Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AN ARTINIAN RING HAVING THE STRONG LEFSCHETZ PROPERTY AND REPRESENTATION THEORY

  • Shin, Yong-Su (Department of Mathematics Sungshin Women's University)
  • Received : 2019.03.06
  • Accepted : 2019.11.25
  • Published : 2020.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

It is well-known that if char𝕜 = 0, then an Artinian monomial complete intersection quotient 𝕜[x1, …, xn]/(x1a1, …, xnan) has the strong Lefschetz property in the narrow sense, and it is decomposed by the direct sum of irreducible 𝖘𝖑2-modules. For an Artinian ring A = 𝕜[x1, x2, x3]/(x16, x26, x36), by the Schur-Weyl duality theorem, there exist 56 trivial representations, 70 standard representations, and 20 sign representations inside A. In this paper we find an explicit basis for A, which is compatible with the S3-module structure.

Keywords

Acknowledgement

Supported by : Sungshin Women's University

This paper was supported by a grant from Sungshin Women's University.

References

  1. W. Fulton, Young Tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997.
  2. W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0979-9
  3. R. Goodman and N. R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications,68, Cambridge University Press, Cambridge, 1998.
  4. 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
  5. 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
  6. J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer-Verlag, New York, 1978.
  7. A. Iarrobino, P. M. Marques, and C. MaDaniel, Jordan type and the Associated graded algebra of an Artinian Gorenstein algebra, arXiv:1802.07383 (2018).
  8. S. J. Kang, Y. R. Kim, and Y. S. Shin, The strong Lefschetz property and representation theory, In parparation.
  9. 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
  10. 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
  11. 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