Classical symmetric varieties and quiver representations

The parametrization of Borel orbits on classical symmetric varieties G / K G/K is traditionally described via the combinatorics of clans, developed by Matsuki, Oshima, Wyser, and Yamamoto. We provide a quiver-theoretic reproof of the indexing sets for G = G L n G = GLₙ or S p 2 n Sp₂ₙ and K K a two-block Levi subgroup using Gabriel’s Theorem. This method reveals an unexpected independence from algebraic closure: while orbit counts for most symmetric varieties are sensitive to the base field, our parametrizations remain valid over arbitrary fields for G L n GL_n , and in characteristic not 2 for S p 2 n Sp₂ₙ . We further reformulate these sets via matchings in corona graphs to derive new enumerative results, including the ultra log-concavity of orbit counts.