A bivariate QQ-polynomial structure for the non-binary Johnson scheme

The notion of multivariate $P$- and $Q$-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate $P$-polynomial association scheme. We show here that it is also a bivariate $Q$-polynomial association scheme for some parameters. This provides, with the $P$-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.Comment: 20 page

Suborbits of (m,k)-isotropic subspaces under finite singular classical groups

AbstractLet Fq2ν+δ+l be one of the (2ν+δ+l)-dimensional singular classical spaces and let G2ν+δ+l,2ν+δ be the corresponding singular classical group of degree 2ν+δ+l. All the (m,k)-isotropic subspaces form an orbit under G2ν+δ+l,2ν+δ, denoted by M(m,k;2ν+δ+l,2ν+δ). Let Λ be the set of all the orbitals of (G2ν+δ+l,2ν+δ,M(m,k;2ν+δ+l,2ν+δ)). Then (M(m,k;2ν+δ+l,2ν+δ),Λ) is a symmetric association scheme. First, we determine all the orbitals and the rank of (G2ν+δ+l,2ν+δ,M(m,k;2ν+δ+l,2ν+δ)), calculate the length of each suborbit. Next, we compute all the intersection numbers of the symmetric association scheme (M(ν+k,k;2ν+δ+l,2ν+δ),Λ), where k=1 or k=l−1. Finally, we construct a family of symmetric graphs with diameter 2 based on M(2,0;4+δ+l,4+δ)

On the Farthest Subconstituent of the q-Johnson Graph Jq(n, k)

The farthest subconstituent of the q-Johnson graph Jq(n, k) is well-known to be iso-morphic to the bilinear forms graph Mk(q) in the case of n = 2k. This fact is generalized for n ≥ 2k