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Integral standard table algebras with a faithful nonreal element of degree 3. (English) Zbl 0958.05140

Let \(\mathbf B\) be a basis of a finite-dimensional associative and commutative algebra \(A\) with identity \(\mathbf 1\) over the complex field \(\mathbf C\). The pair \((A,{\mathbf B})\) is called a table algebra (with table basis \(\mathbf B\)) if \({\mathbf 1}\in {\mathbf B}\) and (TA1) for all \(a,b\in {\mathbf B}\), \(ab=\sum_{c\in {\mathbf B}}\lambda_{abc}c\) with \(\lambda_{abc}\) nonnegative real numbers; (TA2) there is an algebra automorphism \(\overline{\phantom x}\) of \(A\) such that \(\overline {\mathbf B}={\mathbf B}\) (the elements of \(\mathbf B\) which are fixed by \(\overline{\phantom x}\) are called real elements); (TA3) \(\lambda_{ab{\mathbf 1}}\neq 0\) if and only if \(a=\overline b\). There is a unique algebra homomorphism \(|\;|:A\to {\mathbf C}\) such that \(|b|=|\overline b|\in {\mathbf R}^+\) for all \(b\in {\mathbf B}\). The positive real numbers \(\{|b|\}_{b\in {\mathbf B}}\) are called the degrees of \((A,{\mathbf B})\). A table algebra \((A,{\mathbf B})\) is called integral (homogeneous of degree \(\lambda\), standard), if all the structure constants \(\lambda_{abc}\) and all the degrees \(|b|\) are rational integers (if \(|{\mathbf B}|>1\) and \(|b|=\lambda\) for all \(b\in {\mathbf B}\), if \(\lambda_{b\overline b{\mathbf 1}}=|b|\) for all \(b\in {\mathbf B})\). A nonempty set \({\mathbf D}\subseteq {\mathbf B}\) is called a closed (a table) subset if \(\text{Supp}(ab)\subset D\) for all \(a,b\in {\mathbf D}\). For any \(c\in {\mathbf B}\) the set \({\mathbf B}_c=\bigcup_{n\geq 1}\text{Supp}(c^n)\) is a closed subset. An element \(c\in {\mathbf B}\) is called faithful (linear) if \({\mathbf B}_c={\mathbf B}\) (if \(\text{Supp}(c^n)={\mathbf 1}\) for some \(n>0\)). The set of all linear elements of \({\mathbf B}\) is denoted by \(L({\mathbf B})\). Main theorem: Let \((A,{\mathbf B})\) be a standard integral table algebra with a faithful nonreal element \(a\in {\mathbf B}\) of degree 3. If \(L({\mathbf B})\) is trivial, then one of the following holds: (a) \(a\overline a=3\cdot {\mathbf 1}+3b\), where \(|b|=2\) and \(\mathbf B\) is isomorphic to a wreath product of \({\mathbf V}\) and \({\mathbf Z}_m\), where \({\mathbf V}={\mathbf 1},v\), \(v^2=2\cdot {\mathbf 1}+v\). (b) \(a\overline a=3\cdot {\mathbf 1}+b\), where \(|b|=6\) and \(b\in \mathbf B\). (c) \((A,{\mathbf B})\) is a homogeneous standard integral table algebra of degree 3.

MSC:

05E30 Association schemes, strongly regular graphs
20C15 Ordinary representations and characters
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