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Stephen P. Humphries

Publications and source records attributed to Stephen P. Humphries.

11 recordsLinked to original sources
arXiv2026

Schur rings over cyclic groups having Almost Commutative Terwilliger algebras

Terwilliger algebras are subalgebras of a matrix algebra constructed from an association scheme. Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions in the case where the association scheme is commutative. A sixth condition for a Terwilliger algebra coming from a commutative Schur ring to be almost commutative has since been discovered. In this paper we first provide a classification of orbit Schur rings that produce an almost commutative Terwilliger algebra for a finite cyclic group. In particular, we show that the subgroup of automorphisms used to form the orbit Schur ring is either trivial, or the whole automorphism group when the cyclic group has prime power order. If the cyclic group has order $2^n$, these are the only options. If the cyclic group has order $p^n$, for an odd prime $p$, then the automorphism subgroup of order $p^{n-1}$ also works. If the group has a non-prime power order then the only orbit Schur ring that produces an almost commutative Terwilliger algebra comes from the trivial subgroup of the automorphism group. We then give a condition for when a wedge product of Schur rings produces an almost commutative Terwilliger algebra. This allows us to determine exactly when a Schur ring over a cyclic group produces an almost commutative Terwilliger algebra.

math.CO↗
arXiv2025

Almost Commutative Terwilliger Algebras I: The Group Association Scheme

Terwilliger algebras are a subalgebra of a matrix algebra constructed from an association scheme. Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions. In this paper we first determine an equivalent sixth condition for a Terwilliger algebra coming from a commutative Schur ring to be almost commutative. We then provide a classification of which finite groups result in an almost commutative Terwilliger algebra when looking at the group association scheme determined by the conjugacy classes. In particular, we show that all such groups are either abelian, or Camina groups. We then compute the dimension of each Terwilliger algebra, and we also express each of the group association schemes with an almost commutative Terwilliger algebra as a wreath product of the group schemes of finite abelian groups and $1-$class association schemes. Furthermore, we give the non-primary primitive idempotents for each Terwilliger algebra for those groups.

math.RT↗
arXiv2025

Almost Commutative Terwilliger Algebras II: Strong Gelfand Pairs

Terwilliger algebras are a subalgebra of a matrix algebra constructed from an association scheme. In 2010, Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions for a Terwilliger algebra to be almost commutative. In this paper we look at Terwilliger algebras coming from strong Gelfand pairs $(G,H)$ for a finite group $G$. From such a pair, one can create a Terwilliger algebra using the Schur ring of $H-$classes of elements of $G$. We determine all strong Gelfand pairs that give an Almost Commutative Terwilliger algebra.

math.RT↗
arXiv2024

Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification

Terwilliger algebras are a subalgebra of a matrix algebra that are constructed from association schemes over finite sets. In 2010, Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative. In that paper she gave five equivalent conditions for a Terwilliger algebra to be almost commutative. In this paper, we provide a classification of which groups result in an almost commutative Terwilliger algebra when looking at the group association scheme (the Schur ring generated by the conjugacy classes of the group). In particular, we show that all such groups are either abelian, or Camina groups. Following this classification, we then compute the dimension and non-primary primitive idempotents for each Terwilliger algebra of this form for the first three types of groups whose group association scheme gives an almost commutative Terwilliger algebra. The final case will be considered in a second paper.

math.RT↗
arXiv2024

Schur rings over ${\bf {\rm Sp}(n,2)}$ and multiplicity one subgroups

We study commutative Schur rings over the symplectic groups Sp$(n,2)$ containing the class $\mathcal C$ of symplectic transvections. We find the possible partitions of $\mathcal C$ determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp$(n,2)$.

math.GR↗
arXiv2021

Strong Gelfand Pairs of SL(2,p)

A strong Gelfand pair (G,H) is a group G together with a subgroup H such that every irreducible character of H induces a multiplicity-free character of G. We classify the strong Gelfand pairs of the special linear groups SL(2, p) where p is a prime.

math.GR↗
arXiv2021

Difference sets disjoint from a subgroup II: groups of order $4p^2$

We study finite groups $G$ having a normal subgroup $H$ and $D \subset G \setminus H, D \cap D^{-1}=\emptyset,$ such that the multiset $\{ xy^{-1}:x,y \in D\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\it DRAD difference set}). We show that there are no such groups of order $4p^2$, where $p$ is an odd prime.

math.GR↗
arXiv2017

Difference sets disjoint from a subgroup

We study finite groups $G$ having a subgroup $H$ and $D \subset G \setminus H$ such that the multiset $\{ xy^{-1}:x,y \in D\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\it difference set}). We show that $H$ has to be normal, that $|G|=|H|^2$, and that $|D \cap Hg|=|H|/2$ for all $g \notin H$. We show that $H$ is contained in every normal subgroup of prime index, and other properties. We give a $2$-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets.

math.GR↗
arXiv2016

Weak Cayley table groups of some crystallographic groups

For a group $G$, a weak Cayley isomorphism is a bijection $f:G \to G$ such that $f(g_1g_2)$ is conjugate to $ f(g_1)f(g_2)$ for all $g_1,g_2 \in G$. They form a group $\mathcal W(G)$ that is the group of symmetries of the weak Cayley table of $G$. We determine $\mathcal W(G)$ for each of the seventeen wallpaper groups $G$, and for some other crystallographic groups.

math.GR↗
arXiv2015

Commutative Schur Rings Over Symmetric Groups II: The Case n=6

We determine the commutative Schur rings over $S_6$ that contain the sum of all the transpositions in $S_6$. There are eight such types (up to conjugacy), of which four have the set of all the transpositions as a principal set of the Schur ring.

math.GR↗