Representations of elementary abelian p-groups and finite subgroups of fields

Suppose $\mathbb{F}$ is a field of prime characteristic p and E is a finite subgroup of the additive group $(\mathbb{F},+)$. Then E is an elementary abelian p-group. We consider two such subgroups, say E and $E^{\prime }$, to be equivalent if there is an $\alpha \in {\mathbb{F}}^{\times }:=\mathbb{F}?\{0\}$ such that $E=\alpha E^{\prime }$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.     

On Macaulay duals of Hilbert ideals

In this note we present some computational tools to aide in the determination of Macaulay inverses of Hilbert ideals of finite groups and related ideals.     

On the Wronskian combinants of binary forms

Given a sequence A=(A₁,…,A_r) of binary d-ics, we construct a set of combinants C={C_q:0≤q≤r,q≠1}, to be called the Wronskian combinants of A. We show that the span of A can be recovered from C as the solution space of an SL(2)-invariant differential equation. The Wronskian combinants define a projective imbedding of the Grassmannian G(r,S_d), and, as a corollary, any other combinant of A is expressible as a compound transvectant in C.Our main result characterises those sequences of binary forms that can arise as Wronskian combinants; namely, they are the ones such that the associated differential equation has the maximal number of linearly independent polynomial solutions. Along the way we deduce some identities which relate Wronskians to transvectants. We also calculate compound transvectant formulae for C in the case r=3.     

On the coinvariants of modular representations of cyclic groups of prime order

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three-dimensional indecomposable representation.     

The work of Gian-Carlo rota on invariant theory

The purpose of this paper is twofold: first, to explain Gian-Carlo Rotas work on invariant theory; second, to place this work in a broad historical and mathematical context. Rotas work falls under three specific cases: vector invariants, the invariants of binary forms, and the invariants of skew-symmetric tensors. We discuss each of these cases and show how determinants and straightening play central roles. In fact, determinants constitute all invariants in the vector case; for binary forms and skew-symmetric tensors, they constitute all invariants when invariants are represented symbolically. Consequently, we explain the symbolic method both for binary forms and for skew-symmetric tensors, where Rota developed generalizations of the usual notion of a determinant. We also discuss the Grassmann algebra, with its two operations of meet and join, which was a theme which ran through Rotas work on invariant theory almost from the very beginning.To the memory of Gian-Carlo Rota     

Degree bounds—An invitation to postmodern invariant theory

This is an invitation to invariant theory of finite groups; a field where methods and results from a wide range of mathematics merge to form a new exciting blend. We use the particular problem of finding degree bounds to illustrate this.     

On the saturation sequence of the rational normal curve

Let denote the rational normal curve of order d. Its homogeneous defining ideal admits an SL₂-stable filtration J₂⊆J₄⊆…⊆I_C by sub-ideals such that the saturation of each J_2q equals I_C. Hence, one can associate to d a sequence of integers (α₁,α₂,…) which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the α_q, using inter alia the methods of classical invariant theory.     

The Noether numbers for cyclic groups of prime order

The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p−3 conjecture.”     

An algorithm for computing the kernel of a locally finite iterative higher derivation

Let R be a PID. This article gives an algorithm for computing the kernel of a locally finite iterative higher derivation of a finitely generated R-domain.     

On the kernels of some higher derivations in polynomial rings

Let A=R[x₁,…,x_n] be the polynomial ring in n variables over an integral domain R with unit, let D be a rational higher R-derivation on A and let be the extension of D to the quotient field of A. We prove that, if the transcendental degree of the kernel of D over R is not less than n−1, then the quotient field of the kernel of D equals the kernel of . Moreover, when n=2, we give a necessary and sufficient condition for an R-subalgebra of A to be expressed as the kernel of a rational higher R-derivation on A.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Noncommuting selfmaps and invariant approximations

Common fixed-point results are established for a new class of noncommuting selfmaps. We apply them to obtain several invariant approximation results which unify, extend, and complement well-known results.     

On the Dimension of Coinvariants of Permutation Representations

Let be a univariate, separable polynomial of degree n with roots x ₁,…,x _n in some algebraic closure of the ground field . It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on by permutation of the coordinates and each defines a relation P − P(x ₁,…,x _n ) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants has dimension ‖G‖ over . To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σ _n acting via the tautological permutation representation.     

On the multiplicity and the Cohen-Macaulayness of fiber cones of graded algebras

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?_n≥0S_n be a finitely generated standard graded algebra over A. Set S⁺=?_n>0S_n. Denote by F_J(S)=?_n≥0→(S_n/JS_n) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of F_J(S) in terms of minimal reductions of S⁺.     

Star-generating vectors of Rudin's quotient modules

The purpose of this paper is to study a class of quotient modules of the Hardy module H²(Dⁿ)$H^2({\mathbb{D}}^n)$. Along with the two variables quotient modules introduced by W. Rudin, we introduce and study a large class of quotient modules, namely Rudin's quotient modules of H²(Dⁿ)$H^2({\mathbb{D}}^n)$. By exploiting the structure of minimal representations we obtain an explicit co-rank formula for Rudin's quotient modules.     

On the weak topology of Banach spaces over non-archimedean fields

It is known that within metric spaces analyticity and K-analyticity are equivalent concepts. It is known also that non-separable weakly compactly generated (shortly WCG) Banach spaces over R or C provide concrete examples of weakly K-analytic spaces which are not weakly analytic. We study the case which totally differs from the above one. A general theorem is provided which shows that a Banach space E over a locally compact non-archimedean non-trivially valued field is weakly Lindelöf iff E is separable iff E is WCG iff E is weakly web-compact (in the sense of Orihuela). This provides a non-archimedean version of a remarkable Amir-Lindenstrauss theorem.     

Degree bounds for syzygies of invariants

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in (Proc. Amer. Math. Soc. 129 (4) (2001) 955). Here we study minimal free resolutions of invariant rings. For finite linearly reductive groups G it was recently shown in (Adv. Math. 156 (1) (2000) 23, Electron Res. Announc. Amer. Math. Soc. 7 (2001) 5, Adv. Math. 172 (2002) 151) that rings of invariants are generated in degree at most the group order |G|. In characteristic 0 this degree bound is a classical result by Emmy Noether (see Math. Ann. 77 (1916) 89). Given an invariant ring of a finite linearly reductive group G, we prove that the ideal of relations of a minimal set of generators is generated in degree at most ?2|G|.