×

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order. (English) Zbl 1198.13009

Assume that a finite group \(G\) is represented on a vector space \(V\) over a field \(F\) and \(\{x_1,\ldots,x_n\}\) is a basis for \(V^{\ast}=\mathrm{hom}_{F}(V,F)\). Then \(G\) also acts on the symmetric algebra, \(F[V]=F[x_1,\ldots,x_n]\), of \(V^{\ast}\), by \((g\cdot f)(v)=f(g^{-1}v)\). Let us remark that if \(F\) is the prime field \(F_p\), then \(F[V]\) is not the algebra of regular functions on \(V\), because the functions \(x_1\) and \(x_1^p\) coincide on \(V\). The algebra of elements of \(F[V]\) that are left invariant by the action of \(G\) is denoted by \(F[V]^G\). The case is called modular if the characteristic of \(F\) is \(p>0\) and \(p\) divides \(|G|\).
Consider the non-modular case. Then \(F[V]^G\) possesses some nice properties. For example, it is always Cohen-Macaulay. Further, \(F[V]^G\) is a polynomial algebra if and only if \(|G|\) is generated by reflections (group elements fixing a hyperplane of \(V\)). It was established by Noether for \(p=0\) and by J. Fogarty [Electron. Res. Announc. Am. Math. Soc. 7, 5–7 (2001; Zbl 0980.13003)], P. Fleischmann [Adv. Math. 156 (1), 23–32 (2000; Zbl 0973.13003)] for \(p>0\) that \(F[V]^G\) is generated by elements of degree less than or equal to \(|G|\). But in the modular case these results do not hold. There are now several references for modular invariant theory, for example, [see M. D. Neusel and L. Smith, Invariant theory of finite groups. Mathematical Surveys and Monographs. 94. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 0999.13002); H. Derksen and G. Kemper, Computational invariant theory. Encyclopaedia of Mathematical Sciences. Invariant Theory and Algebraic Transformation Groups. 130(1). Berlin: Springer (2002; Zbl 1011.13003)].
In the paper under review \(G=C_p\) is the cyclic group, the characteristic of \(F\) is \(p>0\), and \(G\) acts diagonally on \(V=mV_2=V_2\oplus \cdots \oplus V_2\), where \(V_2\) stands for the 2-dimensional indecomposable representation of \(C_p\). The algebra of invariants \(F[mV_2]^{C_p}\) was first studied by D. R. Richman [Adv. Math. 81, No. 1, 30–65 (1990; Zbl 0715.13002)]. He showed that the algebra is not generated by elements of degree less than \(m(p-1)\), thus demonstrating that the Noether bound on degrees of generators does not hold in the modular case. A set of generators was established by H.E.A. Campbell and I. P. Hughes [Adv. Math. 126, 1–20 (1997; Zbl 0877.13004)]. R. J. Shank and D. L. Wehlau [J. Symbolic Comput. 34 (5), 307–327 (2002; Zbl 1048.13002)] described a minimal generating set for the algebra of invariants.
In the paper under review, a new proof of the result of Shank and Wehlau is given. Moreover, it is shown that the obtained generating set is also a SAGBI basis for \(F[mV_2]^{C_p}\) and a procedure for finding explicit decomposition of \(F[mV_2]\) into a direct sum of indecomposable \(C_p\)-modules is described. As an application, a generating set for the algebra of invariants \(F[mV_2]^{SL_2(F_p)}\) is obtained. Finally, it is shown that \(F[mV_2]^{SL_2(F_p)}\) is generated by elements of degree less than or equal to \((p+m-2)(p-1)\) and this upper bound is explicit.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
20C20 Modular representations and characters

Software:

Magma
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Benson, D. J., Polynomial Invariants of Finite Groups, London Math. Soc. Lecture Note Ser., vol. 190 (1993), Cambridge Univ. Press · Zbl 0864.13001
[2] Bosma, W.; Cannon, J. J.; Playoust, C., The Magma algebra system I: The user language, J. Symbolic Comput., 24, 235-265 (1997) · Zbl 0898.68039
[3] Campbell, H. E.A.; Hughes, I. P., Vector invariants of \(U_2(F_p)\): A proof of a conjecture of Richman, Adv. Math., 126, 1-20 (1997) · Zbl 0877.13004
[4] Campbell, H. E.A.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L., Bases for rings of coinvariants, Transform. Groups, 1, 4, 307-336 (1996) · Zbl 0877.20006
[6] Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math., 77, 778-782 (1955) · Zbl 0065.26103
[7] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms (1992), Springer-Verlag
[8] Coxeter, H. S.M., The product of the generators of a finite group generated by reflections, Duke Math. J., 18, 765-782 (1951) · Zbl 0044.25603
[9] Derksen, H.; Kemper, G., Computational Invariant Theory, Invariant Theory and Algebraic Transformation Groups, I, Encyclopaedia Math. Sci., vol. 130 (2002), Springer-Verlag
[10] Dickson, L. E., A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc., 12, 75-98 (1911) · JFM 42.0136.01
[11] Ellingsrud, G.; Skjelbred, T., Profondeur d’anneaux d’invariants en caractéristique \(p\), Compos. Math., 41, 2, 233-244 (1980) · Zbl 0438.13007
[12] Fleischmann, P., The Noether bound in invariant theory of finite groups, Adv. Math., 152, 1, 23-32 (2000) · Zbl 0973.13003
[13] Fogarty, J., On Noether’s bound for polynomial invariants of a finite group, Electron. Res. Announc. Amer. Math. Soc., 7, 5-7 (2001) · Zbl 0980.13003
[14] Hughes, I. P.; Kemper, G., Symmetric powers of modular representations, Hilbert series and degree bounds, Comm. Algebra, 28, 2059-2088 (2000) · Zbl 0965.13004
[15] Kapur, D.; Madlener, K., A completion procedure for computing a canonical basis of a \(k\)-subalgebra, (Kaltofen, E.; Watt, S., Proceedings of Computers and Mathematics, vol. 89 (1989), MIT), 1-11
[16] Kempe, A., On regular difference terms, Proc. Lond. Math. Soc., 25, 343-350 (1894) · JFM 25.0235.01
[17] Koshy, T., Catalan Numbers with Application (November 2008), Oxford University Press
[18] Neusel, M. D.; Smith, L., Invariant Theory of Finite Groups, Math. Surveys Monogr., vol. 94 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0999.13002
[19] Noether, E., Der Endlichkeitssatz der invarianten endlicher Gruppen, Math. Ann.. (Collected Papers (1983), Springer-Verlag: Springer-Verlag Berlin), 77, 181-184 (1915), reprinted · JFM 45.0198.01
[20] Richman, D. R., On vector invariants over finite fields, Adv. Math., 81, 1, 30-65 (1990) · Zbl 0715.13002
[21] Robbianno, L.; Sweedler, M., Subalgebra Bases, Lecture Notes in Math., vol. 1430 (1990), Springer, pp. 67-87
[22] Serre, J.-P., Groupes finis d’automorphismes d’anneaux locaux réguliers, (Colloque d’Algèbre. Colloque d’Algèbre, Paris, 1967. Colloque d’Algèbre. Colloque d’Algèbre, Paris, 1967, Exp., vol. 8 (1968), Secrétariat Mathématique: Secrétariat Mathématique Paris), 11 pp · Zbl 0200.00002
[23] Shank, R. J., S.A.G.B.I. bases for rings of formal modular seminvariants, Comment. Math. Helv., 73, 4, 548-565 (1998) · Zbl 0929.13001
[24] Shank, R. J.; Wehlau, D. L., Computing modular invariants of \(p\)-groups, J. Symbolic Comput., 34, 5, 307-327 (2002) · Zbl 1048.13002
[25] Shank, R. J.; Wehlau, D. L., Noether numbers for subrepresentations of cyclic groups of prime order, Bull. Lond. Math. Soc., 34, 4, 438-450 (2002) · Zbl 1071.13001
[26] Shephard, G. C.; Todd, J. A., Finite unitary reflection groups, Canad. J. Math., 6, 274-304 (1954) · Zbl 0055.14305
[27] Smith, L., Polynomial Invariants of Finite Groups, Res. Notes Math., vol. 6 (1995), A K Peters Ltd.: A K Peters Ltd. Wellesley, MA · Zbl 0864.13002
[28] Weyl, H., The Classical Groups (1997), Princeton University Press · JFM 65.0058.02
[29] Wilkerson, C. W., A Primer on the Dickson Invariants, Amer. Math. Soc. Contemp. Math. Ser., 19, 421-434 (1983) · Zbl 0525.55013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.