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.”     

Modular Lie representations of groups of prime order

Let K be a field of prime characteristic p and let G be a group of order p. For any finite-dimensional KG-module V and any positive integer n let L ⁿ (V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then L ⁿ (V) can be considered as a KG-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of L ⁿ (V) up to isomorphism. Mathematics Subject Classification (2000): 17B01, 20C20.     

On the existence of semifields of prime power order admitting free automorphism groups

The purpose of this paper is to prove the existence of semifields of order q ⁴ for any odd prime power q = p^r, q > 3, admitting a free automorphism group isomorphic to Z ₂ × Z ₂.     

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 behavior of elements of prime order from Singer cycles in representations of special linear groups

We consider extremal problems for continuous functions that are nonpositive on a closed interval and can be represented as series in Gegenbauer polynomials with nonnegative coefficients. These problems arise from the Delsarte method of finding an upper bound for the kissing number in a Euclidean space. We develop a general method for solving such problems. Using this method, we reproduce results of previous authors and find a solution in the following 11 new dimensions: 147, 157, 158, 159, 160, 162, 163, 164, 165, 167, and 173. The arising extremal polynomials are of a new type.     

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 invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V₂ of the cyclic group, Cp, of order p over a field F of characteristic p, FCp[mV₂]. This ring of invariants was first studied by David Richman (1990) [20] who showed that the ring required a generator of degree m(p−1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes (1997) in [3]. Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants FCp[mV₂]. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for FCp[mV₂]. Further, our results provide a procedure for finding an explicit decomposition of F[mV₂] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cp on V₂ is as the p-Sylow subgroup of SL₂(Fp), we describe a generating set for the ring of invariants F[mV₂]^SL₂(Fp) and show that (p+m−2)(p−1) is an upper bound for the Noether number, for m>2.     

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.     

On the topology of valuation-theoretic representations of integrally closed domains

Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let $A(X)={?}_{V\in X}V$ and $J(X)={?}_{V\in X}{\mathfrak{M}}_V$, where ${\mathfrak{M}}_V$ denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring $A(X)$. We show that if $J(X)\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that $A(X)=A(Y)$. If any countable subset of points is removed from Y, then the resulting set remains a representation of $A(X)$. Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring $A(X)$ with specific focus on topological conditions that guarantee $A(X)$ is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings $A(X)$ where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.     

On collineation groups of a projective plane of prime order

Towards the study of finite projective plane of prime order, the following result is proved in this paper. Let be a projective plane of prime order p and let G be a collineation group of . If p¦|G|, then either is Desarguesian or the maximal maximal normal subgroup of G is not trivial. In particular, is Desarguesian if G does not leave invariant any point or line.Partially supported by NSERC A8460.Partially supported by CNPq.     

On restrictions of modular spin representations of symmetric and alternating groups

Let be an algebraically closed field of characteristic and be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups of and -modules such that the restriction is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where is the Schur's double cover or .