Develop the super-Grassmannian interpretation of the Berezinian correction pairing

Develop the super-Grassmannian geometric interpretation of the matrix-valued pairing \(\Delta\), including its possible identification with a tangent vector to a super-Grassmannian at the point corresponding to \(D\), in the spirit of the classical description of the tangent space of a Grassmannian via homomorphisms between tautological and quotient bundles.

Background

For scalar block size m=n=1m=n=1, the odd off-diagonal entries determine coefficient vectors b,c∈Vb,c\in V, and the correction term Δ\Delta is proportional to the decomposable bivector b∧cb\wedge c. Its projective class therefore gives the Plücker image of the two-plane spanned by bb and cc in Gr(2,V)\mathrm{Gr}(2,V).

For general block sizes, Δ\Delta lies in Mm(K)⊗Λ2(V)M_m(K)\otimes\Lambda^2(V) and can be viewed as a family of Plücker-type data obtained by composing the odd blocks through D0−1D_0^{-1} and antisymmetrizing in the generator space. The paper records this coordinate-level analogy but does not establish the proposed super-Grassmannian geometric interpretation or the tangent-vector identification.

References

We do not develop the resulting super-Grassmannian geometry (e.g.\ an identification of \Delta with a tangent vector to a super-Grassmannian at the point corresponding to D, in the spirit of the classical description of \mathrm{Gr}(2,V)'s tangent space via \mathrm{Hom} of the tautological sub- and quotient bundles) beyond this coordinate-level observation, and leave it, as in the 1|1 note, for future work.

— Explicit Super-Linear Algebra over K[θ1, θ2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size  (2609.05148 - V, 4 Sep 2026) in Remark \ref{rem:plucker}, Section “The pairing \(\Delta\), the alternating form on \(R_{\bar1}\), and a Plücker-coordinate reading”