Errata to ``Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties''

We make a correction to Remark 4.3 and the proof of Theorem 4.2 (Peterson's Theorem) which identifies $qH^*(SL_n/P)$ with the coordinate ring $\mathcal O(\mathcal Y_P)$ of a certain affine stratum of the Peterson variety $\mathcal Y$. Explicitly, we introduce additional coordinates to obtain a complete coordinate system on $B^+w_P B^-/B^-$ and then show that they lie in the defining ideal of the Peterson variety $\mathcal Y_P$, hence play no role in the presentation of $\mathcal O(\mathcal Y_P)$.