Quaternionic equiangular lines
Boumediene Et-Taoui
Abstract
Abstract Let π½ = β, β or β. A p -set of equi-isoclinic n -planes with parameter Ξ» in π½ r is a set of p n -planes spanning π½ r each pair of which has the same non-zero angle arccos <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msqrt> <m:mi>Ξ»</m:mi> </m:msqrt> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \sqrt{\lambda} \end{array}$ . It is known that via a complex matrix representation, a pair of isoclinic n -planes in β r with angle arccos <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msqrt> <m:mi>Ξ»</m:mi> </m:msqrt> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \sqrt{\lambda} \end{array}$ yields a pair of isoclinic 2 n -planes in β 2 r with angle arccos <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msqrt> <m:mi>Ξ»</m:mi> </m:msqrt> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \sqrt{\lambda} \end{array}$ . In this article we characterize all the p -tuples of equi-isoclinic planes in β 2 r which come via our complex representation from p -tuples of equiangular lines in β r . We then construct all the p -tuples of equi-isoclinic planes in β 4 and derive all the p -tuples of equiangular lines in β 2 . Among other things it turns out that the quadruples of equiangular lines in β 2 are all regular, i.e. their symmetry groups are isomorphic to the symmetric group S 4 .