The invariants of the second symmetric power representation of SL_2(F_q)

TL;DR

This paper computes the invariants of the second symmetric power representation of SL_2 over finite fields, revealing they form a hypersurface and the invariant field is purely transcendental, with intermediate results on Sylow p-subgroups.

Contribution

It provides explicit generators for the invariants of the second symmetric power representation of SL_2(F_q), including the structure of the invariant ring and field.

Findings

Invariants form a hypersurface

Invariant field is purely transcendental

Sylow p-subgroup invariants are hypersurfaces

Abstract

For a prime p>2 and q=p^n, we compute a finite generating set for the SL_2(F_q)-invariants of the second symmetric power representation, showing the invariants are a hypersurface and the field of fractions is a purely transcendental extension of the coefficient field. As an intermediate result, we show the invariants of the Sylow p-subgroups are also hypersurfaces.