We show that the set of totally positive unipotent lower-triangular Toeplitzmatrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$.Furthermore we prove a natural cell decomposition for its closure. The proofuses properties of the quantum cohomology rings of the partial flag varietiesof $GL_n(\C)$ relying in particular on the positivity of the structureconstants, which are enumerative Gromov--Witten invariants. We also give acharacterization of total positivity for Toeplitz matrices in terms of the(quantum) Schubert classes. This work builds on some results of Dale Peterson'swhich we explain with proofs in the type $A$ case.
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