Combinatorial realization of the finite-N matrix-harmonic tridiagonal matrices on the fuzzy sphere as intersection matrices of a distance-regular graph or association scheme

Hoppe and Yau (Comm. Math. Phys. 195 (1998) 67-77) showed that replacing the commuting coordinates of spherical harmonics on $S^2$ by $N\times N$ Hermitian matrices obeying $su(2)$-type commutation relations (realized in the $N$-dimensional irreducible representation) produces, at each finite $N$, two dual families of discrete orthogonal polynomials (of dual Hahn type) whose three-term recurrences assemble into tridiagonal matrices $\widetilde{B}^{(m)}$ - the constituents of a discrete Laplacian on $gl(N)$ reproducing the first $N^2$ eigenvalues of the spherical Laplacian. Hoppe and Yau noted in [1] that these tridiagonal structures resemble the intersection matrices of distance-regular graphs and P-Q-polynomial association schemes, while, for the blocks of size $N-m\ge3$, the finite-$N$ feasibility criteria of the standard theory are not met, and left open whether a natural graph or combinatorial object realizes them. The surviving open core: for finite $N$, does there exist a natural combinatorial object whose intersection or eigen structure reproduces the Hoppe-Yau data without violating feasibility conditions - or a proof that none exists in a precisely formulated class.

Background

A fuzzy sphere is a finite-dimensional, noncommutative surrogate for the algebra of functions on the two-sphere: one replaces the commuting coordinate functions $x_1,x_2,x_3$ (with $\vec{x}^2=1$) by $N\times N$ Hermitian matrices $X_1,X_2,X_3$ satisfying $[X_1,X_2]=2iX_3/\sqrt{N^2-1}$ and cyclic permutations, plus $X_1^2+X_2^2+X_3^2=\mathbb{I}$ - an irreducible spin-$s=(N-1)/2$ representation. This matrix regularization was introduced in the context of field theory on the quantized sphere by Grosse, Klimčík and Prešnajder (1996), and matrix harmonics - the matrices $T^{(N)}_{lm}$ obtained by substituting $X_a$ for $x_a$ in the solid spherical harmonics - were shown by Hoppe and Yau to have remarkable finite-$N$ structure. Their matrix elements are Wigner $3j$-symbols (the standard angular-momentum recoupling coefficients of $SU(2)$, i.e. the Clebsch-Gordan coefficients in symmetrized form), $(T^{(N)}_{lm})_{M_1M_2}=\sqrt{2l+1}(-)^{s-M_1}\binom{s\ l\ s}{-M_1\ m\ M_2}$, and the $N^2$ matrices $T^{(N)}_{lm}$ ($0\le l\le N-1$, $|m|\le l$) diagonalize the adjoint-action operator $\Delta_N=\sum_i[S_i,[S_i,\cdot]]$ (a discrete Laplacian on $gl(N)$) with the familiar eigenvalues $l(l+1)$. Evaluating the harmonics on the discrete spectrum of $X_3$ produces finite families of discrete orthogonal polynomials of Chebyshev-Hahn type, $f^{(m)}_{l-m}(j)={}_3F_2(-(l-m),l+m+1,-j;m+1,-(N-1-m);1)$ - special cases of the dual Hahn polynomials, members of the Askey scheme (the standard hierarchy of hypergeometric orthogonal-polynomial families - a nomenclature unrelated to association schemes), many of whose members also arise as Clebsch-Gordan (angular-momentum recoupling) coefficients. Crucially, the transposed evaluation tables are again polynomial families (polynomials of degree $j$ in the dual variable combination $\tilde y(\tilde y+2m+1)$, with $\tilde y=l-m$ running over $0,\dots,N-1-m$), so the two families are dual in the sense familiar from the theory of association schemes. An association scheme is a collection of $0$-$1$ matrices (relations) on a finite vertex set whose span is a commutative algebra (the Bose-Mesner algebra) with a distinguished basis; a distance-regular graph - a graph in which the number of vertices at each distance from a vertex, and the intersection numbers $a_i,b_i,c_j$ of neighbors in adjacent distance classes, are independent of the vertex - is exactly a scheme whose relations are the distance matrices (a P-polynomial scheme), and its first intersection matrix $B_1$ is the tridiagonal matrix with entries built from $a_i,b_i,c_i$ (Brouwer, Cohen and Neumaier's monograph is the standard reference). A scheme is P-polynomial if the eigenvalues of each relation matrix are polynomials in a single generating eigenvalue, and Q-polynomial if the same holds dually for the multiplicities - the structure responsible for orthogonal-polynomial duality. Hoppe and Yau's tridiagonal matrices $\widetilde{B}^{(m)}$ (size $N-m$, indexed by $j=0,\dots,N-1-m$, with off-diagonal entries $-\beta^{(m)}(j+1+m)(N-1-m-j)$ and $-\beta^{(m)}j(N-j)$, diagonal $\alpha^{(m)}+\beta^{(m)}(2j(N-1-m-j)+(m+1)(N-1-m))$, constant row sum, and eigenvalues $\alpha^{(m)}+\beta^{(m)}\,n(n+2m+1)$ for $n=0,\dots,N-1-m$ - equivalently $\alpha^{(m)}+\beta^{(m)}(l(l+1)-m(m+1))$ in the angular-momentum label $l=m,\dots,N-1$) carry precisely this intersection-matrix form [1], and the paper cites a standard distance-regular-graph reference. The abstract framework for such dual pairs of tridiagonal operators with mutually dual eigenbases is the Leonard-pair theory of Terwilliger, in which the dual Hahn family is one of the standard families; and the quantum-probability literature (Hora and Obata) systematically relates Jacobi matrices, interacting Fock spaces and distance-regular graphs. What the finite-$N$ matrix-harmonic construction does not supply, however, is the combinatorial side: no vertex set, no relations, no graph.

Problem Statement

For each integer $N\ge2$ and each $m\in\{0,\dots,N-1\}$, the finite-$N$ matrix harmonics on the fuzzy sphere produce a tridiagonal matrix $\widetilde B^{(m)}$ of size $N-m$, indexed by $j=0,\dots,N-1-m$ (upper off-diagonal $-\beta^{(m)}(j+1+m)(N-1-m-j)$, lower off-diagonal $-\beta^{(m)}j(N-j)$, diagonal $\alpha^{(m)}+\beta^{(m)}(2j(N-1-m-j)+(m+1)(N-1-m))$, constant row sum, simple real spectrum $\alpha^{(m)}+\beta^{(m)}\,n(n+2m+1)$ for $n=0,\dots,N-1-m$ - equivalently $\alpha^{(m)}+\beta^{(m)}(l(l+1)-m(m+1))$ in the angular-momentum label $l=m,\dots,N-1$), together with the dual Hahn polynomial families $f^{(m)}_{l-m}(j)={}_3F_2(-(l-m),l+m+1,-j;m+1,-(N-1-m);1)$ whose evaluation vectors are its eigenvectors, and the ambient $N^2$-dimensional Laplacian $\Delta_N$ whose eigenvalue $l(l+1)$ has multiplicity $2l+1$. The question, as raised by Hoppe and Yau's comparison of these matrices [1] with the intersection matrices of distance-regular-graph theory: (1) does there exist, for finite $N$ and a non-degenerate block of size $N-m\ge3$, a natural graph or combinatorial object - a distance-regular graph, an association scheme, or a recognized generalization such as a multivariate P-polynomial scheme - whose intersection matrices (or eigenmatrices, the matrices collecting the eigenvalues and multiplicities of the relations) coincide with the $\widetilde B^{(m)}$ or a natural renormalization of them, where 'natural renormalization' includes the standard normalizations of intersection-matrix theory (affine shift and rescaling, diagonal similarity)? (2) More generally, can the finite-$N$ families of discrete orthogonal polynomials arising from the matrix harmonics be fitted into the standard theory of distance-regular graphs and P- and Q-polynomial association schemes - matching the spectral pattern, the duality, and the multiplicity structure - without violating the feasibility conditions of that theory (nonnegativity, $c_1=1$, $b_d=0$, integrality of intersection numbers, and integrality of the induced multiplicities - the multiplicity formula yielding the spherical value $2l+1$ only formally at finite $N$)? The realization is sought for a specified non-degenerate $(N,m)$ ($N-m\ge3$), or uniformly over a family of them, within a precisely formulated admissible class of combinatorial objects, and, in the negative direction, the question asks whether non-existence can be established: for (1), the coincidence asked for is that of the intersection or eigen matrices up to natural renormalization; for (2), a match of the spectral pattern, the duality, and the multiplicity structure (the value $2l+1$ on the $N^2$-dimensional space) is asked in addition. Whether the full range of $(N,m)$ admitting a realization can be determined is likewise part of the question.

References

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