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We show that the k-double Schur functions defined by the authors, and thequantum double Schubert polynomials studied by Kirillov and Maeno and byCiocan-Fontanine and Fulton, can be obtained from each other by an explicitrational substitution. The main new ingredient is an explicit computation ofKostant's solution to the Toda lattice in terms of equivariant Schubertclasses.

From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice