DOI QR코드

DOI QR Code

THE KERNELS OF THE LINEAR MAPS OF FINITE GROUP ALGEBRAS

  • Dan Yan (MOE-LCSM School of Mathematics and Statistics Hunan Normal University)
  • Received : 2023.01.02
  • Accepted : 2023.09.01
  • Published : 2024.01.31

Abstract

Let G be a finite group, K a split field for G, and L a linear map from K[G] to K. In our paper, we first give sufficient and necessary conditions for Ker L and Ker L ∩ Z(K[G]), respectively, to be Mathieu-Zhao spaces for some linear maps L. Then we give equivalent conditions for Ker L to be Mathieu-Zhao spaces of K[G] in term of the degrees of irreducible representations of G over K if G is a finite Abelian group or G has a normal Sylow p-subgroup H and L are class functions of G/H. In particular, we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if K is a split field for G.

Keywords

Acknowledgement

The author is very grateful to professor Wenhua Zhao for some useful suggestions. She is also grateful to the Department of Mathematics of Illinois State University, where this paper was partially finished, for hospitality during her stay as a visiting scholar. The author is very grateful to the referee for some useful suggestions and comments.

References

  1. J. J. Duistermaat and W. van der Kallen, Constant terms in powers of a Laurent polynomial, Indag. Math. (N.S.) 9 (1998), no. 2, 221-231. https://doi.org/10.1016/S0019-3577(98)80020-7
  2. A. van den Essen, S. Kuroda, and A. J. Crachiola, Polynomial Automorphisms and the Jacobian Conjecture-new results from the beginning of the 21st century, Frontiers in Mathematics, Birkhauser/Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-60535-3
  3. I. M. Isaacs, Character Theory of Finite Groups, Dover, New York, 1994.
  4. A. Konijnenberg, Mathieu subspaces of finite products of matrix rings, Master's thesis, Radboud University Nijmegen, The Netherlands, 2012.
  5. G. Navarro, Characters and Blocks of Finite Groups, London Mathematical Society Lecture Note Series, 250, Cambridge Univ. Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511526015
  6. D. S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York, 1977.
  7. W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010), no. 7, 1200-1216. https://doi.org/10.1016/j.jpaa.2009.10.007
  8. W. Zhao, Mathieu subspaces of associative algebras, J. Algebra 350 (2012), 245-272. https://doi.org/10.1016/j.jalgebra.2011.09.036
  9. W. Zhao and R. Willems, An analogue of the Duistermaat-van der Kallen theorem for group algebras, Cent. Eur. J. Math. 10 (2012), no. 3, 974-986. https://doi.org/10.2478/s11533-012-0028-4
  10. W. Zhao and D. Yan, Existence of nonzero trace-zero idempotents in the group algebras of finite groups, J. Algebra Appl. (2024).