The Payne q-Clans

Abstract

Suppose that $C = \{ A_t \equiv \left( {\begin{array}{*{20}c} {x_t } \\ 0 \\ \end{array} \begin{array}{*{20}c} {y_t } \\ {z_t } \\ \end{array} } \right):t \in F\}$ is a q-clan for which each of the functions xt, yt, zt is a monomial function. In a rather remarkable paper, T. Penttila and L. Storme [PS98] show that up to the usual equivalence of q-clans, the three known examples are the only ones. Since the two non-classical families exist only for e odd, we assume throughout this chapter that e is odd. Then the three known families have the following appearance. There is some positive integer i for which

$$ C = \{ A_t = \left( {\begin{array}{*{20}c} {t^{\frac{1} {{2i}}} } \\ 0 \\ \end{array} \begin{array}{*{20}c} {t^{\frac{1} {2}} } \\ {t^{\frac{{2i - 1}} {{2i}}} } \\ \end{array} } \right):t \in F\} . $$