Standard Integral Table Algebras Generated by Non-real Element of Small Degree

Abstract

A table algebra is a finite-dimensional associative and commutative algebra over the complex field $\bbfC$ with a distinguished basis $B={b_1=1,b_2,\dots,b_n}$ such that: (1) For all $i,j,m$ we have $b_ib_j=\sumβ_{ijm}b_m$ where $β_{ijm}$ are nonnegative real numbers which are called the structure constants of the algebra. (2) There is an automorphism of $A$, denoted by $-$, whose order divides $2$ with the property that if $b_i\in B$, then $\overline b_i\in B$. In this case $\overline i$ is defined by $b_{\overline i}=\overline b_i$ and $b_i$ is called real if $\overline b_i=b_i$. (3) For all $i,j$ we have $β_{ij1}e 0\Leftrightarrow j=\overline i$.\par Two well-known examples of table algebras relating to a finite group $G$ are as follows: One is the center of the group algebra $\bbfC G$ with basis elements being the class sum of each conjugacy class, and the other is the algebra of class functions $\text{cf}(G,\bbfC)$ with basis elements being the irreducible complex characters of $G$. In both of these algebras there is an algebra homomorphism from $Z(\bbfC G)$ (or $\text{cf}(G,\bbfC)$) to $\bbfC$ with definition $f(\sum_{x\in K}x)=|\text{class}(x)|$ ($f(χ_i)=χ_i(1)$), where $K$ is a conjugacy class in $G$ and $|\text{class}(x)|$ is the size of conjugacy class containing $x\in G$ and $χ_i$ is an irreducible complex character of $G$. Moreover, for both of these table algebras all the structure constants and all values of $f$ are integers. In general it is proved by {\it Z. Arad} and {\it H. I. Blau} [J. Algebra 138, No. 1, 137-185 (1991; Zbl 0790.20015)] that if $A$ is a table algebra, then there is an algebra homomorphism $f\colon A\to\bbfC$ such that $f(b_i)=f(\overline b_i)\in\bbfR^+$ for all $i$, $1\le i\le n$, where $\bbfR^+$ denotes the set of positive real numbers. In this case the positive real numbers $f(b_i)$, $1\le i\le n$, are called the degrees of the algebra. In this way we are led to define an integral table algebra, abbreviated ITA, to be a table algebra $(A,B)$ such that all the structure constants $β_{ijm}$ and all the degrees $f(b_i)$ being rational integers.\par Here we introduce some terminology concerning a table algebra $(A,B)$. A table algebra $(A,B)$ is called homogeneous of degree $λ$ if and only if $|β|\ge 2$ and for some fixed $λ\in\bbfR^+$, $f(b)=λ$ for all $b\in B-{1}$. In the case that $(A,B)$ is ITA and homogeneous it is abbreviated by HITA. Since $B$ is basis for $A$, hence every $x\in A$ can be written as $x=\sum^n_{i=1}λ_ib_i$ where $λ_i\in\bbfC$. The set of all $λ_i>0$ is called the support of $x$ and is denoted by $\text{supp}_B(x)$. For $c\in B$, we define $B_c=\bigcup^\infty_{k=1}\text{supp}_B(c^n)$ and if $B_c=B$, then $c$ is called a faithful element of $B$. If the function $f$ is such that $f(b)=β_{b\overline b1}$ for all $b\in B$, then $B$ is called standard. If $(A,B)$ is an ITA with $B$ standard, then it is called a standard integral table algebra, SITA for short.\par Classification of integral table algebras is an important research problem. The degrees of an ITA is a natural parameter which may be used for classification of ITA. Homogeneous integral table algebras of degree $1$ were classified by {\it Z. Arad} and {\it H. I. Blau} [loc. cit.].\par The classification of HITAs of degree $2$ with a faithful element was obtained by {\it H. Blau} [in Algebra Colloq. 4, No. 4, 393-408 (1997; Zbl 0894.20004)]. A complete classification of the HITAs of degree $3$ with a faithful element provided that the algebra does not contain linear elements was obtained by {\it Z. Arad}, {\it H. I. Blau}, {\it E. Fisman}, {\it V. Miloslavsky}, {\it M. Muzychuk} and {\it B. Xu} [Homogeneous integral table algebras of degree three: A trilogy, Mem. Am. Math. Soc. 684 (2000; Zbl 0958.20010)].\par Since every element of a table algebra is contained in the unique subalgebra generated by this element, hence it is natural to study table algebras generated by a single element. The full classification of the HITAs of degrees $4$ or $5$ is another research problem. In the book under review the authors continue their research about the standard integral table algebras (SITA) generated by a non-real element of degree $4$ or $5$. We remark that in [Beitr. Algebra Geom. 41, No. 1, 33-47 (2000; Zbl 0964.20005)], {\it M.-R. Darafsheh} and {\it A. Rahnamai Barghi} attempted to classify standard HITAs of degree $5$ under the additional condition that the basis set contains a faithful element of width $3$. We recall that the width of an element $b\in B$ is defined to be the support of $b\overline b$.\par The book under review contains 5 chapters which are contributed by several authors. Chapter 1 is introductory and is written by {\it Z. Arad} and {\it M. Muzychuk} describing all the concepts needed to study table algebras. Chapter two is written by {\it Z. Arad}, {\it M. Muzychuk}, {\it H. Arisha} and {\it E. Fisman} and deals with SITAs containing a faithful non-real element of degree $4$. Chapter three is written by {\it Z. Arad}, {\it F. Bunger}, {\it E. Fisman} and {\it M. Muzychuk} and deals with SITAs with a faithful non-real element of degree $5$. Chapter four is written by {\it F. Bunger} and studies SITAs with a faithful real element of degree $5$ and width 3. Finally, in chapter five which is written by {\it M. Hirasaka}, primitive commutative association schemes which contain a connected non-symmetric relation of valency 3 or 4 are classified. This chapter may be viewed as an application of abstraction of Bose-Mesner algebras to ITAs. In this book for all the known SITAs which are generated by a non-real element of degree $k$, the degrees of all basis elements are bounded by a function of $k$.\par This observation leads the authors to make the following conjecture: If a SITA is generated by a non-real element of degree $k$, then there is a function $f\colon\bbfN\to\bbfN$, such that the degrees of all the basis elements of bounded by $f(k)$. Here $\bbfN$ denotes the set of all the natural numbers.