### Abstract

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.

Original language | English |
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Pages (from-to) | 163-202 |

Number of pages | 40 |

Journal | Kyushu Journal of Mathematics |

Volume | 67 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 25 2013 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Nakashima, H., & Nomura, T. (2013). Clans defined by representations of euclidean Jordan algebras and the associated basic relative invariants.

*Kyushu Journal of Mathematics*,*67*(1), 163-202. https://doi.org/10.2206/kyushujm.67.163