From quantum cohomology to algebraic combinatorics: The example of flag manifolds (Q5930079)

This article deals with the quantum cohomology of the classical flag manifold \(\text{GL}(n,\mathbb C)/B\), where \(B\) is the subgroup of upper triangular nonsingular matrices. The cohomology ring can be obtained as a factor ring of the polynomials in \(n\) variables modulo the ideal generated by the elementary symmetric polynomials. A distinguished basis for the classical cohomology ring is given by the Schubert polynomials. The quantum cohomology ring again is a factor ring of the polynomial ring but now with \(n-1\) additional variables, the deformation parameters, modulo the ideal generated by the ``quantum elementary polynomials''. The author relates the quantum Schubert polynomial and some other related basis with problems in algebraic combinatorics. For the proofs and further details he mainly refers to the following publications of the author [Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069) and ``On algebraic and combinatorial properties of Schur and Schubert polynomials'' (Bayreuther Math. Schr. 59) (2000; Zbl 0958.05001)].