Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

References

• Altafi N, Iarrobino A, Khatami L. Complete intersection Jordan types in height two. arXiv:math, AC/1810.00716 (2018), 46 p.
   Google Scholar
• Boij M, Migliore J, Miró-Roig RM, et al. The non-Lefschetz locus. J Algebra. 2018;505:288–320. doi: 10.1016/j.jalgebra.2018.03.006
   Web of Science ®Google Scholar
• Gondim R. On higher Hessians and the Lefschetz properties. J Algebra. 2017;489:241–263. doi: 10.1016/j.jalgebra.2017.06.030
   Web of Science ®Google Scholar
• Gondim R, Zappalà G. Lefschetz properties for Artinian Gorenstein algebras presented by quadrics. Proc Amer Math Soc. 2018;146(3):993–1003. doi: 10.1090/proc/13822
   Web of Science ®Google Scholar
• Guerrieri L. Lefschetz properties of Gorenstein graded algebras associated to the Apéry set of a numerical semigroup. arXiv:math,AC/1712.01569 v.2 (2018), 22 p.
   Google Scholar
• Maeno T, Watanbe J. Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials. Illinois J Math. 2009;53(2):593–603. doi: 10.1215/ijm/1266934795
   Google Scholar
• Yoshida S. The Lefschetz elements of coinvariant algebras of binary polyhedral groups. Tokyo J Math. 2013;36(2):309–336. doi: 10.3836/tjm/1391177973
   Web of Science ®Google Scholar
• Iarrobino A, Macias Marques P, McDaniel C. Artinian algebras and Jordan type. arXiv:math.AC/1802.07383 v. 3 (2018), 35p.
   Google Scholar
• Iarrobino A, Macias Marques P, McDaniel C. Artinian Gorenstein algebras that are free extensions over , and Macaulay duality. arXiv:math.AC/1807.02881 (2018), 23p.
   Google Scholar
• Harima T, Wachi A, Watanabe J. The quadratic complete intersections associated with the action of the symmetric group. Illinois J Math. 2015;59:99–113. doi: 10.1215/ijm/1455203161
   Google Scholar
• Maeno T, Numata Y, Wachi A. Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups. Algebr Represent Theory. 2011;4:625–638. doi: 10.1007/s10468-010-9207-9
   Web of Science ®Google Scholar
• McDaniel C. The strong Lefschetz property for coinvariant rings of finite reflection groups. J Algebra. 2011;331:68–95. doi: 10.1016/j.jalgebra.2010.11.007
   Web of Science ®Google Scholar
• Harima T, Watanabe J. The strong Lefschetz property for Artinian algebras with non-standard grading. J Algebra. 2007;311(2):511–537. doi: 10.1016/j.jalgebra.2007.01.019
   Web of Science ®Google Scholar
• Iarrobino A, Macias Marques P, McDaniel C. Free extensions and Jordan type. arXiv:math.AC/1806.02767 (2018), 33p.
   Google Scholar
• Harima T, Watanabe J. The finite free extension of Artinian K-algebras with the strong Lefschetz property. Rend Semin Mat Univ Padova. 2003;110:119–146.
   Web of Science ®Google Scholar
• Harima T, Watanabe J. Erratum to: ‘The finite free extension of Artinian K-algebras with the strong Lefschetz property’. Rend Semin Mat Univ Padova. 2004;112:237–238.
   Google Scholar
• Harima T, Maeno T, Morita H, et al. The Lefschetz properties. Berlin: Springer; 2013. (Lecture Notes in Mathematics; 2080).
   Google Scholar
• Briançon J. Description de Hilb. Invent Math. 1977;41(1):45–89. doi: 10.1007/BF01390164
   Web of Science ®Google Scholar
• Basili R, Iarrobino A. Pairs of commuting nilpotent matrices, and Hilbert function. J Algebra. 2008;320(3):1235–1254. doi: 10.1016/j.jalgebra.2008.03.006
   Web of Science ®Google Scholar
• Lindsey M. A class of Hilbert series and the strong Lefschetz property. Proc Amer Math Soc. 2011;139(1):79–92. doi: 10.1090/S0002-9939-2010-10498-7
   Web of Science ®Google Scholar
• Migliore J, Nagel U, Schenck H. The Weak Lefschetz property for quotients by quadratic monomials. arXiv:math,AC/1706.05058 (2017), 15 p.
   Google Scholar
• Babson E, Nevo E. Lefschetz properties and basic constructions on simplicial spheres. J Algebraic Combin. 2010;31(1):111–129. doi: 10.1007/s10801-009-0189-9
   Web of Science ®Google Scholar
• Moore JC, Smith L. Hopf algebras and multiplicative fibrations. I. Amer J Math. 1968;90:752–780. doi: 10.2307/2373482
   Web of Science ®Google Scholar
• Smith L, Stong R. Projective bundle ideals and Poincaré Duality Algebras. J Pure Appl Algebra. 2011;215(4):609–627. doi: 10.1016/j.jpaa.2010.06.011
   Web of Science ®Google Scholar
• Smith L. Invariants of coinvariant algebras. Forum Math. 2015;27(6):3425–3437. doi: 10.1515/forum-2013-0151
   Web of Science ®Google Scholar
• Shephard GC, Todd JA. Finite unitary reflection groups. Canad J Math. 1954;6:274–304. doi: 10.4153/CJM-1954-028-3
   Web of Science ®Google Scholar
• Steinberg R. Differential equations invariant under finite reflection groups. Trans Am Math Soc. 1964;112:392–400. doi: 10.1090/S0002-9947-1964-0167535-3
   Google Scholar
• Lehrer GI, Taylor DE. Unitary reflection groups. Cambridge: Cambridge University Press; 2009.
   Google Scholar
• Borel A. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann Math (2). 1953;57:115–207. (French). doi: 10.2307/1969728
   Web of Science ®Google Scholar
• Bernstein IN, Gelfand IM, Gelfand SI. Schubert cells and cohomology of the spaces G/P. Russian Math Surveys. 1973;28(3):1–26. doi: 10.1070/RM1973v028n03ABEH001557
   Google Scholar
• Humphreys JE. Reflection groups and Coxeter groups. Cambridge: Cambridge University Press; 1990. (Cambridge Studies in Advanced Mathematics; 29).
   Google Scholar
• Springer TA. Linear algebraic groups. reprint of the 1998 2nd ed. Basel: Birkhäuser; 2009.
   Google Scholar
• Huybrechts D. Complex geometry. An introduction. Berlin: Springer; 2005. (English).
   Google Scholar
• Griffiths P, Harris J. Principles of algebraic geometry. 2nd ed. New York, NY: John Wiley & Sons Ltd.; 1994. (English).
   Google Scholar
• Taylor DE. Reflection subgroups of finite complex reflection groups. J Algebra. 2012;366:218–234. doi: 10.1016/j.jalgebra.2012.04.033
   Web of Science ®Google Scholar
• Neusel MD. Invariant theory. Providence, RI: American Mathematical Society; 2007. (Student Mathematical Library; 36).
   Google Scholar
• Iarrobino A. Associated graded algebra of a Gorenstein Artin algebra. Providence, RI: American Mathematical Society; 1994. (Amer. Math. Soc. Memoir; 107 #514).
   Google Scholar
• Almkvist G. Partitions into odd, unequal parts. J Pure Appl Algebra. 1985;38:121–126. doi: 10.1016/0022-4049(85)90003-9
   Web of Science ®Google Scholar
• Almkvist G. Representations of and unimodal polynomials. J Algebra. 1987;108:283–309. doi: 10.1016/0021-8693(87)90103-7
   Web of Science ®Google Scholar
• Almkvist G. Proof of a conjecture about unimodal polynomials. J Number Theory. 1989;32(1):43–57. doi: 10.1016/0022-314X(89)90096-6
   Web of Science ®Google Scholar
• Stanley RP. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Graph theory and its applications: East and West. Proceedings of the first China-USA international conference, held in Jinan, China, June 9–20, 1986, New York: New York Academy of Sciences; 1989. p. 500–535.
   Google Scholar
• Hughes JWB. Lie algebraic proofs of some theorems on partitions. In: Zassenhaus H, editor. Number theory and algebra. New York: Academic Press; 1977. p. 135–155.
   Google Scholar
• Odlyzko A, Richmond B. On the unimodality of some partition polynomials. European J Combin. 1982;3(6):69–84. doi: 10.1016/S0195-6698(82)80010-3
   Google Scholar
• Kerber A. Algebraic combinatorics via finite group actions. Mannheim: Bibliographisches Institut; 1991.
   Google Scholar
• Migliore J, Nagel U. Survey article: a tour of the weak and strong Lefschetz properties. J Commut Algebra. 2013;5(3):329–358. doi: 10.1216/JCA-2013-5-3-329
   Web of Science ®Google Scholar
• Meyer DM, Smith L. Poincaré duality algebras, Macaulay's dual systems, and Steenrod operators. Cambridge: Cambridge University Press; 2005.
   Google Scholar