Modular invariance of the character table of the Hamming association scheme \(H(d,q)\) (Q1323870)

The following theorem is proved: Let \(P\) be the character table of the Hamming association scheme \(H(d,q)\) (whose entries are Krawtchouk polynomials). Let \(T= \alpha_ 0\text{ diag}(1,\alpha,\dots,\alpha^ d)\) (a diagonal matrix whose entry in row and column \(i\) equals \(\alpha_ 0\alpha^ i\)), where \(\alpha\) and \(\alpha_ 0\) are defined by \(\alpha^ 2+ (q- 2)\alpha+ 1=0\) and \(\alpha^ 3_ 0= q^{d/2}/(1+(q- 1)\alpha)^ d\). Then \((PT)^ 3= q^{3d/2} I\). This implies that the matrix \(S\) of the fusion algebra at algebraic level obtained from \(H(d,q)\) satisfies the modular invariance property, namely \((ST)^ 3= S^ 2= I\) for the diagonal matrices \(T\). In Appendix I a second, coding theoretic proof of the theorem is presented. Appendix II contains further examples of self-dual \(P\)- and \(Q\)-polynomial association schemes which have the modular invariance property.