Clans defined by representations of euclidean Jordan algebras and the associated basic relative invariants

Starting with a representation φ{symbol} 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 VE, we obtain a clan V ⁰_E with unit element. By computing the determinant of the right multiplication operators of V ⁰ _E, we get an explicit expression of the basic relative invariants of V ⁰_E in terms of the Jordan algebra principal minors of V and the quadratic map associated with φ{symbol}. For the dual clan of V ⁰ _E, we also obtain an explicit expression of the basic relative invariants in a parallel way.