An infinite-dimensional character coincidence between Lie algebras of type B and BC
Steven V Sam, Keller VandeBogert
TL;DR
This work reveals an infinite-dimensional bridge between type B and BC Lie theories by leveraging an isomorphism between O(2m+1) and SpO(2m|1) character rings, enabling Jacobi–Trudi-type descriptions of odd quadric hypersurface rings. The authors construct Z_{V,\mathbf{C}^n}, a functorial, representation-theoretic object whose GL_n-weights reproduce the quadric-structure A = S^\bullet(V)/(q) and whose O(V)×GL_n character matches that of S^\bullet(\mathbf{C}^{2m|1}⊗\mathbf{C}^n), via a Howe dual pair with respect to SPO(2m|1). They develop orthosymplectic Littlewood complexes, compute their homology, and connect these to parabolic Verma modules and Tor groups, yielding explicit equivariant Hilbert series and resolutions. The type D (even quadric) case is shown to exhibit failures of equivariant positivity, tied to Proctor’s odd symplectic phenomena, while a parallel orthosymplectic analogue and conjectural extensions are outlined. Together, these results provide a robust algebraic framework for Jacobi–Trudi structures on quadric rings and open avenues for further interplay between current-algebra, Clifford- and Howe-duality perspectives in the super setting.
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.