Related papers: Leonard pairs having zero-diagonal TD-TD form

Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the eigen-pairs…

Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is…

Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such…

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$,…

Polynomial $n\times n$ matrices $A(\lambda)$ and $B(\lambda)$ over a field $\mathbb F $ are called semi-scalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over the field $\mathbb F $ and an invertible $n\times n$ matrix…

A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero.…

Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is…

Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic…

In this paper, we focus on the equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids, a natural family of graphic matroids associated with the complete tripartite graphs $K_{1,1,n}$. These polynomials were introduced by…

We consider the completeness problem for left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, all of which have Lie algebra of the form $\mathbb{R} \ltimes_A \mathbb{R}^2$, where $A$ is a real $2 \times 2$ matrix…

The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ are a family of discrete orthogonal polynomials involving two real parameters $r$ and $s$. Let $L,L^*$ denote the corresponding Leonard pair. Assume that $r\not=0$ and $r+s=0$. We show that…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy (i)--(iv) below: (i) Each of $A$, $A^*$ is…

We prove the conjecture by Damm and Fassbender that, for any pair $L,M$ of real traceless matrices, there exists an orthogonal $V$ such that $V^{-1} L \, V$ is hollow and $V M V^{-1}$ is almost hollow, where a matrix is hollow if and only…

The notion of factorized $A_2$-Leonard pair is introduced. It is defined as a rank 2 Leonard pair, with actions in certain bases corresponding to the root system of the Weyl group $A_2$, and with some additional properties. The functions…

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered…

The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1,1,n}$, has full automorphism group isomorphic to the hyperoctahedral group whenever $n \ge 2$. In the equivariant setting for this action, we…

The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are…

We study the algebraic sets of pairs of matrices defined by the vanishing of the anti-diagonal as well as the cross-diagonal of their commutator matrix. We prove that, over a field of prime characterisitic, the coordinate ring of the latter…

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal…

Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. Pick a nonzero $q \in \mathbb F$ such that $q^4 \not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah…