4 problems

Let $(\Psi,\mu,w)$ be a labeled root ideal with partition $\mu$, let $j\geq0$, and let $X_w$ be the corresponding Schubert variety. Write $\mathfrak u_\Psi$ for the root subspace a…

Chen–Haiman's conjecture. The Catalan functions $H(\Psi;\mu)$ are Schur positive:

Let $\lambda$ be a partition, let $\operatorname{kskew}(\lambda)$ denote its associated $k$-skew diagram, and let $\Phi(\operatorname{kskew}(\lambda))$ be the Chen-Haiman root idea…

Let $k$-Schur functions be indexed by partitions $u$ with $u1leq k$. Chen-Haiman constructed an ideal of positive roots associated to each such partition, and let…