The relatively universal cover of the natural embedding of the long root geometry for the group $\mathrm{SL}(n+1,\mathbb{K})$

I. Cardinali; L. Giuzzi; A. Pasini

Paper

The relatively universal cover of the natural embedding of the long root geometry for the group $\mathrm{SL}(n+1,\mathbb{K})$

Abstract

The long root geometry

for the special linear group

admits an embedding in the (projective space of) the vector space of the traceless square matrices of order

with entries in the field

, usually regarded as the {\em natural} embedding of

. S. Smith and H. Völklein (A geometric presentation for the adjoint module of

, {\em J. Algebra}, vol. 127) have proved that the natural embedding of

is relatively universal if and only if

is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of

which covers the natural one, but that information is not sufficient to exhaustively describe it. The "if" part of Smith-Völklein's result also holds true for any

, as proved by Völklein in his investigation of the adjoint modules of Chevalley groups (H. Völklein, On the geometry of the adjoint representation of a Chevalley group, {\em J. Algebra}, vol. 127). In this paper we give an explicit description of the relatively universal embedding of

which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to

where

is the transcendence degree of

over its minimal subfield (if

) or the generating rank of

over

(if

). Accordingly, both the "if" and the "only if" part of Smith-Völklein's result hold true for every