The classification of a class of homogeneous integral table algebras of degree five (Q2704392)

A table algebra \(A\) is a finite dimensional commutative and associative algebra with identity over \(\mathbb{C}\) with a basis \(B=\{b_1=1,b_2,\dots,b_n\}\) such that: (i) For \(b_i,b_j\in B\) there exist \(\lambda_{ijm}\in\mathbb{R}^+\cup\{0\}\), called structure constants, such that \(b_ib_j=\sum^n_{m=1}\lambda_{ijm}b_m\). (ii) There exists an algebra automorphism \(-\) of \(A\) whose order divides 2 such that \(b_i\in B\) implies \(\overline b_i\in B\). (iii) For all \(i,j\), \(\lambda_{ij1}\neq 0\) if and only if \(j=\overline i\).NEWLINENEWLINENEWLINEIn this case by \textit{Z. Arad} and \textit{H. I. Blau} [J. Algebra 138, No. 1, 137-185 (1991; Zbl 0790.20015)] there is an algebra homomorphism \(f\colon A\to\mathbb{C}\) such that \(f(b_i)=\overline{f(b_i)}\in\mathbb{R}^+\) for all \(i\), \(1\leq i\leq n\), and \(f(b_i)\) are called the degrees of the table algebra \((A,B)\). If all the structure constants and the degrees of the table algebra are integers then \((A,B)\) is called an integral table algebra (ITA). Now if all the degrees of the non-identity elements in \(B\) are equal, then the integral table algebra \((A,B)\) is called a homogeneous integral table algebra (HITA). For \(a\in A\), if \(a=\sum^n_{m=1}\lambda_mb_m\), then the support of \(a\) is defined to be \(\text{supp}(a)=\{b_k\mid\lambda_k>0\}\) and \(b\in B\) is said to be faithful if \(B=\bigcup^\infty_{i=1}\text{supp}(b^i)\). The width of \(b\in B\) is defined to be the support of \(b\overline b\), and \(b\) is said to be real if \(b=\overline b\).NEWLINENEWLINENEWLINEClassification of HITA is important in studying table algebras. HITA's of degree up to 3 are classified completely by \textit{H. I. Blau, B. Xu, Z. Arad, E. Fisman, V. Miloslavsky}, and \textit{M. Muzychuk} [in Homogeneous integral table algebras of degree three, Mem. Am. Math. Soc. 684 (2000; Zbl 0958.20010)].NEWLINENEWLINENEWLINEIn the present paper the author classifies HITA's of degree five containing a real faithful element of width 2. This is part of the author's Ph. D. Thesis which was done in 1998 at the Tarbiat Modarres University, Tehran, Iran.