ABSTRACT

Totally positive Toeplitz matrices and quantum cohomology

Konstanze Rietsch

A matrix in GL_{n}(ℝ) is called totally positive if all of its minors are positive. For GL_{n} and other simply laced reductive groups, total positivity has been shown by Lusztig [4] to be closely related to canonical bases. We will demonstrate [6] that for the subset X of unipotent upper-triangular Toeplitz matrices in GL_{n} the corresponding notion of total positivity is closely related to properties of quantum cohomology rings of (partial) flag varieties and their Schubert bases. This surprising connection allows us to prove a very elementary to state parametrization of the totally nonnegative part X_{≥0} of X, and show that it has a very natural cell decomposition. The work uses arguments from quantum Schubert calculus [1] [2] for the partial flag varieties Gl_{n}/P and relies in particular on the positivity of the structure constants (Gromov–Witten invariants). Another essential ingredient and part of the original motivation for this work are some remarkable results, or rather a whole point of view, due to Dale Peterson [5] [3] which we will describe in the type A case.

References

[1] I. Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. (1999), no. 3, 485–523.

[2] S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 168 (1997), 565–596.

[3] B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ρ, Selecta Math. (N.S.) 2 (1996), 43–91.

[4] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry: in honor of Bertram Kostant (G. I. Lehrer, ed.), Progress in Mathematics, vol. 123, Birkhaeuser, Boston, 1994, pp. 531–568.

[5] D. Peterson, Quantum cohomology of G/P, Lecture Course, M.I.T., Spring Term, 1997.

[6] K. Rietsch. Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. arXiv.org/abs/math.QA/0112024, 2001.