Kite-free \(P\)- and \(Q\)-polynomial schemes (Q1895824)

Let \(Y= (X, \{R_i\}_{0\leq i\leq d})\) denote a \(P\)-polynomial association scheme. A kite of length \(i\) \((2\leq i\leq d)\) in \(Y\) is a 4- tuple \(xyzu\) \((x,y,z, u\in X)\) such that \((x, y)\in R_1\), \((x, z)\in R_1\), \((y, z)\in R_1\), \((u, y)\in R_{i- 1}\), \((u, z)\in R_{i- 1}\), \((u, x)\in R_i\). The main result of the reviewed paper is the following theorem: Let \(Y\) be a \(P\)- and \(Q\)-polynomial association scheme. Suppose \(Y\) has diameter \(d\geq 3\), and suppose \(Y\) has intersection number \(a_1\neq 0\). Then the statements (1) through (3) below are equivalent: (1) \(Y\) has classical parameters \((d, b, \alpha, \beta)\), and either \(b< -1\), or \(Y\) is a dual polar scheme or a Hamming scheme. (2) \(Y\) has no kites of length 2 and no kites of length 3. (3) \(Y\) has no kites of any length \(i\) \((2\leq i\leq d)\).