Lower degree bounds for modular vector invariants

Madran, Uğur

doi:10.1090/S0002-9939-06-08574-1

Lower degree bounds for modular vector invariants
HTML articles powered by AMS MathViewer

by Uğur Madran PDF

Proc. Amer. Math. Soc. 135 (2007), 987-995 Request permission

Abstract:

Let $G$ be a finite group of order divisible by a prime $p$ acting on an $\mathbb {F}$ vector space $V,$ where $\mathbb {F}$ is the field with $p$ elements and $\dim _{\mathbb {F}} V=n$. Consider the diagonal action of $G$ on $m$ copies of $V.$ This note sharpens a lower bound for $\beta (\mathbb {F}[\oplus _mV]^G)$ for groups which have an element of order $p$ whose Jordan blocks have sizes at most 2.

References

D. J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190, Cambridge University Press, Cambridge, 1993. MR 1249931, DOI 10.1017/CBO9780511565809
H. E. A. Campbell and I. P. Hughes, Vector invariants of $U_2(\mathbf F_p)$: a proof of a conjecture of Richman, Adv. Math. 126 (1997), no. 1, 1–20. MR 1440251, DOI 10.1006/aima.1996.1590
David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2
G. Kemper, Lower degree bounds for modular invariants and a question of I. Hughes, Transform. Groups 3 (1998), no. 2, 135–144. MR 1628445, DOI 10.1007/BF01236430
U. Madran, Modular Vector Invariants. Ph.D. Dissertation, Bilkent Univ., Ankara, to appear. will be available online through Bilkent University Library Thesis Database.
Emmy Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1915), no. 1, 89–92 (German). MR 1511848, DOI 10.1007/BF01456821
E. Noether, Der Endlichkeitssatz der Invarianten endlicher linear Gruppen der Charakteristik $p$. Nachr. Akad. Wiss. Göttingen (1926), 28-35.
David R. Richman, On vector invariants over finite fields, Adv. Math. 81 (1990), no. 1, 30–65. MR 1051222, DOI 10.1016/0001-8708(90)90003-6
David R. Richman, Invariants of finite groups over fields of characteristic $p$, Adv. Math. 124 (1996), no. 1, 25–48. MR 1423197, DOI 10.1006/aima.1996.0076
Larry Smith, Polynomial invariants of finite groups. A survey of recent developments, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211–250. MR 1433171, DOI 10.1090/S0273-0979-97-00724-6
Larry Smith, Polynomial invariants of finite groups, Research Notes in Mathematics, vol. 6, A K Peters, Ltd., Wellesley, MA, 1995. MR 1328644
S. A. Stepanov, Vector invariants of symmetric groups in the case of a field of prime characteristic, Diskret. Mat. 12 (2000), no. 4, 25–38 (Russian, with Russian summary); English transl., Discrete Math. Appl. 10 (2000), no. 5, 455–468. MR 1826176, DOI 10.1515/dma.2000.10.5.455
Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR 1255980, DOI 10.1007/978-3-7091-4368-1