David L. Wehlau

Modular invariant theory

1 First Steps.- 2 Elements of Algebraic Geometry and Commutative Algebra.- 3 Applications of Commutative Algebra to Invariant Theory.- 4 Examples.- 5 Monomial Orderings and SAGBI Bases.- 6 Block Bases.- 7 The Cyclic Group Cp.- 8 Polynomial Invariant Rings.- 9 The Transfer.- 10 Invariant Rings via Localization.- 11 Rings of Invariants which are Hypersurfaces.- 12 Separating Invariants.- 13 Using SAGBI Bases to Compute Rings of Invariants.- 14 Ladders.- References.- Index.

Polarization of Separating Invariants

We prove a characteristic free version of Weyls theorem on polarization. Our result is an exact analogue of Weyls theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

Bases for rings of coinvariants

We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants.

Binary codes and caps

The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in ℙG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere.

Noether numbers for subrepresentations of cyclic groups of prime order

Let W be a finite-dimensional Z/p-module over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W](Z/P), is called the Noether number of the representation, and is denoted by beta(W). A lower bound for beta(W) is derived, and it is shown that if U is a Z/p submodule of W, then beta(U) less than or equal to beta(W). A set of generators, in fact a SAGBI basis, is constructed for k[V2 circle plus V-3](Z/P), where V-n is the indecomposable Z/p-module of dimension n.

Computing Modular Invariants of p-groups

Let V be a finite dimensional representation of a p -group, G, over a field,k , of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, kV ]G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for kV ] G. We use these methods to analyse k 2V3 ]U3where U3is the p -Sylow subgroup ofGL3 (Fp) and 2 V3is the sum of two copies of the canonical representation. We give a generating set for k 2 V3]U3forp= 3 and prove that the invariants fail to be Cohen?Macaulay forp 2. We also give a minimal generating set for kmV2 ]Z/pwere V2is the two-dimensional indecomposable representation of the cyclic group Z/p.

Depth of modular invariant rings

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper1 we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].

Non-Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p ,t he ring of vector invariants of m copies of that representation is not Cohen-Macaulay for m 3. In the second section of the paper we use Poincarseries methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(jGj 1). If the ring of invariants is a hypersurface, the upper bound can be improved tojGj.

Long Binary Linear Codes and Large Caps in Projective Space

We obtain, in principle, a complete classification of all long inextendable binary linear codes. Several related constructions and results are presented.

The transfer in modular invariant theory

Abstract We study the transfer homomorphism in modular invariant theory paying particular attention to the image of the transfer which is a proper non-zero ideal in the ring of invariants. We prove that, for a p -group over F p whose ring of invariants is a polynomial algebra, the image of the transfer is a principal ideal. We compute the image of the transfer for SL n ( F q ) and GL n ( F q ) showing that both ideals are principal. We prove that, for a permutation group, the image of the transfer is a radical ideal and for a cyclic permutation group the image of the transfer is a prime ideal.