Related papers: Combinatorics of cluster structures in Schubert va…

Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives,…

We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…

We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a…

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the…

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron…

Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…

We prove that the duals of the quantum Frobenius morphisms and their splittings by Lusztig are compatible with quantum cluster monomials. After specialisation, we deduce that the canonical Frobenius splittings on flag varieties are…

Starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as the cluster X-varieties, as defined in math.AG/0311245. In particular they are Poisson…

Let G be a connected reductive group and X an equivariant compactifiction of G. In X, we study generalised and opposite generalised Schubert varieties, their intersections called generalised Richardson varieties and projected generalised…

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum…

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several…

It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $GL_n(\mathbb{C})/B$ can be characterized purely combinatorially in terms of Bruhat order on the symmetric group $\mathfrak{S}_n$. In a recent preprint,…

In this article, we construct SL$_k$-friezes using Pl\"ucker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Pl\"ucker embedding. When this cluster…

Naisse and Vaz defined an extension of KLR algebras to categorify Verma modules. We realise these algebras geometrically as convolution algebras in Borel-Moore homology. For this we introduce Grassmannian-Steinberg quiver flag varieties.…

This is a survey article on Richardson varieties and their combinatorics. A Richardson variety is the intersection, inside the flag manifold GL_n/B_+, of a Schubert cell (B_- u B_+)/B_+ and an opposite Schubert cell (B_+ w B_+)/B_+ (or the…

We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in $\mathbb{TP}^5$. This follows from a result of Sturmfels and…

Let L be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories sub(Q) for Q an injective L-module, and we introduce a mutation…

In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials.…

A stratified variety has a Kazhdan-Lusztig atlas if it can be locally modelled with Kazhdan-Lusztig varieties stratified by Schubert varieties in some Kac-Moody flag manifold via stratified isomorphisms. In this paper, we show that the…

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.