arXiv subjects

H. E. A. Campbell

Publications and source records attributed to H. E. A. Campbell.

8 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↗

arXiv2025

Invariants of the finite orthogonal groups in odd dimension and even characteristic

We describe the ring of invariants for the finite orthogonal groups in odd dimension and even characteristic acting on the defining representation. We construct a minimal algebra generating set and describe the relations among the generators. This ring of invariants is shown to be a complete intersection and thus is Cohen-Macaulay. This extends the previous computation of Kropholler, Mohseni Rajaei, and Segal valid over the field of order 2.

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↗

arXiv2018

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 $E'$, to be equivalent if there is an $α\in\mathbb{F}^*:=\mathbb{F}\setminus\{0\}$ such that $E=αE'$. 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.

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

Rings of invariants for modular representations of elementary abelian p-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three dimensional representation of an elementary abelian p-group is a complete intersection.

math.AC↗

arXiv2009

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

In this paper, we study the vector invariants, ${\bf{F}}[m V_2]^{C_p}$, of the 2-dimensional indecomposable representation $V_2$ of the cylic group, $C_p$, of order $p$ over a field ${\bf{F}}$ of characteristic $p$. This ring of invariants was first studied by David Richman \cite{richman} who showed that this 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 in \cite{campbell-hughes}. Later, Shank and Wehlau in \cite{cmipg} 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 ${\bf{F}}[m V_2]^{C_p}$. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for ${\bf{F}}[m V_2]^{C_p}$. Further, our techniques also serve to give an explicit decomposition of ${\bf{F}}[m V_2]$ into a direct sum of indecomposable $C_p$-modules. Finally, noting that our representation of $C_p$ on $V_2$ is as the $p$-Sylow subgroup of $SL_2({\bf F}_p)$, we are able to determine a generating set for the ring of invariants of ${\bf{F}}[m V_2]^{SL_2({\bf F}_p)}$.

math.AC↗