An infinite-dimensional character coincidence between Lie algebras of type B and BC

Abstract

We utilize an isomorphism between the character rings of the odd orthogonal group and the orthosymplectic supergroup to understand equivariant positivity properties of the type B quadric hypersurface ring. Our main result establishes a well-behaved functorial construction of Schur modules “with respect to” the quadric hypersurface ring, an essential fact used by the authors in previous work to construct pure free resolutions. Our techniques combine ideas from commutative algebra, Lie theory, and algebraic geometry to understand representations of orthosymplectic supergroups and their corresponding type B counterparts.