## The character tables of Paige’s simple Moufang loops and their relationship to the character tables of PSL(2,q).(English)Zbl 0682.20050

From the introduction: Let $${\mathcal H}=(X,\{R_ i\}_{0\leq i\leq d})$$ be a (commutative) association scheme. Let $$A_ i$$ be the adjacency matrix with respect to the relation $$R_ i$$, and let $${\mathcal A}=<A_ 0,A_ 1,...,A_ d>=<E_ 0,E_ 1,...,E_ d>$$ be the Bose-Mesner algebra with $$E_ 0,E_ 1,...,E_ d$$ the primitive idempotents. Let $$A_ i=\sum^{d}_{j=0}p_ i(j)E_ j$$. Then the $$(d+1)$$ by $$(d+1)$$ matrix $$P=(p_{ij})$$, whose (i,j)-entry $$p_{ij}$$ is $$p_ j(i)$$, is called the first eigenmatrix of the association scheme $${\mathcal H}$$. We also call it the character table of $${\mathcal H}$$. More generally, let Q be a finite loop or quasigroup. For $$x\in Q$$, there are defined permutations of Q, L(x): $$y\mapsto xy$$, R(x): $$y\mapsto yx$$. Let Gr(Q) be the group generated by all the L(x) and R(x). Then taking the orbits of the group Gr(Q) acting on the set $$Q\times Q$$ as the relations, we get a commutative association scheme. The matrix P is called the character table of the loop (or quasigroup). A loop is called a Moufang loop if the identity $$(xy)(zx)=x((yz)x)$$ is satisfied. A loop Q is called simple if there is no non-trivial loop homomorphism from Q. It is known that there are no finite simple non-associative Moufang loops other than the ones defined by Paige. In this paper we determine the conjugacy classes and the character tables of Paige’s Moufang loops. The reader can notice striking similarities to the conjugacy classes and the character tables of some groups mentioned in the title.
Reviewer: M.Csikós

### MSC:

 20N05 Loops, quasigroups 20C15 Ordinary representations and characters 05B30 Other designs, configurations 20G40 Linear algebraic groups over finite fields
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