Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

A bstract We construct Schubert line defects in the 3d $$ \mathcal{N}=2 $$ N = 2 supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifold X = Fl( k ; n ), generalizing our construction for complete flag manifolds given in a companion paper (part I) [1]. In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d $$ \mathcal{N}=2 $$ N = 2 supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varieties X w ⊆ X in the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of [𝒪 w ] — more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes [ X w ] in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials.

A bstract We construct new half-BPS line defects in 3d $$ \mathcal{N}=2 $$ N = 2 supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds X = Fl( n ). Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target space X and the line defects flow to objects supported on Schubert varieties X w ⊆ X . These Schubert line defects form an important basis of the quantum K-theory of X . They are realized as $$ \mathcal{N}=2 $$ N = 2 supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert variety X w , with the 1d degrees of freedom physically realizing a Bott-Samelson resolution of X w . Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf $$ {\mathcal{O}}_{X_w} $$ O X w as a (double) quantum Grothendieck polynomial, generalizing previous results for X a Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes [ X w ] in the quantum cohomology ring of X .

Abstract We study the multiplicity number of the characteristic cycle of the intersection complex of the matroid Schubert variety. It is shown to be a combinatorial invariant, and can be computed by explicit formulas. We also conjecture that the generalization to an arbitrary matroid is non-negative.

Abstract Brasselet, the second author and Yokura introduced Hodge-theoretic Hirzebruch-type characteristic classes $$IT_{1, *}$$ I T 1 , ∗ , and conjectured that they are equal to the Goresky-MacPherson L -classes for pure-dimensional compact complex algebraic varieties. In this paper, we show that the framework of Gysin coherent characteristic classes of singular complex algebraic varieties developed by the first and third author in previous work applies to the characteristic classes $$IT_{1, *}$$ I T 1 , ∗ . In doing so, we prove the ambient version of the above conjecture for subvarieties in a Grassmannian. Since the homology of Schubert subvarieties injects into the homology of the ambient Grassmannian, this implies the conjecture for all Schubert varieties in a Grassmannian. We also study other algebraic characteristic classes such as Chern classes and Todd classes (or their variants for the intersection cohomology sheaves) within the framework of Gysin coherent characteristic classes.

We state combinatorial formulas for hyperoctahedral group ( 𝔅 n ) character evaluations of the form χ ( C ˜ w BC ( 1 ) ) where C ˜ w BC ( 1 ) ∈ ℤ [ 𝔅 n ] is a type- BC Kazhdan–Lusztig basis element, with w ∈ 𝔅 n corresponding to simultaneously smooth type- B and C Schubert varieties. We also extend the definition of symmetric group codominance to elements of 𝔅 n and show that for each element w ∈ 𝔅 n as above, there exists a BC -codominant element v ∈ 𝔅 n satisfying χ ( C ˜ w BC ( 1 ) ) = χ ( C ˜ v BC ( 1 ) ) for all 𝔅 n -characters χ . Combinatorial structures and maps appearing in these formulas are type- BC extensions of planar networks, unit interval orders, indifference graphs, poset tableaux, and colorings. Using the ring of type- BC symmetric functions, we introduce natural generating functions Y ( C ˜ w BC ( 1 ) ) for the above evaluations. These provide a new type- BC analog of Stanley’s chromatic symmetric functions [Adv. Math. 111 (1995) pp. 166–194].

Abstract Let K be a nonarchimedean local field of characteristic zero with valuation ring R , for instance, $$K=\mathbb {Q}_p$$ K = Q p and $$R=\mathbb {Z}_p$$ R = Z p . We prove a general integral geometric formula for K –analytic groups and homogeneous K –analytic spaces, analogous to the corresponding result over the reals. This generalizes the p –adic integral geometric formula for projective spaces recently discovered by Kulkarni and Lerario, e.g., to the setting of Grassmannians. Based on this, we outline the construction of a nonarchimedean probabilistic Schubert Calculus. For this purpose, we characterize the relative position of two subspaces of $$K^n$$ K n by a position vector, a nonarchimedean analogue of the notion of principal angles, and we study the probability distribution of the position vector for random uniform subspaces. We then use this to compute the volume of special Schubert varieties over K . As a second application of the general integral geometry formula, we initiate the study of random fewnomial systems over nonarchimedean fields, bounding, and in some cases exactly determining, the expected number of zeros of such random systems.

Residual intersections and Schubert varieties

Pieri and Murnaghan–Nakayama type rules for Chern classes of Schubert Cells

Abstract Schubert polynomials are distinguished representatives of Schubert cycles in the cohomology of the flag variety. In the spirit of Bergeron and Sottile, we use the Bruhat order to give $(n-1)!$ different combinatorial formulas for the Schubert polynomial of a permutation in $S_{n}$. By the work of Lenart and Sottile, one extreme of the formulas recover the classical pipedream (PD) formula. We prove the other extreme corresponds to bumpless pipedreams (BPDs). We give two applications of this perspective to view BPDs: using the Fomin–Kirillov algebra, we solve the problem of finding a BPD analogue of Fomin and Stanley’s algebraic construction on PDs; we also establish a bijection between PDs and BPDs using Lenart’s growth diagram, which conjecturally agrees with the existing bijection of Gao and Huang.

The F-pure threshold of a Schubert cycle

The lattice of flats ℒ M of a matroid M is combinatorially well-behaved and, when M is realizable, admits a geometric model in the form of a “Schubert variety of hyperplane arrangement”. In contrast, the lattice of flats of a polymatroid exhibits many combinatorial pathologies and admits no similar geometric model. We address this situation by defining the lattice ℒ P of “combinatorial flats” of a polymatroid P . Combinatorially, ℒ P exhibits good behavior analogous to that of ℒ M : it is graded, determines P when P is simple, and is top-heavy. When P is realizable over a field of characteristic 0, we show that ℒ P is modeled by “the Schubert variety of a subspace arrangement”. Our work generalizes a number of results of Ardila–Boocher and Huh–Wang on Schubert varieties of hyperplane arrangements; however, the geometry of Schubert varieties of subspace arrangements is noticeably more complicated than that of Schubert varieties of hyperplane arrangements. Many natural questions remain open.

Abstract Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

Statement of the problem. The phenomenon of Schubertianism, which is one of the most representative manifestations of the European chamber and vocal tradition, has not yet received the proper interdisciplinary theoretical considering. In most musicological works, it is understudied either as a stylistic category, or as a historical reception of Franz Schubert’s creative work, or as a manifestation of the vocal interpretation of German Lied. At the same time, Schubertianism is much broader – it constitutes a unique cultural and aesthetic code that combines intonation psychology, poetic semantics, performance dramaturgy and the philosophy of subjectivity. The absence of a holistic integral model of this phenomenon determines the need for its comprehensive scientific analysis. Objectives, methods, and novelty of the research. The purpose of the study is to understand Schubertianism systematically as a multidimensional cultural phenomenon that has formed a separate model of musical thinking based on the psychological dramaturgy of Lied, intonational subjectivity and the interpretative tradition of the 19th–20th centuries. The research is based on historical-contextual, structural-analytical, semiotic, hermeneutic and cultural studies approaches, taking into account vocal performance as a component of musical semantics. The scientific novelty lies in revealing the Schubertianism as an integrative intonation and cultural code that encompasses musical text, psychological states, performance models and cultural memory; in defining Lied as a form of musical self-disclosure of subjectivity; and also in emphasizing the decisive role of Dietrich Fischer Dieskau’s performance school in forming the canon of interpretation of Schubert cycles. Research results. As a result of the study, it has been established that Schubertianism functions as a holistic intonation and semantic system, which combines poetic imagery, psychological reflexivity, chamber dramaturgy, and philosophical symbolism. The three F. Schubert’s vocal cycles – «Die schöne Müllerin», «Winterreise» and «Schwanengesang» – form the dramaturgical axis of the lyrical subjectivity: from emotional primordiality to existential brokenness and meditative maturity. The Fischer-Dieskau’s performance paradigm has established microdynamic accuracy, text-centricity, intonation chamber nature and psychological depth as the main principles of interpreting the Lied. Conclusion. Thus, the phenomenon of Schubertianism goes beyond the boundaries of its definition as style and appears as a universal cultural code that determines the modern perception of musical subjectivity, artistic thinking and performance memory.

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.

We prove a formula for the degrees of Ikeda and Naruse’s P-Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an upper bound on the Castelnuovo–Mumford regularity of certain Pfaffian varieties known as vexillary skew-symmetric matrix Schubert varieties. Similar combinatorics additionally yields a new formula for the degree of Grassmannian Grothendieck polynomials and the regularity of Grassmannian matrix Schubert varieties, complementing a 2021 formula of Rajchgot, Ren, Robichaux, St. Dizier, and Weigandt.

Abstract We realise the Bott-Samelson resolutions of type A Schubert varieties as quiver Grassmannians. In order to explicitly describe this isomorphism, we introduce the notion of a geometrically compatible decomposition for any permutation in the symmetric group $$S_n$$ S n . For smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety. To obtain these isomorphisms, we construct a special quiver with relations and investigate two classes of quiver Grassmannians for this quiver.

The theory of Newton–Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties. In this paper, we study Newton–Okounkov bodies of Schubert varieties from the theory of cluster algebras. We construct Newton–Okounkov bodies using specific valuations which generalize extended g-vectors in cluster theory, and discuss how these bodies are related to string polytopes and Nakashima–Zelevinsky polytopes.

Nakada’s colored hook formula is a vast generalization of many important formulae in combinatorics, such as the classical hook length formula and the Peterson’s formula for the number of reduced expressions of minuscule Weyl group elements. In this paper, we use cohomological properties of Segre–MacPherson classes of Schubert cells and varieties to prove a generalization of a cohomological version of Nakada’s formula, in terms of smoothness properties of Schubert varieties. A key ingredient in the proof is the study of a decorated version of the Bruhat graph. Weights of the paths in this graph give the terms in the generalized Nakada’s formula, and the summation over all paths is equal to the equivariant multiplicity of the Chern–Schwartz–MacPherson class of a Richardson variety. Among the applications we mention an algorithm to calculate structure constants of multiplications of Segre–MacPherson classes of Schubert cells, and a skew version of Nakada–Peterson’s formula.

In this paper, we show that the Bruhat order on any sect of a symmetric variety of type $AIII$ is lexicographically shellable. Our proof proceeds from a description of these posets as rook placements in a partition shape which fits in a $p \times q$ rectangle. This allows us to extend an EL-labeling of the rook monoid given by Can to an arbitrary sect. As a special case, our result implies that the Bruhat order on matrix Schubert varieties is lexicographically shellable.