Understanding star-fundamental algebras

Abstract

Star-fundamental algebras are special finite dimensional algebras with involution ∗ over an algebraically closed field of characteristic zero defined in terms of multialternating ∗-polynomials. We prove that the upper-block matrix algebras with involution introduced in Di Vincenzo and La Scala [J. Algebra 317 (2007), pp. 642–657] are star-fundamental. Moreover, any finite dimensional algebra with involution contains a subalgebra mapping homomorphically onto one of such algebras. We also give a characterization of star-fundamental algebras through the representation theory of the symmetric group.