Method of orbit sums in the theory of modular vector invariants
- Institute for Information Transmission Problems Russian Academy of Sciences, Moscow (Russian Federation)
Let F be a field, V a finite-dimensional F-vector space, G{<=}GL{sub F}(V) a finite group, and V{sup m}=V+...+V the m-fold direct sum with the diagonal action of G. The group G acts naturally on the symmetric graded algebra A{sub m}=F[V{sup m}] as a group of non-degenerate linear transformations of the variables. Let A{sub m}{sup G} be the subalgebra of invariants of the polynomial algebra A{sub m} with respect to G. A classical result of Noether [1] says that if charF=0, then A{sub m}{sup G} is generated as an F-algebra by homogeneous polynomials of degree at most |G|, no matter how large m can be. On the other hand, it was proved by Richman [2], [3] that this result does not hold when the characteristic of F is positive and divides the order |G| of G. Let p, p>2, be a prime number, F=F{sub p} a finite field of p elements, V a linear F{sub p}-vector space of dimension n, and H{<=}GL{sub F{sub p}}(V) a cyclic group of order p generated by a matrix {gamma} of a certain special form. In this paper we describe explicitly (Theorem 1) one complete set of generators of A{sub m}{sup H}. After that, for an arbitrary complete set of generators of this algebra we find a lower bound for the highest degree of the generating elements of this algebra. This is a significant extension of the corresponding result of Campbell and Hughes [4] for the particular case of n=2. As a consequence we show (Theorem 3) that if m>n and G{>=}H is an arbitrary finite group, then each complete set of generators of A{sub m}{sup G} contains an element of degree at least 2(m-n+2r)(p-1)/r, where r=r(H) is a positive integer dependent on the structure of the generating matrix {gamma} of the group H. This result refines considerably the earlier lower bound obtained by Richman [3].
- OSTI ID:
- 21267029
- Journal Information:
- Sbornik. Mathematics, Vol. 197, Issue 11; Other Information: DOI: 10.1070/SM2006v197n11ABEH003816; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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