Let F be a field with characteristic 2 and C2α be a cyclic group with order 2α. The ring, which has a Z-basis by finite dimensional indecomposable F[C2α] modules and multiplications by tensor product of F module, is called representation ring and denoted by Rep(F[C2α]). Based on Higman’s work, we further get the Krull dimension and the concrete forms of all prime ideals of Rep(F[C2α]). And then, we prove that Rep(F[C2α]) is a reduced ring and find the minimal primary decomposition of the zero ideal. At last, we prove that Spec(Rep(F[C2α])) is a connected topological space.
LU Xin
,
TANG Guoping
. Krull dimension and primary decomposition of Rep(F[C2α])[J]. Journal of University of Chinese Academy of Sciences, 2025
, 42(2)
: 145
-152
.
DOI: 10.7523/j.ucas.2023.057
[1] Higman D G. Indecomposable representations at characteristic p[J]. Duke Mathematical Journal, 1954, 21(2): 377-381. DOI: 10.1215/s0012-7094-54-02138-9.
[2] Srinivasan B. The modular representation ring of a cyclic p-group[J]. Proceedings of the London Mathematical Society, 1964, 14(4): 677-688. DOI: 10.1112/plms/s3-14.4.677.
[3] Green J A. The modular representation algebra of a finite group[J]. Illinois Journal of Mathematics, 1962, 6(4): 607-619. DOI: 10.1215/ijm/1255632708.
[4] Webb P. A course in finite group representation theory[M]. Cambridge: Cambridge University Press, 2016.
[5] Atiyah M F, MacDonald I G. Introduction to commutative algebra[M]. Reading: Addison-Wesley Publishing Company, 1969.
[6] Anderson F W, Fuller K R. Rings and categories of Modules[M]. 2nd ed. New York: Springer New York, 1992. DOI: 10.1007/978-1-4612-4418-9.
