Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

AbstractLet Γ be a d-bounded distance-regular graph with diameter d⩾3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L′(x,i) by inclusion or reverse inclusion, L′(x,i) is denoted by LO′(x,i) or LR′(x,i). We prove that LO′(x,i) and LR′(x,i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of LO′(x,i)

The structure of some linear transformations

AbstractLet F denote an algebraically closed field and let V denote a finite-dimensional vector space over F. Recently Ito and Terwilliger considered a system of linear transformations A+,A-,A+∗,A-∗ on V which generalizes the notions of a tridiagonal pair and a q-inverting pair. In their paper they mentioned some open problems about this system. In this paper we solve Problem 1.2 with the following results. Let {Vi}i=0d denote the common eigenspaces of A+,A- and let {Vi∗}i=0d denote the common eigenspaces of A+∗,A-∗. We show that each of A+,A-,A+∗,A-∗ is determined up to affine transformation by the sequences {Vi}i=0d; {Vi∗}i=0d. We also show that the following are equivalent: (i) there exists a nonzero bilinear form 〈,〉 on V such that 〈A+u,v〉=〈u,A+v〉 and 〈A+∗u,v〉=〈u,A+∗v〉 for all u,v∈V; (ii) there exist scalars α,α∗,β,β∗ in F with α,α∗ nonzero such that A-=αA++βI and A-∗=α∗A+∗+β∗I; and (iii) both A+,A+∗ and A-,A-∗ are tridiagonal pairs

Two error-correcting pooling designs from symplectic spaces over a finite field

AbstractIn this paper, we construct two classes of t×n,se-disjunct matrix with subspaces in a symplectic space Fq(2ν) and prove that the ratio efficiency t/n of two constructions are smaller than that of D’yachkov et al. (2005) [2]

The Terwilliger algebra of the doubled Odd graph

Let $2.O_{m+1}$ denote the doubled Odd graph on a set of cardinality $2m+1$ for $m\geq 1$. Denote its vertex set by $X$ and fix a vertex $x_0\in X$. Let $\mathcal{A}$ denote the centralizer algebra of the stabilizer of $x_0$ in the automorphism group of $2.O_{m+1}$, and $T:=T(x_0)$ the Terwilliger algebra of $2.O_{m+1}$ with respect to $x_0$. In this paper, we first give a basis of $\mathcal{A}$ by considering the action of stabilizer of $x_0$ on $X\times X$, and also give three subalgebras of $\mathcal{A}$ such that their direct sum is $\mathcal{A}$ as vector space. Next, we describe the decomposition of $T$ for $m\geq 3$ by using all the homogeneous components of $V:=\mathbb{C}^X$. Finally, we show that $\mathcal{A}$ coincides with $T$ based on the above decomposition of $T$. This result tells us that the $2.O_{m+1}$ may be the first example of bipartite but not $Q$-polynomial distance-regular graph for which the above two algebras are equal.Comment: arXiv admin note: substantial text overlap with arXiv:2207.0126