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In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta [6] , but our method of derivation is to use Cherednik's intertwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip [19]. Specializations of the Monk formula's at q = 0 and/or t = 0 provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials. [ABSTRACT FROM AUTHOR]

The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian G (n , m) are the same as multiplication rules for the basis of Schur polynomials in the ring of symmetric polynomials. In this paper, we explain how to recover this somewhat mysterious connection by using the geometric Satake correspondence to put the structure of a representation of G L m on H • (G (n , m)) and comparing it to the Fock space representation on symmetric polynomials. This proof also extends to equivariant Schubert calculus, and gives an explanation of the relationship between torus-equivariant cohomology of Grassmannians and double Schur polynomials. [ABSTRACT FROM AUTHOR]

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand–Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko–Smirnov–Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators. [ABSTRACT FROM AUTHOR]

We consider a conjecture that identifies two types of base point free divisors on M ¯ 0 , n . The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on M ¯ 0 , n to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on M ¯ 0 , n for a large class, and we give sufficient conditions for the non-vanishing of these divisors. [ABSTRACT FROM AUTHOR]

Denote by G (k , n) the Grassmannian of linear subspaces of dimension k in P n . We show that if φ : G (l , n) → G (k , n) is a nonconstant morphism and l ≠ 0 , n - 1 , then l = k or l = n - k - 1 and φ is an isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This information is then used to study the question of (non)existence of nonconstant maps among these varieties, generalizing previous results for projective spaces and Grassmannians. [ABSTRACT FROM AUTHOR]

We develop a combinatorial rule to compute the real geometry of type B Schubert curves S (λ ∙) in the orthogonal Grassmannian O G (n , C 2 n + 1) , which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A [J. Algebraic Combin. 45(1), 191–243 (2017)]. First, using the type B Wroński map studied in [Adv. Math. 224(3), 827–862 (2010)], we show that the real locus of the Schubert curve has a natural covering map to R P 1 , with monodromy operator ω defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute ω without rectifying the skew tableau. The first uses the crystal operators introduced in [Algebr. Comb. 3(3), 693–725 (2020)], while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves. [ABSTRACT FROM AUTHOR]

In this article, we show that the algebraic degree in semidefinite programming can be expressed in terms of the coefficient of a certain monomial in a doubly symmetric polynomial. This characterization of the algebraic degree allows us to use the theory of symmetric polynomials to obtain many interesting results of Nie, Ranestad and Sturmfels in a simpler way. [ABSTRACT FROM AUTHOR]

Denote by G(k, n) the Grassmannian of linear subspaces of dimension k in Pn. We show that if n >m then every morphism &#966: G(k, n) → G(l,m) is constant. [ABSTRACT FROM AUTHOR]

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product J C P ∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for J C P ∞ in both cohomology and K -theory. Here, we study the torus-equivariant cohomology of J C P ∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis. Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety. [ABSTRACT FROM AUTHOR]

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes, J. High Energy Phys. 2021 (2021) 65]. [ABSTRACT FROM AUTHOR]

We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which are solutions to the original overdetermined system. These approaches each use different additional information for this certification, such as liaison, Newton-Okounkov bodies, or intersection theory. They may be used to certify individual solutions, reject nonsolutions, or certify that we have found all solutions. [ABSTRACT FROM AUTHOR]

We compare the cohomology ring of the flag variety F 'n and the Chow cohomology ring of the Gelfand-Zetlin toric variety XGZ. We show that H-.F 'n;Q/is the Poincaré duality quotient of the subalgebra of A-.XGZ;Q/generated by degree 1 elements. We compute these algebras for n D 3 and see that, in general, this subalgebra does not have Poincaré duality. [ABSTRACT FROM AUTHOR]

In this paper we calculate Reidemeister torsion of flag manifold $K/T$ of compact semi-simple Lie group $K=SU_{n+1}$ using Reidemeister torsion formula and Schubert calculus, where $T$ is maximal torus of $K$. We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold $K/T$ of compact semi-simple Lie group $K=SU_{n+1 }$ using root data, and Groebner basis techniques.

The aim of this paper is to explain how to parametrize the equations of the facets of Kirwan polytopes using the notion of Ressayre's pairs. [ABSTRACT FROM AUTHOR]

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions. [ABSTRACT FROM AUTHOR]

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being 0-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furthermore, it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be readily determined. On the other hand, like the Euclidean space, the affine Grassmannian serves as a concrete computational platform on which various distances, metrics, probability densities may be explicitly defined and computed via numerical linear algebra. Moreover, many standard problems in machine learning and statistics—linear regression, errors-in-variables regression, principal components analysis, support vector machines, or more generally any problem that seeks linear relations among variables that either best represent them or separate them into components—may be naturally formulated as problems on the affine Grassmannian. [ABSTRACT FROM AUTHOR]

A spanning configuration in the complex vector space C k is a sequence (W 1 , ⋯ , W r) of linear subspaces of C k such that W 1 + ⋯ + W r = C k . We present the integral cohomology of the moduli space of spanning configurations in C k corresponding to a given sequence of subspace dimensions. This simultaneously generalizes the classical presentation of the cohomology of partial flag varieties and the more recent presentation of a variety of spanning line configurations defined by the author and Pawlowski. This latter variety of spanning line configurations plays the role of the flag variety for the Haglund–Remmel–Wilson Delta Conjecture of symmetric function theory. [ABSTRACT FROM AUTHOR]

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Bürgisser and Lerario, in J Reine Angew Math, 10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617–644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543–571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815–4829, 2002; Proc Am Math Soc 133(10):2835–2844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by "convex hypersurfaces" we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces X 1 , ... , X d k , n ⊂ R P n , where d k , n = (k + 1) (n - k) , are in random position if each one of them is randomly translated by elements g 1 , ... , g d k , n sampled independently from the orthogonal group with the uniform distribution. Denoting by τ k (X 1 , ... , X d k , n ) the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that τ k (X 1 , ... , X d k , n ) = δ k , n · ∏ i = 1 d k , n | Ω k (X i) | | Sch (k , n) | , where δ k , n is the expected degree from [6] (the average number of k-flats incident to d k , n many random (n - k - 1) -flats), | Sch (k , n) | is the volume of the Special Schubert variety of k-flats meeting a fixed (n - k - 1) -flat (computed in [6]) and | Ω k (X) | is the volume of the manifold Ω k (X) ⊂ G (k , n) of all k-flats tangent to X. We give a formula for the evaluation of | Ω k (X) | in terms of some curvature integral of the embedding X ↪ R P n and we relate it with the classical notion of intrinsic volumes of a convex set: | Ω k (∂ C) | | Sch (k , n) | = 4 V n - k - 1 (C) , k = 0 , ... , n - 1. As a consequence we prove the universal upper bound: τ k (X 1 , ... , X d k , n ) ≤ δ k , n · 4 d k , n . Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case k = 1 , n = 3 for every m > 0 we can provide examples of smooth convex hypersurfaces X 1 , ... , X 4 such that the intersection Ω 1 (X 1) ∩ ⋯ ∩ Ω 1 (X 4) ⊂ G (1 , 3) is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions. [ABSTRACT FROM AUTHOR]

The representation theory of rational Cherednik algebras of type A at t = 0 gives rise, by considering supports, to a natural family of smooth Lagrangian subvarieties of the Calogero-Moser space. The goal of this article is to make precise the relationship between these Lagrangian families and Schubert cells in the adelic Grassmannian. In order to do this we show that the isomorphism, as constructed by Etingof and Ginzburg, from the spectrum of the centre of the rational Cherednik algebra to the Calogero-Moser space is compatible with the factorization property of both of these spaces. As a consequence, the space of homomorphisms between certain representations of the rational Cherednik algebra can be identified with functions on the intersection of some Schubert cells. [ABSTRACT FROM AUTHOR]

Define a truncation rt (p) of a polynomial p in { x1,x2,x3,…} as the polynomial with all but the first t variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulas for certain Schubert structure constants in K ( Flags ( C^n)), in particular generalizing those studied by Kogan (2001) in which only cohomology was treated, and those studied by Buch (2002) on the Grassmannian case. The terms in the answer are computed using “marching” operations on permutation diagrams.

J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas, A. Yong. Our proof then combines a saturation theorem of D. Anderson, E. Richmond, A. Yong, a reading order independence property, and É. Tardos' algorithm for combinatorial linear programming. [ABSTRACT FROM AUTHOR]

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes. [ABSTRACT FROM AUTHOR]

The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan- Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming fromthe Chern character. Like the classical rule, both rules are multiplicity-free signed sums. [ABSTRACT FROM AUTHOR]

We prove a determinantal formula that describes the K -theoretic degeneracy loci classes for Grassmann bundles. We further prove Pfaffian formulas for symplectic and odd orthogonal Grassmann bundles. The former generalizes Damon–Kempf–Laksov's determinantal formula, and the latter generalize Pragacz–Kazarian's formulas for the Chow ring. [ABSTRACT FROM AUTHOR]

We define the equivariant Kazhdan–Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank. [ABSTRACT FROM AUTHOR]

In 2005, Knutson–Vakil conjectured a puzzle rule for equivariant $K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem. [ABSTRACT FROM AUTHOR]

Formulating a Schubert problem as the solutions to a system of equations in either Plücker space or in the local coordinates of a Schubert cell usually involves more equations than variables. Using reduction to the diagonal, we previously gave a primal–dual formulation for Schubert problems that involved the same number of variables as equations (a square formulation). Here, we give a different square formulation by lifting incidence conditions which typically involves fewer equations and variables. Our motivation is certification of numerical computation using Smale's α -theory. [ABSTRACT FROM AUTHOR]

We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves $$S(\lambda _\bullet )$$ , which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In Levinson (One-dimensional Schubert problems with respect to osculating flags, 2016, doi:), it was shown that the real geometry of these curves is described by the orbits of a map $$\omega $$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $${\mathbb {RP}}^1$$ , with $$\omega $$ as the monodromy operator. We provide a fast, local algorithm for computing $$\omega $$ without rectifying the skew tableau and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux (Pechenik and Yong in Genomic tableaux, 2016. ), which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of $$S(\lambda _\bullet )$$ . [ABSTRACT FROM AUTHOR]

We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two important conjectures about them: a positivity conjecture, and the agreement with the topologically defined Schubert classes in the smooth case. We prove some special cases of these conjectures. [ABSTRACT FROM AUTHOR]

We give an explicit (new) morphism of modules betweenandand prove (the known result) that the two modules are isomorphic. Our map identifies submodules of the cohomology of the flag variety that are isomorphic to each ofand. With this identification, the map is simply the product within the ring. We use this map in two ways. First, we describe module bases forthat are different from traditional Schubert classes and from each other. Second, we analyze aW-representation onvia restriction to subgroupsWP. In particular, we show that the character of the Springer representation onis a multiple of the restricted representation ofWPon. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Knutson asserting that the Schubert structure constants of the cohomology ring of a two-step flag variety are equal to the number of puzzles with specified border labels that can be created using a list of eight puzzle pieces. As a consequence, we obtain a puzzle formula for the Gromov-Witten invariants defining the small quantum cohomology ring of a Grassmann variety of type A. The proof of the conjecture proceeds by showing that the puzzle formula defines an associative product on the cohomology ring of the two-step flag variety. It is based on an explicit bijection of gashed puzzles that is analogous to the jeu de taquin algorithm but more complicated. [ABSTRACT FROM AUTHOR]