English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let  G  be the universal Chevalley–Demazure group scheme corresponding to a reduced irreducible root system of rank at least 2, and let  R  be a commutative ring. We analyze the linear representations  ρ  :  G ( R ) +  → GL  n   ( K ) over an algebraically closed field  K  of the elementary subgroup  G ( R ) +  ⊂  G ( R ). Our main result is that under certain conditions, any such representation has a standard description; that is, there exist a commutative finite‐dimensional  K ‐algebra  B , a ring homomorphism  f :  R  →  B  with Zariski‐dense image, and a morphism of algebraic groups  σ :  G ( B ) → GL  n   ( K ) such that  ρ  coincides with  σ  ○  F  on a suitable finite index subgroup Γ⊂ G ( R ) + , where  F  :  G ( R ) +  →  G ( B ) +  is the group homomorphism induced by  f . In particular, this confirms a conjecture of Borel and Tits for Chevalley groups over a field of characteristic zero.

On linear representations of Chevalley groups over commutative rings