CLANS DEFINED BY REPRESENTATIONS OF EUCLIDEAN JORDAN ALGEBRAS AND THE ASSOCIATED BASIC RELATIVE INVARIANTS

Abstract

Starting with a representation φ of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in V_E:=E⊕V. By the adjunction of a unit element to V_E, we obtain a clan V_E⁰ with unit element. By computing the determinant of the right multiplication operators of V_E⁰, we get an explicit expression of the basic relative invariants of V_E⁰ in terms of the Jordan algebra principal minors of V and the quadratic map associated with φ. For the dual clan of V_E⁰, we also obtain an explicit expression of the basic relative invariants in a parallel way.