{
  "annotation_id": "b790e39f-5d1e-48f7-b098-8e33847c0e5e",
  "date_created": "2026-03-02T18:01:28.477000Z",
  "date_modified": "2026-03-02T18:01:28.477000Z",
  "file_hash": "29dc328661d338a9cac1c2b3a6db088416f61e845f43b738580e1c2505d01c6a",
  "private": false,
  "record": {
    "abstract": "We study the algebraic aspects of equivariant quantum cohomology algebra of\nthe flag manifold. We introduce and study the quantum double Schubert\npolynomials, which are the Lascoux-Schutzenberger type representatives of the\nequivariant quantum cohomology classes. Our approach is based on the quantum\nCauchy identity. We define also quantum Schubert polynomials as the\nGram-Schmidt orthogonalization of some set of monomials with respect to the\nscalar product, defined by the Grothendieck residue. Using quantum Cauchy\nidentity, we prove that quantum Schubert polynomials are the specialization of\nquantum double Schubert polynomials with second set of variables equals to\nzero, and as a corollary obtain a simple formula for the quantum Schubert\npolynomials. We also prove the higher genus analog of Vafa-Intriligator\u0027s\nformula for the flag manifolds and study the quantum residues generating\nfunction. We introduce the extended Ehresman-Bruhat order on the symmetric\ngroup and formulate the equivariant quantum Pieri rule.",
    "arxiv_id": "q-alg/9610022",
    "authors": [
      "Anatol N. Kirillov",
      "Toshiaki Maeno"
    ],
    "categories": [
      "q-alg",
      "math.QA"
    ],
    "title": "Quantum double Schubert polynomials, quantum Schubert polynomials and   Vafa-Intriligator formula",
    "url": "https://arxiv.org/abs/q-alg/9610022"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "ca0b94b9-99fc-4d5d-812e-a697ae446f45",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}