Structure constants of Peterson Schubert calculus

We give an explicit, positive, and type-uniform formula for all equivariant structure constants of the Peterson Schubert calculus in arbitrary Lie types, using only the Cartan matrix of the corresponding root system $Φ$. As an application, we derive a type-uniform formula for the mixed $Φ$-Eulerian numbers.

💡 Research Summary

This paper solves a long‑standing problem in the Schubert calculus of Peterson varieties by providing a completely explicit, positive, and type‑uniform formula for all equivariant structure constants of the Peterson Schubert calculus, together with a uniform expression for mixed Φ‑Eulerian numbers.

The authors begin by recalling the Peterson variety Pet G as a remarkable subvariety of the flag variety G/B associated to a complex semisimple group G. The variety carries a natural action of a one‑dimensional sub‑torus S ⊂ T, and its S‑equivariant cohomology H_S^*(Pet G) has a distinguished basis {p_I | I⊂Δ} consisting of Peterson Schubert classes. Earlier work (Goldin–Mihalcea–Singh, Drellich, etc.) proved that the structure constants c^K_{I,J} defined by
p_I·p_J = Σ_K c^K_{I,J} p_K
are polynomials in the equivariant parameter t with non‑negative integer coefficients, but no uniform closed formula was known.

Using the Borel‑type presentation of H_S^*(Pet G) due to Harada–Horiguchi–Masuda, the authors first express each class p_I as a single monomial:

p_I = det(C_I)·|W_I|·∏_{i∈I} θ_i,

where C_I is the Cartan sub‑matrix indexed by I, |W_I| is the order of the corresponding parabolic subgroup, and θ_i are the equivariant first Chern classes of the fundamental line bundles.

The main theorem (Theorem 1.3) then gives the equivariant structure constant as

c^K_{I,J}= det(C_I) det(C_J) |W_K| |W_I|^{-1} |W_J|^{-1} det(C_K) ·