2014/09/15 by Kazumasa Nomura, Nomura, Kazumasa
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.RA
paper · pdf · doi:10.48550/arxiv.1409.4333
arXiv admin note: substantial text overlap with arXiv:1408.2180
arxiv created 2014/09/15 · openalex publication_date 2014/09/15 · arxiv updated 2014/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix an algebraically closed field \mathbbF and an integer d ≥ 3. Let V be a vector space over \mathbbF with dimension d+1. A Leonard pair on V is an ordered pair of diagonalizable linear transformations A: V → V and A^* : V → V, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let \vi\i=0d (resp. \v^*i\i=0d) be such an eigenbasis for A (resp. A^*). For 0 ≤ i ≤ d define a linear transformation Ei : V → V such that Ei vi=vi and Ei vj =0 if j ≠ i (0 ≤ j ≤ d). Define E^*i : V → V in a similar way. The sequence Φ=(A, \Ei\i=0d, A^*, \E^*i\i=0d) is called a Leonard system on V with diameter d. With respect to the basis \vi\i=0d, let \þi\i=0d (resp. \a^*i\i=0d) be the diagonal entries of the matrix representing A (resp. A^*). With respect to the basis \v^*i\i=0d, let \θ^*i\i=0d (resp. \ai\i=0d) be the diagonal entries of the matrix representing A^* (resp. A). It is known that \θi\i=0d (resp. \þ^*i\i=0d) are mutually distinct, and the expressions (θi-1-θi+2)/(θi-θi+1), (θ^*i-1-θ^*i+2)/(θ^*i - θ^*i+1) are equal and independent of i for 1 ≤ i ≤ d-2. Write this common value as β+ 1. In the present paper we consider the "end-entries" θ0, θd, θ^*0, θ^*d, a0, ad, a^*0, a^*d. We prove that a Leonard system with diameter d is determined up to isomorphism by its end-entries and β if and only if either (i) β≠ ± 2 and qd-1 ≠ -1, where β=q+q-1, or (ii) β= ± 2 and Char(\mathbbF) ≠ 2.