arXiv subjects

Emily Gullerud

Publications and source records attributed to Emily Gullerud.

4 recordsLinked to original sources

arXiv2026

Synthetic Buildings for Finite Groups

We introduce synthetic buildings for each finite group $G$ and prime $p$. These are $G$-simplicial complexes, among which are the $p$-subgroups complex of K.S. Brown, and also the buildings of finite groups of Lie type in characteristic $p$. Synthetic buildings are useful because of special properties, including providing a formula for the cohomology of $G$. Our goal is to describe the kinds of synthetic buildings that can arise, and ideally to classify them. The \textit{minimal} synthetic buildings play a special role and for most groups there are not very many of these. However, we show that it is possible to find sequences of finite groups with increasingly many minimal synthetic buildings as we move along the sequence, and also groups with minimal synthetic buildings of different dimensions. It is also possible to find groups with infinitely many non-minimal synthetic buildings of a given dimension. In other situations, we show that there are no minimal synthetic buildings of equivariant homotopy type distinct from those of Brown's complex and the complex consisting of a single point. We make a number of conjectures.

math.GR↗

arXiv2022

Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence

We explore a certain family $\{A_n\}_{n=1}^{\infty}$ of $n \times n$ tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of this family, we find a closed form solution. The coefficients of these characteristic polynomials turn out to involve the diagonal entries of Pascal's triangle in a tantalizingly predictive manner. Lastly, we explore a relation between the eigenvalues of various members of the family. More specifically, we give a sufficient condition on the values $m,n \in \mathbb{N}$ for when $\texttt{spec}(A_m)$ is contained in $\texttt{spec}(A_n)$. We end the paper with a number of open questions, one of which intertwines our characteristic polynomials with the Fibonacci sequence in an intriguing manner involving ellipses.

math.CO↗

arXiv2020

An Euler phi function for the Eisenstein integers and some applications

The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[ρ]$ where $ρ$ is the primitive third root of unity $e^{2πi/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[ρ]/(θ)$ for any given Eisenstein integer $θ$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[ρ]/(γ^n)\right)^\times$ are cyclic where $γ$ is prime in $\mathbb{Z}[ρ]$ and $n \in \mathbb{N}$, thereby generalizing well-known results of similar applications in the integers and some lesser known results in the Gaussian integers. As another application, we prove that the celebrated Euler-Fermat theorem holds for the Eisenstein integers.

math.NT↗

arXiv2018

Generating Bézout trees for Pythagorean pairs

Relatively prime pairs of integers can be represented as nodes in three way branching trees. We construct trees of Bézout coefficients which correspond to the relatively prime pairs in the aforementioned trees. As one application, we compare the Bézout coefficients in these trees to those returned by the gcd function in Matlab. As another application, we use these trees to decrease the computation time required to create computer generated hyperbolic wallpaper designs.

math.NT↗