arXiv subjects

Ian Seong

Publications and source records attributed to Ian Seong.

7 recordsLinked to original sources

arXiv2025

The nucleus of the Grassmann graph $J_q(N,D)$

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with vertex set $X$ that consists of the $D$-dimensional subspaces of $\mathcal{V}$. Two vertices are adjacent whenever their intersection has dimension $D-1$. Fix a vertex $x$ in $X$. The Terwilliger algebra $T=T(x)$ of $J_q(N,D)$ with respect to $x$ is the subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^* = A^*(x)$. It is known that an irreducible $T$-module $W$ has certain parameters called the endpoint $r$, the dual endpoint $t$, and the diameter $d$. The displacement of $W$ is defined to be the integer $r+t-D+d$. Let $\mathcal{N}=\mathcal{N}(x)$ denote the span of all irreducible $T$-modules with displacement 0. We call $\mathcal{N}$ the nucleus of $J_q(N,D)$ with respect to $x$. In this paper, we study the structure of $\mathcal{N}$. Specifically, we present a formula for the dimension of $\mathcal{N}$, construct two explicit bases for $\mathcal{N}$, and describe the action of $A$ and $A^*$ on these bases. To obtain these results, we use the projective geometry $P_q(N)$, consisting of all subspaces of $\mathcal{V}$, as a key tool.

math.CO↗

arXiv2025

Counting edges of different types in a local graph of a Grassmann graph

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick a vertex $y$. In this paper we define three types of edges in $X$, namely type $0$, type $+$, and type $-$; for adjacent vertices $w,z$ such that $\partial(w,y)=\partial(z,y)$, the type of the edge $wz$ depends on the subspaces $w+z,w,z,w\cap z$ and their intersections with $y$. Pick a vertex $x$ such that $1<\partial(x,y)<k$. Let $\Gamma(x)$ denote the local graph of $x$ in $J_q(n,k)$. Our general goal is to count the number of edges in $\Gamma(x)$ for each type. Consider a two-vertex stabilizer $\text{Stab}(x,y)$ in $GL(V)$; it is known that the $\text{Stab}(x,y)$-action on $\Gamma(x)$ has five orbits. Pick two orbits $\mathcal{O},\mathcal{N}$ that are not necessarily distinct; for a given $w\in \mathcal{O}$, we find the number of vertices in $z\in \mathcal{N}$ such that the edge $wz$ has (i) type $0$, (ii) type $+$, (iii) type $-$. To find these numbers, we use many results that involve a projective geometry $P_q(n)$, which is the set of all subspaces of $V$.

math.CO↗

arXiv2024

A generalization of the Askey-Wilson relations using a projective geometry

In this paper, we present a generalization of the Askey-Wilson relations that involves a projective geometry. A projective geometry is defined as follows. Let $h>k\geq 1$ denote integers. Let $\mathbb{F}_{q}$ denote a finite field with $q$ elements. Let $\mathcal{V}$ denote an $(h+k)$-dimensional vector space over $\mathbb{F}_{q}$. Let the set $P$ consist of the subspaces of $\mathcal{V}$. The set $P$, together with the inclusion partial order, is a poset called a projective geometry. We define a matrix $A\in \text{Mat}_{P}(\mathbb{C})$ as follows. For $u,v\in P$, the $(u,v)$-entry of $A$ is $1$ if each of $u,v$ covers $u\cap v$, and $0$ otherwise. Fix $y\in P$ with $\dim y=k$. We define a diagonal matrix $A^*\in \text{Mat}_{P}(\mathbb{C})$ as follows. For $u\in P$, the $(u,u)$-entry of $A^{*}$ is $q^{\dim(u\cap y)}$. We show that \begin{align*} &A^2A^{*}-\bigl(q+q^{-1}\bigr)AA^{*}A+A^{*}A^{2}-\mathcal{Y}\bigl(AA^{*}+A^{*}A\bigr)-\mathcal{P} A^{*}=\Omega A+G, \newline &A^{*2}A-\bigl(q+q^{-1}\bigr) A^*AA^*+AA^{*2}=\mathcal{Y}A^{*2}+\Omega A^{*}+G^{*}, \end{align*} where $\mathcal{Y}, \mathcal{P}, \Omega, G, G^*$ are matrices in $\text{Mat}_{P}(\mathbb{C})$ that commute with each of $A, A^*$. We give precise formulas for $\mathcal{Y}, \mathcal{P}, \Omega, G, G^*$.

math.CO↗

arXiv2024

Some orbits of a two-vertex stabilizer in a Grassmann graph

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of $J_q(n,k)$ such that $1<\partial(x,y)<k$. Let $\text{Stab}(x,y)$ denote the subgroup of $GL(V)$ that stabilizes both $x$ and $y$. In this paper, we investigate the orbits of $\text{Stab}(x,y)$ acting on the local graph $\Gamma(x)$. We show that there are five orbits. By construction, these five orbits give an equitable partition of $\Gamma(x)$; we find the corresponding structure constants. In order to describe the five orbits more deeply, we bring in a Euclidean representation of $J_q(n,k)$ associated with the second largest eigenvalue of $J_q(n,k)$. By construction, for each orbit its characteristic vector is represented by a vector in the associated Euclidean space. We compute many inner products and linear dependencies involving the five representing vectors.

math.CO↗

arXiv2023

Using a Grassmann graph to recover the underlying projective geometry

Let $n,k$ denote integers with $n>2k\geq 6$. Let $\mathbb{F}_q$ denote a finite field with $q$ elements, and let $V$ denote a vector space over $\mathbb{F}_q$ that has dimension $n$. The projective geometry $P_q(n)$ is the partially ordered set consisting of the subspaces of $V$; the partial order is given by inclusion. For the Grassmann graph $J_q(n,k)$ the vertex set consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. The graph $J_q(n,k)$ is known to be distance-regular. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick two vertices $x,y$ in $J_q(n,k)$ such that $1<\partial(x,y)<k$. The set $P_q(n)$ contains the elements $x,y,x\cap y,x+y$. In our main result, we describe $x\cap y$ and $x+y$ using only the graph structure of $J_q(n,k)$. To achieve this result, we make heavy use of the Euclidean representation of $J_q(n,k)$ that corresponds to the second largest eigenvalue of the adjacency matrix.

math.CO↗

arXiv2019

The closure-complement-frontier problem in saturated polytopological spaces

Let $X$ be a space equipped with $n$ topologies $\tau_1,...,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $\mathcal{KF}_n$ generated by $\{k_i,f_i:1\leq i\leq n\}\cup\{c\}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=\frac{5}{24}n^4+\frac{37}{12}n^3+\frac{79}{24}n^2+\frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,\tau_1,...,\tau_n)$ and a subset $A\subseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $\mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $\mathcal{KF}_2$.

math.GN↗

arXiv2014

Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope

We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. Starting from this LP relaxation, our main results assert that for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices of the input graph. Our lower bounds are nearly tight. Our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for t = O(n). This is one of the very few known examples of an intractable combinatorial optimization problem for which LP-based algorithms induce a constant approximation ratio, still lift-and-project LP and SDP tightenings of the same LP have unbounded IGs. We also show that the SDP that has given the best algorithm known for t-PVC has integrality gap n/t on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, one of our main contributions is that we make explicit of a new and simple methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system).

cs.DS↗