Abstract The open projected Richardson varieties form a stratification for the partial flag variety $G/P$. We compare the Segre–MacPherson classes of open projected Richardson varieties with those of the corresponding affine Schubert cells by pushing or pulling them to the affine Grassmannian. In the Grassmannian case, the open projected Richardson varieties are well known as open positroid varieties. We obtain symmetric functions that represent the Segre–MacPherson classes of open positroid varieties, constructed explicitly in terms of pipe dreams for affine permutations.

In this article, we provide characterizations of toric Richardson varieties across all types through three distinct approaches: 1) poset theory, 2) root theory, and 3) geometry.

We show the existence of cluster A \mathcal {A} -structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig’s coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc [Adv. Math. 300 (2016), pp. 190–228] on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.

Abstract Let $Z\subset \operatorname{Fl}(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside $\operatorname{Fl}(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.

Leclerc's conjecture on a cluster structure for type A Richardson varieties

We study the effect of Feigin’s flat degeneration of the type A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {A}$$\\end{document} flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some degenerate Schubert varieties (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin’s Gröbner degeneration) with Richardson varieties in higher rank partial flag varieties.

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-totally nonnegative flag variety has a cellular decomposition into totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson varieties. Moreover, each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U^{-}$\\end{document} for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

Let [Formula: see text] and [Formula: see text] be positive coprime integers with [Formula: see text]. Let [Formula: see text] denote the subgroup of diagonal matrices in [Formula: see text]. We study the GIT quotient of Richardson varieties [Formula: see text] in the Grassmannian Gr[Formula: see text] by [Formula: see text] with respect to a [Formula: see text]-linearized line bundle [Formula: see text] corresponding to the Plücker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety [Formula: see text] to be smooth.

The isomorphism problem for Grassmannian Schubert varieties

Abstract A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalise several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalisation, in the flag variety case, of a theorem of Knutson–Woo–Yong; we also generalise this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill–Noether varieties on elliptic curves, and a conjectural generalisation to higher genus curves.

Toric Richardson varieties of Catalan type and Wedderburn–Etherington numbers

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing the coordinate ring at a suitable set of Plücker coordinates. We prove that these Plücker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.

Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular points. Further, we have computed the exact number of singular points of the GIT quotient of $G_{2,n}.$

A Birkhoff–Bruhat atlas for partial flag varieties

Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studying Gröbner degenerations of their corresponding ideals. These degenerations are parametrised by block diagonal matching fields in the sense of Sturmfels [33]. We associate a weight vector to each block diagonal matching field and study its corresponding initial ideal. In particular, we characterise when such ideals are toric, hence providing a family of toric degenerations for Richardson varieties. Given a Richardson variety $$X_{w}^v$$ and a weight vector $$\mathbf{w}_\ell $$ arising from a matching field, we consider two ideals: an ideal $$G_{k,n,\ell }|_w^v$$ obtained by restricting the initial of the Plücker ideal to a smaller polynomial ring, and a toric ideal defined as the kernel of a monomial map $$\phi _\ell |_w^v$$ . We first characterise the monomial-free ideals of form $$G_{k,n,\ell }|_w^v$$ . Then we construct a family of tableaux in bijection with semi–standard Young tableaux which leads to a monomial basis for the corresponding quotient ring. Finally, we prove that when $$G_{k,n,\ell }|_w^v$$ is monomial-free and the initial ideal in $$_{\mathbf{w}_\ell }(I(X_w^v))$$ is quadratically generated, then all three ideals in $$_{\mathbf{w}_\ell }(I(X_w^v))$$ , $$G_{k,n,\ell }|_w^v$$ and ker $$(\phi _\ell |_w^v)$$ coincide, and provide a toric degeneration of $$X_w^v$$ .

For any simple, simply connected algebraic group [Formula: see text] of type [Formula: see text] and [Formula: see text] and for any maximal parabolic subgroup [Formula: see text] of [Formula: see text], we provide a criterion for a Richardson variety in [Formula: see text] to admit semistable points for the action of a maximal torus [Formula: see text] with respect to an ample line bundle on [Formula: see text].

We study the GIT quotient of the minimal Schubert variety in the Grassmannian admitting semistable points for the action of maximal torus T, with respect to the T-linearized line bundle and show that this is smooth when When n = 7 and r = 3 we study the GIT quotients of all Richardson varieties in the minimal Schubert variety. This builds on work by Kumar [21], Kannan and Sardar [18], Kannan and Pattanayak [17], and Kannan et al. [16]. It is known that the GIT quotient of is projectively normal. We give a different combinatorial proof.

Abstract In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs. This result generalizes a theorem of Scott [Proc. Lond. Math. Soc. (3) 92 (2006) 345–380] for the Grassmannian, and proves a folklore conjecture for Schubert varieties that has been believed by experts since Scott's result [Proc. Lond. Math. Soc. (3) 92 (2006) 345–380], though the statement was not formally written down until Muller–Speyer explicitly conjectured it [Proc. Lond. Math. Soc. (3) 115 (2017) 1014–1071]. To prove this conjecture we use a result of Leclerc [Adv. Math. 300 (2016) 190–228] who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety can be identified with cluster algebras. Our proof also uses a construction of Karpman [J. Combin. Theory Ser. A 142 (2016) 113–146] to build plabic graphs associated to reduced expressions. We additionally generalize our result to the setting of skew‐Schubert varieties; the latter result uses generalized plabic graphs, that is, plabic graphs whose boundary vertices need not be labeled in cyclic order.

Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(ℂn) arising naturally at the intersection of geometry, representation theory, and combinatorics. Recent results of Abe, Horiguchi, Masuda, Murai, and Sato as well as of Abe, DeDieu, Galetto, and Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand–Zetlin polytope GZ(λ) for λ = (λ1, λ2, …, λn). In the main results of this paper we use and generalize tools developed by Anderson and Tymoczko, by Kiritchenko, Smirnov, and Timorin, and by Postnikov in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand–Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the αi:= λi − λi+1. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial (n − 1)-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in Flag(ℂn) as a sum of the cohomology classes of a certain set of Richardson varieties.

Total positivity for the Lagrangian Grassmannian