Relationship between Jacobi–Trudi structures, homotopy Lie algebras, and deviations

Determine the precise relationship between Jacobi–Trudi structures on Koszul algebras, the homotopy Lie algebra π^•(A) (whose enveloping algebra is Ext_A^•(k,k)), and the deviations ε_i(A) that measure the graded dimensions of π^i(A). Establish how total positivity constraints on Hilbert functions correspond to constraints on deviations and clarify whether there exists a canonical isomorphism A^{⊗ n} ≅ U(π_{C^n}^•(A))/I_{C^n} for general n beyond the known cases.

Background

The paper proposes that Jacobi–Trudi structures for algebras A may be realized via quotients of enveloping algebras of ‘fattened’ homotopy Lie algebras π{C^n}^•(A), suggesting an isomorphism of the form A^{⊗ n} ≅ U(π{C^n}^•(A))/I_{C^n}. This is known for n=1 and, in some special cases, for n=2 (e.g., certain isotropic Grassmannians).

Moreover, for totally positive Hilbert functions, the authors note that the deviations ε_i(A) determine the Hilbert function, implying numerical constraints. However, they explicitly state that the exact conceptual and structural relationship among Jacobi–Trudi structures, homotopy Lie algebras, and deviations remains unclear, motivating a precise characterization.