Characterizing completely regular codes from an algebraic viewpoint

Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD–D canonical form and the LB–UB canonical form

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension.A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A : V \to V$ and $A^* : V \to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $\mathbb{F}$-linear map $K: V \to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$,define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$.For each solution $B, B^*$, we describe the corresponding companion $K = A-B$.

In Linear algebra, the concept of Leonard pair (LP) was motivated by the theory of Q-polynomial distance-regular graphs. In this dissertation, we will first give a brief introduction to LPs and to two closely-related classes of objects: (i) bipartite Leonard pairs (BLPs) and (ii) almost bipartite Leonard pairs (ABLPs). Taking these as departure points, we will introduce a new class of object - doubly almost bipartite Leonard pairs (DABLPs). The primary aim of our work is to fully classify (up to isomorphism) this new family. In addition, since there is known to be a natural correspondence between Leonard pairs and families of orthogonal polynomials, we reveal which families of orthogonal polynomials correspond to the DABLPs. Several related objects, such as Leonard triples, modular Leonard triples, spin Leonard pairs, and near-bipartite Leonard pairs have corresponding notions for the doubly almost bipartite case. These analogous objects are also defined and briefly explored.

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations and that satisfy (i) and (ii) below. There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing is diagonal. There exists a basis for V with respect to which the matrix representing is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. In the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers , , , and the dual eigenvalues . In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the and . For the second characterization, the focus is on the , , and .

The universal DAHA of type [formula omitted] and Leonard pairs of q-Racah type

A characterization of Leonard pairs using the parameters [formula omitted

Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs

By introducing a duality operator in coherent algebras (i.e. adjacency algebras of coherent configurations) we give a new interpretation to Delsarte's duality theory for association schemes. In particular we show that nonnegative matrices and positive semidefinite matrices, (0, 1)-matrices and distance matrices, regular graphs and spherical 2-designs, distance regular graphs and Delsarte matricesare pairs of dual objects. Several “almost dual” properties which are not yet fully understood are also reported.

On a New Family of (P and Q)-polynomial Schemes

Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

Matroids With Weighted Bases and Feynman Integrals

The subconstituent algebra of a distance-regular graph; thin modules with endpoint one

Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let ? be a Galois ring of characteristic 4,X a cyclotomic scheme over ? with classD and ? the Terwilliger algebra ofX, then we show that most of the irreducible ?-modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible ?-modules using Jacobi sums.

We report the results obtained in the study of Alain Connes noncommutative spectral geometry construction focusing on its essential ingredient of the algebra doubling. We show that such a two-sheeted structure is related with the gauge structure of the theory, its dissipative character and carries in itself the seeds of quantization. From the algebraic point of view, the algebra doubling process has the same structure of the deformed Hops algebra structure which characterizes quantum field theory.