Geometric interpretation of the polynomials in Theorem 3.1

Ascertain whether the symmetric polynomials appearing in Theorem 3.1 (specifically the polynomials that encode the quadratic and cubic Casimir actions on Schur modules) admit a general geometric interpretation for arbitrary rank r, beyond the special cases noted in the paper.

Background

The paper identifies coefficients governing the transformation of logarithmic Chern classes under Schur functors with actions of quadratic and cubic Casimir elements of sl_r. In special cases, these coefficients admit geometric or representation-theoretic interpretations (e.g., dimensions of spaces of sections on Grassmannians when r=4, and a Cartan power for g=so8 when r=3).

However, the authors note that they lack a general explanation or interpretation valid for all ranks, motivating a broader investigation into geometric meanings of the polynomials arising in Theorem 3.1.