arXiv subjects
Publications and source records attributed to Jeff York Ye.
We show that various partitions of the canonical basis of quantum groups constructed by Lusztig and by Kashiwara coincide. Using this partition, we show that the subset corresponding to open Richardson varieties equals to the intersection of the subsets corresponding to the Schubert cells. We also show that in type $A$, the weights from open Richardson varieties are saturated in the corresponding Bruhat interval polytope.
We construct cluster structures on the strata $\mathring{S}_w$ of a stratification of the symmetric space $SL_n/SO_n$ over $\mathbb{C}$. To accomplish this, we study foldings of upper cluster algebras and show that the cluster structures on $SL_n/SO_n$ can be obtained from those on $SL_n$ via folding. As a corollary, we show that these cluster structures are compatible with the De Concini Poisson structures, and we construct cluster structures on the variety of symmetric matrices $\text{Sym}_n$.
We show that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra. As an application, we construct cluster structures on a large class of flat reductive monoids. After localization, we obtain cluster structures on any connected reductive group whose commutator group is simply connected.
We show coordinate rings of open Richardson varieties are upper cluster algebras for any symmetrizable Kac--Moody type. We further show the coordinate rings of (generalized) open Richardson varieties on the twisted product of flag varieties are upper cluster algebras for any symmetrizable Kac--Moody type. This includes, as special cases, reduced double Bruhat cells, Bott-Samelson varieties, braid varieties. Our results generalize various results by Casals--Gorsky--Gorsky--Le--Shen--Simental and Galashin--Lam--Sherman-Bennett--Speyer in finite types.
In this paper, we give a sufficient and necessary condition for a regular element of a quantum cluster algebra $\mathcal{O}_q(\mathcal{X})$ to be universally polynomial. This resolves several conjectures by the first author on the polynomiality of the cluster realization of quantum group generators in different families of positive representations.