Polynomial table algebras and their covering numbers (Q1902138)

A table algebra \((A,\mathbb{B})\) with \(\mathbb{B}=\{\chi_0=1_A,\chi_1,\dots,\chi_k\}\) is a polynomial table algebra if there is an algebra isomorphism \(\psi:A\to\mathbb{C}[\lambda]/(f(\lambda))\), \(f(\lambda)\in\mathbb{C}[\lambda]\), such that the degrees of \(\psi(\chi_0),\psi(\chi_1),\dots,\psi(\chi_k)\) in \(\mathbb{C}[\lambda]/(f(\lambda))\) are distinct. The covering number \(cn(\mathbb{B})\) of \((A,\mathbb{B})\) is the least positive integer \(m\) such that \(\text{supp}(\chi^m_i)=\mathbb{B}\) for all \(\chi_i\in\mathbb{B}\), \(i\neq 0\), if such \(m\) exists. If \(cn(\mathbb{B})=2k\), we show \((A,\mathbb{B})\) is a polynomial table algebra and, by a suitable reordering of \(\chi_i\) if necessary, the first intersection matrix is tridiagonal as follows, \[ \begin{pmatrix} 0 & 1\\ b_0 & 0 & c_2 \\ & b_1 & 0 & c_3 \\ & & \ddots & \ddots & \ddots & \\ & & & b_{k-2} & 0 & c_k \\ & & & & b_{k -1} & a_k \end{pmatrix} , \] where \(b_i>0\), \(c_j>0\), \(a_k>0\).