Algebro — Geometric applications of schur s- and q-polynomials

Quantum cohomology of the Lagrangian Grassmannian

“…From the analysis in [P,Sect. 6] and [PR] we obtain that the map which sends Q λ (X) to σ λ for all λ ∈ D n extends to a surjective ring homomorphism φ : Λ n → H * (LG, Z) with kernel generated by the relations Q i,i (X) = 0 for 1 i n; the morphism φ is evaluation on the Chern roots of the tautological rank n quotient bundle over LG.…”

Schubert Polynomials for the Classical Groups

“…Implicit in our results are new proofs of theorems of Pragacz [15] identifying the Schubert polynomials for isotropic Grassmannians-varieties consisting of subspaces isotropic with respect to a symplectic form on C 2n or a symmetric form on C 2n or C 2n +!. Pragacz shows that these Schubert polynomials are Schur Q-or P-functions, in the same sense that the Schubert polynomials for ordinary Grassmannians are ordinary Schur functions.…”

Positivity of Thom polynomials II: the Lagrange singularities

“…We also recall from [14] their cohomological interpretation in terms of Schubert classes for the Lagrangian Grassmannian.…”

“…The family of Q-functions was invented and investigated in [15] when studying Lagrangian degeneracy loci. It is modelled on the classical Schur Q-functions (see, e.g., [14]). More precisely, for a strict partition I, the Schur Q-function of a vector bundle E is obtained from Q I by the substitution…”