The Projective Geometry of Freudenthal's Magic Square

Abstract

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial. ᮊ

Classification of multipartite systems featuring only $| W\rangle $ and $| {GHZ}\rangle $ genuine entangled states

“…The orbit structure of the projectivized Hilbert spaces P(H) with the SLOCC groups G of Table II is fully provided by Ref 25 . In particular the authors show that, except for G = SL 2 (C) and H = Sym 3 (C 2 ) (three bosonic qubits), there are exactly 4 orbits.…”

“…The variety σ + (X) is the closure of points of type |ψ + |χ where |ψ and |χ are two separable states which do not form a generic pair (see Ref 25 for the description of the isotropic condition satisfied by this pair (|ψ , |χ )). The smooth points of σ + (X) are therefore identified with biseparable states.…”

“…This will be emphasized in Appendix A 2 when we refer to a uniform geometric parametrization of the varieties of separable states given by Ref 25 .…”

“…The algebraic varieties of Table II and Manivel investigated the geometry of those varieties in a series of papers [25][26][27] . Their goal was to establish new connections between representation theory and algebraic geometry.…”

Flexibility of Schubert classes

“…It is a 27-dimensional cominuscule homogeneous variety. We refer the reader to [IM], [LM1], [LM2], [LM3] and [NS] for more details on the discussion in this section.…”