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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study a coproduct in type: \(A\) quantum open Toda lattice in terms of a coproduct in the shifted Yangian of \(\mathfrak{sl}_2\). At the classical level this corresponds to the multiplication of scattering matrices of Euclidean \(SU(2)\) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra. Yangians; Toda lattice; affine Grassmannian Finkelberg, M.; Kamnitzer, J.; Pham, K.; Rybnikov, L.; Weekes, A., Comultiplication for shifted Yangians and quantum open Toda lattice, Adv. Math., 327, 349-389, (2018) Groups and algebras in quantum theory and relations with integrable systems, Geometry and quantization, symplectic methods, Selfadjoint operator theory in quantum theory, including spectral analysis, Quantum dynamics and nonequilibrium statistical mechanics (general), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Lagrangian submanifolds; Maslov index, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Comultiplication for shifted Yangians and quantum open Toda lattice | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a description of the singular locus of a Schubert variety X, in the flag variety G/B, where G is a classical group and B is a Borel subgroup. The singular locus is determined by using standard monomial theory as developed in ''Geometry of G/B'' [by the authors and \textit{C. Musieli}; part I-IV in Collect. Publ. C. P. Ramanujan and Papers in his Mem., Tata Inst. Fundam. Res., Stud. Math. 8, 207-239 (1978); Proc. Indian Acad. Sci., Sect. A 87, No.2, 1-54 (1978); ibid. 88, No.2, 93-177 (1979); ibid. 88, No.4, 279-362 (1979; Zbl 0447.14010-14013); part V (to appear)]. A consequence of this theory is the determination of the ideal defining X in G/B, using which, we are able to write the Jacobian matrix Jw,\(\tau\) (here w is given by \(X=X(w)\) and \(e_{\tau}\) is the T-fixed point in G/B corresponding to \(\tau\), \(\tau\leq w\), T being a maximal torus contained in B) in the affine neighborhood \(U^-_{\tau}\cdot\tau \) of \(e_{\tau}\), where \(U^-_{\tau}=\tau U^-\tau^{-1}\), \(U^-\) being the unipotent part of the Borel subgroup of G opposite to B. Evaluating Jw,\(\tau\) at \(e_{\tau}\), we obtain the dimension of \(Z_{w,\tau}\), the Zariski tangent space to X(w) at \(e_{\tau}\). Denoting by \(\{\) \(p(\lambda\),\(\mu)\}\) the weight vectors as given by standard monomial theory, let \(R(w,\tau)=\{\beta\in \tau (\Delta^+)\quad there\quad exists\quad a\quad p(\lambda,\mu),\quad such\quad that\quad\quad w\ngeq\lambda \quad and\quad X_{- \beta}p(\lambda,\mu)=cp(\tau),c\in k^*\}.\) Then we have the main theorem \(\dim Z_{w,\tau}=N-\# R(w,\tau)\) where \(N=\# \Delta^+=\# \{positive\quad roots\}).\) In particular we have X(w) is smooth at \(e_{\tau}\) if and only if \(N-\# R(w,\tau)=\upharpoonright (w),\) the length of w. dimension of Zariski tangent space; singular locus of a Schubert variety; flag variety; standard monomial theory; Jacobian matrix; weight vectors C.S. Seshadri : Normality of Schubert variety . Proceeding de ''Algebraic Geometry'' (Bombay, Avril 1984). Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Singular locus of a Schubert variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let the vector bundle \(\mathcal{E}\) be a deformation of the tangent bundle over the Grassmannian \(G(k, n)\). We compute the ring structure of sheaf cohomology valued in exterior powers of \(\mathcal{E}\), also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [the authors, Commun. Math. Phys. 352, No. 1, 135--184 (2017; Zbl 06705482)] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples. quantum cohomology; sheaf cohomology; Grassmannians Guo, J., Lu, Z., Sharpe, E.: Classical sheaf cohomology rings on Grassmannians. arXiv:1605.01410 Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Classical sheaf cohomology rings on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Matrix Schubert varieties are the closures of the orbits of \(B \times B\) acting on all \(n \times n\) matrices, where \(B\) is the group of invertible lower triangular matrices. Extending work of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)], \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from \textit{A. Knutson} and \textit{E. Miller}'s [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] and can be used to give new proofs of some of their results, as we explain at the end of this article. Schubert varieties; Gröbner bases; Grothendieck polynomials; simplicial complexes Combinatorial aspects of algebraic geometry, Combinatorial aspects of simplicial complexes, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Permutations, words, matrices Gröbner geometry for skew-symmetric matrix Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials From the bijective correspondence between standard tableaux T of shape \(\lambda\) and left LR (Littlewood-Richardson) words Lof weight ##img## (conjugate of \(\lambda )\), we show that generic points of orbital variety ##img## and set ##img## of subspaces stable under a nilpotent linear endomorphism describe each other. Here, ##img## is the word tableau associated to L. As a result, orderings defined by inclusions of closures correspond (Theorem A). Jordan type; orbital variety; Littlewood-Richardson tableau Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory Correspondence between orbital varieties and sets of subspaces stable under nilpotent endomorphism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this work, \(\tau\)-functions of the Kadomtsev-Petviashvili hierarchy are studied in terms of abelian group actions on finite dimensional Grassmannians. These functions are viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal, and Wilson. A determinantal formula of Gekhtman and Kasman involving exponentials of finite-dimensional matrices is shown to follow naturally from such reductions. All reduced flows of exponential type generated by matrices with arbitrary nondegenerate Jordan forms are derived, both in the Grassmannian setting and within the fermionic operator formalism. A slightly more general determinantal formula involving resolvents of the matrices generating the flow, valid on the big cell of the Grassmannian, is also derived. An explicit expression is deduced for the Plücker coordinates appearing as coefficients in the Schur function expansion of the \(\tau\)-function. finite Grassmannians; fermionic operators; Shur function Balogh, F.; Fonseca, T.; Harnad, J., Finite dimensional KP tau functions. I. finite grassmanians, J. Math. Phys., 55, 8, 083517, (2014) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Grassmannians, Schubert varieties, flag manifolds Finite dimensional Kadomtsev-Petviashvili {\(\tau\)}-functions. I: Finite Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack. Grassmannian; Schubert problems; continuation; geometric Littlewood-Richardson rule; homotopies; numerical Schubert calculus; path following; polynomial system F. Sottile, R. Vakil, and J. Verschelde, \textit{Solving Schubert problems with Littlewood-Richardson homotopies}, Proc. ISSAC 2010 (Stephen M. Watt, ed.), ACM, 2010, pp. 179-186. Symbolic computation and algebraic computation, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Solving Schubert problems with Littlewood-Richardson homotopies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The ring of cyclic quasi-symmetric functions is introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; they arise as toric \(P\)-partition enumerators, for toric posets \(P\) with a total cyclic order. The associated structure constants are determined by cyclic shuffles of permutations. For every non-hook shape \(\lambda \), the coefficients in the expansion of the Schur function \(s_\lambda\) in terms of fundamental cyclic quasi-symmetric functions are nonnegative. The theory has applications to the enumeration of cyclic shuffles and SYT by cyclic descents. enumeration of cyclic shuffles; cyclic shuffles of permutations Symmetric functions and generalizations, Enumeration in graph theory, Grassmannians, Schubert varieties, flag manifolds Cyclic quasi-symmetric functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights \(\lambda\) and \(\mu\) for a Lie algebra \(\mathfrak g\), which we will assume is simply laced. In this paper, we relate the category \(\mathcal O\) over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category \(\mathcal O\) is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category \(\mathcal O\) for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category \(\mathcal O\) are in canonical bijection with a product monomial crystal depending on the choice of parameters.
This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand-Tsetlin modules of \(U(\mathfrak{gl}_n )\) and its associated W-algebras. Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations, Quantum groups (quantized enveloping algebras) and related deformations On category \(\mathcal O\) for affine Grassmannian slices and categorified tensor products | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we focus on the \(H_2\) optimal model reduction methods of coupled systems and ordinary differential equation (ODE) systems. First, the \(\varepsilon\)-embedding technique and a stable representation of an unstable differential algebraic equation (DAE) system are introduced. Next, some properties of manifolds are reviewed and the \(H_2\) norm of ODE systems is discussed. Then, the \(H_2\) optimal model reduction method of ODE systems on the Grassmann manifold is explored and generalized to coupled systems. Finally, numerical examples demonstrate the approximation accuracy of our proposed algorithms. model reduction; \(H_2\) optimality; coupled systems; manifolds Numerical methods for differential-algebraic equations, Grassmannians, Schubert varieties, flag manifolds \(H_2\) optimal model reduction of coupled systems on the Grassmann manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well-known that the classical Weyl character formula for irreducible representations of a compact Lie group is a consequence of the classical Lefschetz fixed point formula applied to the corresponding generalized flag variety. In the context of Arakelov geometry, a fixed point formula of Lefschetz type has recently been formulated and proved by \textit{K. Köhler} and \textit{D. Roessler} [Invent. Math. 145, 333-396 (2001; Zbl 0999.14002)]. Again by applying that formula to generalized flag varieties (now over Spec(\(\mathbb Z\))), the authors present, in the paper under review, a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over \(\text{Spec}(\mathbb Z)\) [see \textit{J. C. Jantzen}, ``Representations of algebraic groups'' (1987; Zbl 0654.20039)], except for the three exceptional cases \(G_2\), \(F_4\) and \(E_8\). The proof involves the computation of the equivariant Ray Singer analytic torsion associated with certain vector bundles on the corresponding complex generalized flag variety.
In the special case the flag variety is Hermitean symmetric, this computation has been carried out by \textit{K. Köhler} in a previous paper [J. Reine Angew. Math. 460, 93-116 (1995; Zbl 0811.53050)]. In the general case, the authors decompose the flag variety into Hermitean symmetric flag varieties by various fibrations and inductively apply a special case of a formula of \textit{X. Ma} [Ann. Inst. Fourier 50, 1539-1588 (2000; Zbl 0964.58025)], which relates the equivariant analytic torsion of the total space of a fibration to the equivariant analytic torsion of its base and its fibre. The authors in fact give a proof of this special case based on the arithmetic Lefschetz formula again. In the final chapter of the paper under review, the authors use the Jantzen sum formula to derive explicit formulae for the global height of ample line bundles on an arbitrary generalized flag variety. This way they recover formulas for projective spaces proved by \textit{H. Gillet} and \textit{C. Soulé} [Ann. Math. (2) 131, 163-203 (1990; Zbl 0715.14018) and 205-238 (1990; Zbl 0715.14006)], and for quadrics proved by \textit{J. Cassaigne} and \textit{V. Maillot} [J. Number Theory 83, 226-255 (2000; Zbl 1001.11027)]. integral representations of Chevalley schemes; Jantzen sum formula; Arakelov geometry; generalized flag variety; equivariant Ray-Singer torsion; Hermitean symmetric space; arithmetic Lefschetz formula Kai Köhler and Damian Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. I. Statement and proof, Invent. Math. 145 (2001), no. 2, 333 -- 396. , https://doi.org/10.1007/s002220100151 K. Köhler and D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. II. A residue formula, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 81 -- 103 (English, with English and French summaries). Christian Kaiser and Kai Köhler, A fixed point formula of Lefschetz type in Arakelov geometry. III. Representations of Chevalley schemes and heights of flag varieties, Invent. Math. 147 (2002), no. 3, 633 -- 669. Arithmetic varieties and schemes; Arakelov theory; heights, Grassmannians, Schubert varieties, flag manifolds, Determinants and determinant bundles, analytic torsion, Representation theory for linear algebraic groups, Heights A fixed point formula of Lefschetz type in Arakelov geometry. III: Representations of Chevalley schemes and heights of flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar \(\mathbb{CP}^{n - 1}\) and Grassmannian models. They naturally arise in the description of continuum limits of spin chains, and their phase structure is sensitive to the values of the topological angles, which are determined by the representations of spins in the chain. Gapless phases can in certain cases be explained by the presence of discrete 't Hooft anomalies in the continuum theory. We also discuss integrable flag manifold sigma models, which provide a generalization of the theory of integrable models with symmetric target spaces. These models, as well as their deformations, have an alternative equivalent formulation as bosonic Gross-Neveu models, which proves useful for demonstrating that the deformed geometries are solutions of the renormalization group (Ricci flow) equations, as well as for the analysis of anomalies and for describing potential couplings to fermions. sigma model; spin chain; flag manifold; integrable model; Haldane conjectures; coherent states; geometric quantization; 't Hooft anomaly; \(\mathrm{PSU}(n)\) bundle; Gross-Neveu model; Ricci flow; supersymmetric sigma model Special quantum systems, such as solvable systems, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Grassmannians, Schubert varieties, flag manifolds, Coherent states, Geometry and quantization, symplectic methods, Supersymmetry and quantum mechanics, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices, Constrained dynamics, Dirac's theory of constraints, Renormalization group methods in equilibrium statistical mechanics Flag manifold sigma models. Spin chains and integrable theories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we study the \(T_{\mathrm w}\)-equivariant cohomology of the weighted Grassmannians \(\mathrm{wGr}(d,n)\) introduced by \textit{A. Corti} and \textit{M. Reid} [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 141--163 (2002; Zbl 1060.14071)], where \(T_{\mathrm w}\) is the \(n\)-dimensional torus that naturally acts on \(\mathrm{wGr}(d,n)\). We introduce the equivariant weighted Schubert classes and, after we show that they form a basis of the equivariant cohomology, we give an explicit formula for the structure constants with respect to this Schubert basis. We also find a linearly independent subset \(\{\mathrm wu_1,\dots,\mathrm wu_{n-1}\}\) of \(\mathrm{Lie}(T_w)^\ast\) such that those structure constants are polynomials in \(\mathrm wu_i\)'s with nonnegative coefficients, up to a permutation on the weights. weighted Grassmannians; equivariant weighted Schubert classes; equivariant cohomology; structure constants Abe, H; Matsumura, T, Equivariant cohomology of weighted Grassmannians and weighted Schubert classes, Int. Math. Res. Not. IMRN, 9, 2499-2524, (2015) Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Equivariant cohomology of weighted Grassmannians and weighted Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we obtain a Kähler structure on the tangent bundle of a Finsler manifold of positive constant flag curvature. We show that there does not exist any Finsler metric such that this structure became locally symmetric or Einstein manifold. Similar results are obtained on a tube around zero section in the tangent bundle in the case of a Finsler manifold of negative constant flag curvature.{
\copyright 2010 American Institute of Physics} Peyghan, E.; Tayebi, A.: A Kähler structure on Finsler spaces with non-zero constant flag curvature, J. math. Phys. 51, 1-11 (2010) Quantization of the gravitational field, String and superstring theories in gravitational theory, Global differential geometry of Finsler spaces and generalizations (areal metrics), Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds, Grassmannians, Schubert varieties, flag manifolds, Special Riemannian manifolds (Einstein, Sasakian, etc.) A Kähler structure on Finsler spaces with nonzero constant flag curvature | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the \(F\)-rationality of matrix Schubert varieties. Although it is known that such varieties are \(F\)-regular (hence \(F\)-rational) by the global \(F\)-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott-Samelson resolution of Schubert varieties. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities. Grassmannians, Schubert varieties, flag manifolds, Linkage, complete intersections and determinantal ideals, Combinatorial aspects of commutative algebra, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Complete intersections On the \(F\)-rationality and cohomological properties of matrix Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(F_k(X)\subset \mathbb G(k, n)\) the \textit{Fano scheme} parameterizing \(k\)-dimensional linear subspaces contained in a variety \(X \subset \mathbb P^n\).
The paper under review concerns the case in which \(X\) is a complete intersection of multi-degree \(\mathbf d := (d_1 ,\dotsc , d_s )\), with
\(1\le s \le n - 2\) and \(\prod_{i=1}^s d_i\ge 2\) and the main result of it extends a recent result by Riedl and Yang on the case of hypersurfaces (see [\textit{E. Riedl} and \textit{D. Yang}, J. Differ. Geom. 116, No. 2, 393--403 (2020; Zbl 1448.14042)], Theorem 3.3) which asserts that a very general hypersurface \(X\subset \mathbb P^n\) of degree \(d\) such that \(n \le \frac{(d+1)(d+2)}{6}\), then \(F_1 (X)\) contains no rational curves.
The result is the following (Theorem 1.2): Let \(X \subset \mathbb P^n\) be a very general complete intersection of multi-degree \(\mathbf d := (d_1 ,\dotsc , d_s )\), with \(1\le s \le n - 2\) and \(\prod_{i=1}^s d_i> 2\). Let \(1\le k\le n - s - 1\) be an integer. If
\[n\le k-1+\frac{1}{k+2}\sum_{i=1}^s\binom{d_i+k+1}{k-1},\]
then \(F_k (X)\) contains neither rational nor elliptic curves.
Moreover, in Section 4, Theorem 4.1, the case of the quadrics -- i.e. \(\prod_{i=1}^s d_i= 2\) -- is considered, obtaining that \(F_k(X)\) is empty if \(k>\lfloor\frac{n-s}{2} \rfloor\) and if \(k\le \lfloor\frac{n-s}{2} \rfloor\) then \(F_k(X)\) has one or two components, which are rationally connected. complete intersections; parameter spaces; Fano schemes; rational and elliptic curves Complete intersections, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, Rational and unirational varieties On Fano schemes of linear spaces of general complete intersections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The irreducible representations of \(\text{Gl}(n,\mathbb{C})\) can be described by Schur functors, the composition of which defines plethysm. Its understanding is an important problem of invariant theory, as well as in relation with the representations of symmetric groups. In this paper, we address the problem geometrically. Through a generalization of the classical Veronese or Segre embeddings, we construct embeddings of flag manifolds into other flag manifolds, on which plethysm can be interpreted in terms of sections of suitable line bundles. We infer the existence of natural filtrations of plethysm, which readily implies different properties of its multiplicities: vanishing conditions, growth, asymptotic behavior. In particular, we discuss the possibility to describe, thanks to our construction, the moment-polytopes attached to the asymptotics of plethysm. plethysm; Schur functors; representations of symmetric groups; flag manifolds; Gauss maps L. Manivel. ''Applications de Gauss et pléthysme''. Ann. Inst. Fourier 47 (1997), pp. 715--773. DOI. 12 Emmanuel Briand, Amarpreet Rattan and Mercedes Rosas Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups Gauss maps and plethysm | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the present paper, we prove that the toric ideals of certain \(s\)-block diagonal matching fields have quadratic Gröbner bases. Thus, in particular, those are quadratically generated. By using this result, we provide a new family of toric degenerations of Grassmannians. matching fields; toric ideals; Gröbner bases; Grassmannians; toric degenerations; SAGBI bases Commutative rings defined by binomial ideals, toric rings, etc., Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Grassmannians, Schubert varieties, flag manifolds Quadratic Gröbner bases of block diagonal matching field ideals and toric degenerations of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We associate to every matroid \(M\) a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of \(M\), in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We also introduce a \(q\)-deformation of the Möbius algebra of \(M\), and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples. matroid; Orlik-Terao algebra; intersection cohomology; Kazhdan-Lusztig theory B. Elias, N. Proudfoot, and M. Wakefield, The Kazhdan-Lusztig polynomial of a matroid, \textit{Adv.} \textit{Math.}, 299 (2016), 36--70.Zbl 1341.05250 MR 3519463 Combinatorial aspects of representation theory, Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups The Kazhdan-Lusztig polynomial of a matroid | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties. We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibers of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic \(\sigma\)-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of \textit{X. He} and \textit{M. Rapoport} in [Manuscr. Math. 152, No. 3--4, 317--343 (2017; Zbl 1432.14023)] this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions. While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure. We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide. reduction of Shimura varieties; affine Deligne-Lusztig varieties; Newton stratification; Hodge-Newton decomposition Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Linear algebraic groups over local fields and their integers, Grassmannians, Schubert varieties, flag manifolds, Witt vectors and related rings Fully Hodge-Newton decomposable Shimura varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple connected complex Lie group. The additive eigencone \(\overline{\Gamma}_n(G)\) is a polyhedral cone containing the set of solutions to the additive eigenvalue problem, a generalization of the Hermitian eigenvalue problem. The additive eigencone is functorial, and for certain subgroups satisfies a stronger functoriality property: the eigencone of the subgroup is determined by the inequalities of the larger eigencone. \textit{P. Belkale} and \textit{S. Kumar} [J. Algebr. Geom. 19, No. 2, 199--242 (2010; Zbl 1233.20040)] first studied this property for subgroups invariant under a diagram automorphism of \(G\). We study a new class of subgroups arising from centralizers of torus elements that have the strong eigencone functoriality property. Linear algebraic groups over the reals, the complexes, the quaternions, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Maximal rank subgroups and strong functoriality of the additive eigencone | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the complex Grassmannian \(\mathrm{Gr} (k, n)\) of \(k\)-dimensional subspaces of \(\mathbb{C}^n\). There is a natural inclusion \(i_{n, r} : \mathrm{Gr} (k, n) \hookrightarrow \mathrm{Gr} (k, n + r)\). Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion \(i_{n, r}\) in the space of mappings from \(\mathrm{Gr} (k, n)\) to \(\mathrm{Gr} (k, n + r)\) for \(r \geq 1\) and in particular to show that the cohomology of \(\text{map} (\mathrm{Gr} (n, k), \mathrm{Gr} (n, k + r); i_{n, r})\) contains a truncated algebra \(\mathbb{Q} [x] / x^{r + n + k^2 - n k} \), where \(|x| = 2\), for \(k \geq 2\) and \(n \geq 4\). complex Grassmannian; Sullivan models; cohomology algebra Rational homotopy theory, Classification of homotopy type, Function spaces in general topology, Grassmannians, Schubert varieties, flag manifolds, Fiber bundles in algebraic topology Rational cohomology algebra of mapping spaces between complex Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review studies equivariant Schubert calculus for general (oriented) cohomology theories. The classical (cohomology, \(K\) theory) level of such questions is in the crossroads of geometry, representation theory, and algebraic combinatorics. It turns out that for more general cohomology theories key geometric objects are not necessarily the classical Schubert varieties, but their Bott-Samelson resolutions. The main results of the paper can be interpreted as a GKM description of the general cohomology of these Bott-Samelson resolutions. Namely, the authors describe the torus fixed point restrictions of a distinguished basis. Their formulas imply the injectivity of the fixed-point localization map. Moreover, they describe the image of that map in the expected GKM style. The authors also provide connection to more standard point-of-views: they provide formulas for the ``Schubert classes'' as opposed to the ``Bott-Samelson classes'', that is they calculate the push-forward images of the distinguished basis, that now live in the general cohomology of flag varieties. The results promise applications in the emerging branch of geometric representation theory that studies general cohomologies. Bott-Samelson variety; flag variety; equivariant oriented cohomology; restriction formula Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Connective \(K\)-theory, cobordism, Bordism and cobordism theories and formal group laws in algebraic topology, Homology and cohomology of homogeneous spaces of Lie groups, Equivariant cobordism On equivariant oriented cohomology of Bott-Samelson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author generalizes the crystal operator formula of \textit{P. Baumann} and \textit{S. Gaussent} [Represent. Theory 12, 83-130 (2008; Zbl 1217.20028)] to the case of double Mirković-Vilonen (shortly MV) cycles; this works in all untwisted affine cases. The formula is applied to the case of type A double MV cycles, by providing an explicit isomorphism of crystals between the set of double MV cycles and the Naito-Sagaki-Saito (shortly NSS) crystal for \(\widehat{sl}_n\) described by \textit{S. Naito} et al. [Springer Proc. Math. Stat. 40, 361-402 (2013; Zbl 1355.17013)]. As a consequence, it is given a new proof of the fact that the NSS crystal is isomorphic to the \(B(\infty)\)-crystal; the result is new for \(n=2\). MV cycles; affine Kac-Moody groups; crystals; Fock space; complex connected reductive groups; affine Grassmannians; Mirković-Vilonen cycles; canonical bases Muthiah, D., Double MV cycles and the naito-sagaki-Saito crystal, Adv. Math., (2013) Linear algebraic groups over the reals, the complexes, the quaternions, Kac-Moody groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations Double MV cycles and the Naito-Sagaki-Saito crystal. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Integrable hierarchies arising from Schrödinger equations with energy-dependent potentials are found to determine flows on the strata of the Grassmannian different from the big cell. As a consequence they have wide classes of solutions associated with the zero sets of KdV \(\tau\)-functions. The group-theoretical description of these hierarchies from the point of view of Birkhoff factorization theorem is given. Birkhoff strata; \(\tau\)-functions; Schrödinger equations with energy-dependent potentials; Grassmannian Mañnas M, J. Geo. Phys. 29 pp 13-- (1999) KdV equations (Korteweg-de Vries equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Grassmannians, Schubert varieties, flag manifolds Hidden hierarchies of KdV type on Birkhoff strata | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Based on the basis theorem of Bruhat--Chevalley and the formula for multiplying Schubert classes obtained by \textit{H. Duan} [Invent. Math. 159, No. 2, 407--436 (2005; Zbl 1077.14067)] and programmed by \textit{H. Duan} and \textit{X. Zhao} [Int. J. Algebra Comput. 16, 1197--1210 (2006; Zbl 1107.14047)], the authors introduce a new method computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). In this paper they demonstrate this technique to descrtibe the Chow rings of the seven generalized Grassmannians (resp. the seven homogeneous spaces of rank one) for the exceptional Lie groups \(F_4\), \(E_6\), \(E_7\) and \(E_8\). flag manifolds; Schubert varieties; Chow ring H. Duan, X. Zhao, Appendix to ''The Chow rings of generalized Grassmannians'', arXiv:math.AG/0510085 . (Equivariant) Chow groups and rings; motives, Grassmannians, Schubert varieties, flag manifolds The Chow rings of generalized Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the product \(\mathcal{G}:=\prod_{i=1}^m \text{Gr}(k_i,V\otimes W)\) (resp. \(\mathcal{G}':=\prod_{i=1}^m \text{Gr}(V\otimes W,k_i)\)) of linear subspaces of \(V \otimes W\) of dimension \(k_i\) (resp. quotient linear spaces of \(V\otimes W\) of dimension \(k_i\)) where \(V\) and \(W\) are two fixed vector spaces over complex numbers. For a set of positive integers \(\theta_i\), we consider the ample line bundle \(L=\bigotimes_{i=1}^m \prod^*_i(\mathcal{O}_{\text{Gr}(k_i,V\otimes W)}(\theta_i))\) (resp. \(L'=\bigotimes_{i=1}^m \prod^*_i(\mathcal{O}_{\text{Gr}(V\otimes W,k_i)}(\theta_i))\)) which admits a unique \(\text{SL}(V)\)-linearization. There is a diagonal action of \(\text{SL}(V)\) on \(\mathcal{G}\) (resp. \(\mathcal{G}'\)) by operation on the factor \(V\). Note that for \(m=1\), \(\mathcal{G}'\) was studied by C. Simpson in order to construct the moduli space of coherent sheaves.
In this paper, the author investigates the GIT stability with respect to this action. This leads the author to define the normalized total weighted dimension of the configuration \(K=\{K_i\}\in \mathcal{G}\) with respect to \(\theta=\{\theta_i\}\) by
\[
\mu_{\theta}(K)=\frac{1}{\dim{V}}\sum_i \theta_i \dim{K_i}
\]
and the configuration is said \(\mu\)-semistable (resp. \(\mu\)-stable) with respect to the weights \(\theta_i\) if for every subspace \(H\) of \(K\), \(\mu_{\theta}(H) \leq \mu_{\theta}(K)\) resp. \(<).\)
The GIT stability of \(K\) with respect to \(L\) is equivalent to its \(\mu\)-stability with respect to the same weights. One defines polystable configurations as direct sum of a finite number of stable subconfigurations of the same normalized total weighted dimension. \newline Assume now that \(\dim{W}=1\). Inspired \textit{S.K. Donaldson} [J. Differ. Geom. 59, No.~3, 479--522 (2001; Zbl 1052.32017)], \textit{H. Luo} [J. Differ. Geom. 49, No.~3, 577--599 (1998; Zbl 1006.32022)], \textit{X. Wang} [Math. Res. Letter No.~2--3, 393--411 (2002; Zbl 1011.32016)] and using the formalism of moment maps, the author introduces a notion of balanced metric on \(V\). Generalizing the work of B. Totaro and A. Klyachko, he proves that the existence of a hermitian balanced metric is equivalent to the polystability of the configuration.
Another result of the paper is about a generalized Gelfand-MacPherson correspondence. Consider a configuration of vector subspaces \((V_1,\dots ,V_m)\in \text{Gr}(k_1,n)\times \cdots \times \text{Gr}(k_m,n)\), and denote \(U^0_{n,(k_1,..,k_m)}\) the space of matrices \(M\) of size \(n\times \sum_i k_i\) such that \(M\) and each of its \(n\times k_i\) block \(M_i\) are of maximum rank. On \(U^0_{n,(k_1,\dots ,k_m)}\) there are two groups acting, one is \(\text{SL}(n)\) and the other one is
\[
G_{k_1,\dots ,k_m}=S(\text{GL}(k_1)\times\cdots \times \text{GL}(k_m)) \subset \text{SL}(k_1+\dots +k_m).
\]
Note that quotienting \(U^0_{n,(k_1,..,k_m)}\) by \(G_{k_1,\dots ,k_m}\) we get \(X=\text{Gr}(k_1,n)\times \dots \times \text{Gr}(k_m,n),\)
while quotienting \(U^0_{n,(k_1,\dots ,k_m)}\) by \(\text{SL}(n)\) gives \(Y=\text{Gr}(n,k_1+\dots +k_m).\)
Finally there is a homeomorphism between the orbit spaces \(X/\text{SL}(n)\) and \(Y/G_{k_1,\dots ,k_m}\). In the GIT setup, it is obtained, following the approach of M. Kapranov, a natural isomorphism between \(X^{\text{ss}}//\text{SL}(n)\) and \(Y^{\text{ss}}//G_{k_1,\dots ,k_m}\) for any choice of \(\theta\). Actually, it is expected an isomorphism between the Chow quotients of the two actions.
Moreover, inspired from the lines of [\textit{I. V. Dolgachev} and \textit{Y. Hu}, Publ. Math., Inst. Hautes Étud. Sci. 87, 5--56 (1998; Zbl 1001.14018)], the author studies the dependence on \(\theta\) of the moduli. This allows him to give an example of a manifold (precisely \(X=\prod_{i=1}^m \text{Gr}(2,\mathbb{C}^4)\)) for which there are no top chambers in the \(\text{SL}(V)\)-ample cone of \(X\).
Using the classical Grothendieck's embedding of the Quot scheme into the Grassmannian, some of the previous results can be extended to the case of configurations of coherent quotient sheaves. In the case of configurations of vector bundles, a notion of balanced metric is defined and its existence is a necessary and sufficient condition for the polystability of the configuration in the Quot scheme, which leads to a partial generalisation of [\textit{X. Wang}, Math. Res. Letter No. 2--3, 393--411 (2002; Zbl 1011.32016)] and [\textit{D.H. Phong} and \textit{J. Sturm}, Commun. Anal. Geom. 11, No. 3, 565--597 (2003; Zbl 1098.32012)]. balanced metric; Grassmannian; GIT stability; stable configurations; moment map Hu, Y, Stable configurations of linear subspaces and quotient coherent sheaves, Q. J. Pure. Appl. Math., 1, 127-146, (2005) Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds Stable configurations of linear subspaces and quotient coherent sheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X=G/P\) be a Schubert variety, defined as the quotient of the Lie group \(G\) by a parabolic subgroup \(P\). \(X\) is minuscule when \(P\) is associated with a weight \(w\) such that \(<a^*, w>\leq 1\), for any positive root \(a\) with respect to a Borel subgroup. A first example of minuscule Schubert variety is the set of lines meeting a fixed line in \(\mathbb P^3\).
The author studies the set \(\Hom_\alpha(\mathbb P^1,X)\) of maps whose image lies in a fixed class \(\alpha\in A_1(X)\). In other words, he studies the variety of rational curves in a given class \(\alpha\). If \(U\) is the dense orbit on \(X\) under the stabilizer of \(X\), then \(U\) is homogeneous, and the variety of maps \(\mathbb P^1\to U\) can be studied using general results on homogeneous varieties. \(X-U\) has codimension at least \(2\) and the author proves that, after a deformation, one may assume that the image of a map \(f\in \Hom_\alpha(\mathbb P^1,X)\) lies in \(U\). Using this fact, the author proves that the irreducible components of \(\Hom_\alpha(\mathbb P^1,X)\) are parametrized by the classes \(\beta\in \text{Pic}(U)^*\) such that \(\Hom_\beta(\mathbb P^1,U)\) is non empty and \(\beta\) maps to \(\alpha\) under the natural morphism \( \text{Pic}(U)^*\to A_1(X)\). Schubert varieties Perrin N., Rational curves on minuscule Schubert varieties, J. Algebra, 2005, 294(2), 431--462 Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Rational curves on minuscule Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials To each simple Lie algebra \(\mathfrak g\) and an element \(w\) of the corresponding Weyl group \textit{C. De Concini, V. G. Kac} and \textit{C. Procesi} [Stud. Math., Tata Inst. Fundam. Res. 13, 41--65 (1995; Zbl 0878.17014)] associated a subalgebra \(U^w_- \) of the quantized universal enveloping algebra \(U_q(\mathfrak g)\), which is a deformation of the universal enveloping algebra \(U(n_- \cap w(n_+))\) and a quantization of the coordinate ring of the Schubert cell corresponding to \(w\). The torus invariant prime ideals of these algebras were classified by \textit{G. Cauchon} and the author [Represent. Theory 14, 645--687 (2010; Zbl 1233.17011)], and the author [Proc. Lond. Math. Soc. (3) 101, No. 2, 454--476 (2010; Zbl 1229.17020)]. These ideals were also explicitly described in the author's cited paper. They index the the Goodearl-Letzter strata of the stratification of the spectra of \(U^w_-\) into tori. In this paper, we derive a formula for the dimensions of these strata and the transcendence degree of the field of rational Casimirs on any open Richardson variety with respect to the standard Poisson structure [\textit{K. R. Goodearl} and \textit{T. H. Lenagan}, J. Algebra 260, No. 2, 657--687 (2003; Zbl 1059.16035)]. prime spectrum; Goodearl-Letzter stratification; quantum nilpotent algebras; Poisson structures on flag varieties M. Yakimov, Strata of prime ideals of De Concini--Kac--Procesi algebras and Poisson geometry, In: New Trends in Noncommutative Algebra, Contemp. Math., 562, 265 278, Amer. Math. Soc., Providence, RI, 2012. Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups, Grassmannians, Schubert varieties, flag manifolds Strata of prime ideals of De Concini-Kac-Procesi algebras and Poisson geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A natural partition of flag superspaces into Schubert supercells indexed by Weyl supergroups is described. The closures and the singularities of these Schubert cells are closely studied. To resolve these singularities the authors generalize the classical Bott-Samelson construction. flag superspaces; Schubert supercells Manin, Y.I., Voronov, A.A.: Supercellular partitions of flag superspaces. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh. VINITI Moscow \textbf{32}, 27-70 (1988) Supermanifolds and graded manifolds, Complex supergeometry, Supervarieties, Grassmannians, Schubert varieties, flag manifolds Supercell partitions of flag superspaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(u\) be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at \(u\) with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold. Linear algebraic groups over the reals, the complexes, the quaternions, Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Total positivity in Springer fibres | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincaré polynomials of the moduli spaces in low degrees. Bialynicki-Birula decomposition; Poincaré polynomial Fine and coarse moduli spaces, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds Poincaré polynomials of moduli spaces of stable maps into flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In J. Differ. Geom. 20, 389--431 (1984; Zbl 0565.17007), \textit{S. Kumar} described the Schubert classes which are the dual to the closures of the Bruhat cells in the flag varieties of the Kac-Moody groups associated to the infinite dimensional Kac-Moody algebras. These classes are indexed by affine Weyl groups and can be chosen as elements of integral cohomologies of the homogeneous space \(\widehat{L}_{\text{pol}} G_{\mathbb C}/ \widehat{B}\) for any compact simply connected semi-simple Lie group \(G\). Later, \textit{S. Kumar} and \textit{B. Kostant} [Adv. Math. 62, 187--237 (1986; Zbl 0641.17008)] gave explicit cup product formulas of these classes in the cohomology algebras by using the relation between the invariant-theoretic relative Lie algebra cohomology theory (using the representation module of the nilpotent part) with the purely nil-Hecke rings. These explicit product formulas involve some BGG-type operators \(A^i\) and reflections. Using some homotopy equivalences, we determine cohomology ring structures of \(LG/T\) where \(LG\) is the smooth loop space on \(G\). Here, as an example we calculate the products and explicit ring structure of \(LSU_2/T\) using these ideas.
Note that these results grew out of a chapter of the author's thesis [On the complex cobordism of flag varieties associated to loop groups, PhD thesis, University of Glasgow (1998)]. divided-power algebras; Schubert classes; Kac-Moody groups; Kac-Moody algebras; BGG-type operators; homotopy equivalences; cohomology ring C. Özel, ``On the cohomology ring of the infinite flag manifold LG/T,'' Turkish Journal of Mathematics, vol. 22, no. 4, pp. 415-448, 1998. Cohomology of Lie (super)algebras, Homology and cohomology of homogeneous spaces of Lie groups, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment On the cohomology ring of the infinite flag manifold \(LG/T\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over \(\text{GL}_{n}({\mathbb C})\) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of this result in terms of roots. We conclude with a section on open questions. J. S. Tymoczko. ''Linear conditions imposed on flag varieties''. Amer. J. Math. 128 (2006), pp. 1587--1604.DOI. Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Classical groups (algebro-geometric aspects), Linear algebraic groups over the reals, the complexes, the quaternions Linear conditions imposed on flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials algebraic geometry; systems theory; McMillan degree; Kronecker indices; transfer functions; system invariants; Grassmann manifold; Grothendieck invariants R. Hermann and C. Martin, ''Application of algebraic geometry to systems theory: Part II: The McMillan degree and Kronecker indices as topological and holomorphic invariants,''SIAM J. Contr. Optimiz.,16, 743--755 (1978). Algebraic methods, Vector and tensor algebra, theory of invariants, Multivariable systems, multidimensional control systems, Grassmannians, Schubert varieties, flag manifolds Applications of algebraic geometry to systems theory: the McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathcal F}\) be a globally generated irreducible homogeneous vector bundle over \(\text{Gr}(k,n)\), the Grassmannian of \(k\)-dimensional quotients of \({\mathbb C}^n\). Then a general section of \({\mathcal F}\) vanishes on a manifold \(X\) whose canonical divisor is a multiple of the hyperplane section in the Plücker embedding. When \(X\) is a Fano manifold (i.e. when the multiple is negative), the author proves that \(X\) is then projectively normal with respect to the Plücker embedding, and that small deformations of \(X\) are also obtained as zero sections of the vector bundle \({\mathcal F}\). In case \(X\) has dimension four and its Picard group is generated by the anticanonical divisor, the same result is proved if the only assumption on \({\mathcal F}\) is that it is a direct sum of irreducible homogeneous vector bundles. The proof is based on the use of Bott's theorem to check the vanishing of suitable cohomology of certain homogeneous vector bundles on \(\text{Gr}(k,n)\) obtained from \({\mathcal F}\). Fano manifold; Grassmannian; fourfold; Picard group; homogeneous vector bundles Küchle, O, Some properties of Fano manifolds that are zeros of sections in homogeneous vector bundles over Grassmannians, Pac. J. Math., 175, 117-125, (1996) Fano varieties, \(4\)-folds, Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Some properties of Fano manifolds that are zeros of sections in homogeneous vector bundles over Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0695.00010.]
A line complex C is a hypersurface in the Grassmannian \(G=G(1,3)\) of lines in \({\mathbb{P}}^ 3\). A planar pencil L is the set of lines in a given plane in \({\mathbb{P}}^ 3\), through a given point. Because G is a hypersurface in \({\mathbb{P}}^ 5\), we know that C is the restriction to G of some hypersurface H in \({\mathbb{P}}^ 5\); denote by n the degree of H. Because L embeds as a line in \({\mathbb{P}}^ 5\), we may expect a general L to meet C in n distinct points, and this holds in characteristic 0 if C is smooth.
This paper investigates when multiple intersections appear, and examines the locus of pencils which give multiple intersections. In this context, \textit{H. Schubert} [``Kalkül der abzählenden Geometrie'' (1879, reprint Berlin 1979; Zbl 0417.51008); p. 269-270] gave 43 intersection formulas, 18 of which the author proved rigorously in Commun. Algebra 16, No.11, 2363-2385 (1988; Zbl 0661.14039), using her stationary multiple point theory. - Used alone, that theory imposes some restrictive genericity conditions; here the author deploys Hilbert scheme techniques of \textit{P. Le Barz} [Adv. Math. 64, 87-117 (1987; Zbl 0621.14031)] to obtain significantly better results. line complex; hypersurface in the Grassmannian; multiple intersections; stationary multiple point; Hilbert scheme Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Schubert's coincidence formulas for line complexes and the contribution of embedded planar pencils | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the \(p\)-smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition. modular representation theory; equivariant cohomology; moment graphs; constructible sheaves; tilting modules; Schubert varieties; \(p\)-smooth locus Fiebig, P. & Williamson, G., Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. \textit{Ann. Inst. Fourier (Grenoble)}, 64 (2014), 489-536. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Classical groups (algebro-geometric aspects), Modular representations and characters, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Intersection homology and cohomology in algebraic topology, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Parity sheaves, moment graphs and the \(p\)-smooth locus of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A loop group LG is the group of maps of the circle \(S^ 1\) into some topological group G (with group law coming from pointwise multiplication in G). The book under review is devoted to the study of structure and representations of LG in the case when G is either a compact or a complex Lie group.
Groups similar to LG enter mathematics in several points. Their Lie algebras form a class of Kac-Moody Lie algebras, the so-called affine algebras. In the last few years these algebras were extensively studied, and their deep relations with various branches of mathematics (combinatorics, finite groups) and physics (quantum field theory, especially string models) were displayed [see, for example, Vertex operators in mathematics and physics, \textit{J. Lepowsky}, \textit{S. Mandelstam} and \textit{I. M. Singer} (eds.) (Publ., Math. Sci. Res. Inst. 3) (1985; Zbl 0549.00013)].
The present book differs from the other sources in that it mostly adopts analytic and geometric, rather that algebraic and combinatoric, approaches.
The first part of the book studies the group LG itself. After the introduction (Chapter I) and a survey of the results about finite- dimensional representations of Lie groups (Chapter 2) the authors give in Chapter 3 general facts about infinite-dimensional Lie groups and consider LG from this viewpoint. In Chapter 4 they study one of the most important properites of loop groups, namely the existence of natural central extensions by the circle group T; these extensions are sometimes more important than loop groups themselves. In this chapter the extensions are constructed and studied by differential geometric methods. Chapter 5 contains a brief survey of Kac-Moody Lie algebras as Lie algebras of loop groups.
Chapter 6 is one of the main chapters in the first part of the book. In this chapter LG is represented as a group of operators in an appropriate Hilbert space, namely in the space \(H=L^ 2(S^ 1,V)\) of \(L^ 2\)- functions on the circle with values in some finite-dimensional representation of G; LG acts in H pointwise. The idea (coming from quantum field theory) is to consider the polarization of H, i.e. the decomposition \(H=H_+\oplus H_-\) where \(H_+\) (resp. \(H_-)\) is the space of functions with vanishing negative (resp. positive) Fourier coefficients. Properties of operators from LG with respect to this decomposition form a very interesting and important part of the theory.
Another very important concept in the first part of the book is the notion of the Grassmannian Gr(H) of a polarized Hilbert space H, introduced in Chapter 7. The authors study the canonical (determinant) line bundle on Gr(H), Schubert cell decomposition of Gr(H), etc.
Chapter 8 introduces the fundamental homogeneous space X of LG that is defined as LG/G where \(G\subset LG\) is considered as the subgroup of constant loops. Two main properties of X are as follows. First, X can be considered as a (infinite-dimensional) complex manifold via the identification \(X=LG_{{\mathbb{C}}}/L_+G_{{\mathbb{C}}}\) where \(G_{{\mathbb{C}}}\) is the complexification of G and \(L_+G_{{\mathbb{C}}}\) consists of boundary values of analytic mappings of the disk \(| z| <1\) into \(G_{{\mathbb{C}}}\). Second, X can be canonically imbedded into Gr(H), thus inheriting from Gr(H) the stratification by Schubert cells and other nice features.
The remaining Chapters 9-14 deal with the representation theory of loop groups. We will not describe the content of these chapters and restrict ourselves to giving their titles: Ch. 9: Representation theory. Ch. 10: The fundamental representation. Ch. 11: The Borel-Weil theory. Ch. 12: The spin representation. Ch. 13: ''Blips'' or ''vertex'' operators. Ch. 14: The Kac character formula and the Bernstein-Gel'fand-Gel'fand resolution. determinant line bundle; vertex operators; loop group; Kac-Moody Lie algebras; affine algebras; infinite-dimensional Lie groups; central extensions; circle group; Grassmannian; polarized Hilbert space; Schubert cell decomposition; homogeneous space; complex manifold; Borel-Weil theory; spin representation; Kac character formula; Bernstein-Gel'fand- Gel'fand resolution A. Pressley and G. Segal, \textit{Loop Groups}, Clarendon Press, Oxford (1986). Infinite-dimensional Lie groups and their Lie algebras: general properties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Research exposition (monographs, survey articles) pertaining to topological groups, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), Grassmannians, Schubert varieties, flag manifolds, Harmonic analysis on homogeneous spaces, Homogeneous complex manifolds Loop groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct examples of surfaces of general type with \(p_g=1,q=0\) and \(K^2=6\). We use as key varieties Fano fourfolds and Calabi-Yau threefolds that are zero section of some special homogeneous vector bundle on Grassmannians. We link as well our construction to a classical Campedelli surface, using the Pfaffian-Grassmannian correspondence. Calabi-Yau threefolds; Grassmannian; surfaces of general type Surfaces of general type, Calabi-Yau manifolds (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Surfaces of general type with \(P_G = 1, Q = 0, K^2 = 6\) and Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper, we construct these semi-toric degenerations using quasi-valuations and triangulations of Newton-Okounkov bodies. distributive lattice; Hibi variety; standard monomial theory; toric degeneration; Newton-Okounkov body; Grassmann variety Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) From standard monomial theory to semi-toric degenerations via Newton-Okounkov bodies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected, reductive, linear algebraic group over \(\mathbb{C}\), split over \(\mathbb{R}\). Let \(T\) be an \(\mathbb{R}\)-split maximal torus in \(G\), and \(B^+,B^-\) a pair of opposite Borel subgroups containing \(T\). Let \(\mathcal B\) be the variety of Borel subgroups and \({\mathcal B}^*:=(B^+w_0B^+)(B^-w_0B^-)\), where \(w_0\) is the longest element in \(W\), the Weyl group of \(G\). In this paper, the author determines the number of connected components of \({\mathcal B}^*\). Further the author computes the Euler characteristic of compactly supported cohomology for the variety \(B^-wB^-\cap B'vB'\), where \(v,w\in W\), and \(B'=uB^-u^{-1}\). connected reductive linear algebraic groups; maximal tori; Borel subgroups; Weyl groups; numbers of connected components; Euler characteristic; compactly supported cohomology Rietsch, K.: Intersections of Bruhat cells in real flag varieties. Int. Math. Res. Not. 13, 623--640 (1997) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Cohomology theory for linear algebraic groups Intersections of Bruhat cells in real flag varieties. Appendix by G. Lusztig | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Iwahori-Hecke algebra \(H_n\) associated to the symmetric group \(S_n\) has a faithful representation as an algebra of operators of the polynomial algebra in \(n\) variables such that the symmetric polynomials are treated as scalars under the action of \(H_n\). These symmetrizing operators are the Newton divided differences and their deformations. The simple operators satisfy the Yang-Baxter relations. The authors give several expressions of the operators corresponding to a maximal permutation and recover the generalized Euler-Poincaré characteristic defined by Hirzebruch in the geometry of flat manifolds. Restricting the action of the Hecke algebra to weight spaces, the authors also recover one of the usual descriptions of its representations. They also obtain \(q\)-idempotents and give a \(q\)-analogue of the Specht representations as orbits of products of \(q\)-Vandermonde functions. The authors study different constructions of irreducible representations corresponding to hook partitions and describe them in terms of Kazhdan-Lustig graphs. This interpretation is applied to the diagonalization of the Hamiltonian of a quantum spin chain with the quantum superalgebra \(U_q ({\mathfrak su} (1/1))\) as a symmetry algebra. Iwahori-Hecke algebra; symmetric group; Newton divided differences; Yang- Baxter relations; Euler-Poincaré characteristic; flat manifolds; irreducible representations; hook partitions; Kazhdan-Lustig graphs; quantum spin chain; quantum superalgebra; symmetry algebra Duchamp, G., Krob, D., Lascoux, A., Leclerc, B., Scharf, T., Thibon, J.Y.: Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras. Publ. RIMS \textbf{31}, 179-201 (1995) Combinatorial aspects of representation theory, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Applications of linear algebraic groups to the sciences Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Suppose that \(K\) is a complete field with respect to a non-Archimedean valuation \(\nu:K\rightarrow \mathbb{R}\cup \{-\infty\}\) which induces an absolute value \(|\cdot |=\exp(-\nu(\cdot))\) on \(K\) and \(X\) is an algebraic variety over \(K\). The analytification of \(X\) which is denoted by \(X^{\mathrm{an}}\) is the set of all multiplicative seminorms on the ring of regular functions \(K[X]\) of \(X\) which are compatible with \(\nu\). If we take an embedding of \(X\) into an affine space and take the coordinatewise valuation map, we get the tropicalization of \(X\). Forming the tropicalization by this way depends on the particular choice of the embedding. The inverse limit of all tropicalizations which is now independent of the coordinates is shown to be homeomorphic to \(X^{\mathrm{an}}\) by \textit{S. Payne} [Math. Res. Lett. 16, No. 2-3, 543--556 (2009; Zbl 1193.14077)] This is one of the strong interrelations between the two concepts. Many natural questions arise in this manner as the structure of the maps between \(X^{\mathrm{an}}\) and the tropicalization of \(X\) for different varieties \(X\). If there exists a continuous section of the tropicalization map which is from \(X^{\mathrm{an}}\) to the tropicalization of \(X\) then the tropicalization is called faithful. If \(X\) is a curve, the faithful tropicalization problem is discussed in [\textit{M. Baker} et al., ``Nonarchimedean geometry, tropicalization, and metrics on curves'', \url{arxiv:1104.0320}]. They show that, for every finite embedded subgraph \(\Gamma\) of non-leaves of \(X^{\mathrm{an}}\) in the \(\mathbb{R}\)-tree model of \(X^{\mathrm{an}}\), there is a tropicalization of \(X\) such that \(\Gamma\) maps isometrically onto it. One may ask for a faithful tropicalization when the variety is higher dimensional. In the paper under review, this question is answered for Grassmannian of planes. The main result of the paper is given as Theorem 1 which shows that there exists a continuous section of the tropicalization map which is a map from \(\mathrm{Gr}(2,n)^{\mathrm{an}}\) to the tropical \(\mathrm{Gr}(2,n)\) where \(\mathrm{Gr}(2,n)\) is the Grassmannian of 2-planes in n-space. Hence they show that the tropical \(\mathrm{Gr}(2,n)\) is homeorphic to a closed subset of \(\mathrm{Gr}(2,n)^{\mathrm{an}}\).
The paper is clearly written and well-organized. Although it is self-contained, it may be good to have an idea about the papers [\textit{M. Baker} et al., ``Nonarchimedean geometry, tropicalization, and metrics on curves'', \url{arxiv:1104.0320}. (2011)] and [\textit{D. Speyer} and \textit{B. Sturmfels}, Adv. Geom. 4, No. 3, 389--411 (2004; Zbl 1065.14071)] before reading this paper. After a clear introduction, Berkovich analytic space is defined, where they also give two examples \({(\mathbb{A}_K^n})^{\mathrm{an}}\) (also they give skeleton map here) and \({(\mathbb{P}_K^n})^{\mathrm{an}}\). Then they define the tropicalization in a very general setting. In the next section they give the definition of \(\mathrm{Gr}(2,n)\) by using the Plücker embedding and they define the tropicalization of \(\mathrm{Gr}(2,n)\). A technical proof of the main theorem is given in the next section. In the following sections they give the fibers of the tropicalization map and piecewise linear structures on tropical \(\mathrm{Gr}(2,n)\) and \(\mathrm{Gr}(2,n)^{\mathrm{an}}\). In the last section they show an application to view the Petersen graph inside the analytification of the quotient of an open subvariety of \(\mathrm{Gr}(2,5)\) by a torus action. tropical geometry; Berkovich spaces; Grassmannians; tropical Grassmannians; analytification of Grassmannians; space of phylogenetic trees Maria Angelica Cueto, Mathias Häbich & Annette Werner, ``Faithful tropicalization of the Grassmannian of planes'', Math. Ann.360 (2014) no. 1-2, p. 391-437 , Grassmannians, Schubert varieties, flag manifolds, Rigid analytic geometry, Topology of analytic spaces Faithful tropicalization of the Grassmannian of planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Hilbert scheme \(\text{Hilb}^4 \mathbb{P}^3\) of zero-dimensional subschemes of length 4 in \(\mathbb{P}^3\) is singular along the smooth locus parametrising fat points of maximal embedding dimension, and the transverse singularity is the cone over the Grassmannian \(G (2,6)\) in its Plücker embedding. The author gives a less computational proof of this fact using the \(PGL (3)\)-action on the Grassmannian and on the versal deformation of the fat point in question.
In the second section the ranks of the Chow groups of \(\text{Hilb}^4 \mathbb{P}^3\) and the Betti numbers of its desingularisation are determined. Betti numbers of desingularisation; fat points of maximal embedding dimension; Chow groups Katz, S.: The desingularization of hilb4p3 and its Betti numbers. Zero-dimensional schemes, 231-242 (1994) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Topological properties in algebraic geometry The desingularization of \(\text{Hilb}^ 4 \mathbb{P}^ 3\) and its Betti numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) denote an adjoint semi-simple group over a field. Following \textit{C. E. Contou-Carrère} [Adv. Math. 71, 186--231 (1988; Zbl 0688.14046)], we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the building associated to the group \(G\). We then determine the fibre of this resolution thanks to the combinatorics of the building. Grassmannians, Schubert varieties, flag manifolds, Groups with a \(BN\)-pair; buildings The fibre of the Bott-Samelson resolution | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. In particular, the number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach). These equations may be used to compute Galois and monodromy groups of intersections of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of \textit{K. Eriksson} and \textit{S. Linusson} [Adv. Appl. Math. 25, No. 2, 194--211 (2000; Zbl 0957.05112); ibid. 212--227 (2000; Zbl 0957.05113)], and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a Littlewood-Richardson coefficient to be 0. We describe pathologies of Eriksson and Linusson's permutation array varieties (failure of existence, irreducibility, equidimensionality, and reducedness of equations), and define the more natural permutation array schemes. In particular, we give several counterexamples to the Realizability Conjecture based on classical projective geometry. Finally, we give examples where Galois/monodromy groups experimentally appear to be smaller than expected. Sara Billey and Ravi Vakil, Intersections of Schubert varieties and other permutation array schemes, Algorithms in algebraic geometry, IMA Vol. Math. Appl., vol. 146, Springer, New York, 2008, pp. 21 -- 54. Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Intersections of Schubert varieties and other permutation array schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semi-simple simply connected linear algebraic group over an algebraically closed field \(k\), and let \(P \subset G\) be a parabolic subgroup. Wahl's conjecture is that for \(G/P\) the Gaussian map (which is a map in sheaf cohomology induced by the diagonal embedding \(G/P \to G/P \times G/P\)) is a surjection. The conjecture is known to hold over the complex numbers, and the authors are interested in the case that the field has positive characteristic.
The first author with \textit{V. B. Mehta} and \textit{A. J. Parameswaran} [J. Algebra 208, 101--128 (1998; Zbl 0955.14006)] showed that Wahl's conjecture holds in odd characteristic if there is a splitting of \(G/P \times G/P\) compatibly splitting the diagonal copy of \(G/P\) in \(G/P \times G/P\) with maximal multiplicity. \textit{V. B. Mehta} and \textit{A. J. Paramsewaran} [Int. J. Math. 8, 495--498 (1997; Zbl 0914.14021)] showed for \(G = SL_n\) and \(P\) being any maximal parabolic that such a splitting exists. Note that all Grassmannians arise as such a \(G/P\).
In this work, the authors show that the argument of Mehta and Parameswaran also holds for the symplectic and orthogonal Grassmannians and deduce the validity of Wahl's conjecture in these cases (for odd characteristic). That is, for the case when \(G\) is the symplectic group \(Sp_{2n}\) with \(P = P_n\) or \(G\) is the special orthogonal group \(SO_{2n}\) with \(P = P_n\). The main result shown here is that for either such \(G\) (with \(p\) odd in the symplectic case) and a Borel subgroup \(B \subset P \subset G\), the \(B\)-canonical splitting of \(G/B\) has maximal multiplicity along \(P/B\). Wahl's conjecture; Frobenius splitting; canonical splitting; maximal multiplicity; diagonal splitting; Grassmannians V. Lakshmibai, K. N. Raghavan, P. Sankaran, Wahl's conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians, Cent. Eur. J. Math. 7 (2009), no. 2, 21423. Grassmannians, Schubert varieties, flag manifolds, Mappings of semigroups, Linear algebraic groups over arbitrary fields Wahl's conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This work is about degenerate flag varieties of type \(A_n\) and \(C_n\) over \(\mathbb{Z}\). Given an irreducible module \(V(\lambda)\) for a complex Lie algebra, in 2010 Evgeny Feigin defined the degenerate flag variety \(Fl(\lambda)^a\) as the closure of a certain highest weight orbit in the projectivization of a degenerate version \(V(\lambda)^a\) of \(V(\lambda)\).
In the case of type \(A_n\) and \(C_n\) the authors have previously shown (see [\textit{G. Cerulli Irelli} and \textit{M. Lanini}, Int. Math. Res. Not. 2015, No. 15, 6353--6374 (2015; Zbl 1349.14157)]) that \(Fl(\lambda)^a\) can be realized as Schubert varieties in a partial flag variety of the same type and bigger rank. In this work they show that \(V(\lambda)^a\) itself is isomorphic to a Demazure representation for a group of type respectively \(A_{2n}\) and \(C_{2n}\). As corollary they redemonstrate the previous result. Furthermore their results are characteric free.
In the case of \(\mathrm{SL}_n(\mathbb{C})\) the result is as follow. Let \(\mathcal{N}^-\), \(\mathcal{H}^-\), \(\mathcal{N}^+\), \(\mathcal{B}\) the Lie subalgebra of \(\mathcal{SL}_n(\mathbb{C})\) of strictly lower triangular, diagonal strictly upper triangular and upper triangular matrices. Let \(\tilde{\mathcal{N}}^-\), \(\tilde{\mathcal{H}}^-\), \(\tilde{\mathcal{N}}^+\) and \(\tilde{\mathcal{B}}\) be the correspond Lie subalgebra of \(\mathcal{SL}_{2n}\) of lower triangular, diagonal and upper triangular matrices. Let \(\mathcal{N}\) be the unipotent subgroup of lower triangular matrices with determinant 1.
Consider the following subalgebra of \(\mathrm{Lie}(\mathrm{SL}_{2n})\):
\[
\mathcal{N}^a=\Bigg\{ \begin{pmatrix}0&A\\
0&0\end{pmatrix}\mid A\in \mathcal{N}\Bigg\}
\]
and the following subgroup of \(\mathrm{SL}_{2n}\):
\[
\mathcal{N}^a=\Bigg\{ \begin{pmatrix}\mathbf{1}&A\\
0&\mathbf{1}\end{pmatrix} \mid A\in \mathcal{N}\Bigg\}.
\]
\(\mathcal{N}^a\) is an abelianization of \(\mathcal{N}\) and \(\mathcal{N}^a\) is an abelianization of \(\mathcal{N}\).
The embedding \(\mathcal{N}^{-,a}\rightarrow \tilde{b}\) induces an embedding of the enveloping algebra of \(\mathcal{N}^{-,a}\) into \(U(\tilde{b})\). Let \(\tilde{V}(\mu)\) be a irreducible representation for \(\mathrm{SL}_{2n}\) with highest weight \(\mu\) and let \(w\) be an element of the Weyl group. The Demazure submodule \(\tilde{V}(\mu)_{w\mu}\) is the cyclic \(U(\tilde{b})\)-module generated by a weight vector of weight \(w\mu\).
The PBW filtration on \(U(\mathcal{N}^-)\) induces a filtration on the cyclic \(U(\mathcal{N}^-)\)-module \(V(\lambda)=U(\mathcal{N}^-).v_\lambda\) and the associated graded space \(V^a(\lambda)=gr(V(\lambda))\) becomes a module for the associated graded algebra
\[
S^\bullet(\mathcal{N}^-) =gr U(\mathcal{N}^-)\cong S^\bullet(\mathcal{N}^{-,a} ).
\]
The action of \(n^{-,a}\) on \(V^a(\lambda)\) can be integrated to an action of \(N^{-,a}\). The closure of the orbit \(N^{-,a}.[V_\lambda]\subset P(V^a(\lambda))\) is called the degenerate flag variety \(Fl(\lambda)^a\).
Let \(\lambda^*\) be the dual dominant weight of \(\lambda\) and let \(\Psi:h^*\rightarrow \tilde{h}^*\) be the linear map who sends i-th fundamental weight to the 2i-fundamental weight. After defining explicitly an element \(\tau\in \tilde{W}\) of the Weyl group of \(\mathrm{SL}_{2n}\) the authors prove the following results:
i) The Demazure submodule \(\tilde{V}(\psi(\lambda^*))_\tau\) is isomorphic to the abelianized module \(V^a(\lambda)\) as \(n^{-,a}\) module.
ii) The Schubert Variety \(X(\tau)\subset \mathbb{P}(\tilde{V}(\psi(\lambda^*))_\tau)\) is isomorphic to the degenerate Flag variety \(F^a(\lambda)\). Schubert varieties; degenerate flag varieties; Demazure modules Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, Representation theory for linear algebraic groups, Linear algebraic groups over global fields and their integers Degenerate flag varieties and Schubert varieties: a characteristic free approach | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper studies the so-called higher associated hypersurfaces of a projective variety via the notion of coisotropy. For a \(k\)-dimensional projective variety \(X\) in \(\mathbb{P}^n\), the \(i\)-th associated hypersurface of \(X\) consists of (the Zariski closure of) all \((n-k-1+i)\)-dimensional linear spaces in \(\mathbb{P}^n\) that meet \(X\) at a smooth point non-transversely, which is a subvariety of a Grassmannian. Historically, the cases \(i = 0\) and \(i=1\) have been studied as the Chow and Hurwitz form of \(X\), respectively.
A main result of this paper is a new and direct proof of a characterization (due originally to Gel'fand, Kapranov and Zelevinsky) of such hypersurfaces in the Grassmannian. Namely, a hypersurface in the Grassmannian is the associated hypersurface of some (irreducible) projective variety iff it is coisotropic, i.e. every normal space at a smooth point of the hypersurface is spanned by rank 1 homomorphisms. Since the notion of coisotropy does not depend on the underlying projective variety, this provides an intrinsic description of all higher associated hypersurfaces (hence the term coisotropic hypersurfaces).
In addition, many other results on coisotropic hypersurfaces are given: e.g. the coisotropic hypersurfaces of the projective dual of \(X\) are the reverse of those of \(X\), and the degrees of these are precisely the polar degrees of \(X\). It is also shown that hyperdeterminants are precisely the coisotropic hypersurfaces associated to Segre varieties. Finally, equations for the Cayley variety of all coisotropic forms of a given degree are given, inside Grassmannians of lines. The author has also written a Macaulay2 package to explicitly realize computation of coisotropic hypersurfaces. Chow form; hyperdeterminant; polar degree; associated hypersurface; Grassmannian Solving polynomial systems; resultants, Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Symbolic computation and algebraic computation Coisotropic hypersurfaces in Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Iwahori-Hecke algebra of the symmetric group is the convolution algebra of \(\text{GL}_n\)-invariant functions on the variety of pairs of complete flags over a finite field. In this paper the author considers the convolution on the space of triples of two flags and a vector. The resulting algebra \(R_n\), which he calls the mirabolic Hecke algebra, had originally been described by Solomon. The author gives a new presentation for \(R_n\) and shows that it is a quotient of a cyclotomic Hecke algebra as defined by Ariki and Koike. He recovers the results of Siegel about the representations of \(R_n\), and uses Jucys-Murphy elements to describe the center and to give a \(\mathfrak{gl}_\infty\)-structure on the Grothendieck group of the category of its representations. Finally, he also outlines a strategy towards a proof of the conjecture that the mirabolic Hecke algebra is a cellular algebra. Iwahori-Hecke algebras; convolution algebras; mirabolic Hecke algebras; Jucys-Murphy elements Rosso, D, \textit{the mirabolic Hecke algebra}, J. Algebra, 405, 179-212, (2014) Hecke algebras and their representations, Group actions on varieties or schemes (quotients), Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds The mirabolic Hecke algebra. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For algebraic symmetric spaces a good theory of embeddings has been developed by the last two authors having in mind applications to classical enumerative geometry. Here we compute the cohomology by first computing equivariant cohomology.
The results generalize both the theory of toric varieties and that of the flag varieties. algebraic symmetric spaces; embeddings; enumerative geometry; equivariant cohomology; toric varieties; flag varieties E. Bifet, C. De Concini, C. Procesi, \textit{Cohomology of regular embeddings}, Adv. Math. \textbf{82} (1990), 1-34. Classical real and complex (co)homology in algebraic geometry, Embeddings in algebraic geometry, Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds Cohomology of regular embeddings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\text{Gr}(k, n)\) be the Grassmannian. The quantum multiplication by the first Chern class \(c_1(\text{Gr}(k,n))\) induces an endomorphism \(\hat{c}_1\) of the finite-dimensional vector space \(\mathrm{QH^*(Gr}(k, n))_{|q = 1}\) specialized at \(q = 1\). Our main result is a case that a conjecture by Galkin holds. It states that the largest real eigenvalue of \(\hat{c}_1\) is greater than or equal to \(\dim \mathrm{Gr}(k, n)+1\) with equality if and only if \(\text{Gr}(k, n) = \mathbb{P}^{n-1}\) algebraic geometry; conjecture O; quantum cohomology Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Positive matrices and their generalizations; cones of matrices, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Galkin's lower bound conjecture holds for the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(p_1, p_2,\dots,p_r\) be positive integers such that \(p_1+p_2 +\dots+p_r=n\leq m\). The author defines two kinds of compact analytical (real and complex) manifolds, denoted by \(G_{(p_1,p_2,\dots,p_r;m)}(F)\) and \(G^*_{(p_1,p_2,\dots,p_r;m)}(F)\), where \(F\) is the field of real or complex numbers. These manifolds consist of \(n\times m\) matrices in certain canonical form. In special case when \(p_1=\dots =p_r=1\) these manifolds are the ordinary Grassmann manifolds \(G_{(r,m)}(F)\). generalized Grassmann manifold; Grassmannian Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds Generalization of the Grassmann manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study affine Grassmannians for ramified triality groups. These groups are of type \(^3D_4\), so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebras. Using these composition algebras, we give descriptions of the affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space. affine Grassmannians; triality; octonions; composition algebras Grassmannians, Schubert varieties, flag manifolds, Group schemes, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Simple, semisimple, reductive (super)algebras, Linear algebraic groups over arbitrary fields, Loop groups and related constructions, group-theoretic treatment Affine Grassmannians for triality groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(C\) be a smooth projective curve and \(W\) a symplectic bundle over \(C\). Let \(LQ_e(W)\) be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves \(E \subset W\) of degree \(e\). We give a closed formula for intersection numbers on \(LQ_e(W)\). As a special case, for \(g\geq 2\), we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [\textit{Y. I. Holla}, Math. Ann. 328, No. 1--2, 121--133 (2004; Zbl 1065.14042)]. Lagrangian quot scheme; symplectic vector bundle; Lagrangian Grassmannian; Gromov-Witten invariant; Vafa-intriligator formula Vector bundles on curves and their moduli, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) Counting maximal Lagrangian subbundles over an algebraic curve | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We review the construction of families of projective varieties, in particular Calabi-Yau threefolds, as quasilinear sections in weighted flag varieties. We also describe a construction of tautological orbibundles on these varieties, which may be of interest in heterotic model building. M. I. Qureshi and B. Szendroi \({ref.surNamesEn}, Calabi-Yau threefolds in weighted flag varieties,, \)Adv. High Energy Physics\(, 2012, (2012)\) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Topology and geometry of orbifolds Calabi-Yau threefolds in weighted flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper yields a description of the structure of certain (irreducible) components of Springer fibers associated to closed \(K\)-orbits in the flag variety of a complex classical group \(G\) for the pairs
\[
(G,K) = (\mathrm{Sp}(2n), \mathrm{Sp}(2p)\times\mathrm{Sp}(2q)) \quad \text{and} \quad (G,K) = (\mathrm{SO}(2n), \mathrm{GL}(n)), \tag{*}
\]
where \(p + q = n\), which arise from the real Lie groups \(\mathrm{Sp}(p,q)\) and \(\mathrm{SO}^*(2n)\) and fall, respectively, in the two cases referred to as type C and type D. Analogous results for other pairs were established in two previous papers by the same authors [Represent. Theory 12, 403--434 (2008; Zbl 1186.22017); J. Algebra 345, No. 1, 109--136 (2011; Zbl 1254.22007)] and here we use the terminology and notation of the review of the second one. The authors give an algorithm to construct a generic element \(f\) in \(\mathfrak n^-\cap\mathfrak p\) for the pairs \((G,K)\) in (*) and define a sequence \(((G_0,K_0),(G_2,K_2),\dots,(G_{2m},K_{2m}))\) of group pairs, where \(G_0 = G\), \(K_0 = K\), \(G_{2j}\subset G_{2(j-1)}\), \(K_{2j} = K\cap\,G_{2j}\) for \(1\leq j\leq m\), as well as sequences \((Q_0,Q_2,\dots,Q_{2m})\), \((L_0,L_2,\dots,L_{2m})\), where \(Q_{2j}\) is a parabolic subgroup of \(K_{2j}\) and \(L_{2j}\) is the (reductive) Levi factor of \(Q_{2j}\); they also define a suitable reductive subgroup \(L_{2j+1}\) of \(K_{2j}\) and, only for type C, reductive subgroups \(\widehat{L}_{2j+1}, \widehat{\widehat{L}}_{2j+1}\) of \(L_{2j+1}\) for \(0\leqslant j\leqslant m\). Then they prove the main theorem of this paper: \(\gamma_{\mathcal Q}^{-1}(f)\) is the closure of
\[
\left(\prod_{j=0}^mZ^{(j)}\right)M_mM_{m-1}\dots M_1M_0\cdot\mathfrak b, \; \text{where} \; M_j = \begin{cases}\widehat{L}_{2j+1}\widehat{\widehat{L}}_{2j+1}L_{2j} &\; \text{for type C}\\ L_{2j+1}L_{2j} &\; \text{for type D}\end{cases}
\]
and \(Z^{(j)}\) is a suited subgroup of \(Z_K(f)\). flag variety; nilpotent cone; Springer fiber; associated cycle; discrete series representation Barchini, L.; Zierau, R.: Components of Springer fibers associated to closed orbits for the symmetric pairs \((Sp(n),Sp(p){\times}Sp(q))\) and \((SO(2n),GL(n))\) I, J. pure appl. Algebra 217, No. 10, 1807-1824 (2013) Semisimple Lie groups and their representations, Linear algebraic groups over the reals, the complexes, the quaternions, Coadjoint orbits; nilpotent varieties, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Components of Springer fibers associated to closed orbits for the symmetric pairs \((\mathrm{Sp}(2n), \mathrm{Sp}(2p)\times\mathrm{Sp}(2q))\) and \((\mathrm{SO}(2n), \mathrm{GL}(n))\). I. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the enumerative significance of the \(s\)-pointed genus zero Gromov-Witten invariant on a homogeneous space \(X\). For that, we give an interpretation in terms of rational curves on \(X\). Gromov-Witten invariants; Grassmannians; Schubert varieties; classical problems Martín, López, A.: Gromov-Witten invariants and rational curves on Grassmannians, J. Geom. Phys., 61, 213-216, (2011) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Gromov-Witten invariants and rational curves on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Any totally positive \((k+m)\times n\) matrix induces a map \(\pi_+\) from the positive Grassmannian \(\mathrm{Gr}_+(k,n)\) to the Grassmannian \(\mathrm{Gr}(k,k+m)\), whose image is the \textit{amplituhedron} \(\mathcal{A}_{n,k,m}\) and is endowed with a top-degree form called the \textit{canonical form} \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\). This construction was introduced by \textit{N. Arkani-Hamed} and \textit{J. Trnka} [J. High Energy Phys. 2014, No. 10, Paper No. 030, 33 p. (2014; Zbl 1468.81075)], where they showed that \(\mathbf{\Omega }(\mathcal{A}_{n,k,4})\) encodes scattering amplitudes in \(\mathcal{N}=4\) super Yang-Mills theory. One way to compute \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\) is to subdivide \(\mathcal{A}_{n,k,m}\) into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of \(\mathcal{A}_{n,k,4}\). However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes [\textit{L. J. Billera} and \textit{B. Sturmfels}, Ann. Math. (2) 135, No. 3, 527--549 (1992; Zbl 0762.52003); \textit{I. M. Gelfand} et al., Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser (1994; Zbl 0827.14036)], the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of \(\mathcal{A}_{n,k,m}\) in order to establish the notion of \textit{secondary amplituhedron}. For this purpose, we first extend the projection \(\pi_+\) to a rational map \(\pi :\mathrm{Gr} (k,n)\dashrightarrow \mathrm{Gr} (k,k+m)\) and provide a concrete birational parametrization of the fibers of \(\pi\). We then use this to explicitly describe a rational top-degree form \(\omega_{n,k,m}\) (with simple poles) on the fibers and compute \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\) as a summation of certain residues of \(\omega_{n,k,m}\). As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when \(n-k-1=m\) (even). We show that, in this case, each fiber of \(\pi\) is parametrized by a projective space and its volume form \(\omega_{n,k,m}\) has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the \textit{Jeffrey-Kirwan residue} computes \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\) from the fiber volume form \(\omega_{n,k,m}\). In particular, we give conceptual proofs of the statements of \textit{L. Ferro} et al. [J. Phys. A, Math. Theor. 52, No. 4, Article ID 045201, 25 p. (2019; Zbl 1422.81148)]. Finally, we propose a more general framework of \textit{fiber positive geometries} and analyze new families of examples such as fiber polytopes and Grassmann polytopes. amplituhedron; canonical form; triangulations of polytopes; fiber polytopes; Grassmann polytopes Polytopes and polyhedra, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Yang-Mills and other gauge theories in quantum field theory, Grassmannians, Schubert varieties, flag manifolds Triangulations and canonical forms of amplituhedra: a fiber-based approach beyond polytopes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a general smooth curve of genus \(g\) defined over an algebraically closed field of characteristic \(0\). Fix a general \((p,q)\in X\times X\). Fix positive integers \(d\), \(r\) and two ramification profiles \(\alpha =(a_0,\dots ,a_r)\in \mathbb {N}^{r+1}\), \(\beta = (b_0,\dots ,b_r)\in \mathbb {N}^{r+1}\). Let \(G^{r,\alpha,\beta}_d(X,p,q)\) denote the set of all \(g^r_d\)'s on \(X\) with ramification profile \(\alpha\) at \(p\) and ramification profile \(\beta\) at \(q\). The author computes the Euler characteristic of \(G^{r,,\alpha,\beta}_d(X,p,q)\) using Eisenbud-Harris theory of limit linear series and the refinements of it given by Osserman and Murray-Osserman. To get their formula they count set-valued tableaux. These combinatorial ideas have inspired other papers [\textit{T. Matsumura}, Ann. Comb. 24, No. 1, 55--67 (2020; Zbl 1435.05225)]. Euler characteristic; limit linear series; Brill-Noether varieties; general genus \(g\) curve; set-valued tableaux Special divisors on curves (gonality, Brill-Noether theory), Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Euler characteristics of Brill-Noether varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The representation theory for infinite dimensional classical groups is relatively well understood and the same is true for infinite symmetric groups, but this is not the case for groups of infinite matrices over finite fields \(\mathrm{GL}(\infty,\mathbb{F}_q)\). In the paper under review the author gives some steps towards the development of this theory. There are several attempts to extend the well known techniques for studying \(\mathrm{GL}(n,\mathbb{F}_q)\) to the groups \(\mathrm{GL}(\infty,\mathbb{F}_q)\), here the author chooses the way through inverse limits of homogeneous spaces (c.f. \textit{D. Pickrell} [J. Funct. Anal. 70, 323--356 (1987; Zbl 0621.28008)]).
Let \(\ell\) be the direct sum of countable copies of \(\mathbb{F}_q\) and let \(\ell^\diamond\) be the direct product of countable copies of \(\mathbb{F}_q\). The author consider the linear space \(\ell\oplus\ell^\diamond\) equipped with a natural topology coming from the extension of the discrete topology of \(\mathbb{F}_q\). Set \(\mathrm{GL}(\ell\oplus\ell^\diamond)\) for the group of continuous invertible linear operators in \(\ell\oplus\ell^\diamond\), this group has a natural homomorphism \(\theta\) into \(\mathbb{Z}\) whose kernel is denoted by \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)\). Take \(\mathcal{M}\) as the set of subspaces of \(\ell\oplus\ell^\diamond\) of the form \(v\oplus vT\), for some infinite matrix \(T\) over \(\mathbb{F}_q\). A subspace \(M\) of \(\ell\oplus\ell^\diamond\) is said semi-infinite if there exists \(L\in\mathcal{M}\) such that \(\alpha(L)=\dim\left(L/(L\cap M)\right)\) and \(\beta(L)=\dim\left(M/(L\cap M)\right)\), are both finite. The relative dimension of \(L\) is then defined as \(\dim(L)=\alpha(L)-\beta(L)\). The primary target of the paper is to understand the set \(\mathrm{Gr}^\alpha\) of subspaces \(L\in\mathcal{M}\) of relative dimension \(\alpha\), these subspaces are called the semi-infinite Grassmannians and we have that \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)\) acts transitively on each \(\mathrm{Gr}^\alpha\).
In the paper it is shown that there exists a unique, up to a scalar factor, and finite \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)-\)invariant Borel measure \(\mu\) on \(\mathrm{Gr}^0\). When \(\mu\) is normalized by the condition \(\mu(\mathcal{M})=1\), then
\[
\mu(\mathrm{Gr}^0)=\prod_{j=1}^{\infty}(1-q^{-j})^{-1}.
\]
Given \(L\in \mathrm{Gr}^0\), it is defined \(\Sigma_L=\{K\in \mathrm{Gr}^0\mid \dim\left(L/(L\cap K)\right)=\dim\left(K/(L\cap K)\right)=1\}\), and it is shown that there exists a unique probability measure \(\nu_L\) on \(\Sigma_L\) invariant with respect to the stabilizer of \(L\) in \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)\). Then the operator
\[
\Delta f(L)=\int\limits_{\Sigma_L}f(K)d\nu_L(K),
\]
is a bounded self adjoint \(\mathrm{GL}^0(\ell\oplus\ell^\diamond)-\)intertwining operator in \(L^2(\mathrm{Gr}^0,\mu)\) with spectrum \(\{1,q^{-1},q^{-2},\ldots\}\). Finally the author gives the invariant functions in terms of the Al-Salam--Carlitz orthogonal polynomials, and describes a measure on the flags of \(\ell\oplus\ell^\diamond\). infinite-dimensional groups; infinite matrices over finite fields; Grassmanians in representation theory; invariant measures; Al-Salam--Carlitz polynomials; \(q-\)Hahn polynomials; flags Neretin, YA, The space \(L\)\^{}\{2\} on semi-infinite Grassmannian over finite field, Adv. Math., 250, 320-350, (2014) Grassmannians, Schubert varieties, flag manifolds, Representations of infinite symmetric groups, General groups of measure-preserving transformations, Group structures and generalizations on infinite-dimensional manifolds, Linear algebraic groups over finite fields, Finite ground fields in algebraic geometry The space \(L^2\) on semi-infinite Grassmannian over finite field | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article contains some algebraic tools that can be used to make computations in the cohomology ring of Lagrangian flag manifolds, and Lagrangian degeneracy loci. The main tool is the study of several operators on a certain basis, which is orthonormal under a scalar product. This basis is useful in studying Schubert classes in Lagrangian manifolds.
The article also contains some simple proofs of previously known results, for example of the Giambelli-type formula for maximal Lagrangian Schubert classes. Lagrangian flag manifolds; Lagrangian degeneracy loci; Schubert polynomials; Giambelli-type formula Lascoux, A; Pragacz, P, Operator calculus for \({\widetilde{Q}}\)-polynomials and Schubert polynomials, Adv. Math., 140, 1-43, (1998) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Determinantal varieties, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Operator calculus for \(\widetilde{Q}\)-polynomials and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce coplactic raising and lowering operators \(E^{\prime}_i\), \(F^{\prime}_i\), \(E_i\), and \(F_i\) on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of ``doubled crystal'' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur \(Q\)-functions. We also give a new criterion for such tableaux to be ballot. combinatorial crystals; shifted Young tableaux; symmetric function theory; orthogonal Grassmannian Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus A crystal-like structure on shifted tableaux | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be an adjoint simple algebraic group of inner type over a field \(F\) and let \(X\) be a twisted flag variety, i.e. \(X\) is a projective \(G\)-homogeneous variety over \(F\).
This paper deals with the problem of decomposing the Chow motive \(M(X)\) of \(X\) when \(G\) is anisotropic. In this case no general decomposition methods are known, except several particular cases of quadrics, Severi-Brauer varieties and exceptional varieties of type \(F_4\).
Here the authors provide methods that allow to decompose the motives of some anistropic twisted flag \(G\)-varieties, where the root system of \(G\) is of types \(A_n\), \(B_n\), \(C_n\), \(G_2\) and \(F_4\), i.e. it has a Dynkin diagram that does not branch.
The following are the main results in this direction contained in the paper.
1) The motive of the variety \(X= X(1,\dots,n)\) of complete flags is isomorphic to
\[
M(X)\simeq\bigoplus_i M(SB(A)(i))^{\oplus a_i},
\]
where \(0\leq i\leq n(n-1)/2\), \(A\) denotes a central simple algebra over \(F\) and \(SB(A)\) the corresponding Severi-Brauer variety. Here \(a_i\) are the coefficients of the polynomial
\[
\phi_n(Z)= \prod_{k=2,\dots, n} (z^k- 1)/(z-1)
\]
2) For the ``incidence variety'' \(X(1,n)\) one has:
\[
M(X(1,n))\simeq\bigoplus_{i= 0,\dots, n-1} M(SB(A)(i)).
\]
3) Let \(SB_2(A)\) be a generalized Severi-Brauer variety for a division algebra \(A\) of degree 5 then there is an isomorphism
\[
M(SB_2(A))\simeq M(SB(B))\oplus M(SB(B))(2),
\]
where \(B\) is a division algebra Brauer-equivalent to the tensor square \(A^{\otimes 2}\).
4) The Krull-Schmidt theorem fails for the motives of projective homogeneous \(G\)-varieties when \(G= \text{PGL}_1(A)\) and \(A\) is a division algebra of degree 5.
Note that 4) provides a counterexample to the uniqueness of a direct sum decomposition in the category of Chow motives with integral coefficients. Chow motive; anisotropic projective homogeneous variety Calmès, B.; Petrov, V.; Semenov, N.; Zainoulline, K., Chow motives of twisted flag varieties, Compos. Math., 142, 4, 1063-1080, (2006) Motivic cohomology; motivic homotopy theory, Homology and cohomology of homogeneous spaces of Lie groups, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Chow motives of twisted flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a reductive group and \(G^{\vee}\) be its Langlands dual. The geometric Satake isomorphism is an identification of the category of finite dimensional representations of \(G^{\vee}\) with the category of equivariant perverse sheaves on the affine Grassmannian of \(G\). Under this isomorphism, simple \(G^{\vee}\)-modules can be identified with certain intersection cohomology groups of varieties that are obtained from orbits of the affine Grassmannian. This relationship allows one to compute certain intersection cohomology groups via the representation theory of \(G^{\vee}\). The goal of this paper is to begin to develop a comparable theory for affine Kac-Moody groups which will be continued in forthcoming work.
For \(G\) semisimple and simply connected, let \(G_{\text{aff}}\) denote the corresponding (untwisted) affine Kac-Moody group and \(G_{\text{aff}}^{\vee}\) denotes its dual. The idea presented in the paper is that the integrable representations of \(G_{\text{aff}}^{\vee}\) should be related to the geometry of certain moduli spaces of \(G\)-bundles on affine two-space. The authors nicely describe the intuition behind the program and then introduce precisely the varieties and \(G\)-bundles of interest. They make a number of conjectures about how things should behave, with the main conjecture involving intersection cohomology. The conjectures are verified for representations of level \(k\) where \(k\) is sufficiently large. Slightly weaker results are obtained for level one representations for arbitrary \(G\) and for arbitrary levels when \(G\) is the special linear group. Grassmannians; geometric Satake isomorphism; reductive group; Kac-Moody group; Langlands duality; intersection cohomology; moduli space; Uhlenbeck space A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian. I. Transversal slices via instantons on Ak-singularities, \textit{Duke Math. J.}, 152 (2010), no. 2, 175--206. Zbl 1200.14083 MR 2656088 Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Kac-Moody groups, Geometric Langlands program: representation-theoretic aspects Pursuing the double affine Grassmannian. I: Transversal slices via instantons on \(A_k\)-singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Studying integration along Bott-Samelson cycles the author proves the following beautiful explicit formula for the degree of the Schubert variety \(X_w\) in the flag manifold \(G/T\) associated to an element \(w\) in the Weyl group:
Let \(b_w\) be the Kähler sequence and \(A_w\) the Cartan matrix then,
\[
\sum \frac{k!b_1^{r_1}\cdots b_k^{r_k}}{r_1!\cdots r_k!} T_{A_w}(x_1^{r_1}\cdots x_k^{r_k}), \quad k=I(w),
\]
where the sum is over all \(r_1+\cdots +r_k=k\) with \(r_i\geq 0\), and \(r_1+\cdots +r_i\leq i\), and where, for each \(k\times k\)-matrix \(A=(a_{ij})\), the homomorphism \(T_A:\mathbb Z[x_1,\dots, x_k]^{(k)}\to \mathbb Z\) on the homogeneous polynomials of degree \(k\) is defined recursively by:
(1) For \(h\in \mathbb Z[x_1,\dots, x_{k-1}]^{(k)}\) we have \(T_A(h)=0\).
(2) If \(k=1\) then \(T_A(x_1)=1\).
(3) For \(h\in \mathbb Z[x_1,\dots, x_{k-1}]^{(k-r)}\) with \(r\geq 1\) we have
\[
T_A(hx_k^r) =T{A'}(h(a_{1,k}x_1+\cdots +a_{k-1,k}x_{k-1})^{r-1}),
\]
where \(A'\) is the matrix obtained from \(A\) deleting the \(k\)'th column and the \(k\)'th row.
The author also provides two generalizations of this result. Bott-Samelson cycles; Cartan Numbers; Schubert varieties; degrees; compex structures; Weyl group; Kaehler sequence; Cartan matrix; flag manifolds Duan, H.: The degree of a Schubert variety. Advances in Mathematics 180(1), 112--133 (2003) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus The degree of a Schubert variety. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For part I and II see ibid. 67, 295-302 (1979) and 68, 56-62 (1980; 457.14010 and Zbl 0464.14001).] It is proved that the class of the Jacobian variety in the Chow ring A(V) of a non-singular irreducible variety V is determined by the Chow ring \(A(F(n+1))\) of the flag variety \(F(n+1)\) for a sufficiently large n and by the Gysin homomorphism \(\rho_*: A(V^{\Delta})\to A(V)\) where \(\rho:V^{\Delta}\to V\) is the tangent flag bundle. Jacobian variety; Chow ring; flag variety Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Tangent flag bundles and Jacobian varieties. III | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this article is to continue the authors' earlier work [``Schubert unions in Grassmann varieties'', Finite Fields Appl. 13, No. 4, 738--750 (2007; Zbl 1136.14036)] on certain subsets of the Grassmann varieties \(G(\ell,m)\) over a finite field \(F\) with applications to the theory of error-correcting codes. The case \(\ell=2\) was of particular importance.
The first three sections set the background, mostly based on their previous work. The later sections raise and partially answer questions on Schubert unions, especially in the case \(\ell \geq 3\). In particular, they investigate which Schubert unions have the maximal number of \(F\)-rational points.
This is a fairly technical paper and the interested reader is advised to read the authors' earlier paper first. Hansen, J. P.; Johnsen, T.; Ranestad, K.: Grassmann codes and Schubert unions, Séminaires et congrès 21, 103-121 (2009) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Geometric methods (including applications of algebraic geometry) applied to coding theory Grassmann codes and Schubert unions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula due to \textit{A. S. Buch, A. Kresch, H. Tamvakis}, and \textit{A. Yong} [Duke Math. J. 122, 125--143 (2004; Zbl 1072.14067)], which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained. Schubert polynomial; Young tableau; rc-graph; crystal graph; Kohnert diagram Lenart, C.: A unified approach to combinatorial formulas for Schubert polynomials. J. Algebr. Comb. 20, 263--299 (2004) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory A unified approach to combinatorial formulas for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This sequel to ``A combinatorial theory of higher-dimensional permutation arrays'' [Adv. Appl. Math. 25, No. 2, 194-211 (2000; Zbl 0957.05110 above)] describes the relative position for more than two flags by use of permutation arrays, or equivalently a decomposition of the product manifold indexed by such arrays. Two partial orders are also defined, one on all permutation arrays, and one on those arrays realized by flags. It is seen that in specific cases these coincide to yield the Bruhat order, or a poset isomorphic to the partition lattice.
However, many open questions are also posed, the most prevalent being whether every permutation array can be realized by flags. This difficult conjecture has only been confirmed in a few cases, but since no counterexamples have been found to date, there is hope. flag manifold; permutation arrays; Bruhat order; partition lattice Eriksson, K.; Linusson, S.: A decomposition of F\$ell(n)\(d indexed by permutation arrays. Adv. appl. Math. 25, 212-227 (2000)\) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices A decomposition of \(\text{Fl}(n)^d\) indexed by permutation arrays | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the classical problems in invariant theory is the study of binary quantics. The main object was to give an explicit description and study the geometric properties of \(SL_2\) quotients of the projective space for a suitable choice of linearization. The aim of this paper is to begin the study of a natural generalization of this classical question.
Let \(k\) be an algebraically closed field. Let \(G\) be a semisimple algebraic group over \(k\), \(T\) a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\), \(N\) the normalizer of \(T\) in \(G\), \(W= N/T\), the Weyl group. Consider the quotient variety \(N\setminus \setminus G/B\). In fact the aim is to study more generally the variety \(N\setminus \setminus G/Q\), where \(Q\) is any parabolic subgroup of \(G\) containing \(B\).
We study torus quotients of these homogeneous spaces. We classify the Grassmannians for which semi-stable = stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring of \(T\) invariants of the homogeneous coordinate ring of the Grassmannian \(G_{2,n}\) (\(n\) odd) over the ring generated by \(R_1\), the first graded part of the ring of \(T\) invariants.
We prove the following results:
(a) the varieties \(T\setminus \setminus G/Q\) and \(N\setminus \setminus G/Q\) are Frobenius split and as an application the vanishing theorems for higher cohomologies of these varieties;
(b) as a part of result (a), we prove the vanishing of the higher cohomology groups for the binary quantics;
(c) for the line bundle \(L\) on \(G_{r,n}\) associated to the fundamental weight \(\varpi_r\), \((G_{r,n})_T^{ss} (L)= (G_{r,n})_T^s(L)\) if and only if \(r\) and \(n\) are coprime;
(d) existence of smooth projective varieties as quotients of certain \(G/Q\) modulo a maximal torus \(T\) (in the case of \(G= SL_n)\);
(e) for \(n\) odd, a partial result about \(R_1\) generation of the graded ring \(k\widehat {[G_{2,n}]}^T= \bigoplus_{d\geq 0}R_d\). torus quotients of homogeneous spaces; Grassmannian; vanishing theorems Kannan, S. S., Torus quotients of homogeneous spaces, \textit{Proc. Indian Acad. Sci. (Math. Sci.)}, 108, 1, 1-12, (1998) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Vanishing theorems in algebraic geometry Torus quotients of homogeneous spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The lectures are based on the results of our research originated six years ago in [Geometric and unipotent crystals, in: Alon, N. (ed.) et al., GAFA 2000. Visions in mathematics -- Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part I. Basel: Birkhäuser, 188--236 (2000; Zbl 1044.17006)] and continued in [Geometric and unipotent crystals. II: From unipotent bicrystals to crystal bases, in: Etingof, Pavel (ed.) et al., Israel mathematical conference proceedings. Quantum groups. Proceedings of a conference in memory of Joseph Donin, Haifa, Israel, July 5--12, 2004. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 433, 13--88 (2007; Zbl 1154.14035)].
1. First lecture: Geometric and unipotent crystals;
2. Second lecture: Positive geometric crystals and crystal bases. Berenstein, A., Kazhdan, D.: Lecture notes on geometric crystals and their combinatorial analogues. Combinatorial aspect of integrable systems. MSJ Mem. vol. 17, pp. 1--9, Mathematical Society of Japan, Tokyo (2007) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Lecture notes on geometric crystals and their combinatorial analogues | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In their work on the infinite flag variety, \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula.
\textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono [loc. cit.]. Grothendieck polynomials; bumpless pipe dreams; alternating sign matrices Combinatorial aspects of algebraic geometry, Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Graph polynomials, Combinatorial aspects of representation theory, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Bumpless pipe dreams and alternating sign matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Plabic graphs are intimately connected to the positroid stratification of the positive Grassmannian. The duals to these graphs are quivers, and it is possible to associate to them cluster algebras. For the top-cell graph of \(\mathrm{Gr}^+(k, n)\), this cluster algebra is the homogeneous coordinate ring of the corresponding positroid variety. We prove that the same statement holds for plabic graphs describing lower dimensional cells. In this way we obtain a map from the positroid strata onto cluster subalgebras of \(\mathrm{Gr}^+(k, n)\). We explore some of the consequences of this map for tree-level scattering amplitudes in \(N = 4\) super Yang-Mills theory. scattering amplitudes; differential and algebraic geometry Paulos, MF; Schwab, BUW, Cluster algebras and the positive Grassmannian, JHEP, 1410, 31, (2014) Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Supersymmetric field theories in quantum mechanics, Yang-Mills and other gauge theories in quantum field theory Cluster algebras and the positive Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\text{Fl}({\mathbb C}^n)\) be the flag variety i.e. the variety of complete flags \(0=V_0\subset V_1\subset \cdots \subset V_n=\mathbb C^n.\) The classes of Schubert structure sheaves \([{\mathcal O}_{X_{\pi}}]\) constitute an additive \(\mathbb Z\)-basis of the \(K\)-theory ring \(K(\text{Fl} ({\mathbb C}^n)).\) \(X_\pi\) denotes here the Schubert variety corresponding to a permutation \(\pi \in {\mathcal S}_n.\) The Schubert constants are defined by the following formula:
\[
[{\mathcal O}_{X_{\pi}}]\cdot [{\mathcal O}_{X_{\rho}}]= {\sum}_{\pi \in {\mathcal S}_n} {\mathcal C}^{\pi}_{{\sigma},{\rho}}[{\mathcal O}_{X_{\pi}}].
\]
This formula stabilizes with respect to inclusions \({\mathcal S}_n\hookrightarrow {\mathcal S}_{n+1}\) therefore the Schubert constants may be defined for \(({\sigma}, {\rho},{\pi})\in {\mathcal S}_{\infty }^3\). The analogous problem for the Grassmannians was settled in the cohomology case by the Littlewood-Richardson rule and in the case of the \(K\)-theory by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)]. The main result of the authors is a substraction-free formula for the family they call truncation Schubert problems. This formula has some nice specializations. It allows one to compute the \(K\)-theory generalizations of numbers considered by \textit{M. Kogan} [Int. Math. Res. Not. 15, 765--782 (2001; Zbl 0994.05150)] and the \(K\)-theory Littlewood-Richardson coefficients considered by Buch [loc. cit.]. The proof is combinatorial. Schubert problem; \(K\)-theory Knutson, A.; Yong, A., A formula for K-theory truncation Schubert calculus, Int. Math. Res. Not., 70, 3741-3756, (2004) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes A formula for \(K\)-theory truncation Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood-Richardson rule. Given three Schubert varieties \(S_{1}, S_{2}, S_{3}\) with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra.
The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of selfadjoint elements \(a,b,c\) with \(a+b+c=0\) in the factor \(\mathcal R^{\omega}\) are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if \(x,y,z\) are selfadjoint elements of such a factor and \(x+y+z=0\), then there exist selfadjoint \(a,b,c \in \mathcal R^{\omega}\) such that \(a+b+c=0\) and \(a\) (respectively, \(b,c\)) has the same eigenvalue distribution as \(x\) (respectively, \(y,z\)). A (`complete') matricial form of this result is known to imply an affirmative answer to an embedding question formulated by Connes.
The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of \(n\times n\) complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds. Schubert variety; hive; honeycomb; factor Bercovici, H., Collins, B., Dykema, K., Li, W.S., Timotin, D.: Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor. J. Funct. Anal. 258, 1579--1627 (2010) Classification of factors, Grassmannians, Schubert varieties, flag manifolds, Eigenvalue problems for linear operators, Symmetric functions and generalizations Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \(V\) is a vector space of finite dimension \(m\) over a finite field \({\mathbb F}_q\) with a fixed \(2\)-tensor, generalizing previous work in [\textit{J. Carrillo-Pacheco} and \textit{F. Zaldívar}, Des. Codes Cryptography 60, No. 3, 291--298 (2011; Zbl 1225.94028)], the authors have been systematically studying linear codes associated to subvarieties of the Grassmann variety given by isotropic subspaces of the vector space \(V\) with respect to a given symplectic or orthogonal tensor. In the paper under review, they consider the case when the finite field is of the form \({\mathbb F}_{q^2}\), for \(q\) a prime power, in which case it has the involution \(\sigma:{\mathbb F}_{q^2}\rightarrow {\mathbb F}_{q^2}\) given by \(\sigma(a)=a^q\), and thus they may assume that the vector space \(V\) has a nondegenerate \(\sigma\)-sesquilinear hermitian form \(\omega\) of Witt index \(n\), which implies that \(m=2n\) or \(m=2n+1\) and any totally isotropic subspace of \(V\) has dimension \(k \leq n\). For the Grassmannian \(\text{Gr}(k,V)\) embedded in the projective space \({\mathbb P}(\bigwedge^kV)\) by the Plücker map, the authors consider the \textit{Hermitian Grassmannian} subvariety \(H_{m,k}\subseteq \text{Gr}(k,V)\) consisting of all totally isotropic subspaces of dimension \(k\) of \(V\). Moreover, they further restrict their interest in this paper on the linear codes defined by the Hermitian varieties for \(k=2\), that is for the varieties \(H_{m,2}\). Their main result gives explicit formulas for the parameters of these codes: word length, dimension and minimum distance. Using Chow's theorem [\textit{W.-L. Chow}, Ann. of Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] that characterizes the group of automorphisms of a Grassmann variety, the authors obtain the monomial automorphism group of the code associated to the Hermitian variety \(H_{m,2}\). Grassmann variety; Grassmann code; Hermitian variety; Hermitian code Cardinali, I.; Giuzzi, L., Line Hermitian Grassmann codes and their parameters, preprint Grassmannians, Schubert varieties, flag manifolds, Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory) Line Hermitian Grassmann codes and their parameters | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies the conditions a Fano threefold V of genus 1 must satisfy in order to allow an embedding in the Grassmannian G(1,4) of lines in \({\mathbb{P}}^ 4\). He proves that there exists a unique relation involving the numerical invariants of V, which is satisfied by 13 of the 18 different Fano threefolds classified by Iskovskih. Moreover for some of them he gives an explicit embedding. Fano threefold; embedding in the Grassmannian \(3\)-folds, Families, moduli of curves (analytic), Grassmannians, Schubert varieties, flag manifolds, Embeddings in algebraic geometry Fano threefolds in G(1,4) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A plabic graph is a planar bicolored graph embedded in a disk, which satisfies some combinatorial conditions. Postnikov's boundary measurement map takes the space of positive edge weights of a plabic graph G to a positroid cell in a totally nonnegative Grassmannian. In this note, we investigate plabic graphs which are symmetric about a line of reflection, up to reversing the colors of vertices. These symmetric plabic graphs arise naturally in the study of total positivity for the Lagrangian Grassmannian. We characterize various combinatorial objects associated with symmetric plabic graphs, and describe the subset of a Grassmannian which can be realized by symmetric weightings of symmetric plabic graphs. total positivity for the Lagrangian Grassmannian; Postnikov's boundary measurement Karpman, R.; Su, Y., Combinatorics of symmetric plabic graphs, (2015), Preprint Planar graphs; geometric and topological aspects of graph theory, Coloring of graphs and hypergraphs, Grassmannians, Schubert varieties, flag manifolds Combinatorics of symmetric plabic graphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simply-connected, semi-simple algebraic group over an algebraic closed field, let \(T\) be a maximal torus of \(G\) and let \(B\) be a Borel subgroup of \(G\) containing \(T\). The authors are interested to determine what Schubert varieties of a projective homogeneous variety \(G/P\) contain a semistable point with respect to the action of \(T\) and to a fixed ample line bundle \(\mathcal{L}\). In this work there are two main results. In the first part of this article the authors restrict themselves to the case where \(G\) is a simple group of type \(B\), \(C\) or \(D\) and \(P\) is a maximal parabolic subgroup. With these hypotheses they classify the minimal elements \(w\in W/W_{P}\) such that \(X(w)_{T}^{ss}(\mathcal{L})\neq\emptyset\). Here \(X(w)=\overline{BwP/P}\) is the Schubert variety associated to \(w\). The authors affirm that, when \(G\) exceptional or \(P\) non-maximal, the same problem is more complicated.
In the second part of this work, the authors classify the Coxeter elements \(\tau\) of \(W\) such that there is a non-trivial line bundle \(\mathcal{L}\) on \(G/B\) with \(X(\tau)_{T}^{ss}(\mathcal{L})\neq\emptyset\). An element of \(W\) is a Coxeter element if it can be written as a product of distinct simple reflections. The authors are interested to such elements for the following reason: a Schubert variety \(X(w)\) contains a (rank \(G\))-dimensional \(T\)-orbit if and only if \(w\geq \tau\) for some Coxeter element \(\tau\). In this part, they do not make special assumption on \(G\).
Let \(\chi\) be the \(B\)-character associated to a fixed globally generated line bundle \(\mathcal{L}\). Supposing that \(\chi\) belongs to the root lattice, the authors prove that \(X(w)_{T}^{ss}(\mathcal{L})\neq\emptyset\) if and only if \(w\chi\leq0\). This fact allows them to prove their main theorems in a combinatorial way.
Remark that in the Proposition 3 (and only in that Proposition) the authors use a definition of \(\omega\geq0\) different from the usual one. They say that a weight \(\omega\) is greater or equal to 0 if it can written as a positive linear combination of simple roots. In particular, they do not assume that the coefficients are integral. semistable points; line bundle; Coxeter element; Schubert varieties Kannan, S. S.; Pattanayak, S. K., Torus quotients of homogeneous spaces-minimal dimensional Schubert varieties admitting semi-stable points, \textit{Proc. Indian Acad. Sci. (Math. Sci.)}, 119, 4, 469-485, (2009) Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds Torus quotients of homogeneous spaces --- minimal dimensional Schubert varieties admitting semi-stable points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author obtains a beautiful, closed formula for the rank of the incidence matrix of points and hyperplane sections of a Grassmann variety over a finite field. This is done by relating this dimension to a modification of the usual Hilbert function of a variety. Grassmann variety; Hilbert function; \(p\)-rank; incidence matrix of points; hyperplane sections; finite field Moorhouse, G.E.: Some \(P\)-ranks related to finite geometric structures. In: Johnson, N. (ed.) Mostly Finite Geometries, pp 353-364. Marcel Dekker, Inc., New York (1997) Linear codes and caps in Galois spaces, Grassmannians, Schubert varieties, flag manifolds, Finite ground fields in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Some \(p\)-ranks related to geometric structures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The manifold \(\mathcal Fl_n\) of complete flags in the \(n\) dimensional vector space \(\mathbb C^n\) over the complex numbers is an object that, by its various definitions, is an object in the intersection of algebra and geometry. On the one hand it can be expressed as the quotient \(B\backslash \text{GL}_n\) of all invertible \(n\times n\)-matrices by its subgroup of lower triangular matrices, and on the other hand as fibers of certain bundles constructed universally from complex vector bundles. Combinatorics enter into the study via the cohomology ring \(H^\ast(\mathcal Fl_n) =H^\ast(\mathcal Fl_n;\mathbb Z)\) with integer coefficients, that can be described as the quotient of the polynomial ring \(\mathbb Z[x_1,\dots, x_n]\) modulo the ideal generated by all non-constant homogeneous functions invariant under permutations of \(x_1,\dots, x_n\). This ring is a free abelian group of rank \(n!\) with basis given by monomials dividing \(\prod_{i=1}^{n-1}x_i^{n-i}\). The ring also has a much more geometric basis given by the Schubert classes \([X_w]\) in the cohomology ring \(H^\ast(\mathcal Fl_n)\).
This article makes an important contribution to bridging the algebra and combinatorics of Schubert polynomials with the geometry of Schubert varieties. It brings new perspectives to problems in commutative algebra concerning ideals generated by minors of generic matrices, and provides a geometric context in which polynomial representatives for Schubert classes are uniquely singled out with no choices but a Borel subgroup of the general linear group \(\text{GL}_n\mathbb C\) in such a way that it is geometrically obvious that these representatives have nonnegative coefficients.
One of the main ideas in the article is to translate ordinary cohomological statements concerning Borel orbit closures on the flag manifold \(\mathcal Fl_n\) into equivariant-cohomological statements concerning double Borel orbit closures on the \(n\times n\) matrices \(M_n\). To be more precise, the preimage \(\widetilde X_w\subseteq \text{GL}_n\mathbb C\) of a Schubert variety \(X_w\in \mathcal Fl_n\) is an orbit closure for the action of the product \(B\times B^+\) of the lower and upper triangular subgroups of \(\text{GL}_n\mathbb C\) acting by multiplication on the left and right. When \(\overline X_w\subseteq M_n\) is the closure of \(\widetilde X_w\) and \(T\) is the torus in \(B\), the \(T\)-equivariant cohomology class \([\overline X_w]_T\in H_T^\ast(M_n)\) is the polynomial representative. It has positive coefficients because there is a \(T\)-equivariant flat (Gröbner) degeneration of \(\overline X_w\) to \(\mathcal L_w\) that is a union of coordinate subspaces \(L\subseteq M_n\). Each subspace \(L\subseteq \mathcal L_w\) has an equivariant cohomology class \([L]_T\in H_T^\ast(M_n)\) that is a monomial in \(x_1,\dots, x_n\), and the sum of these is \([\overline X_w]_T\). The formula is \([\overline X_w]_T = [\mathcal L_w]_T =\sum_{L\in \mathcal L_w}[L]_T\). More importantly, the authors identify a particularly natural degeneration of the matrix Schubert variety \(\overline X_w\) with a reduced and Cohen-Macaulay limit \(\mathcal L_w\) in which the subspaces have combinatorial interpretations and coincides with known combinatorial formulas for Schubert polynomials.
Instead of using equivariant classes associated to closed subvarieties of non-compact spaces the authors develop their theory in the context of multidegrees. The equivariant considerations for matrix Schubert varieties \([\overline X_w]\subseteq M_n\) are then done as multigraded commutative algebra for the Schubert determinantal ideals \(I_w\) cutting out the varieties \(\overline X_w\).
The Gröbner geometry of Schubert polynomials introduced provides a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. In fact the authors describe, for every matrix Schubert variety \(\overline X_w\);
(1) its multidegree and Hilbert series in terms of Schubert and Grothendieck polynomials
(2) a Gröbner basis consisting of minors in its defining ideal \(I_w\)
(3) the Stanley-Reisner complex \(\mathcal L_w\) of its initial ideal \(J_w\), which they prove is Cohen-Macaulay
(4) an inductive irredundant algorithm of weak Bruhat order for listing the facets of \(\mathcal L_w\).
The authors introduce a powerful inductive method that they call Bruhat induction, for working with determinantal ideals and their initial ideals. Bruhat induction as well as the derivation of the main theorems concerning Gröbner geometry rely on results concerning positivity of torus-equivariant cohomology classes represented by subschemes and shellability of certain simplicial complexes that reflect the nature of reduced subwords of words in Coxeter generators for Coxeter groups. The latter technique gives a new perspective, from simplicial topology, of the combinatorics of Schubert and Grothendieck polynomials.
Among the most important applications of the work is the geometrically positive formulae for Schubert polynomials, and connections with Fulton's theory of degeneracy loci, relations between multidegrees and \(K\)-polynomials on \(n\times n\) matrices with classical cohomological theories on the flag manifold, and comparisons with the commutative algebra of determinantal ideals. Stanley-Reisner complex; Coxeter group; Bruhat order; Cohen-Macaulay ideal; initial ideal; Bruhat group; equivariant cohomology; divided differences; Bruhat induction Knutson, [Knutson and Miller 05] A.; Miller, E., Gröbner Geometry of Schubert Polynomials., Ann. Math. (2), 161, 3, 1245-1318, (2005) Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Group actions on posets and homology groups of posets [See also 06A09] Gröbner geometry of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the vector bundle \(R(G)\) formed by all curvature tensors corresponding to torsion free connections on a smooth manifold from the point of view of representation theory. The authors consider the case when a manifold admits a Grassmann structure, i.e., its tangent bundle decomposes into a tensor product of two vector bundles with fibers of positive dimension. In particular, they demonstrate how simple submodules of \(R(G)\) correspond to certain classes of connections on a manifold with a Grassmann structure and study these classes of connections in detail. holonomy; curvature tensor; Grassmann structure Issues of holonomy in differential geometry, Grassmannians, Schubert varieties, flag manifolds Holonomy, geometry and topology of manifolds with Grassmann structure | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A class of algebras that unify a variety of calculations in the representation theory of classical groups is discussed. Because of their relation to the classical Pieri Rule, these algebras are called double Pieri algebras. A generalization of the standard monomial theory of Hodge is developed for double Pieri algebras, that uses pairs of semistandard tableaux, rather than a single one. SAGBI theory and toric deformation are key tools. The deformed double Pieri algebras are described using a doubled version of Gelfand--Tsetlin patterns. The approach allows the discussion to avoid dealing with relations between generators. Pieri rule; classical groups; SAGBI theory; standard monomials; toric deformation Semisimple Lie groups and their representations, Actions of groups on commutative rings; invariant theory, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Grassmannians, Schubert varieties, flag manifolds, Vector and tensor algebra, theory of invariants, Representation theory for linear algebraic groups Pieri algebras and Hibi algebras in representation theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(F=(F_1,\dots,F_n)\) be a maximal flag of subspaces of \({\mathbb R}^n\), i.e. \(F_i\) is a subspace of \({\mathbb R}^n\) of dimension \(i\) and \(F_i\subset F_{i+1}\). The set of all such flags is isomorphic to the flag variety \(SL_n({\mathbb R})/B\) and the subset \(U_F\) of all flags transversal to the given flag \(F\), is an open Schubert cell. In a previous paper, the authors proved that the number of connected components of the intersection \(U_F\cap U_G\) for two transversal flags \(F\) and \(G\) is equal to the number of orbits of some finite group \(G\) acting on a certain \({\mathbb F}_2\)-vector space \(V\). For \(n=3\), \(4\) and \(5\) this number is equal to \(6\), \(20\) and \(52\) respectively. For \(n>5\) it was conjectured, that the number of \(G\)-orbits in \(V\) should be equal to \(3\cdot 2^{n-1}\).
In this new paper, the authors give a detailed description of the action of \(G\) on \(V\), and in particular they prove their conjecture on the number of connected components of the intersection of two opposite open Schubert cells. Schubert cells; flags; number of orbits; action of finite group B. Shapiro, M. Shapiro, and A. Vainshtein, ''Skew-symmetric vanishing lattices and intersections of Schubert cells,'' Internat. Math. Res. Notices, 11(1998), 563--588 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Group actions on varieties or schemes (quotients) Skew-symmetric vanishing lattices and intersections of Schubert cells | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Finding a combinatorial rule for the multiplication of Schubert polynomials is a long-standing problem. We give a combinatorial proof of the extended Pieri rule, which says how to multiply a Schubert polynomial by a complete or elementary symmetric polynomial, and describe some observations in the direction of a general rule. Schubert polynomials; Pieri rule; symmetric polynomial Assaf, S., Bergeron, N., Sottile, F.: On the multiplication of Schubert polynomials. In preparation Symmetric functions and generalizations, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds On the multiplication of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A smooth complex projective variety is called \(\mathcal{B}\)-regular if it admits an algebraic action of the upper triangular Borel subgroup \(\mathcal{B}\subset \text{SL}(2)\) such that the unipotent radical in \(\mathcal{B}\) has a unique fixed point. An important example of a regular variety is the variety of full flags \(G/B\), where \(G\) is a semisimple group and \(B\) is a Borel subgroup in \(G\); here \(\mathcal{B}\) is associated with a regular nilpotent element \(e\in\text{Lie}(G)\).
It was proved by \textit{M.~Brion} and \textit{J. B.~Carrell} [Mich. Math. J. 52, No. 1, 189--203 (2004; Zbl 1084.14044)] that the equivariant cohomology algebra of a \(\mathcal{B}\)-regular variety \(X\) is the coordinate ring of a remarkable affine curve in \(X\times\mathbb{P}^1\). The main result of this paper uses this fact to classify the \(\mathcal{B}\)-invariant subvarieties \(Y\) of a \(\mathcal{B}\)-regular variety \(X\) for which the restriction map \(\imath_Y : H^*(X) \to H^*(Y)\) is surjective. This result may be applied to Schubert subvarieties, certain nulpotent Hessenberg varieties (including the Peterson variety) and certain Springer fibres in \(G/B\). regular varieties; subvarieties; flag varieties; Schubert subvarieties; equivariant cohomology J. B. Carrell and K. Kaveh, On the equivariant cohomology of subvarieties of a \({\mathfrak B}\) -regular variety, Transform. Groups 13 (2008), 495-505. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds On the equivariant cohomology of subvarieties of a \(\mathfrak{B}\)-regular variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [Comment. Math. Helv. 80, No. 1, 1-27 (2005; Zbl 1095.16008)], \textit{A. Maffei} proved a certain relationship between quiver varieties of type \(A\) and the geometry of partial flag varieties over the nilpotent cone. This relation was conjectured by Nakajima, and Nakajima proved his conjecture for a simple case. In Maffei's proof, the key step was a reduction of the general case of the conjecture to the simple case treated by Nakajima through a certain isomorphism. In this paper, we study properties of this isomorphism. quiver varieties of type \(A\); partial flag varieties; nilpotent cones; Maffei isomorphism Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds Remarks on the Maffei's isomorphism. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Subword complexes were defined by \textit{A. Knutson} and \textit{E. Miller} [Adv. Math. 184, No. 1, 161--176 (2004; Zbl 1069.20026)] for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. \textit{S. Assaf} and \textit{D. Searles} [ibid. 306, 89--122 (2017; Zbl 1356.14039)] defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres. Gröbner degenerations of matrix Schubert varieties; Stanley symmetric functions Combinatorial aspects of simplicial complexes, Symmetric functions and generalizations, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Slide complexes and subword complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper computes the characteristic cycles of intersection cohomology sheaves on transversal slices in a double affine Grassmannian. vanishing cycles; Poisson varieties; Uhlenbeck spaces; double affine Grassmannian Finkelberg, M. V.; Kubrak, D. V., Vanishing cycles on Poisson varieties, Funct. Anal. Appl.. Funct. Anal. Appl., Funktsional. Anal. i Prilozhen., 49, 2, 70-78, (2015), Translation of Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Poisson manifolds; Poisson groupoids and algebroids, Vanishing theorems in algebraic geometry Vanishing cycles on Poisson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{P. Balmer} [K-Theory 19, No. 4, 311--363 (2000; Zbl 0953.18003)] extends the definition of the classical Witt ring of a smooth variety \(X\) over a field \(k\) in a conceptual way to a graded group \(W^*(X)\), provided that \(2\) is invertible in \(k\). For more information on this construction and related topics the reader might consult the overview article [\textit{P. Balmer}, ``Witt groups'', E. Friedlander (ed.) et al., Handbook of \(K\)-theory. Vol. 1 and 2. Berlin: Springer. 539--579 (2005; Zbl 1115.19004)]. When \(k\) is the field of complex numbers \(\mathbb{C}\) the above graded ring \(W^*(X)\) is the natural domain of a ring homomorphism into the quotient ring \(KO^*(X)/rKU^*(X)\), where \(r\) denotes the forgetful homomorphism from complex to real \(K\)-theory. In the paper under review the author shows that this homomorphism is an isomorphism in the case where \(X=G_{m,n}\) is the Grassmann variety of complex \(m\)-dimensional \(\mathbb{C}\)-linear subspaces in \(\mathbb{C}^{m+n}\).
The author first determines the two groups and then analyzes the homomorphism from \(W^*(X)\) to \(KO^*(X)/rKU^*(X)\). A computation of the Witt group \(W^*(X)\) indeed is carried out in much more general form by \textit{P. Balmer} and \textit{B. Calmès} [``Witt groups of Grassmann varieties'', J. Algebr. Geom. 21, No. 4, 601642 (2012; Zbl 1273.14098)]. However, in the present paper the author presents a different approach which is based on an analysis of the so-called Gersten-Witt complex. The Gersten-Witt complex in fact gives rise to a spectral sequence which convergences to the desired Witt ring \(W^*(X)\) (as was carried out by \textit{P. Balmer} and \textit{C. Walter} [Ann. Sci. Éc. Norm. Supér. (4) 35, No. 1, 127--152 (2002; Zbl 1012.19003)]), and in the paper under review the author investigates a further spectral sequence, which was introduced by Pardon and Gille (e.g. see \textit{S. Gille} [J. Pure Appl. Algebra 208, No. 2, 391--419 (2007; Zbl 1127.19005)]), in order to compute the necessary input for the aforementioned spectral sequence and then proceeds from there. The ring \(KO^*(G_{m,n})/rKU^*(G_{m,n})\) on the other hand is calculated via the classical Atiyah-Hirzebruch spectral sequence, based on some computational input from \textit{S. Hara} [J. Math. Kyoto Univ. 31, No. 2, 487--493 (1991; Zbl 0735.19006)] and \textit{A. Kono} and \textit{S. Hara} [J. Math. Kyoto Univ. 31, No. 3, 827--833 (1991; Zbl 0748.57015)], respectively. Using various intermediate results obtained along the way the actual proof for the fact that the homomorphism is an isomorphism comes out quite naturally at the end. Grassmann variety; \(K\)-theory; Witt group; Gersten-Witt complex N. Yagita, A note on the Witt group and the \(KO\)-theory of complex Grassmannians , J. \(K\)-theory 9 (2012), 161-175. Witt groups of rings, Topological \(K\)-theory, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Grassmannians, Schubert varieties, flag manifolds A note on the Witt group and the \(KO\)-theory of complex Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct a set of derived equivalences corresponding to mutations of an exceptional collection on the relevant Grassmannian, and give a mirror symmetry interpretation, following a physical analysis of \textit{R. Eager} et al. [Chin. Ann. Math., Ser. B 38, No. 4, 901--912 (2017; Zbl 1373.32019)]. Calabi-Yau threefolds; stringy Kähler moduli; derived category; derived equivalence; matrix factorizations; Landau-Ginzburg model; Pfaffian; Grassmannian Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Derived categories, triangulated categories, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Stringy Kähler moduli for the Pfaffian-Grassmannian correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate Gärdenfors' notion of ``meeting of minds'' in our geometric setting. semantics; vector space model; algebraic geometry Natural language processing, Grassmannians, Schubert varieties, flag manifolds Semantic spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Counting the number of algebraic curves and varieties subject to various conditions is one basic problem in the enumerative algebraic geometry, and Schubert calculus is a systematic and effective theory to solve such problems. It was developed by Schubert, and his most comprehensive and accessible exposition of this theory is given in this book.
Right from the beginning, the theory of Schubert calculus has attracted the attention of many great mathematicians. For example, Hilbert proposed a rigorous justification of Schubert calculus as the 15th problem in his famous list of 23 problems. Recent developments in string theory have contributed to solutions of some outstanding problems in enumerative geometry, and, hence, greatly renewed interest in this subject.
The English translation of this classic by Schubert will be most valuable and interesting to both beginners and experts in enumerative geometry in order to learn how Schubert thought about the problems and how he proposed to solve them, in particular to appreciate the freshness of the subject under development. As Schubert put it: this book ``should acquaint the reader with the ideas, problems and results of a new area of geometry'' and ``should teach the handling of a peculiar calculus that enables one to determine in an easy and natural way a great many of those geometric numbers and relations between singularity numbers.''
See the review of the 1979 reprint edition of the 1879 original in [Zbl 0417.51008]. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, Real and complex geometry, Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Collected or selected works; reprintings or translations of classics, History of geometry The calculus of enumerative geometry. Translated from the German by Wolfgang Globke | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The connectedness theorem of Fulton and Hansen states that, given an irreducible complete variety \(X\) and a morphism \(\varphi\colon X\to \mathbb{P}^n\times\mathbb{P}^n\) such that the image has dimension strictly bigger than \(n\), then the inverse image by \(\varphi\) of the diagonal is connected. It was suggested by \textit{W. Fulton} and \textit{R. Lazarsfeld} [in: Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 26-92 (1981; Zbl 0484.14005)] that a numerical condition may replace the choice of the diagonal in the connectedness result. In the present article such a numerical condition is given and a considerable extension of the Fulton-Hansen result is obtained. For example, it is shown that the connectedness result holds if the diagonal is replaced by any subvariety of the product \(\mathbb{P}^n\times \mathbb{P}^n\) of dimension \(n\) which dominates both coordinates. The numerical condition is also extended to the case of Grassmannians, and several interesting results are obtained, among these Hansen's results on maps into products of Grassmannians. inverse image of the diagonal; connectedness; Grassmannians; Fano varieties Debarre, Théorèmes de Connexité pour les Produits d'Espaces Projectifs et les Grassmanniennes, Amer. J. Math. 118 pp 1347-- (1996) Topological properties in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Fano varieties Connectivity theorems for the products of projective spaces and the Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We enumerate smooth and rationally smooth Schubert varieties in the classical finite types \(A\), \(B\), \(C\), and \(D\), extending Haiman's enumeration for type \(A\). To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of ``nearly-maximally labelled'' staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors showing that all finite-type rationally smooth Schubert varieties have a complete BP decomposition to finish the enumeration. flag manifold; Schubert varieties; smoothness; rational smoothness; Billey-Postnikov decomposition; Coxeter group; enumeration; generating function Edward Richmond and William Slofstra, Staircase diagrams and enumeration of smooth Schubert varieties. J. Combin. Theory Ser. A 150 (2017), 328--376. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Staircase diagrams and enumeration of smooth Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A holomorphic conformal structure on a complex manifold \(X\) is an everywhere non-degenerate section \(g \in H^ 0(S^ 2 \Omega^ 1_ X(N))\) for some line bundle \(N\). In this paper, we show that if \(X\) is a projective complex \(n\)-dimensional manifold with non-numerically effective \(K_ X\) and admits a holomorphic conformal structure, then \(X \cong \mathbb{Q}^ n\). This in particular answers affirmatively a question of \textit{S. Kobayashi} and \textit{T. Ochiai} [Tôhoku Math. J., II. Ser. 34, 587-629 (1982; Zbl 0508.32007)]. They asked if the same holds assuming \(c_ 1(X) > 0\). As a consequence, we also show that any projective conformal manifold with an immersed rational null geodesic is necessarily a smooth hyperquadric \(\mathbb{Q}^ n\). projective conformal manifold; holomorphic conformal structure; null geodesic Ye, Yun-Gang, Extremal rays and null geodesics on a complex conformal manifold, International Journal of Mathematics, 5, 1, 141-168, (1994) Other complex differential geometry, Grassmannians, Schubert varieties, flag manifolds Extremal rays and null geodesics on a complex conformal manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathrm{Gr}^\circ(k,n) \subset \text{Gr}(k,n)\) denote the open positroid stratum in the Grassmannian. We define an action of the extended affine \(d\)-strand braid group on \(\mathrm{Gr}^\circ (k,n)\) by regular automorphisms, for \(d\) the greatest common divisor of \(k\) and \(n\). The action is by quasi-automorphisms of the cluster structure on \(\mathrm{Gr}^\circ (k,n)\), determining a homomorphism from the extended affine braid group to the cluster modular group for \(\mathrm{Gr}(k,n)\). We also define a quasi-isomorphism between the Grassmannian \(\mathrm{Gr}(k,rk)\) and the Fock-Goncharov configuration space of \(2r\)-tuples of affine flags for \(\mathrm{SL}_k\). This identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. \textit{S. Fomin} and \textit{P. Pylyavskyy} [Adv. Math. 300, 717--787 (2016; Zbl 1386.13062)] proposed a description of the cluster combinatorics for \(\mathrm{Gr}(3,n)\) in terms of \textit{G. Kuperberg}'s [Commun. Math. Phys. 180, No. 1, 109--151 (1996; Zbl 0870.17005)] basis of non-elliptic webs. As our main application, we prove many of their conjectures for \(\mathrm{Gr}(3,9)\) and give a presentation for its cluster modular group. We establish similar results for \(\mathrm{Gr}(4,8)\). These results rely on the fact that both of these Grassmannians have finite mutation type. cluster algebra; Grassmannian; braid group; quasi-homomorphism; web Cluster algebras, Grassmannians, Schubert varieties, flag manifolds Braid group symmetries of Grassmannian cluster algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The differential equation on \(Z\) is considered in the paper as a hypersurface \(H\subseteq{\mathcal D}_k^rZ\) in a kind of iterated Grassmannian of the tangent bundle: \(D^0_kZ=Z\), \(D^1_kZ= \text{Gr}_k TZ,\dots,D^r_k Z\subset \text{Gr}_k TD_n^{r-1}Z\), defined in a way that one can call \(D_k^rZ\) the infinitesimal data of dimension \(k\) and order \(r\). Natural analogues of Schubert cycles \(C^r_i\) are then defined in \(D_k^r\mathbb{P}^n\), and it is shown that \(C^r_0\), \(C^r_1, \dots,C_r^r\) is a basis for \(A^1D_k^r \mathbb{P}^n\), the group of divisors of \(D_k^r \mathbb{P}^n\). For a subvariety \(S\subset \mathbb{P}^n\) the divisor of solutions of \(H\) is the push-forward to \(\mathbb{P}^n\) of the intersection \(D^r_kS\cap H\). It is shown how to find its degree if the decomposition of \(H\) with respect to \(C_0^r, C_1^r, \dots, C^r_r\) and some numerical characteristics of \(D^r_kS\) are known. divisor of solutions of a differential equation; Grassmannians of a tangent bundle; intersection numbers; Schubert cycles; divisor of solutions Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves Degree of the divisor of solutions of a differential equation on a projective variety | 0 |
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