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David L. Wehlau

Publications and source records attributed to David L. Wehlau.

14 recordsLinked to original sources
arXiv2026

When are permutation invariants Cohen-Macaulay?

Over a field of characteristic 0, every ring of invariants of a finite group is Cohen-Macaulay. This is not true for fields of positive characteristic. We consider permutation representations and their invariant rings over fields $\mathbb{F}_p$ of prime order. We give an efficient algorithm which for any given permutation representation, determines those primes $p$ for which the invariant ring over $\mathbb{F}_p$ is Cohen-Macaulay, using linear algebra over $\ZZ$. A generalization of the classical discriminant associated to the alternating group is defined for subgroups of certain finite unitary complex reflection groups.

math.AC↗
arXiv2026

Invariants of Finite Orthogonal Groups of Plus Type in Odd Characteristic

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining characteristic. In both cases we construct minimal algebra generating sets and describe the relations among the generators. Both rings of invariants are shown to be complete intersections and thus are Cohen-Macaulay. We expect the techniques we use will generalise to give a systematic computation for rings of invariants for all of the finite classical groups in odd characteristic.

math.AC↗
arXiv2020

Finite Planes, Zigzag Sequences, Fibonacci Numbers, Artin's Conjecture and Trinomials

We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a new family of polynomials which exhibit a number of special properties. These polynomials satisfy a three term recursion and are closely related to zigzag zero-one sequences. Interpreting the polynomials for the "prime" 1 yields the classical Morgan-Voyce polynomials, which form twoorthogonal families of polynomials and which have applications in the study of electrical resistance. Study of the general polynomials reveals deep connections with the Fibonacci series, the order of appearance of prime numbers in the Fibonacci sequence, the order of elements in cyclic groups, Artin's conjecture on primitive roots and the factorization of trinomials over finite fields.

math.NT↗
arXiv2019

Modular invariants of finite gluing groups

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic $p$-group acting on an elementary abelian $p$-group.

math.AC↗
arXiv2017

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 ${\rm GL}(n, \mathbb{F}_q)$, ${\rm SL}(n, \mathbb{F}_q)$ or ${\rm U}(n, \mathbb{F}_q)$ 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 homogenous invariant polynomials $\{\ell_1,\ell_{2},\dots,\ell_{(m+d)n}\}\subseteq \mathbb{F}_q[mW\oplus d W^*]^G$ such that $\mathbb{F}_q(mW\oplus d W^*)^G=\mathbb{F}_q(\ell_1,\ell_2,\dots,\ell_{(m+d)n})$ for all cases except when $md=0$ and $G={\rm GL}(n, \mathbb{F}_q)$ or ${\rm SL}(n, \mathbb{F}_q)$.

math.AC↗
arXiv2017

Symmetric Complete Intersections

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals, and describe formulas for the graded characters of the corresponding quotient rings.

math.AC↗
arXiv2016

Degrees of regular sequences with a symmetric group action

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

math.AC↗
arXiv2016

Hilbert Functions of $\mathfrak S_n$-Stable Artinian Gorenstein Algebras

We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.

math.AC↗
arXiv2013

The Second Main Theorem Vector for the modular regular representation of $C_2$

We study the ring of invariants for a finite dimensional representation $V$ of the group $C_2$ of order 2 in characteristic $2$. Let $σ$ denote a generator of $C_2$ and $\{x_1,y_1 \dots, x_m,y_m\}$ a basis of $V^*$. Then $σ(x_i) = x_i$, and $σ(y_i) = y_i + x_i$. To our knowledge, this ring (for any prime $p$) was first studied by David Richman in 1990. He gave a first main theorem for $(V_2, C_2)$, that is, he proved that the ring of invariants when $p=2$ is generated by $\{x_i, N_i = y_i^2 + x_iy_i, tr(A) | 2 \le |A| \le m\}$ where $A \subset \{0,1\}^m$, $y^A = y_1^{a_1} y_2^{a_2} \cdots y_m^{a_m}$ and $tr(A) = y^A + (y_1+x_1)^{a_1}(y_2+x_2)^{a_2} \cdots (y_m+x_m)^{a_m}.$ In this paper, we prove the second main theorem for $(V_2, C_2)$, that is, we show that all relations between these generators are generated by relations of type I: $\sum_{I \subset A } x^I tr(A-I)$ and of type II: $tr(A) tr(B) = \sum_{L < I} x^{I-L} N^L tr(I-L+J+K) + N^I \sum_{L < J} x^{J-L} tr(L+K)$ for all $m$. We also derive relations of type III which are simpler and can be used in place of the relations of type II.

math.RT↗
arXiv2012

Resolutions of 2 and 3 dimensional rings of invariants for cyclic groups

Let $G$ be the cyclic group of order $n$ and suppose ${\bf F}$ is a field containing a primitive $n^\text{th}$ root of unity. We consider the ring of invariants ${\bf F}[W]^G$ of a three dimensional representation $W$ of $G$ where $G \subset \text{SL}(W)$. We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of $F[W]^G$. The case where $W$ is any two dimensional representation of $G$ is also handled.

math.AC↗
arXiv2012

Weitzenböck derivations of nilpotency 3

We consider a Weitzenböck derivation $Δ$ acting on a polynomial ring $R=K[ξ_1,ξ_2,...,ξ_m]$ over a field $K$ of characteristic 0. The $K$-algebra $R^Δ= \{h \in R \mid Δ(h) = 0\}$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for $Δ$ acting on $R_1$, the degree 1 component of $R$, has only Jordan blocks of size 2. He conjectured (\cite{N}) that a certain set generates $R^Δ$ in that case. Recently Koury (\cite{Kh}), Drensky and Makar-Limanov (\cite{DM}) and Kuroda (\cite{K}) have given proofs of Nowicki's conjecture. Here we consider the case where the Jordan matrix for $Δ$ acting on $R_{1}$ has only Jordan blocks of size at most 3. Here we use combinatorial methods to give a minimal set of generators $\mathcal G$ for the algebra of constants $R^Δ$. Moreover, we show how our proof yields an algorithm to express any $h \in R^Δ$ as a polynomial in the elements of $\mathcal G$. In particular, our solution shows how the classical techniques of polarization and restitution may be used to augment the techniques of SAGBI bases to construct generating sets for subalgebras.

math.RA↗
arXiv2011

Invariants for the Modular Cyclic Group of Prime Order via Classical Invariant Theory

Let $F$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $F$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\leq n_1,n_2,..., n_k \leq p-1$ such that $V \cong \oplus_{i=1}^k V_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $\SL_2(\C)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R.J. Shank, which reduces the computation of the ring of $C_p$-invariants $F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p}$ to the computation of the classical ring of invariants (or covariants) $\C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\SL_2(\C)}$. This shows that the problem of computing modular $C_p$ invariants is equivalent to the problem of computing classical $\SL_2(\C)$ invariants. This allows us to compute for the first time the ring of invariants for many representations of $C_p$. In particular, we easily obtain from this generators for the rings of vector invariants $F[m V_2]^{C_p}$, $F[m V_3]^{C_p}$ and $F[m V_4]^{C_p}$for all $m \in \N$. This is the first computation of the latter two families of rings of invariants.

math.RA↗
arXiv2004

Complete caps in projective space which are disjoint from a subspace of codimension two

Working over the field of order 2 we consider those complete caps (maximal sets of points with no three collinear) which are disjoint from some codimension 2 subspace of projective space. We derive restrictive conditions which such a cap must satisfy in order to be complete. Using these conditions we obtain explicit descriptions of complete caps which do not meet every hyperplane in at least 5 points. In particular, we determine the set of cardinalities of all such complete caps in all dimensions.

math.CO↗