d’Algèbre (Paris, 1967) Secrétariat mathématique, Paris, 1968, pp. 11 (French). MR 0234953
- Müfit Sezer, Constructing modular separating invariants, J. Algebra 322 (2009), no. 11, 4099–4104. MR 2556140, DOI 10.1016/j.jalgebra.2009.07.011
- Müfit Sezer, Decomposing modular coinvariants, J. Algebra 423 (2015), 87–92. MR 3283710, DOI 10.1016/j.jalgebra.2014.08.059
- Müfit Sezer and R. James Shank, On the coinvariants of modular representations of cyclic groups of prime order, J. Pure Appl. Algebra 205 (2006), no. 1, 210–225. MR 2193198, DOI 10.1016/j.jpaa.2005.07.003
- Müfit Sezer and R. James Shank, Rings of invariants for modular representations of the Klein four group, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5655–5673. MR 3458394, DOI 10.1090/tran/6516
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
- Larry Smith, On a theorem of R. Steinberg on rings of coinvariants, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1043–1048. MR 1948093, DOI 10.1090/S0002-9939-02-06629-7
- Larry Smith, On alternating invariants and Hilbert ideals, J. Algebra 280 (2004), no. 2, 488–499. MR 2089248, DOI 10.1016/j.jalgebra.2004.03.027
- Robert Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392–400. MR 167535, DOI 10.1090/S0002-9947-1964-0167535-3
Additional Information
- Müfit Sezer
- Affiliation: Department of Mathematics, Bilkent University, Ankara, 06800, Turkey
- MR Author ID: 703561
- Email: sezer@fen.bilkent.edu.tr
- Wenliang Zhang
- Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Illinois 60607
- MR Author ID: 805625
- Email: wlzhang@uic.edu
- Received by editor(s): May 26, 2015
- Received by editor(s) in revised form: February 22, 2016
- Published electronically: July 21, 2016
- Additional Notes: The first author was supported by a grant from Tübitak:114F427
The second author was partially supported by NSF grants DMS #1247354, #1405602 - Communicated by: Harm Derksen
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5113-5120
- MSC (2010): Primary 13A50
- DOI: https://doi.org/10.1090/proc/13245
- MathSciNet review: 3556257