Paul Terwilliger

activity

20242026

Q

-polynomial structures, and quantum groups

1 citations · 3 across the 11 of their papers we have counts for

collaborators

12 papers

math.CO2026

Variations on a circular Hessenberg pair

Kazumasa Nomura, Paul Terwilliger

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. A Hessenberg pair is an ordered pair of diagona…

math.CO2026

Circular Hessenberg pairs and the tridiagonal relations

Kazumasa Nomura, Paul Terwilliger

A square matrix is said to be Hessenberg whenever each entry below the subdiagonal is zero, and each entry on the subdiagonal is nonzero. A Hessenberg matrix is called circular whe…

math.CO2026

Strengthening the balanced set condition for the distance-regular graph of the bilinear forms

Paul Terwilliger, Jason Williford

We consider a distance-regular graph $Γ = (X, R)$ called the bilinear forms graph $H_{q} (D, N - D)$ ; we assume $N > 2 D \geq 6$ and $q \neq = 2$ . We show that $Γ$ satisfies the followi…

math.CO2025

An equitable partition for the distance-regular graph of the bilinear forms

Paul Terwilliger, Jason Williford

We consider a type of distance-regular graph $Γ = (X, R)$ called a bilinear forms graph. We assume that the diameter $D$ of $Γ$ is at least $3$ . Fix adjacent vertices $x,y \…

math.CO2025

Raising and lowering maps for tridiagonal pairs

Paul Terwilliger

Let $V$ denote a nonzero finite-dimensional vector space. A tridiagonal pair on $V$ is an ordered pair $A, A^{*}$ of maps in $End (V)$ such that (i) each of $A, A^{*}$ is diagonal…

math.CO2025

2-Homogeneous bipartite distance-regular graphs and the quantum group $U_{q}^{'} (so_{6})$

Paul Terwilliger

We consider a 2-homogeneous bipartite distance-regular graph $Γ$ with diameter $D \geq 3$ . We assume that $Γ$ is not a hypercube nor a cycle. We fix a $Q$ -polynomial ordering of th…

math.RA2024

The nucleus of the Johnson graph $J (N, D)$

Kazumasa Nomura, Paul Terwilliger

In this paper, we describe the nucleus of the Johnson graph $Γ = J (N, D)$ with $N > 2 D$ . Let $X$ denote the vertex set of $Γ$ . Let $A \in Mat_{X} (C)$ denote the adjace…

math.QA2024

The $S_{3}$ -symmetric $q$ -Onsager algebra and its Lusztig automorphisms

Paul Terwilliger

The $q$ -Onsager algebra $O_{q}$ is defined by two generators and two relations, called the $q$ -Dolan/Grady relations. In 2019, Baseilhac and Kolb introduced two automorphisms of $O_q…

math.CO2024★ 1 cited

The nucleus of a $Q$ -polynomial distance-regular graph

Paul Terwilliger

Let $Γ$ denote a $Q$ -polynomial distance-regular graph with diameter $D \geq 1$ . For a vertex $x$ of $Γ$ the corresponding subconstituent algebra $T = T (x)$ is generated by the adjace…

math.CO2024★ 1 cited

Projective geometries, $Q$ -polynomial structures, and quantum groups

Paul Terwilliger

In 2023 we obtained a $Q$ -polynomial structure for the projective geometry $L_{N} (q)$ . In the present paper, we display a more general $Q$ -polynomial structure for $L_{N} (q)$ . Our new…

math.CO2024★ 1 cited

The $S_{3}$ -symmetric tridiagonal algebra

Paul Terwilliger

The tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the $q$ -Onsager algebra,…

math.CO2024

The Norton-balanced condition for $Q$ -polynomial distance-regular graphs

Kazumasa Nomura, Paul Terwilliger

Let $Γ$ denote a $Q$ -polynomial distance-regular graph, with vertex set $X$ and diameter $D \geq 3$ . The standard module $V$ has a basis ${x^∣ x \in X}$ , whe…

Variations on a circular Hessenberg pair

Circular Hessenberg pairs and the tridiagonal relations

Strengthening the balanced set condition for the distance-regular graph of the bilinear forms

An equitable partition for the distance-regular graph of the bilinear forms

Raising and lowering maps for tridiagonal pairs

2-Homogeneous bipartite distance-regular graphs and the quantum group Uq′​(so6​)

The nucleus of the Johnson graph J(N,D)

The S3​-symmetric q-Onsager algebra and its Lusztig automorphisms

The nucleus of a Q-polynomial distance-regular graph

Projective geometries, Q-polynomial structures, and quantum groups

The S3​-symmetric tridiagonal algebra

The Norton-balanced condition for Q-polynomial distance-regular graphs

2-Homogeneous bipartite distance-regular graphs and the quantum group $U_{q}^{'} (so_{6})$

The nucleus of the Johnson graph $J (N, D)$

The $S_{3}$ -symmetric $q$ -Onsager algebra and its Lusztig automorphisms

The nucleus of a $Q$ -polynomial distance-regular graph

Projective geometries, $Q$ -polynomial structures, and quantum groups

The $S_{3}$ -symmetric tridiagonal algebra

The Norton-balanced condition for $Q$ -polynomial distance-regular graphs