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http://dx.doi.org/10.4134/JKMS.j180751

MODULAR JORDAN TYPE FOR 𝕜[x, y]/(xm, yn) FOR m = 3, 4  

Park, Jung Pil (Faculty of Liberal Education Seoul National University)
Shin, Yong-Su (Department of Mathematics Sungshin Women's University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.2, 2020 , pp. 283-312 More about this Journal
Abstract
A sufficient condition for an Artinian complete intersection quotient S = 𝕜[x, y]/(xm, yn), where 𝕜 is an algebraically closed field of a prime characteristic, to have the strong Lefschetz property (SLP) was proved by S. B. Glasby, C. E. Praezer, and B. Xia in [3]. In contrast, we find a necessary and sufficient condition on m, n satisfying 3 ≤ m ≤ n and p > 2m-3 for S to fail to have the SLP. Moreover we find the Jordan types for S failing to have SLP for m ≤ n and m = 3, 4.
Keywords
Jordan types; strong Lefschetz property; weak Lefschetz property; Hilbert function;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 T. Harima, J. Migliore, U. Nagel, and J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras, J. Algebra 262 (2003), no. 1, 99-126. https://doi.org/10.1016/S0021-8693(03)00038-3   DOI
2 D. G. Higman, Indecomposable representations at characteristic p, Duke Math. J. 21 (1954), 377-381. http://projecteuclid.org/euclid.dmj/1077465741   DOI
3 J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9, Springer-Verlag, New York, 1978.
4 A. Iarrobino, P. M. Marques, and C. McDaniel, Artinian Algebras and Jordan type, arXiv:math.AC/1802.07383 (2018).
5 K. Iima and R. Iwamatsu, On the Jordan decomposition of tensored matrices of Jordan canonical forms, Math. J. Okayama Univ. 51 (2009), 133-148.
6 A. C. Aitken, The Normal Form of Compound and Induced Matrices, Proc. London Math. Soc. (2) 38 (1935), 354-376. https://doi.org/10.1112/plms/s2-38.1.354   DOI
7 D. Cook, II, The Lefschetz properties of monomial complete intersections in positive characteristic, J. Algebra 369 (2012), 42-58. https://doi.org/10.1016/j.jalgebra.2012.07.015   DOI
8 S. P. Glasby, C. E. Praeger, and B. Xia, Decomposing modular tensor products, and periodicity of Jordan partitions', J. Algebra 450 (2016), 570-587. https://doi.org/10.1016/j.jalgebra.2015.11.025   DOI
9 J. A. Green, The modular representation algebra of a finite group, Illinois J. Math. 6 (1962), 607-619. http://projecteuclid.org/euclid.ijm/1255632708   DOI
10 T. Harima, T. Maeno, H. Morita, Y. Numata, A.Wachi, and J.Watanabe, The Lefschetz properties, Lecture Notes in Mathematics, 2080, Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-38206-2
11 J. Migliore and U. Nagel, Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra 5 (2013), no. 3, 329-358. https://doi.org/10.1216/JCA-2013-5-3-329   DOI
12 Y. R. Kim and Y.-S. Shin, An Artinian point-configuration quotient and the strong Lefschetz property, J. Korean Math. Soc. 55 (2018), no. 4, 763-783. https://doi.org/10.4134/JKMS.j170035   DOI
13 D. E. Littlewood, Polynomial Concomitants and Invariant Matrices, J. London Math. Soc. 11 (1936), no. 1, 49-55. https://doi.org/10.1112/jlms/s1-11.1.49   DOI
14 S. Lundqvist and L. Nicklasson, On the structure of monomial complete intersections in positive characteristic, J. Algebra 521 (2019), 213-234. https://doi.org/10.1016/j.jalgebra.2018.11.024   DOI
15 R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168-184. https://doi.org/10.1137/0601021   DOI
16 L. Nicklasson, The strong Lefschetz property of monomial complete intersections in two variables, Collect. Math. 69 (2018), no. 3, 359-375. https://doi.org/10.1007/s13348-017-0209-3   DOI
17 W. E. Roth, On direct product matrices, Bull. Amer. Math. Soc. 40 (1934), no. 6, 461-468. https://doi.org/10.1090/S0002-9904-1934-05899-3   DOI
18 B. Srinivasan, The modular representation ring of a cyclic p-group, Proc. London Math. Soc. (3) 14 (1964), 677-688. https://doi.org/10.1112/plms/s3-14.4.677   DOI
19 J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, in Commutative algebra and combinatorics (Kyoto, 1985), 303-312, Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987. https://doi.org/10.2969/aspm/01110303   DOI