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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 Let \(G_{k,n}(\mathbb{C})\) for \(2\leq k<n\) denote the Grassmann manifold of \(k\)-dimensional vector subspaces of \(\mathbb{C}^n\). In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the \(G\)-sequence of the inclusion \(G_{2,n} (\mathbb{C})\rightarrowtail G_{2,n+r}(\mathbb{C})\) for \(r\geq 1\). Grassmann manifold; evaluation subgroups Rational homotopy theory, Homology and cohomology of homogeneous spaces of Lie groups, Differentiable structures in differential topology, Grassmannians, Schubert varieties, flag manifolds Derivations of a Sullivan model and the rationalized \(G\)-sequence | 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 introduce certain families of pairwise commuting elements in the Yang-Baxter group \(\mathcal I\mathcal B(\text{B}_n)\) or in the bracket algebra \(\mathcal B\mathcal E(\text{B}_n)\), which conjecturally generate commutative subalgebras in \(\mathcal B\mathcal E(\text{B}_n)\) isomorphic to the Grothendieck ring of the flag variety of type \(\text{B}_n\). The corresponding results/conjectures for the flag varieties of the other classical type root systems can be obtained from those for the type \(\text{B}\) after certain specializations. One of the main results states that the image of the given construction under the natural epimorphism of \(\mathcal B\mathcal E(\text{B}_n)\) to the Nichols-Woronowicz algebra of type \(\text{B}\) is isomorphic to the Grothendieck ring of the flag variety of type \(\text{B}_n\). A similar construction is also presented for the root system of type \(\text{G}_2\). cohomology ring; Yang--Baxter equation; flag variety; Grothendieck ring Kirillov, A., Maeno, T.: On some noncommutative algebras related with K-theory of flag varieties. IRMN \textbf{60}, 3753-3789. Preprint RIMS, 2005 Grassmannians, Schubert varieties, flag manifolds, Noncommutative algebraic geometry On some noncommutative algebras related to \(K\)-theory of flag varieties. 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 Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we show the purity for domains consisting of sets that are weakly separated from a pair of ``generic'' sets \(I\) and \(J\). Our proof also gives a simple formula for the rank of these domains in terms of \(I\) and \(J\). This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables. weak separation; purity conjecture; cluster distance; mutation sequences Grassmannians, Schubert varieties, flag manifolds, Cluster algebras Weak separation, pure domains and cluster distance | 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 \(V_1,\dots,V_k\) and \(W\) be finite dimensional vector spaces over an infinite field, and \(f: V_1\times \dots\times V_k \rightarrow W\) be a \(k\)-linear map. The author proves that \(\dim\overline{\text{Im} f}\leq \dim V_1+\dots+\dim V_k-k+1\), and if \(f\) is multilinear with \(\text{dim}V_1=\ldots =\text{dimV}_k=n\), then \(\text{dim} \overline{\text{Im} f} \leq (n-k)k+1\) where \(\overline{A}\) denotes the Zariski closure of \(A\). multilinear function; decomposable tensor; algebraic variety; dimension inequalities Multilinear algebra, tensor calculus, Grassmannians, Schubert varieties, flag manifolds Dimension inequalities for the closure of the image of a multilinear function | 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 well-known Givental's construction describes a Landau-Ginzburg (LG) model for a complete intersection \(Y\) in a toric variety as a complete intersection in a torus equipped with a function on the torus called the superpotential. Oscillating integrals associated to the LG model are mirror to Gromov-Witten invariants of \(Y\). In the spirit of Givental's construction, LG models for complete intersections in Grassmannians (and partial flag varieties) are described by \textit{V. V. Batyrev} et al. [Nucl. Phys., B 514, No. 3, 640--666 (1998; Zbl 0896.14025); Acta Math. 184, No. 1, 1--39 (2000; Zbl 1022.14014)] via toric degenerations, where the LG models are given as certain complete intersections in complex tori equipped with superpotentials. This paper announces that for a smooth Fano complete intersection \(Y\) in a Grassmannian of planes, the LG model constructed in [loc. cit.] is birational to a torus and the superpotential can be given as a Laurent polynomial. The authors expect that this result can be generalized to complete intersections in an arbitrary Grassmannian. Landau-Ginzburg models; Fano complete intersections in Grassmannians Przyjalkowski, V. V.; Shramov, C. A., On weak Landau-Ginzburg models for complete intersections in Grassmannians, Uspekhi Mat. Nauk, 69, 6-420, 181-182, (2014) Mirror symmetry (algebro-geometric aspects), Fano varieties, Complete intersections, Grassmannians, Schubert varieties, flag manifolds On weak Landau-Ginzburg models for complete intersections 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 paper under review is devoted to a thorough study of parabolic subroot systems of a complex simple Lie algebra. The authors give various descriptions of maximal subroot systems of a given simple Lie algebra and their important characteristics (Theorems 3.1, 3.2, and 3.3). In the second part of the paper, some applications of the obtained results to the study of generalized flag manifolds are considered (Theorems 4.1 and 4.2). complex simple Lie algebra; parabolic subroot system; generalized flag manifold Root systems, Simple, semisimple, reductive (super)algebras, Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds Parabolic subroot systems and their applications | 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{N. Metropolis} and \textit{G.-C. Rota} [Adv. Math. 50, 95-125 (1983; Zbl 0545.05009)] studied the necklace polynomials, and were lead to define the necklace algebra as a combinatorial model for the classical ring of Witt vectors (which corresponds to the multiplicative formal group law \(X+ Y- XY\)). We define and study a generalized necklace algebra, which is associated with an arbitrary formal group law \(F\) over a torsion free ring \(A\). The map from the ring of Witt vectors associated with \(F\) to the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials. We present a combinatorial interpretation for these polynomials in terms of words on a given alphabet. The actions of the Verschiebung and Frobenius operators, as well as of the \(p\)-typification idempotent are described and interpreted combinatorially. A \(q\)-analogue and other generalizations of the cyclotomic identity are also presented. In general, the necklace algebra can only be defined over the rationalization \(A\otimes \mathbb{Q}\). Nevertheless, we show that for the family of formal group laws over the integers \(F_q(X, Y)= X+ Y- qXY\), \(q\in\mathbb{Z}\), we can define the corresponding necklace algebras over \(\mathbb{Z}\). We classify these algebras, and define isomorphic ring structures on the groups of Witt vectors and the groups of curves associated with the formal group laws \(F_q\). The \(q\)-necklace polynomials, which turn out to be numerical polynomials in two variables, can be interpreted combinatorially in terms of so-called \(q\)-words, and they satisfy an identity generalizing a classical one. \(\copyright\) Academic Press. necklace algebra; ring of Witt vectors; necklace polynomials; Frobenius operators; cyclotomic identity; formal group laws; isomorphic ring structures Lenart, C.: Formal group-theoretic generalization of the necklace algebra, including a q-deformation. J. Algebra 199, 703--732 (1998) Enumerative combinatorics, \(q\)-calculus and related topics, Formal groups, \(p\)-divisible groups, Algebraic combinatorics Formal group-theoretic generalizations of the necklace algebra, including a \(q\)-deformation | 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 prove the conjectures of \textit{W. Graham} and \textit{S. Kumar} [Int. Math. Res. Not. 2008, Article ID rnn 093, 43 p. (2008; Zbl 1185.14043)] and \textit{S. Griffeth} and \textit{A. Ram} [Eur. J. Comb. 25, No. 8, 1263--1283 (2004; Zbl 1076.14068)] about the equivariant \(K\)-theory of generalized flag varieties \(G/P\). Several special cases of these conjectures have been studied by \textit{W. Fulton} and \textit{A. Lascoux} [Duke Math. J. 76, No. 3, 711--729 (1994; Zbl 0840.14007)], \textit{H. Pittie} and \textit{A. Ram} [Electron. Res. Announc. Am. Math. Soc. 5, No. 14, 102--107 (1999; Zbl 0947.14025)] and \textit{O. Mathieu} [J. Pure Appl. Algebra 152, No. 1-3, 231--243 (2000; Zbl 0978.22016)].
The cohomology ring of the homogeneous space \(G/P\) has the following positivity property (here \(G\) is a complex linear algebraic group): it has an additive basis formed by classes of algebraic subvarieties, the Schubert varieties. Using the transitive group action, as pioneered by Kleiman, these varieties can be translated generically; subsequently intersecting them yields cycles whose multiplicities, i.e., the structure constants, are positive by virtue of being algebraic.
This positivity extends beyond ordinary cohomology. Graham generalized it to torus equivariant cohomology of \(G/P\), confirming conjectures of Billey and Peterson. In this context the coefficients are polynomials; positivity means that, expressed as polynomials in the simple roots, the coefficients are nonnegative integers.
Moreover, \textit{M. Brion} [J. Algebra 258, No. 1, 137--159 (2002; Zbl 1052.14054)] proved the positivity for ordinary \(K\)-theory, after it had been conjectured by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)]. In this context, positivity means sign alternation: if the dimension of a subvariety differs from the expected dimension by \(i\) in a given intersection product, then the sign of the coefficient on its class is \((-1)^i\). In equivariant \(K\)-theory, the previous conjectures makes precise the notion of positivity for polynomials in terms of which the alternation is phrased. The author proves the previous conjectures using an appropriate generalization of Kleiman transversality. Ordinary Kleiman transversality concerns the movement of subvarieties of \(G/P\) into general position using the transitive group action. This has consequences for non-equivariant cohomology theories because translation preserves rational equivalence. Equivariantly, on the other hand, translation alters the classes of cycles.
Thus, the authors' transversality principle takes place on (finite-dimensional approximations) of Borel mixing spaces of \(G/P\) and of the \(S\)-subvarieties of \(G/P\) (here \(S\) is a torus of \(G\) with some positivity conditions). The non-equivariant homological invariants of the approximation \(\mathcal{X}\) of the Borel mixing space of \(G/P\) are the equivariant invariants of \(G/P\). This space \(\mathcal{X}\) is a bundle over a product of sufficiently large projective spaces with fiber \(G/P\).
There is a group scheme, called the mixing group, which acts on \(\mathcal{X}\) with finitely many orbits. This action is derived from the structure of \(\mathcal{X}\) as a bundle such that the fibers and the basis have transitive automorphism groups.
What makes things simpler in cohomology, as opposed to \(K\)-theory, is that cohomology only requires knowledge on a Zariski open subset. In Anderson's proof [Positivity in the cohomology of flag bundles (after Graham). Preprint, \url{arXiv:math.AG/0711.0983}] of Graham equivariant cohomology positivity each of the relevant cohomology computations is carried out by intersection that occurs in one cell of a paving of the mixed space. As an appropriate group action is transitive, ordinary Kleiman transversality suffices. One must then push down to the base of the mixing space, but this operation transfers the positivity to the resulting class.
In \(K\)-theory, one cannot restrict to an open cell, but must work with closed subvarieties where the group actions is not transitive. Moreover, pushing forward to the base can have higher direct images, a priori causing some negative coefficients. The obstacle of non-transitivity is dealt with by using a result of \textit{S. J. Sierra} [Algebra Number Theory 3, No. 5, 597--609 (2009; Zbl 1180.14048)]. Getting around the second obstacle is accomplished by stipulating rational singularities, taking the cue from Brion's phrasing of the results in ordinary \(K\)-theory [\textit{M. Brion}, J. Algebra 258, No. 1, 137--159 (2002; Zbl 1052.14054)]). Then, the authors modify Brion's geometric argument that brings Kawamata-Viehweg vanishing to bear. The computation of coefficients is reduced to an Euler characteristic using the homological transversality, then it is shown that this class is a sum of at most one term with well-defined sign by using this vanishing. flag variety; equivariant \(K\)-theory; Borel mixing space Dave Anderson, Stephen Griffeth & Ezra Miller, ``Positivity and Kleiman transversality in equivariant \(K\)-theory of homogeneous spaces'', J. Eur. Math. Soc. (JEMS)13 (2011) no. 1, p. 57-84 Equivariant \(K\)-theory, Grassmannians, Schubert varieties, flag manifolds Positivity and Kleiman transversality in equivariant \(K\)-theory 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 In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. These varieties arise in representation theory, algebraic geometry, and combinatorics. We give a connectedness criterion for semisimple Hessenberg varieties that generalizes a criterion given by Anderson and Tymoczko. It also generalizes results of Iveson in type \(A\) which prove that all Hessenberg varieties satisfying this criterion are connected. We then show that nilpotent Hessenberg varieties are rationally connected. Hessenberg varieties; affine paving; rationally connected Precup, Martha, The connectedness of Hessenberg varieties, J. Algebra, 437, 34-43, (2015) Classical groups (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds The connectedness of Hessenberg 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 Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional \(F_4\)-homogeneous manifold associated to a short simple root. smooth Schubert varieties; rational homogeneous manifolds; variety of minimal rational tangents; standard embeddings; Cartan-Fubini extension Grassmannians, Schubert varieties, flag manifolds, Homogeneous complex manifolds, Differential geometry of homogeneous manifolds Standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 | 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 smooth complex variety and let \(E_1\to \cdots \to E_{n-1} \to E_n\to F_n \to F_{n-1}\to \cdots \to F_1\) be a sequence of vector bundles and morphism over \(X\) such that \(\text{rank}(F_i) =\text{rank} (E_i)\) for \(1\leq i\leq n\). For any permutation \(w\in S_{n+1}\) let \(\Omega_w\) be the degeneracy locus \(\{x\in X| \text{rank}(E_q(x)\to F_p(x)) \leq r_w(p,q), \forall 1 \leq p,q\leq n\}\), where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i) \leq q\). \textit{W. Fulton} [Duke Math. J. 96, No.~3, 575--594 (1999; Zbl 0981.14022)] gave a formula for the cohomology class of \(\Omega_w\) in \(H^\ast(X,\mathbb Z)\) as a universal Schubert polynomial in the Chern classes of the vector bundles involved, when the maps are sufficiently general.
The main result of the present article is:
For \(w\in S_{n+1}\) we have
\[
[\mathcal O_{\Omega_w}] =\sum (-1)^{\l(u_1\cdots u_{2n-1}w)} G_{u_1}(E_2-E_1)\cdots G_{u_n}(F_n-E_n)\cdots G_{u_{2n-1}}(F_1-F_2)
\]
in \(K(X)\), where the sum is over all factorizations \(w=u_1\cdots u_{2n-1}\) in the degenerate Hecke algebra such that \(u_i \in S_{\min(i,2n-1)+1}\) for each \(i\), and \(G_u(E-E')\) is the stable Grothendieck polynomial of the permutation \(u\).
This result generalizes a previous result of the authors [Duke Math. J. 122, No.~1, 125--143 (2004; Zbl 1072.14067)].
\textit{A. Buch} [Duke Math. J. 115, No.~1, 75--103 (2002; Zbl 1052.14056)] proved the quiver formula
\[
[\mathcal O_{\Omega_w}] =\sum_\lambda c_{w,\lambda}^{(n)} G_{\lambda^1}(E_2-E_1)\cdots G_{\lambda^n}(F_n-E_n)\cdots G_{\lambda^{2n-1}} (F_1-F_2)
\]
where the sum is over finitely many sequences of partitions \(\lambda=(\lambda^1,\dots, \lambda^{2n-1})\) and where the \(c_{w,\lambda}^{(n)}\) are quiver coefficients and \(G_\alpha =G_{w_\alpha}\) is the stable Grothendieck polynomial for the Grassmannian permutation \(w_\alpha\) corresponding to \(\alpha\). The main result of the present article, together with a result of \textit{A. Lascoux} [Transition on Grothendieck polynomials. Physics and Combinatorics, 2000 (Nagoya), World Sci. Publishing, River Edge, HJ, 2001, 164--179)], proves that these coefficients have alternating signs. In fact, define integers \(a_{w,\beta}\) such that \(G_w=\sum a_{w,\beta}G_\beta\) where the sum is over all permutations \(\beta\). Then the main result is equivalent to the explicit combinatorial formula for quiver coefficients:
\[
c_{w,\lambda}^{(n)} =(-1)^{| \lambda| -\l(w)} \sum_{u_1\cdots u_{2n-1}=w} | a_{u_1,\lambda^1} a_{u_2,\lambda^2}\cdots a_{u_{2n-1},\lambda^{2n-1}}|.
\]
The proof of the main result is based on a special case of this formula proved by \textit{A. Buch} [loc. cit.] together with a Cauchy identity given by \textit{A. N. Kirillov} [J. Math. Sci., New York 121, No.~3, 2360--2370 (2004); translation from Zap. Nauchn. Semin. POMI 283, 123--139 (2001; Zbl 1063.05134)].
As a consequence of their result the authors obtain new formulas for the double Grothendieck polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 295, 629--633 (1982; Zbl 0542.14030)]. degeneracy loci; Schubert polynomials; quiver variety; quiver coefficient; words \beginbarticle \bauthor\binitsA. S. \bsnmBuch, \bauthor\binitsA. \bsnmKresch, \bauthor\binitsH. \bsnmTamvakis and \bauthor\binitsA. \bsnmYong, \batitleGrothendieck polynomials and quiver formulas, \bjtitleAmer. J. Math. \bvolume127 (\byear2005), no. \bissue3, page 551-\blpage567. \endbarticle \OrigBibText ----, Grothendieck polynomials and quiver formulas , Amer. J. Math. 127 (2005), no. 3, 551-567. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Combinatorial aspects of representation theory, \(K\)-theory of schemes Grothendieck polynomials and quiver formulas | 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 an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of \textit{K. Motegi} and \textit{K. Sakai} [J. Phys. A, Math. Theor. 46, No. 35, Article ID 355201, 26 p. (2013; Zbl 1278.82042)] to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by \textit{P. J. McNamara} [Electron. J. Comb. 13, No. 1, Research paper R71, 40 p. (2006; Zbl 1099.05078)]. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations. integrable colored six-vertex model; Lindström-Gessel-Viennot lemma Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Double Grothendieck polynomials and colored lattice models | 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 an integrable colored vertex model whose partition function is a double Grothendieck polynomial and relate it to bumpless pipe dreams. This gives a new proof of recent results of \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)]. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of \textit{K. Motegi} and \textit{K. Sakai} [J. Phys. A, Math. Theor. 46, No. 35, Article ID 355201, 26 p. (2013; Zbl 1278.82042)] to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. We then obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by \textit{P. J. McNamara} [Electron. J. Comb. 13, No. 1, Research paper R71, 40 p. (2006; Zbl 1099.05078)]. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations. Grothendieck polynomial; colored lattice model; vexillary permutation Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Double Grothendieck polynomials and colored lattice models | 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 new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products of Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare this second type to other constructions of Grothendieck polynomials within the more general context of double Grothendieck polynomials and the closely related \({\mathcal H}\)-polynomials. Our methods are based upon the geometry of permutation patterns. \beginbarticle \bauthor\binitsC. \bsnmLenart, \bauthor\binitsS. \bsnmRobinson and \bauthor\binitsF. \bsnmSottile, \batitleGrothendieck polynomials via permutation patterns and chains in the Bruhat order, \bjtitleAmer. J. Math. \bvolume128 (\byear2006), no. \bissue4, page 805-\blpage848. \endbarticle \OrigBibText Cristian Lenart, Shawn Robinson, and Frank Sottile, Grothendieck polynomials via permutation patterns and chains in the Bruhat order , Amer. J. Math. 128 (2006), no. 4, 805-848. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Grothendieck polynomials via permutation patterns and chains in the Bruhat order | 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 flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] can be expressed by a Jacobi-Trudi-type determinant formula generalizing the work of \textit{T. Hudson} and \textit{T. Matsumura} [Eur. J. Comb. 70, 190--201 (2018; Zbl 1408.14030)]. We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide. Grothendieck polynomials; flagged set-valued tableaux; vexillary permutations; Jacobi-Trudi formula Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory and commutative rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Flagged Grothendieck 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 Stable Grothendieck polynomials can be viewed as a \(K\)-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi-type identities, and associated Fomin-Greene operators. symmetric functions; Grothendieck polynomials; Schur polynomials Yeliussizov, D., Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., 45, 295-344, (2017) Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Duality and deformations of stable Grothendieck 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 study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant \(K\)-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)] and \textit{R. Vakil} [Ann. Math. (2) 164, No. 2, 371--422 (2006; Zbl 1163.05337)]. Grassmannian permutations Combinatorial aspects of algebraic geometry, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Groups acting on specific manifolds Littlewood-Richardson coefficients for Grothendieck polynomials from integrability | 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 Grothendieck polynomials, introduced by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Lect. Notes Math. 996, 118--144 (1983; Zbl 0542.14031)], are certain \(K\)-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by \textit{B. Wyser} and \textit{A. Yong} [Transform. Groups 22, No. 1, 267--290 (2017; Zbl 1400.14130)], represent the \(K\)-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the \(K\)-theoretic Schur \(P\)-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain ``Grassmannian'' orbit closures. Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory, etc., Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry On some properties of symplectic Grothendieck 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 goal of the present paper is to extend the mitosis algorithm, originally developed by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] for the case of Schubert polynomials, to the case of Grothendieck polynomials. In addition we will also use this algorithm to construct a short combinatorial proof of Fomin-Kirillov's formula for the coefficients of Grothendieck polynomials. Grothendieck polynomials; pipe dreams; mitosis algorithm Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals Mitosis algorithm for Grothendieck 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 give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula [\textit{P. Magyar}, Comment. Math. Helv. 73, No. 4, 603--636 (1998; Zbl 0951.14036)] for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial. Schubert polynomials; Demazure operator formula Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An orthodontia formula for Grothendieck 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 dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the \(K\)-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2. symmetric functions; reverse plane partitions; Bender-Knuth involutions Symmetric functions and generalizations, Combinatorial aspects of representation theory, Partitions of sets, Grassmannians, Schubert varieties, flag manifolds Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions | 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 dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the \(K\)-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2. dual stable Grothendieck polynomials; symmetric functions; Schur functions; plane partitions; Young tableaux Galashin P, Grinberg D and Liu G 2016 Refined dual stable Grothendieck polynomials and generalized Bender--Knuth involutions \textit{Electron. J. Comb.}23 3--14 Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions | 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 investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the \(K\)-theory of flag varieties (in type \(A\)). The Grothendieck polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 295, 629--633 (1982; Zbl 0542.14030)] serve as polynomial representatives for \(K\)-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in \(n\) variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the \(\beta\)-Grothendieck polynomials of \textit{S. Fomin} and \textit{A. N. Kirillov} [``Grothendieck polynomials and the Yang-Baxter equation'', in: Proceedings of the sixth conference in formal power series and algebraic combinatorics, DIMACS. Piscataway, NJ. 183--189 (1994)], representing classes in connective \(K\)-theory, and we state our results in this more general context. glide polynomials; combinatorial Littlewood-Richardson rule Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry Decompositions of Grothendieck 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 study the double Grothendieck polynomials of Kirillov-Naruse for the symplectic and odd orthogonal Grassmannians. These functions are explicitly written as Pfaffian sum form and are identified with the stable limits of fundamental classes of the Schubert varieties in torus equivariant connective \(K\)-theory of these isotropic Grassmannians. We also provide a combinatorial description of the ring formally spanned be the double Grothendieck polynomials. equivariant \(K\)-theory; isotropic Grassmannians; Schubert class; Pfaffian Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory Double Grothendieck polynomials for symplectic and odd 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 We define a quantum analogue of the Grothendieck ring of finite dimensional modules of a quantum affine algebra of simply laced type. The construction is based on perverse sheaves on a variety related to quivers. We get also a new geometric construction of the tensor category of finite dimensional modules of a finite dimensional simple Lie algebra of type A-D-E. Varagnolo, M.; Vasserot, E., Perverse sheaves and quantum Grothendieck rings, (Studies in Memory of Issai Schur. Studies in Memory of Issai Schur, Chevaleret/Rehovot, 2000. Studies in Memory of Issai Schur. Studies in Memory of Issai Schur, Chevaleret/Rehovot, 2000, Progr. Math., vol. 210, (2003), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 345-365 Quantum groups (quantized enveloping algebras) and related deformations, Geometric invariant theory Perverse sheaves and quantum Grothendieck rings | 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 extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial on the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it. Schur functions; Schubert polynomials; conjecture of Fomin and Kirillov; Grothendieck polynomial Lenart, C., Noncommutative Schubert calculus and Grothendieck polynomials.Adv. Math., 143 (1999), 159--183. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Noncommutative Schubert calculus and Grothendieck 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 Schubert polynomials are the building blocks of several cohomological degeneracy locus formulas. Their role in \(K\)-theory is played by so-called Grothendieck polynomials. In particular, ``stable'' versions of Schubert and Grothendieck polynomials turn up naturally, for example in degeneracy locus problems associated with certain quiver representations.
The paper under review studies the expansion of stable Grothendieck polynomials in the basis of stable Grothendieck polynomials for partitions. This generalizes the result of Fomin-Green in the cohomological setting, where the analogues of stable Grothendick polynomials for partitions are the Schur polynomials. The existence of such a finite, integer linear combination expansion was proved by Buch, and the sign of the coefficients were determined by Lascoux. In this paper the authors give a new, non-recursive combinatorial rule for the coefficients. Namely, they prove that the coefficients, up to explicit sign, count the number of increasing tableau of a given shape, with an associated word having explicit combinatorial properties stemming from the combinatorics of the 0-Hecke monoid. The main ingredient of the proof is a generalized, so-called Hecke insertion algorithm.
The main application showed in the paper is a \(K\)-theoretic analogue of the factor sequence formula of Buch-Fulton for the cohomological quiver polynomials (of equioriented type A). Grothendieck polynomials; \(K\)-theory; 0-Hecke monoid; insertion algorithm; factor sequence formula Buch, A.; Kresch, A.; Shimozono, M.; Tamvakis, H.; Yong, A., Stable Grothendieck polynomials and \textit{K}-theoretic factor sequences, Math. Ann., 340, 359-382, (2008) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], \(K\)-theory of schemes, Symmetric functions and generalizations Stable Grothendieck polynomials and \(K\)-theoretic factor sequences | 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 prove a formula for double Schubert and Grothendieck polynomials, specialized to two re-arrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials. double Schubert polynomials Buch, Anders S.; Rimányi, Richárd, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 339, 1, 1-4, (2004) Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Specializations of Grothendieck 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 symmetric Grothendieck polynomials generalize Schur polynomials and are Schur-positive by degree. Combinatorially this is manifested as the generalization of semistandard Young tableaux by set-valued tableaux. We define a (weak) symmetric \(P\)-Grothendieck polynomial which generalizes \(P\)-Schur polynomials in the same way. Combinatorially this is manifested as the generalization of shifted semistandard Young tableaux by a new type of tableau which we call shifted multiset tableaux. symmetric Grothendieck polynomials; semistandard Young tableaux; set-valued tableaux Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds \(P\)-Schur positive \(P\)-Grothendieck 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 crystal is a directed graph whose encoded information mirror that of the highest weight theory of a root system. Their importance relies on that it reduces problems about representations of Kac-Moody Lie algebras to analogous problems but in a purely combinatorial context; and conversely. References introducing crystals are, from a combinatorial point of view, [\textit{D. Bump} and \textit{A. Schilling}, Crystal bases. Representations and combinatorics. Hackensack, NJ: World Scientific (2017; Zbl 1440.17001); \textit{P. Hersh} and \textit{C. Lenart}, Math. Z. 286, No. 3--4, 1435--1464 (2017; Zbl 1371.05315)]; from the algebraic side, see [\textit{J. Hong} and \textit{S.-J. Kang}, Introduction to quantum groups and crystal bases. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1134.17007)].
Here, the authors associate a type A crystal on the set of \(321\)-avoiding Hecke factorizations; for an expanded version of the content of this paper see [\textit{J. Morse} et al., Electron. J. Comb. 27, No. 2, Research Paper P2.29, 48 p. (2020; Zbl 1441.05237)]. More references: [\textit{M. Albert} et al., Eur. J. Comb. 78, 44--72 (2019; Zbl 1414.05004); \textit{M. Bóna}, Combinatorics of permutations. Boca Raton, FL: CRC Press (2012; Zbl 1255.05001)]. The authors also define a new insertion from decreasing factorizations to pairs of semistandard Yount tableaux, and prove several properties; in particular, this new insertion intertwines with the crystal operators. Everything is related with the combinatorics of Young tableaux.
Additional references: [\textit{M. Gillespie} et al., Algebr. Comb. 3, No. 3, 693--725 (2020; Zbl 1443.05183); \textit{Y.-T. Oh} and \textit{E. Park}, Electron. J. Comb. 26, No. 4, Research Paper P4.39, 19 p. (2019; Zbl 1428.05329); \textit{S. Assaf} and \textit{E. K. Oğuz}, Sémin. Lothar. Comb. 80B, 80B.26, 12 p. (2018; Zbl 1411.05272); \textit{J.-H. Kwon}, Handb. Algebra 6, 473--504 (2009; Zbl 1221.17017); \textit{G. Benkart} and \textit{S.-J. Kang}, Adv. Stud. Pure Math. 28, 21--54 (2000; Zbl 1027.17009); \textit{T. H. Baker}, Prog. Math. 191, 1--48 (2000; Zbl 0974.05080); \textit{G. Cliff}, J. Algebra 202, No. 1, 10--35 (1998; Zbl 0969.17010); \textit{P. Littelmann}, J. Algebra 175, No. 1, 65--87 (1995; Zbl 0831.17004); \textit{A. Puskás}, Assoc. Women Math. Ser. 16, 333--362 (2019; Zbl 1416.05300); \textit{V. I. Danilov} et al., Algebra 2013, Article ID 483949, 14 p. (2013; Zbl 1326.05045); \textit{T. Lam} and \textit{P. Pylyavskyy}, Sel. Math., New Ser. 19, No. 1, 173--235 (2013; Zbl 1260.05043); \textit{V. Genz} et al., Sel. Math., New Ser. 27, No. 4, Paper No. 67, 45 p. (2021; Zbl 07383344); \textit{N. Jacon}, Electron. J. Comb. 28, No. 2, Research Paper P2.21, 16 p. (2021; Zbl 1475.20009); \textit{T. Shoji} and \textit{Z. Zhou}, J. Algebra 569, 67--110 (2021; Zbl 1481.17024)]. 321-avoiding permutatiions; crystal bases; Hecke insertion; Hecke monoid; semistandard Young tableaux; Stembridge crystal Combinatorial aspects of representation theory, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Directed graphs (digraphs), tournaments Crystal for stable Grothendieck 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 Let \(( \mathfrak{g},\mathsf{g})\) be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with \(\mathsf{g}\) being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories \(\mathscr{C}_{\mathfrak{g}}\) and \(\mathscr{C}_{\mathsf{g}}\) of finite-dimensional representations over the quantum loop algebras of \(\mathfrak{g}\) and \(\mathsf{g} \), respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced \(\mathfrak{g} \). In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [\textit{D. Hernandez}, Adv. Math. 187, No. 1, 1--52 (2004; Zbl 1098.17009)]) for simple modules in remarkable monoidal subcategories of \(\mathscr{C}_{\mathfrak{g}}\) for any non-simply-laced \(\mathfrak{g} \), and for any simple finite-dimensional modules in \(\mathscr{C}_{\mathfrak{g}}\) for \(\mathfrak{g}\) of type \(\text{B}_n \). In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of \(T\)-systems, and also we generalize the isomorphisms of [\textit{D. Hernandez} and \textit{B. Leclerc}, J. Reine Angew. Math. 701, 77--126 (2015; Zbl 1315.17011); \textit{D. Hernandez} and \textit{H. Oya}, Adv. Math. 347, 192--272 (2019; Zbl 1448.17019)] to all \(\mathfrak{g}\) in a unified way, that is, isomorphisms between subalgebras of the quantum group of \(\mathsf{g}\) and subalgebras of the quantum Grothendieck ring of \(\mathscr{C}_{\mathfrak{g}} \). Finite-dimensional groups and algebras motivated by physics and their representations, Relationship to Lie algebras and finite simple groups, Simple, semisimple, reductive (super)algebras, Grothendieck groups, \(K\)-theory, etc., Affine algebraic groups, hyperalgebra constructions, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups Isomorphisms among quantum Grothendieck rings and propagation of positivity | 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 double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by \textit{A. N. Kirillov} [``On double Schubert and Grothendieck polynomials for classical groups'', Preprint, \url{arXiv:1504.01469}] and identify them with the polynomials defined by \textit{T. Ikeda} and \textit{H. Naruse} [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)] for the case of maximal Grassmannian permutations. We also give geometric interpretation of them in terms of algebraic localization map and give explicit combinatorial formulas. A.N. Kirillov, H. Naruse, Construction of double Grothendieck polynomials of classical type using Id-Coxeter algebras, preprint. Symmetric functions and generalizations Construction of double Grothendieck polynomials of classical types using idcoxeter algebras | 1 |
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{A. S. Buch} and \textit{R. Rimányi} [C. R., Math., Acad. Sci. Paris 339, No. 1, 1--4 (2004; Zbl 1051.14062)] proved a formula for a specialization of double Grothendieck polynomials based on the Yang-Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by \textit{A. Woo} and \textit{A. Yong} [Am. J. Math. 134, No. 4, 1089--1137 (2012; Zbl 1262.13044)] by constructing a Gröbner basis for the Kazhdan-Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators. Buch-Rimányi formula; double Grothendieck polynomial; specialization Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A note on specializations of Grothendieck 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 Given a vector bundle \(V\) of rank \(n\), the Grothendieck ring of classes of vector bundles of the relative flag manifold \({\mathcal F}(V)\) is generated by the classes \(a_1,\dots,a_n\) of the so-called tautological line bundles on \({\mathcal F}(V)\). The structure sheaf of a Schubert variety in \({\mathcal F}(V)\), having a finite resolution by vector bundles, can be expressed as a (Laurent) polynomial in the \(a_i\), \(1/a_i\). Explicit representatives \(G_\sigma\), \(\sigma\in{\mathfrak S}_n\) were defined by the author and \textit{M.-P. Schützenberger} [in: Invariant theory. Lect. Notes Math. 996, 118--144 (1983; Zbl 0542.14031) and C. R. Acad. Sci., Paris, 295, 629--633 (1982; Zbl 0542.14030)] under the name ``Grothendieck polynomials''. We describe how general Grothendieck of polynomials are related to those for Grassmann manifolds, which themselves are deformations of Schur functions. The geometry of Grassmann varieties is well understood thanks to Schubert, Giambelli and their successors. It is hoped that relating the Schubert subvarieties of a flag manifold to those of a Grassmannian will be of some help to understand those varieties, the singularities of which we still do not know how to describe. Schubert polynomials; 0-Hecke algebra; Grassmann manifolds A. Lascoux, Transition on Grothendieck polynomials, Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 164 -- 179. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of representation theory Transition of Grothendieck 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 \textit{R. A. Gustafson} and \textit{S. C. Milne} [Adv. Math. 48, 177--188 (1983; Zbl 0516.33015)] proved an identity which can be used to express a Schur function \(s_\mu(x_1, x_2, \ldots, x_n)\) with \(\mu = (\mu_1, \mu_2, \ldots, \mu_k)\) in terms of the Schur function \(s_\lambda(x_1, x_2, \ldots, x_n)\), where \(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)\) is a partition such that \(\lambda_i = \mu_i + n - k\) for \(1 \leq i \leq k\). On the other hand, \textit{L. M. Fehér} et al. [Comment. Math. Helv. 87, No. 4, 861--889 (2012; Zbl 1267.14069)] found an identity which relates \(s_\mu(x_1, x_2, \ldots, x_n)\) to the Schur function \(s_\lambda(x_1, x_2, \ldots, x_\ell)\), where \(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_\ell)\) is a partition obtained from \(\mu\) by removing some of the largest parts of \(\mu\). L. M. Fehér et al. [loc. cit.] gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Fehér-Némethi-Rimányi identity. factorial Grothendieck polynomial; Schur function; identity Combinatorial identities, bijective combinatorics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Identities on factorial Grothendieck 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 factorial flagged Grothendieck polynomials are defined by flagged set-valued tableaux of \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)]. We show that they can be expressed by a Jacobi-Trudi type determinant formula, generalizing the work of \textit{T. Hudson} and the first author [Eur. J. Comb. 70, 190--201 (2018; Zbl 1408.14030)]. As an application, we obtain alternative proofs of the tableau and the determinant formulas of vexillary double Grothendieck polynomials, which were originally obtained by Knutson et al. [loc. cit.] and Hudson and the first author [loc. cit.] respectively. Furthermore, we show that each factorial flagged Grothendieck polynomial can be obtained by applying \(K\)-theoretic divided difference operators to a product of linear polynomials. factorial Grothendieck polynomials; flagged partitions; flagged set-valued tableaux; vexillary permutations; Jacobi-Trudi formula; double Grothendieck polynomials Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry Factorial flagged Grothendieck 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 give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by \textit{T. Hudson} et al. [Adv. Math. 320, 115--156 (2017; Zbl 1401.19008)] by \textit{T. Hudson} and \textit{T. Matsumura} [``Segre classes and Kempf-Laksov formula in algebraic cobordism'', Preprint, \url{arXiv:1602.05704}]. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; Grassmannians; Schubert varieties Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory of schemes, Algebraic cycles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] An algebraic proof of determinant formulas of Grothendieck 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 question of when two skew Young diagrams produce the same skew Schur function has been well studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the \(K\)-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons. symmetric functions; Grothendieck polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Coincidences among skew stable and dual stable Grothendieck 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 notion of set-valued Young tableaux was introduced by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by flagged set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram \(D(w)\) of a permutation \(w\), a set-valued Rothe tableau of shape \(D(w)\) is a filling of finite nonempty subsets of positive integers into the squares of \(D(w)\) such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by flagged set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to \textit{T. Matsumura} [J. Algebr. Comb. 49, No. 3, 209--228 (2019; Zbl 1416.05297)]. Employing the properties of tableau complexes given by Knutson et al. [loc. cit.], we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations. Grothendieck polynomial; Schubert polynomial; 1432-avoiding permutation; set-valued Rothe tableau; tableau complex Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of commutative algebra, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Set-valued Rothe tableaux and Grothendieck 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 For a given skew shape, we build a crystal graph on the set of all reverse plane partitions that have this shape. As a consequence, we get a simple extension of the Littlewood-Richardson rule for the expansion of the corresponding dual stable Grothendieck polynomial in terms of Schur polynomials. dual stable Grothendieck polynomials; reverse plane partitions; crystal operators; Littlewood-Richardson rule Galashin, P., A Littlewood-Richardson rule for dual stable Grothendieck polynomials Symmetric functions and generalizations, Partitions of sets, Classical problems, Schubert calculus A Littlewood-Richardson rule for dual stable Grothendieck 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 basic result in the \(K\)-theory of algebraic varieties is the computation of the \(K\)-groups for projective space bundles. In this article the authors compute \(K_0\) for non-commutative analogues of projective space bundles.
The motivation comes from the problem of classifying non-commutative surfaces. This is in its early stages, but the non-commutative analogues of ruled surfaces play a central role. An intersection theory is essential in this study, this can be developed by defining an intersection multiplicity as a bilinear \(\mathbb{Z}\)-valued form, the Euler form, on the Grothendieck group of the surface. The authors give a formula for the Grothendieck group for quantum ruled surfaces, and this is used to show that the associated intersection theory gives natural analogues of the commutative results: If \(X\) is a smooth commutative projective curve and \(f:\mathbb{P}(\mathcal{E})\rightarrow X\) a quantum ruled surface over \(X\), then the fibers do not meet and a section meets a fiber exactly once.
The last two sentences represent the main results of the article, but on the way getting there, a lot of other interesting results and theories appears.
First of all, the terminology and notation generalizes the commutative scheme theory: For abelian categories \(L\) and \(K\), the category \(\text{BIMOD}(K,L)\) of weak \(K\)-\(L\)-bimodules is the opposite of the category of left exact functors \(L\to K.\) If \(\mathcal F\) is is a weak \(K\)-\(L\)-bimodule, the notation is \(\text{Hom}_L(\mathcal F,-)\) for the corresponding left exact functor. \(\mathcal F\) is called a bimodule if \(\text{Hom}_L(\mathcal F,-)\) has a left adjoint, denoted \(-\otimes_K\mathcal F.\) This terminology leads to generalizations of algebras \(A\) as objects in \(\text{BIMOD}(K,K)\), and an \(A\)-module as an element \(M\in K\) together with a morphism \(M\otimes_K A\rightarrow M\) in \(\text{BIMOD}(\text{Ab},K)\) making the obvious diagrams commute.
A quasi-scheme \(X\) is then a Grothendieck category, called \(X\) when thought of as a geometric object, \(\text{Mod}X\) when thought about as a category. The authors define graded \(X\)-modules and graded \(X\)-algebras, leading directly to the quasi-scheme \(\text{Proj}_{\text{nc}}A=\text{GrMod}A/ \text{Tors}A\). Then definitions of \(\mathcal{O}_X\)-bimodule algebras follows, and finally the definition of the analogue of a \(\mathbb{P}^n\)-bundle over a scheme \(X\), the quantum \(\mathbb{P}^n\)-bundles.
This last definition makes the study of quantum ruled surfaces possible, leading up to the main results mentioned earlier. The sections of the rest of the article are ``Connected graded algebras over a quasi-scheme'' including a nice generalization of Nakayama's lemma, ``Flat connected graded \(X\)-algebras'', ``Grothendieck groups'' including the explicit expressions of the \(K_0\)-groups, and finally ``Applications to quantum projective space bundles over a commutative scheme'' defining the intersection theory and proving its main results. non-commutative surface; noncommutative bundle; intersection theory; Grothendieck group; noncommutative algebraic geometry I. Mori, and, S. P. Smith, The Grothendieck group of a quantum projective space bundle, submitted. Noncommutative algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory in geometry The Grothendieck group of a quantum projective space bundle | 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 \(\{f_1,f_2, \dots,f_n\}\) be a regular sequence of polynomials in \(\mathbb{C} [z_1,z_2, \dots,z_n]\) and let \(I:= \langle f_1,f_2, \dots, f_n \rangle\) be the ideal generated by \(f_1,f_2,\dots,f_n\) in \(\mathbb{C}[z_1,z_2, \dots, z_n]\). Let \(V\) be the subvariety in \(X:=\mathbb{C}^n\) defined as a simultaneous zeros of \(f_1,f_2, \dots,f_n\). We define \({\mathcal H}^n_{[V]} ({\mathcal O}_X): =\lim_{k\to \infty} {\mathcal E}xt ({\mathcal O}_X/I^k,{\mathcal O}_X)\) where \({\mathcal O}_X\) is the sheaf of germs of holomorphic functions. \(H^n_{[V]}({\mathcal O}_X)\) and \(\text{Ext} ({\mathcal O}_X/I^k, {\mathcal O}_X)\) are global sections of \({\mathcal H}^n_{[V]} ({\mathcal O}_X)\) and \({\mathcal E}xt ({\mathcal O}_X)/I^k,{\mathcal O}_X)\), respectively. We also define \(\Sigma:= \{\varphi\in H^n_{[V]} ({\mathcal O}_X)\mid f\varphi =0,\;\forall f\in I\}\) and \(\sigma_F: =i\left[ \begin{smallmatrix} 1\\ f_1f_2 \dots f_n\end{smallmatrix} \right]\) where \(i\) is the natural inclusion map \({\mathcal E}xt ({\mathcal O}_X/I^k, {\mathcal O}_X) \to H^n_{[V]} ({\mathcal O}_X)\) and \(\left[ \begin{smallmatrix} 1\\ f_1f_2 \dots f_n \end{smallmatrix} \right]\) is the Grothendieck symbol. Let \({\mathcal H}^n_{[A]} ({\mathcal O}_X)\) be the sheaf of algebraic local cohomology, supported on a point \(A\in X\), with coefficients in the sheaf \(\Omega^n_X\) of germs of holomorphic \(n\)-forms. Let \({\mathcal R}es_A:{\mathcal H}^n_{[A]} ({\mathcal O}_X) \to\mathbb{C}_A\) be the local residue map. Let \(R\) denote the Grothendieck residue pairing \(R:\mathbb{C}[z_1z_2,\dots,z_n]/I \times\Sigma \to\mathbb{C}\) defined by \(R([f],\varphi) :={\mathcal R}es_V (f(z)\varphi (z)dz):= \sum_{A\in V}{ \mathcal R}es_V (f(z)\varphi (z)dz)\) where \([f]\) stands for the coset \(f+I\in\mathbb{C} [z_1,z_2, \dots, z_n]/I\). Let \(\{[b_1], [b_2],\dots, [b_m]\}\) be a basis of the \(\mathbb{C}\)-vector space \(\mathbb{C} [z_1,z_2, \dots,z_n]/I\).
The present author considers the construction problem of the dual basis of \(\{[b_1], [b_2], \dots,[b_m]\}\) with respect to \(R(-,-)\) and the computation problem of the linear form \(\tau_i(f): =R([f], \chi_j)\) and gives solutions to some examples by using the computer algebra system. Hermite-Jacobi formula; Grothendieck duality Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Grothendieck duality and Hermite-Jacobi formula | 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 paper under review the author investigates Grothendieck rings appearing in real geometry, most notably for arc-symmetric sets, and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by \textit{C. McCrory} and \textit{A. Parusiński} [Math. Sci. Res. Inst. Publ. 58, 121--160 (2011; Zbl 1240.14012)]. Let \(S \subseteq \mathbb{R}^n\) be a semialgebraic set. A semialgebraically constructible function on \(S\) is an integer valued function that can be written as a finite sum \(\sum_{i \in I} m_i 1_{S_i}\), where for each \(i \in I\), \(m_i\) is an integer and \(1_{S_i}\) is the characteristic function of a semialgebraic subset \(S_i\) of \(S\). Semialgebraically constructible functions form a commutative ring that will be denoted by \(F(S)\). Dealing with real algebraic sets rather than semialgebraic ones, the push-forward along a regular mapping of the characteristic function of a real algebraic set can not be expressed in general as a linear combination of characteristic functions of real algebraic sets. Nevertheless, the set of all such push-forwards forms a subring \(A(S)\) of \(F(S)\), which is endowed with the same operations. \textit{R. Cluckers} and \textit{F. Loeser} noticed in the introduction of [Invent. Math. 173, No. 1, 23--121 (2008; Zbl 1179.14011)] that \(F(S)\) is isomorphic to the relative Grothendieck ring of semialgebraic sets over \(S\), the push-forward corresponding to the composition with a semialgebraic mapping. The aim of the paper is to continue the analogy further in order to relate the rings of algebraically constructible functions and Nash constructible functions to Grothendieck rings appearing in real geometry. If \(S\) is a real algebraic variety, the author shows that the relative Grothendieck ring \(K_0(\mathbb{R}Var_S)\) of real algebraic varieties over \(S\) maps surjectively to the ring \(A(S)\) of algebraically constructible functions, together with an analogous statement for the ring of Nash constructible functions \(N(S)\). applications of methods of K-theory; constructible functions; semialgebraic sets Applications of methods of algebraic \(K\)-theory in algebraic geometry On Grothendieck rings and algebraically constructible 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 The Schur polynomials \(s_{\lambda }\) are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For \(\rho = (n, n-1, \dots , 1)\) a staircase shape and \(\mu \subseteq \rho\) a subpartition, the Stembridge equality states that \(s_{\rho /\mu } = s_{\rho /\mu^T}\). This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials \(G_{\lambda }\), and the dual stable Grothendieck polynomials \(g_{\lambda }\), developed by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)], \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)], are variants of the Schur polynomials and describe the \(K\)-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood-Richardson rule, we prove that \(G_{\rho /\mu } = G_{\rho /\mu^T}\) and \(g_{\rho /\mu } = g_{\rho /\mu^T}\), the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials. Stembridge equality; Grothendieck polynomial; Young tableau; Hopf algebra Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connections of Hopf algebras with combinatorics The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck 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 {Let \(Fl_n\) be the manifold of complete flags in the \(n\)-dimensional vector space \(\mathbb C^n\). Inspired from ideas from string theory, recently the concept of quantum cohomology ring \(QH^*(X,\mathbb Z)\) of a Kähler algebraic manifold \(X\) has been defined. Then
\[
QH^*(Fl_n,\mathbb Z)\cong H^*(Fl_n,\mathbb Z) \otimes\mathbb Z[q_1,\dots,q_{n-1}],
\]
where \(H^*(X,\mathbb Z)\) is the usual cohomology ring of \(Fl_n\) and \(q_1,\dots,q_{n-1}\) are formal variables (deformation parameters). So, the additive structures of the two cohomology rings are essentially the same. The multiplicative structure of \(H^*(X,\mathbb Z)\) can be recuperated from the multiplicative structure of \(QH^*(X,\mathbb Z)\) by taking \(q_1=\cdots=q_{n-1}=0\). The structure constants for the quantum cohomology are the 3-point Gromov-Witten invariants of genus zero. Recently, Givental, Kim and Ciocan-Lafontaine found a canonical isomorphism
\[
QH^*(X,\mathbb Z)\cong\mathbb Z[q_1,\dots,q_{n-1}][x_1,\dots,x_n]/I_n^q,
\]
where \(x_1,\dots,x_n\) are variables and \(I_n^q\) is a certain ideal which can be explicitly described. This isomorphism extends an old isomorphism of Borel for the ordinary cohomology ring. The next problem naturally arising in the theory of quantum cohomology of the flag manifolds is to find an algebraic/combinatorial method for computing the structure constants of quantum multiplication in the basis of Schubert classes (the Gromov-Witten invariants). The aim of the paper under review is to solve this problem completely.} flag varieties; Schubert varieties; quantum cohomology ring; complete flags; Gromov-Witten invariants S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials , J. Amer. Math. Soc., 168 (1997), 565--596. JSTOR: Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum 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 ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous \(K\)-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model. Schur polynomials; directed last-passage percolation model Symmetric functions and generalizations, Interacting random processes; statistical mechanics type models; percolation theory, Combinatorial probability, Grassmannians, Schubert varieties, flag manifolds Dual Grothendieck polynomials via last-passage percolation | 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 a type \(A_n\) crystal structure on semistandard set-valued tableaux, which yields a new formula and proof for the Schur positivity of symmetric Grothendieck polynomials. For single rows and columns, we construct a \(K\)-theoretic analog of crystals and new interpretation of Lascoux polynomials. We relate our crystal structures to the 5-vertex model using Gelfand-Tsetlin patterns. Grothendieck polynomial; Crystal; Lascoux polynomial; set-valued tableau Combinatorial aspects of representation theory, Classical problems, Schubert calculus Crystal structures for symmetric Grothendieck 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 symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type \(A_n\) crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials. \(A_n\) crystal structure; Schur polynomials; K theory of Grassmannians; Schubert classes Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Quantum groups (quantized enveloping algebras) and related deformations, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Crystal structures for symmetric Grothendieck 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 If \(Y\) is a geometrically connected scheme of finite type over a field \(k\), we have a surjective morphism (as profinite groups): \(\text{ pr}_Y \pi _1(Y,*) \rightarrow G_k \), where \(*\) is a suitable geometric point, \(\pi _1\) is the étale fundamental group and \(G_k\) is the absolute Galois group \(\text{Gal} (k^{\text{sep}}/k)\). Grothendieck conjectured that when \(k\) is finitely generated over \({\mathbb Q}\) and under suitable assumptions on \(Y\) (\(Y\) anabelian), the data \((\pi _1 (Y,*),\text{ pr}_Y)\) determine (functorially) the isomorphism class of \(Y\). One case where this conjecture is supposed to be true is for hyperbolic curves, i.e. when \(Y\) is a smooth curve with \(2-2g-n<0\), where \(g\) is the genus of the smooth compactification \(X\) of \(Y\) and \(n\) is the cardinality of \(S(\overline k)\), with \(S = X-Y\). In this paper the conjecture is shown to be true for affine (i.e. such that \(n>0\)) hyperbolic curves. More precisely what is shown is : Let \(U_1\) and \(U_2\) be two affine hyperbolic curves over a field \(k\) finitely generated over \({\mathbb Q}\), then if there exists an isomorphism \({\mathcal F}: \pi _1 (U_1,*) \rightarrow \pi_1(U_2,*)\) with \(\text{ pr}_{U_1}=\text{ pr}_{U_2}\circ {\mathcal F}\), \(U_1\) is isomorphic to \(U_2\) (as schemes over \(k\)). The proof of the main theorem requires a lot of work and different techniques (study of the function fields and the Galois groups and their connections, study how to recover properties of the curve from data on the fundamental groups); this leads to results interesting per se, like the ones about affine (not necessarily hyperbolic) curves or about curves over discrete valuation rings. hyperbolic curves; fundamental group; Galois group; Grothendiek conjecture Tamagawa A., The Grothendieck conjecture for affine curves, Compos. Math. 109 (1997), no. 2, 135-194. Coverings of curves, fundamental group, Global ground fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry The Grothendieck conjecture for affine curves | 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 As a generalization of skew Schur functions, the refined dual stable Grothendieck polynomials \(\tilde{g}_{\lambda/\mu}(x;t)\) can be easily defined using reverse plane partitions. Motivated by the Jacobi-Trudi formula and its dual for Schur functions, Grinberg conjectured a Jacobi-Trudi type formula for \(\tilde{g}_{\lambda/\mu}(x;t)\) for any skew partition \(\lambda/\mu\). The case for \(\mu=\emptyset\) has been confirmed by \textit{D. Yeliussizov} [J. Algebr. Comb. 45, No. 1, 295--344 (2017; Zbl 1355.05263)]. In the paper under review, the author completely proved Grinberg's conjecture. Comparing the elegant proof of the classical Jacobi-Trudi formula, the author's proof of Grinberg's conjecture is highly nontrivial, which relies on two bijections due to \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)] on RSE-tableaux and requires tedious analysis of the properties of these bijections on extended skew RSE-tableaux. As remarked by the author, Grinberg's conjecture was also independently proved by \textit{A. Amanov} and \textit{D. Yeliussizov} [`Determinantal formulas for dual Grothendieck polynomials'', Preprint, \url{arXiv:2003.03907}]. Jacobi-Trudi formula; plane partition; Grothendieck polynomial; Young tableau Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Jacobi-Trudi formula for refined dual stable Grothendieck 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 the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations, the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nägelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove also an analog of the Giambelli and the Billey-Jockusch-Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321-avoiding permutations. Anatol N. Kirillov, Quantum Schubert polynomials and quantum Schur functions, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 385 -- 404. Dedicated to the memory of Marcel-Paul Schützenberger. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Quantum Schubert polynomials and quantum Schur 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 We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way, we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occurring in the result is actually an interval of the Bruhat order. double Grothendieck polynomials; key polynomials; 0-Hecke algebra; sorting operators; Bruhat order; Pieri formula Pons, V.: Interval structure of the Pieri formula for Grothendieck polynomials, Internat. J. Algebra comput. 23, 123-146 (2013) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Interval structure of the Pieri formula for Grothendieck 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 give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in \textit{M. Chan} et al. [Trans. Am. Math. Soc. 370, No. 5, 3405--3439 (2018; Zbl 1380.05007)], and it generalizes a formula of \textit{C. Lenart} [Ann. Comb. 4, No. 1, 67--82 (2000; Zbl 0958.05128)] and a recent result of \textit{V. Reiner} et al. [J. Comb. Theory, Ser. A 158, 66--125 (2018; Zbl 1391.05269)] to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials. Schur functions; Grothendieck polynomials; insertion algorithms; set-valued tableaux; Brill-Noether theory Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Combinatorial relations on skew Schur and skew stable Grothendieck 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 algebraic group \(G=\text{SL}_{1,{\mathcal A}}\) is considered for an Azumaya algebra \(\mathcal A\) over a semilocal regular ring \(A\) of geometric type. It is proved that any principal homogeneous \(G\)-space that becomes trivial over the field of quotients \(K\) of \(A\) is in fact trivial by itself. This result gives a positive answer to a conjecture of Grothendieck. reduced norms; \(K\)-groups; Gersten complexes; principal homogeneous spaces; Azumaya algebras; semilocal regular rings Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Grothendieck groups, \(K\)-theory, etc., Regular local rings, Homogeneous spaces and generalizations On a Grothendieck conjecture for Azumaya 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 It is well-established that Schur functions are related to the cohomology of Grassmannians. They can be seen as particular cases of Schubert polynomials, which describe the cohomology of flag varieties. Moreover, Schubert polynomials are generalized by Grothendieck polynomials which are defined via \(K\)-theory rather than cohomology. While semi-standard tableaux give a combinatorial description of Schur functions, \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] has shown that set-valued tableaux give a combinatorial description of (stable) Grothendieck polynomials.
This idea of considering set-valued (rather than integer valued) objects was further extended to the theory of P-partition by \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)]. The main goal of the present work is to give an ``enriched'' analog of their results, which means that the underlying combinatorics is similar to that appearing in the theory of Schur \(P\)- and \(Q\)-functions (where two copies of \(\mathbb{N}\) are used as labels in the combinatorial objects). This is motivated by \textit{J. R. Stembridge}'s theory of enriched \(P\)-partitions [Trans. Am. Math. Soc. 349, No. 2, 763--788 (1997; Zbl 0863.06005)].
The authors consider the generating functions of their enriched set-valued \(P\)-partitions (in the same way as Schur functions can be seen as generating functions of semi-standard tableaux). What they obtain are symmetric functions that in some sense generalize \textit{T. Ikeda} and \textit{H. Naruse}'s shifted stable Grothendieck polynomials [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)]. Along the way, they consider various related Hopf algebras, such as an algebra of labeled posets and some subalgebras of quasisymmetric functions. symmetric functions; quasisymmetric functions; Hopf algebras; posets; Grothendieck polynomials Partitions of sets, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Enriched set-valued \(P\)-partitions and shifted stable Grothendieck 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 It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, \(K\)-theoretic deformations of Schur polynomials. Grothendieck polynomials; total positivity; symmetric functions; combinatorial \(K\)-theory Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Positive specializations of symmetric Grothendieck 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 Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial \(K\)-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show that the Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case. symmetric Grothendieck polynomials; Newton polytopes Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Symmetric functions and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Newton polytopes and symmetric Grothendieck 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 In the paper under review, the authors give an explicit formula for the degree of the Grothendieck polynomial \(\mathfrak{G}_w\) of a Grassmannian permutation \(w\) in the symmetric group \(\mathfrak{S}_n\). Their method uses a formula of [\textit{C. Lenart}, Ann. Comb. 4, No. 1, 67--82 (2000; Zbl 0958.05128)] that expresses \(\mathfrak{G}_w\) in terms of Schur polynomials. The authors then use their degree formula to give an explicit formula for the Castelnuovo-Mumford regularity of (the homogeneous coordinate ring of) the Grassmannian matrix Schubert variety \(X_w\) associated to \(w\). The authors also give a counterexample to a conjectured formula, as well as a corrected formula, for the regularity of (the affine coordinate rings of) standard open patches of certain Grassmannian Schubert varieties appearing in [\textit{M. Kummini} et al., Pac. J. Math. 279, No. 1--2, 299--328 (2015; Zbl 1342.14103)]. They further give a conjectured formula for the regularity of standard open patches of arbitrary Grassmannian Schubert varieties. Castelnuovo-Mumford regularity; Grothendieck polynomial; Schubert variety Linkage, complete intersections and determinantal ideals, Classical problems, Schubert calculus, Combinatorial aspects of commutative algebra Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity | 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 presents formulas for the quantum multiplication of classes of Schubert varieties: These formulas are ``quantum analogues'' of the classical Littlewood-Richardson rule and the Kostka numbers, which describe the expansion of products of Schur functions. The basic algorithm involves rim hooks of Young diagrams: Quantum Kostka numbers are shown to be the numbers of tableaux satisfying a certain condition, an efficient algorithm for computing the quantum Littlewood-Richardson numbers is derived. The natural isomorphism of the quantum cohomology rings of Grassmannians \(\text{QH}^*(\text{Gr}(l,l+k))\) and \(\text{QH}^*(\text{Gr}(k,l+k))\) leads to dual versions of these results. Finally, the relation between the quantum cohomology ring and the Verlinde (or fusion) algebra is described. partitions; Young diagrams; rim hooks; Schur functions; Pieri rule; Littlewood-Richardson rule; Kostka numbers; quantum cohomology ring; Schubert varieties; tableaux; Grassmannians Bertram, Aaron; Ciocan-Fontanine, Ionuţ; Fulton, William, Quantum multiplication of Schur polynomials, J. Algebra, 219, 2, 728-746, (1999) Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Quantum multiplication of Schur 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 In order to probe nonregularity of noetherian k-schemes X, groups like \(NK_ 0(X)=coker (K_ 0(X)\to K_ 0(X\times {\mathbb{A}}^ 1))\) and \(K_{-1}\) (a similar cokernel) have been defined. Take in particular \(k={\mathbb{C}}\) and for X a normal quasi-projective surface. It is shown that, if R is the semilocal ring of the singular locus, then there is a surjection \(K_{-1}(X)\twoheadrightarrow C1(\hat R)/C1(R)\) where \(\hat R\) is the completion of R and C1 the ideal class group. If, on the other hand, \(Y\to X\) is a resolution of singularities, the reduced exceptional divisor E of which has smooth components and normal crossings, and if, moreover, Pic(nE)\(\twoheadrightarrow Pic(E)\) is not an isomorphism for some \(n>1\), then \(NK_ 0(X)\) has infinite rank. Examples are given. Laurent polynomial rings; algebraic K-theory; \(NK_ 0\) V. Srinivas, Grothendieck groups of polynomial and Laurent polynomial rings, Duke Math. J. 53 (1986), no. 3, 595 -- 633. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Grothendieck groups, \(K\)-theory and commutative rings Grothendieck groups of polynomial and Laurent polynomial rings | 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 moment of its discovery, the Jones polynomial of a knot has been linked to quantum physics. The main discovery, made by E. Witten, was that it is related to quantum field theory, which unfortunately lacks a mathematical foundation. But already in Witten's work it was noted that the Jones polynomial is related to quantum mechanics. In this paper we discuss progress made in the study of the Jones polynomial from the point of view of quantum mechanics. This study reduces to the understanding of the quantization of the moduli space of flat SU\((2)\)-connections on a surface with the Chern-Simons Lagrangian. We outline some background material, then present the particular example of the torus, in which case it is known that the quantization in question is the Weyl quantization. The paper concludes with a possible application of this theory to the study of the fractional quantum Hall effect, an idea originating in the works of Moore and Read. Jones polynomial; topological quantum field theory; moduli spaces of flat connections; quantization Topological field theories in quantum mechanics, Quantum groups and related algebraic methods applied to problems in quantum theory, Knots and links in the 3-sphere, Applications of global analysis to structures on manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Quantization in field theory; cohomological methods, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Jones' polynomial and the quantum mechanics | 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 explain the relationship between the trace map that occurs in Grothendieck duality from \textit{J. Lipman's} version [Dualizing sheaves, differential and residues on algebraic varieties, Astérisque 117 (1984; Zbl 0562.14003)] and its counterpart (the integral) in the de Rham theory.
Theorem. On a smooth \(n\)-dimensional complete variety \(X\) over \(\mathbb{C}\) the trace map \(\widetilde\theta_X: H^n(X,\Omega^n_X)\to \mathbb{C}\) arising from Lipman's version of Grothendieck duality agrees with
\[
(2\pi i)^{-n}(- 1)^{n(n-1)/2} \int_X: H^{2n}_{\text{DR}}(X,\mathbb{C})\to \mathbb{C}
\]
under the Dolbeault isomorphism. Grothendieck duality; de Rham integral; trace map Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Integral representations; canonical kernels (Szegő, Bergman, etc.), Schemes and morphisms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories The Grothendieck trace and the de Rham integral | 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 normal algebraic variety is \(\mathbb Q\)-Gorenstein if \(rK_X\) is Cartier for some integer \(r>0\) (or equivalently if \((\omega _X^{\otimes r})^{**}\) is locally free). Here \(K_X\) denotes a canonical divisor and \(\omega _X=\mathcal O _X(K_X)\) denotes the canonical bundle. A flat family \(\mathcal X \to C\) of algebraic varieties over a smooth curve \(C\) is a \(\mathbb Q\)-Gorenstein deformation of \(X\) if \(X\) is isomorphic to the fiber over a closed point, \(rK_{\mathcal X /C}\) is Cartier for some integer \(r>0\) and \(rK_{\mathcal X /C}|_X\sim rK_X\). These notions naturally arise and play an important role in the context of the minimal model program and classification of higher dimensional algebraic varieties. It is often useful and important to generalize these definitions to more general situations, for example to families of non-normal varieties over higher dimensional varieties. In this paper, the authors consider the notions of \(\mathbb Q\)-Gorenstein schemes for locally Noetherian schemes with dualizing complexes and \(\mathbb Q\)-Gorenstein morphisms for flat morphisms locally of finite type between locally Noetherian schemes. The authors then prove several useful results for \(\mathbb Q\)-Gorenstein morphisms such as for example the stability under base change and composition, as well as infinitesimal and valuative criteria for a morphism to be \(\mathbb Q\)-Gorenstein. Grothendieck duality; \(\mathbb{Q}\)-Gorenstein morphism Local structure of morphisms in algebraic geometry: étale, flat, etc., Families, moduli, classification: algebraic theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Schemes and morphisms Grothendieck duality and \(\mathbb{Q}\)-Gorenstein morphisms | 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 \(U_q(\mathfrak{sl}_n)\)-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also give an extension of Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions. canonical Grothendieck function; crystal; quantum group; multiset-valued tableau; hook-valued tableau; valued-set tableau Combinatorial aspects of representation theory, Combinatorial identities, bijective combinatorics, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations Crystal structures for canonical Grothendieck 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 Altman A. and Kleiman S., Introduction to Grothendieck duality theory, Lecture Notes in Math. 146, Springer-Verlag, Berlin 1970. Étale and other Grothendieck topologies and (co)homologies, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Riemann-Roch theorems, Duality theorems for analytic spaces, Sheaf cohomology in algebraic topology Introduction to Grothendieck duality 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 We show that Lapointe-Lascoux-Morse \(k\)-Schur functions (at \(t=1\)) and Fomin-Gelfand-Postnikov quantum Schubert polynomials can be obtained from each other by a rational substitution. This is based on Kostant's solution of the Toda lattice and Peterson's work on quantum Schubert calculus. Lapointe-Lascoux-Morse \(k\)-Schur functions; Fomin-Gelfand-Postnikov quantum Schubert polynomials Lam, Thomas; Shimozono, Mark, From double quantum Schubert polynomials to \(k\)-double Schur functions via the Toda lattice, (2011) Symmetric functions and generalizations, Classical problems, Schubert calculus From quantum Schubert polynomials to \(k\)-Schur functions via the 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 The obituary discusses Grothendieck's contributions to modern algebraic geometry between the late 1950's and 1970. It discusses the development of schemes over general commutative rings together with their functorial interpretation and presents the motivations behind Grothendieck topologies and the crystalline topology defined on categories with corresponding cohomology theories, comparison theorems and the theory of motives. Grothendieck; algebraic geometry; flatness; scheme; Grothendieck topologies, crystalline topology Illusie, L.; Raynaud, M., Grothendieck and algebraic geometry, Asia Pac. Math. Newsl., 5, 1-5, (2015) Biographies, obituaries, personalia, bibliographies, History of algebraic geometry Grothendieck and algebraic 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 Hecke-Grothendieck polynomials were introduced by \textit{A. N. Kirillov} [SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016; Zbl 1334.05176)] as a common generalization of Schubert polynomials, dual \(\alpha\)-Grothendieck polynomials, Di Francesco-Zinn-Justin polynomials, etc. Kirillov conjectured that the coefficients of every generalized Hecke-Grothendieck polynomial are nonnegative combinations of certain parameters. Here we prove a weak version of Kirillov's conjecture, that is, under certain conditions, every Hecke-Grothendieck polynomial has only nonnegative integer coefficients. In particular, the proof of this weak version of Kirillov's conjecture serves as a unified proof for the fact that all the Schubert polynomials, dual \(\alpha\)-Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials have only nonnegative coefficients. Hecke-Grothendieck polynomial; Gröbner-Shirshov basis; positivity Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds A weak version of Kirillov's conjecture on Hecke-Grothendieck 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 show that the reduced point variety of a quantum polynomial algebra is the union of specific linear subspaces in \(\mathbb{P}^n\); we describe its irreducible components and give a combinatorial description of the possible configurations in small dimensions. quantum polynomial rings; noncommutative projective geometry; ring theory; torus actions Belmans, P., De Laet, K., Le Bruyn, L. (2015). The point variety of quantum polynomial rings, arXiv preprint arXiv:1509.07312. Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry The point variety of quantum polynomial rings | 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 \(k\) be a perfect field and \(\Lambda\) a coefficient ring. \textit{V. Voevodsky} [Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)] constructed a triangulated category of motives, denoted \(\mathbf{DM}(k, \Lambda)\). One of the basic features of this theory is the comparison with the usual étale theory. Specifically, let \(n\) be an integer prime to the characteristic of \(k\), (with \(\Lambda = \mathbb{Z}/n\mathbb{Z}\)) we have an equivalence of categories
\[
\mathsf{L}\iota^\ast:\mathbf{D}(\mathbf{Sh}_{\text{ét}} (\mathrm{Et}/k, \mathbb{Z}/n\mathbb{Z})) \cong\mathbf{DM}(k, \mathbb{Z}/n\mathbb{Z})
\]
due to Suslin and Voevodsky (see [\textit{C. Mazza} et al., Lecture notes on motivic cohomology. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (2006; Zbl 1115.14010)]). In other words, triangulated motives agree with complexes of étale sheaves with torsion coefficients prime to the characteristic of \(k\).
The purpose of this work is to generalize this construction to a general base scheme replacing \(\mathrm{Spec}(k)\). Unfortunately, the evident comparison functor is not an equivalence in the relative case due to the more complicated behavior of the étale version of the relative triangulated category of motives \(\mathbf{DM}^{\text{ét}}(S, \Lambda)\). Here, the author proposes to work with the equivalent category \(\mathbf{DA}^{\text{ét}}(S, \Lambda)\) of étale relative triangulated motives \textit{without transfers}, that it is shown (in appendix B) is equivalent to \(\mathbf{DM}^{\text{ét}}(S, \Lambda)\). To this end the author proves that under certain reasonable conditions on the base scheme \(S\) there is an equivalence of categories
\[
\mathsf{L}\iota^\ast:\mathbf{D}(\mathbf{Sh}_{\text{ét}} (\mathrm{Et}/S, \mathbb{Z}/n\mathbb{Z})) \cong\mathbf{DA}^{\text{ét}}(S, \mathbb{Z}/n\mathbb{Z}).
\]
It is worthwhile to note that the hypothesis about \(S\) are on the cohomological dimensions of its geometric points. It follows from Gabber's results that any excellent scheme \(S\) satisfies these conditions. The proof of this result and its preliminaries takes roughly the first half of the paper.
In the next sections, the author addresses several properties of this equivalence. First, the fundamental question of the compatibility of this equivalence with Grothendieck's \textit{six operations}, namely the functors \(f^\ast\), \(f_\ast\), \(f^!\) and \(f_!\), associated to a quasi-projective morphism \(f\) together with the bifunctorial closed structure given by \(-\otimes-\) and \(\underline{\mathsf{Hom}}(-,-) \). Further, this equivalence is extended to a rational étale realization funtor that takes the form
\[
\mathfrak{R}^{\text{ét}}_S:\mathbf{DA}^{\text{ét}}_{\text{ct}} (S, \mathbb{Q}) \longrightarrow\hat{\mathbf{D}}^{\text{ét}}_{\text{ct}} (S, \mathbb{Q}_\ell),
\]
where \(\mathbf{DA}^{\text{ét}}_{\text{ct}} (S, \mathbb{Q})\) denotes the subcategory of \textit{constructible} triangulated motives with rational coefficients and \(\hat{\mathbf{D}}^{\text{ét}}_{\text{ct}} (S, \mathbb{Q}_\ell)\) denotes the derived category of \(\ell\)-adic sheaves over \(S\). The result holds for \(S\) excellent and the functor is again compatible with Grothendieck's six operations.
In the final part the author studies the compatibility of the étale realization functor with the formalism of vanishing cycles. Let \(S\) be the spectrum of a Henselian discrete valuation ring and \(f: X \to S\) be a quasi-projective \(S\)-scheme. Let \(\eta\) and \(\sigma\) denote the generic and special points of \(S\) and \(X_{\eta}\) and \(X_{\sigma}\) its corresponding fibers in \(X\). There are functors
\[
\Upsilon_f, \Psi^{\text{mod}}_f: \mathbf{DA}^{\text{ét}} (X_{\eta}, \Lambda) \longrightarrow\mathbf{DA}^{\text{ét}} (X_{\sigma}, \Lambda)
\]
such that, for \(M\) in \(\mathbf{DA}^{\text{ét}} (X_{\eta}, \Lambda)\), \(\Upsilon_f(M)\) is the part with unipotent monodromy of the nearby motive of \(M\) and \(\Psi^{\text{mod}}_f(M)\) corresponds to its part with tame monodromy. Whenever the same primes are invertible over \(\Lambda\) and over \(S\) and there is a uniform bound for the cohomological dimensions with respect to the relevant primes, the functor \(\mathfrak{R}^{\text{ét}}_S\) commutes with the corresponding variants of the nearby cycles functor. Also, the compatibility of the action of monodromy on these objects is analyzed.
The results of the paper build on previous work of the author in [Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I. Paris: Société Mathématique de France (2007; Zbl 1146.14001)]. The article is a continuation of this work and the author suggests the readers to have a copy at hand when reading it. Further prerequisites are: techniques of resolution of singularities (following Hironaka, de Jong and Gabber), Röndigs and Østvær's rigidity theorem, results by Gabber on étale cohomology, and Quillen's algebraic \(K\)-theory motivic homotopy spectrum as developed by Morel-Voevodsky.
An appendix gives an alternative treatment of an intermediate step for the proof of the first main theorem which is contained in section 2. The one in appendix C is shorter but depends on \textit{F. Morel}'s results on motivic spheres (see [\(\mathbb A^1\)-algebraic topology over a field. Berlin: Springer (2012; Zbl 1263.14003)]). motive; étale cohomology; étale realization; Grothendieck six operations; nearby cycles formalism Ayoub, J., La réalisation étale et les opérations de Grothendieck, Ann. Sci. Éc. Norm. Supér. (4), 47, 1, 1-145, (2014) Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), (Equivariant) Chow groups and rings; motives, Étale and other Grothendieck topologies and (co)homologies The étale realization and the Grothendieck operations | 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 Given a smooth projective variety \(X\) over \( \bar{\mathbb Q} \) consider \( X^{\text{an}}_{\mathbb C}\), the compact complex analytic manifold determined by \(X\). Grothendieck's period conjecture in codimension \(k\), \(GPC^k(X)\), is defined here to be the criterion that an element \(\alpha\) in the algebraic de Rham cohomology group \(H^{2k}_{\text{dR}}(X / \bar{\mathbb Q} )\) is in fact the class of some algebraic cycle of codimension \(k\) in \(X\), with rational coefficients, if and only if for all rational homology classes \(\gamma\) in \(H_{2k}(X^{\text{an}} _{\mathbb C}, \mathbb Q)\) the corresponding period is rational, namely it is \( {(2\pi i)^{-k} } \int_\gamma \alpha \in \mathbb Q\). The paper is a sequel of \textit{J.-B. Bost}'s article [Notre Dame J. Formal Logic 54, No. 3--4, 377--434 (2013; Zbl 1355.11074)], where \(GPC^1(X)\) was proved for abelian varieties. Here we find several new results, one of them being the stability of \(GPC^1\) under products, so \(GPC^1\) holds for products of curves and abelian varieties. The method of absolute Hodge classes and a generalization of the Kuga-Satake correspondence are then used to prove that \(GPC^1\) is valid also for the case of symplectic manifolds with second Betti number at least \(4\). The variety of lines on a cubic fourfold \(Y\) is such a symplectic manifold, this fact yields as a consequence the result that \(GPC^2(Y)\) holds. The paper contains moreover several useful explanations and discussions, dealing for instance with the relations between the Hodge conjecture and \(GPC\). De Rham cohomology; Hodge theory; Hodge conjecture; periods Transcendental methods, Hodge theory (algebro-geometric aspects), de Rham cohomology and algebraic geometry, Arithmetic ground fields for abelian varieties Some remarks concerning the Grothendieck period conjecture | 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 algebraic aspects of the equivariant quantum cohomology algebra of the flag manifolds. We introduce and study the quantum double Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x,~y)\), which are the Lascoux-Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x)\) as the Gram-Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using the quantum Cauchy identity, we prove that \(\widetilde{\mathfrak{S}}_w(x)=\widetilde{\mathfrak{S}}_w(x,y)|_{y=0}\) and as a corollary we obtain a simple formula for the quantum Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x)=\partial_{ww_0}^{(y)}\widetilde{\mathfrak{S}}_{w_0}(x,~y)|_{y=0}\). We also prove the higher genus analog of Vafa-Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann-Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule. quantum double Schubert polynomials; Ehresmann-Bruhat graph; quantum Pieri's rule; cohomology; flag manifold Kirillov, A. N.; Maeno, T.: Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-intriligator formula. Discrete math. 217, No. 1-3, 191-223 (2000) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula | 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 main concerns of this multifaceted paper is that of dealing with the quantum cohomology of flag varieties parametrizing inclusions of subspaces of a given vector space, as well as to supply an innovative description of the product structure of the affine Grassmannian \(\mathrm{Gr}\) associated to the group \(\mathrm{SL}_n({\mathbb C})\). Just not to loose the less experienced reader, we recall that the \(\mathrm{Gr}\) we alluded above is in fact the quotient of the group of all \(n\times n\) unimodular square matrices with entries in the ring of Laurent series modulo the action of the subgroup of unimodular square matrices with entries in the ring of formal power series with complex coefficients. The new sharp description that the authors offer for the cohomology \(H^*(\mathrm{Gr})\) and homology \(H_*(\mathrm{Gr})\), whose details are richly displayed in Section 5, definitely shed light of the intimate relationship of the subject, the so-called affine Schubert calculus, with the rich, celebrated and still mysterious theory of Macdonald polynomials. The latter is about some of the richest objects in mathematics. As the authors themselves declare in a enthusiastically inspiring introduction, Macdonald polynomials do not occur just in combinatorics, but also in the theory of double affine Hecke algebras, quantum relativistic systems, diagonal harmonics and Hilbert schemes of points in the plane.
The origin of the noble story told by the authors in the paper, has to do with a basic and a fundamental question. Nearly every professional mathematician is aware that symmetric polynomials admit a basis of Schur polynomials, parametrized by partitions of non-negative integers. Although we shall not recall here their definition, Macdonald polynomials may be seen as symmetric polynomials depending on two extra parameters, say \(t\) and \(q\), and it is then natural to wonder about the transition matrix relating them with the more familiar Schur polynomials. The still open conjecture is that the entries of the transition matrix, a generalization of the so called Kostka-Foulkes polynomial, are polynomials with non-negative integer coefficients. In other words, Macdonald polynomials relate positively to Schur polynomials, a conjecture that inspired many more researches whose output has been the dramatic emerging of the relationship with the affine Schubert calculus. The authors so come to cope with the problem of a more flexible description of the homology and cohomology of \(\mathrm{Gr}\) by introducing a clever new combinatorial tool, which is of crucial importance in all the paper, called Affine Bruhat Countertableaux (ABC). Their generating functions form a basis of \(H^*(\mathrm{Gr})\) and everything leads to a refinement of the Kotska-Foulkes polynomials. The authors deal also with the problem of providing a closer description of the constants structure of the quantum cohomology of flag varieties, where Gromov-Witten invariants related with the art of counting rational curves in homogeneous varieties. There are many more beautiful and interesting features, in this paper, that deserve to be discussed, but this at the price of giving a more detailed account of the fine technical combinatorial tools masterly employed by the authors. This cannot be evidently done in a review, but we can end it by quickly describing the organization. Let us start from the abstract: it already contains the juice of the article and say the reader what it can be found inside. The introduction is simply as beautiful as exciting and is enhanced by the second section where there is a useful interesting account of the related literature. The preliminaries are collected in Section 3: here the reader can be made aware with selected tools from the theory of symmetric functions, explain the basic vocabulary related with Ferrer diagrams, horizontal strips, addable corners, extremal cells and so on. Section 4 enters into the deep core of the paper, being devoted to the affine Pieri's rule, described in terms of sophisticated but versatile combinatorics. The explicit representative of Schubert classes is provided in this section, where the ABC order is also introduced. More relations with Macdonald polynomials are collected in Section 6. The rich reference list is still preceded by an appendix where the output of a Sage routine is displayed to check a conjecture on the equality of two kinds of symmetric functions and by section 7, eventually devoted to the quantum cohomology of the flags. Macdonald polynomials; Hall-Littlewood polynomials; affine Schubert calculus; quantum Schubert calculus; type-A affine Weyl group; affine Grassmannian; Gromov-Witten invariants; Bruhat order; weak \(k\)-Pieri rule; \(k\)-tableaux; affine Bruhat counter-tableaux Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Enumerative combinatorics Quantum and affine Schubert calculus and Macdonald 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 It is well known that both the classical Galois theory and the topological theory of coverings can be formulated as an equivalence between a category of ``subobjects'' and a category of ``actions''. In this paper, there are discussed variants of Grothendieck's axioms characterizing Galois theories in several contexts corresponding to categories of transitive/not-necessarily-transitive (continuous) actions of a discrete/profinite group (or of even a monoid). Especially, the authors give a presentation of \textit{A. Grothendieck}'s fundamental theorem in SGA1 [Séminaire de géométrie algébrique, 1960/61, Lect. Notes Math. 224 (1971; Zbl 0234.14002); exposé V] as a ``passing into the limit'' of Galois's Galois theory.
At the end, several possible extensions of the authors' results in view of Giraud's theorem on topoi, Joyal's spatial groups etc. are illustrated. Galois category; Grothendieck topos; strict epimorphism; Galois theory; coverings Dubuc, E. J.; De La Vega, C. S.: On the Galois theory of Grothendieck. Bol. acad. Nac. cienc. Cordoba 65, 111-136 (2000) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Categories of topological spaces and continuous mappings [See also 54-XX], Separable extensions, Galois theory On the Galois theory of Grothendieck | 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 equivariant quantum cohomology ring \(QH^*_T(Fl({\underline n}))\) of the \(m\)-step partial flag variety in \({\mathbb C}^n\) is studied. This ring is an algebra over \(\Lambda[{\underline q}]={\mathbb Z}[t_1,t_2,\dots,t_n,q_1,q_2,\dots,q_m]\) and can be thought of as a deformation of the usual equivariant cohomology ring which is an algebra over \(\Lambda={\mathbb Z}[t_1,t_2,\dots,t_n]\). Additively \(QH^*_T(Fl({\underline n}))\simeq H^*_T(Fl({\underline n}))\otimes \Lambda[{\underline q}]\). For the parameters \({\underline q}=0\) the algebra structure was computed by Borel. The Schubert classes form a basis of the \(\Lambda[{\underline q}]\)-module \(QH^*_T(Fl({\underline n}))\). The authors give a Giambelli type formula for Schubert classes in terms of the \textit{quantum elementary polynomials}. The resulting \textit{equivariant quantum Schubert polynomials} are specializations of \textit{W. Fulton}'s \textit{universal Schubert polynomials} [Duke Math. J. 96, No. 3, 575--594 (1999; Zbl 0981.14022)] and has already appeared in non-equivariant situation [\textit{I. Ciocan-Fontanine}, Duke Math. J. 98, No. 3, 485--524 (1999; Zbl 0969.14039)]. As a by-product a presentation of \(QH^*_T(Fl({\underline n}))\) was obtained; originally this is a result of [\textit{B. Kim}, Int. Math. Res. Not. 1996, No. 17, 841--851 (1996; Zbl 0881.55007)]. The proof relies on the moving lemma with respect to the mixing group. The paper contains a comprehensive account of the history of recent developments of the Schubert calculus. Schubert calculus; flag varieties; quantum equivariant cohomology; Giambelli formula Anderson, D.; Chen, L.: Equivariant quantum Schubert polynomials. Adv. math. 254, 300-330 (2014) Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Equivariant homology and cohomology in algebraic topology Equivariant quantum 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 study the ``combinatorial anabelian geometry'' that governs the relationship between the dual semi-graph of a pointed stable curve and various associated profinite fundamental groups of the pointed stable curve. Although many results of this type have been obtained previously in various particular situations of interest under unnecessarily strong hypotheses, the goal of the present paper is to step back from such ``typical situations of interest'' and instead to consider this topic in the abstract - a point of view which allows one to prove results of this type in much greater generality under very weak hypotheses. S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math J. 59 (2007), 455-479. Coverings of curves, fundamental group, Arithmetic ground fields for curves A combinatorial version of the Grothendieck conjecture | 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 following, let \(K\) be the field of fractions of a complete discrete valuation ring with separably closed residue field \(k\) with \(\mathrm{char}(k)=p>0\), and let \(A\) be an abelian variety over \(K\). We denote \(A'=\mathrm{Pic}^0_A\). By Grothendieck's theory, there exists a natural pairing \(\langle \cdot, \cdot \rangle_A: \Phi_{A,K} \times \Phi_{A',K} \rightarrow \mathbb{Q}/\mathbb{Z}\) where \(\Phi_{A,K}\) and \(\Phi_{A',K}\) denote the component groups of the Néron models of \(A\) and \(A'\), respectively. Due to work of Bégueri, Bertapelle, Bosch, McCallum, Lorenzini and Werner, the pairing is known to be perfect in many cases. The pairing is conjectured to be perfect if \(k\) is perfect. There are counter examples which show that the pairing is not perfect in general.
The main result of this article extends an earlier result of Bosch and Lorenzini. Let \(C\) be a smooth geometrically connected proper curve of genus \(\geq 1\) over \(K\). We denote the Jacobian of \(C\) by \(J=J(C)\). Let \(\varphi:J \rightarrow J'\) be the canonical principal polarization of \(J\). It is convenient to identify \(J\) with \(J'\) via the morphism \(- \varphi\). This identification has a continuation to Néron models, and hence induces an isomorphism \(\Phi_{J,K} \rightarrow \Phi_{J',K}\). As a consequence, Grothendieck's pairing induces a symmetric pairing \(\langle \cdot, \cdot \rangle_{J,K}:\Phi_{J,K} \times \Phi_{J,K} \rightarrow \mathbb{Q}/\mathbb{Z}\) on the component group \(\Phi_{J,K}\) of the Néron model of \(J(C)\). The main result of Lorenzini reads as follows.
Theorem. Assume that \(C\) has a proper flat regular integral model \(\mathcal{C}\) such that the irreducible components of the reduction \(\mathcal{C}_k\) are smooth, \(\mathcal{C}_k^{\mathrm{red}}\) has normal crossings and the multiplicities of the irreducible components have no proper common factor. If there exist two distinct irreducible components of \(\mathcal{C}_k\) with strictly positive intersection number and coprime multiplicities, then the pairing \(\langle \cdot, \cdot \rangle_{J,K}\) is perfect.
The proof of the above theorem makes use of an earlier result of \textit{S.~Bosch} and \textit{D.~Lorenzini}, Invent. Math. 148, No. 2, 353--396 (2002; Zbl 1061.14042) Th.4.6] which says that, under the theorem's assumptions, for the pairing \(\langle \cdot , \cdot \rangle_{J,K}\) to be perfect it suffices that \(C(K) \not= \emptyset\). The idea of proof is that this situation can be achieved over a carefully chosen base extension \(L\) of \(K\). Consider the morphism \(\gamma:\Phi_{J,K} \rightarrow \Phi_{J,L}\) which is induced by base change. Let \(e_{L/K}\) denote the ramification index of the extension \(L/K\). By the formula \(\big\langle \gamma(x), \gamma(y) \big\rangle_{J,L} = e_{L/K} \cdot \big\langle x,y \big\rangle_{J,K}\) for all \(x,y \in \Phi_{J,K}\), one can relate the perfectness of the pairing \(\langle \cdot , \cdot \rangle_{J,K}\) to the one of \(\langle \cdot , \cdot \rangle_{J,L}\), provided that \(e_{L/K}\) is invertible modulo the group orders of \(\Phi_{J,K}\) and \(\Phi_{J,L}\). In fact, Lorenzini proves that one can choose \(L=K \big( \pi^{1/n} \big)\), where \(\pi\) denotes a uniformizer of \(K\) and where \(n\) is a large enough prime. Here the assumption on the existence of intersecting components with coprime multiplicities comes into play. Grothendieck pairing; component groups; duality Lorenzini, D, Grothendieck's pairing for Jacobians and base change, J. Number Theory, 128, 1448-1457, (2008) Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Curves over finite and local fields, Arithmetic ground fields for curves Grothendieck's pairing for Jacobians and base change | 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 extends the approach to the construction of parametric families of \(r\)-functions as generating functions for weighted Hurwitz numbers which are initiated by him and Guay-Paquet and were extended to other cases by him and others. The general method is developed and used to derive an infinite parametric family of \(2D\) Toda \(r\)-functions of hypergeometric type depending also on an additional pair \((q,t)\) of quantum deformation parameters entering in the definition of the scalar product. For specific choices of the parameters defining the weight generating function the author gives specialized versions of the quantum weighted Hurwitz numbers. By making other specializations involving particular values for the pair \((q,t)\) or their limits reduce the Macdonald or Schur or Hall-Littlewood or Jack polynomials. hypergeometric functions; Macdonald polynomials; generating functions; enumerative combinatorics Harnad, J., Quantum Hurwitz numbers and Macdonald polynomials, J. Math. Phys., 57, 113505, (2016) Bessel and Airy functions, cylinder functions, \({}_0F_1\), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Coverings of curves, fundamental group, Exact enumeration problems, generating functions Quantum Hurwitz numbers and Macdonald 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 note briefly reviews, in the context of mirror symmetry, examples of polarized variations of mixed Hodge structures that produce limiting mixed Hodge structures of Hodge-Tate type in the large complex structure limit. Open, local and closed cases are addressed, as well as relations between them, the focus is on integral structures intrinsic to the A-model, in particular to quantum cohomology.
A number of open problems motivated by the examples are stated and discussed. One is to give a general derivation of homological mirrors for Abel-Jacobi maps, which would bring Beilinson's conjectures to bear on the arithmetic of Gromov-Witten invariants. Another is to use single log divergent periods of Hori-Vafa mirrors to prove integrality properties of open invariants. This is illustrated in the case of \(K_{\mathbb{P}^2}\), and related to the authors' work on higher algebraic cycles of toric hypersurfaces. variation of Hodge structures; Hodge-Tate type; Gromov-Witten invariants; Hori-Vafa mirrors; log divergent periods Variation of Hodge structures (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects) Algebraic cycles and local quantum cohomology | 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 Grothendieck conjecture is a conjecture that the arithmetic fundamental group of a hyperbolic algebraic curve completely determines the algebraic structure of the curve. Let \(X\) be an algebraic variety defined over a field \(K\), and let \(\pi_1(X)\) denote the ``arithmetic'' fundamental group of \(X\). Let \(\text{Gal}(K):=\text{Gal}(\overline K/K)\) denote the absolute Galois group of \(K\) where \(\overline K\) is the separable closure of \(K\). There is an exact sequence of groups
\[
1 \longrightarrow \pi_1(X_{\overline K})\longrightarrow \pi_1(X) @>\text{pr }x>> \text{Gal}(K)\longrightarrow 1
\]
which gives rise to a homomorphism called an outer Galois representation: \(\rho_X: \text{Gal}(K)\to \text{Out}(\pi_1(X_{\overline K}))\). \textit{A. Grothendieck} [``La longue marche à travers de la théorie de Galois'' (1981), in preparation by J. Malgoire and the articles in ``Geometric Galois actions. 1'', Lond. Math. Soc. Lect. Note Ser. 242, 5--48 and 49--58 (1997; Zbl 0901.14001 and Zbl 0901.14002)] formulated a collection of conjectures, the so-called anabelian conjectures, for algebraic varieties \(X\) over base fields \(K\), where \(X\) are anabelian and \(K\) are finitely generated over the prime field.
(GC1) The Fundamental Conjecture: An anabelian algebraic variety \(X\) over a field \(K\) which is finitely generated over the prime field may be reconstituted from the structure of \(\pi_1(X)\) as a topological group equipped with its associated surjection \(\text{pr}_X: \pi_1(X)\to \text{Gal}(K)\).
For algebraic curves in characteristic \(0\), there is a more precise formulation of the conjecture.
(GC2) The Hom Conjecture: For hyperbolic algebraic curves \(X, Y\) over a field \(K\) which is finitely generated over \({\mathbb Q}\), the natural map
\[
\Hom_K(X,Y)\to \Hom_{\text{Gal}(K)}(\pi_1(X),\pi_1(Y))/\sim
\]
defines a bijective correspondence between dominant \(K\)-morphisms and equivalence classes of \(\text{Gal}(K)\)-compatible open homomorphisms.
(GC2) claims that open homomorphisms of the arithmetic fundamental groups always comes from algebro-geometric morphisms. (GC2) is similar to the Tate conjecture [proved by \textit{G. Faltings}, Invent. Math. 73, 349--366 (1983; Zbl 0588.14026)] for abelian varieties and one-dimensional étale homology groups. (However, it was pointed out in this article that the Grothendieck conjectures (GC1), (GC2) are different in essential ways from the Tate conjecture.)
(GC3) The Section Conjecture: Let \(X\) be a hyperbolic algebraic curve over a field \(K\) which is finitely generated over \({\mathbb Q}\). Then every section homomorphism \(\alpha: \text{Gal}(K)\to \pi_1(X)\) of the projection \(\text{pr}_X:\pi_1(X)\to \text{Gal}(K)\) arises either from a \(K\)-rational point of \(X\), or from the \(K\)-rational points at infinity of \(X\).
The three authors of the paper under review were able to prove the conjectures (GC1) and (GC2). The conjecture (GC3) is still unsolved. This article reports on how the authors have succeeded in proving the Grothendieck conjecture.
\textit{Y. Ihara} [Ann Math. (2) 123, 43--106 (1986; Zbl 0595.12003)] and \textit{G. W. Anderson} and \textit{Y. Ihara} [Ann. Math. (2) 128, No. 2, 271--293 (1988; Zbl 0692.14018) and Int. J. Math. 1, No. 2, 119--148 (1990; Zbl 0715.14021)] investigated the pro-\(\ell\) outer Galois representation on \(\pi_1(X)\), for each prime \(\ell\), associated to a genus zero curve \(X:={\mathbb{P}}^1-\Lambda\) where \(\Lambda\subset{\mathbb{P}}^1(K)\) is a finite set containing \(0,1,\infty\). They gave a description of the subfield \(K_X^{(\ell)}\) of \(\overline K\) which arises naturally from the pro-\(\ell\) outer Galois representation \(\rho_X^{(\ell)}\) by means of a system of ``numbers'' obtained from the set \(\Lambda\) of ramification points.
However, further works were needed for an efficient book-keeping of these ``numbers''. This was accommodated by a series of papers by Nakamura translating the pro-\(\ell\) outer Galois representations on \(\pi_1(X)\) into a group-theoretic language involving Galois ``permutations of the pro-cusp points'' over \(\Lambda\) distributed in the ``rim'' of the pro-\(\ell\) universal covering of \({\mathbb{P}}^1-\Lambda\). This reduced the task of understanding the outer Galois representation to that of more accessible Galois permutations of the cusps. This idea was powerful enough to control inertia groups and decomposition groups. The first breakthrough by \textit{H. Nakamura} [J. Reine Angew. Math. 405, 117--130 (1990; Zbl 0687.14028)] was to prove the finiteness theorem that there are only finitely many subsets \(\Lambda\subset{\mathbb{P}}^1(K)\) that give rise to the same arithmetic fundamental group and that fundamental group already determines a curve of the form \({\mathbb{P}}^1-\{4\) points\}. Further, Nakamura reduced the problem of reconstructing (from their arithmetic fundamental groups) curves \({\mathbb{P}}^1-\{n\) points\} of genus \(0\) to the case where \(n=4\). Consequently \textit{H. Nakamura} [J. Reine Angew. Math. 411, 205--210 (1990; Zbl 0702.14024)] was able to prove that the hyperbolic algebraic curves of genus \(0\) over a field which is finitely generated over \({\mathbb Q}\) may be reconstituted from their arithmetic fundamental groups, thereby establishing (GC1) in this case. Nakamura's method also gave a group-theoretic characterization of those section homomorphisms in (GC3), which are also discussed in later work of Tamagawa and Mochizuki.
Now we describe Tamagawa's contribution. Let \(k\) be a finite field, and let \(X\) be a nonsingular affine curve over \(k\). \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135-194 (1997; Zbl 0899.14007)] addressed the analogue of the Grothendieck conjecture in positive characteristic. One of his main results was that the scheme \(X\) may be recovered from \(\pi_1(X)\). Tamagawa's proof was modeled on the work of \textit{K. Uchida} [Ann. Math. (2) 106, 589--598 (1977; Zbl 0372.12017)] and the key steps were
(i) the group-theoretic characterization of the decomposition groups of each closed point of \(X^*\) (the nonsingular compactification of \(X\)),
(ii) the reconstruction of the multiplicative group \(k(X)^{\times}\) and
(iii) the reconstruction of the additive structure on \(k(X)=k(X)^{\times}\cup \{0\}\).
In addition if \(X\) is hyperbolic, then the results are still valid for \(\pi_1(X)^{\text{tame}}\) (which is a quotient of \(\pi_1(X)\)). Further, the isomorphism version of (GC2) for affine hyperbolic curves over fields which are finitely generated over \({\mathbb Q}\) may be derived from the results about the tame fundamental group of affine hyperbolic curves over finite field. Let \(K\) be a number field, and \(X\) be an affine hyperbolic curve over \(K\). The main idea of Tamagawa is to show how to recover group-theoretically the tame fundamental group of the reduction of \(X\) at each finite prime of \(K\) from the arithmetic fundamental group of \(X\) itself. This established the fact that the isomorphism version of (GC2) for affine hyperbolic curves over number fields may be derived from the results about tame fundamental groups of affine hyperbolic curves over finite fields. If \(X_1\) and \(X_2\) are two affine hyperbolic curves over a number field \(K\) such that \(\pi_1(X_1)\simeq \pi_1(X_2)\) over Gal\((K)\), then there is an induced isomorphism \(\pi_1^{\text{tame}}((X_1)_{k_v})\simeq \pi_1^{\text{tame}}((X_2)_{k_v})\) at almost all of the primes \(v\) of \(K\). Thus there is an isomorphism \((X_1)_{k_v}\simeq (X_2)_{k_v}\). From the hyperbolicity, one has \(\text{Isom}(X_1,X_2)\simeq \text{Isom}((X_1)_{k_v}, (X_2)_{k_v})\), and finally this implies that \(X_1\simeq X_2\).
Mochizuki was the one to supply the final piece of the work to settle the conjectures (GC1) and (GC2) in the affirmative, which we now describe briefly. \textit{S. Mochizuki} in his series of papers [J. Math. Sci., Tokyo 3, No. 3, 571--627 (1996; Zbl 0889.11020); ``The local pro-\(p\) Grothendieck conjecture for hyperbolic curves'', RIMS Preprint 1045 (Kyoto Univ. 1995); Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019); ``A Grothendieck conjecture-type result for certain hyperbolic surfaces'' (to appear)] introduced a totally new way of looking at the Grothendieck conjecture. A priori, the conjecture was formulated for objects defined over global fields. Mochizuki's striking idea was to look at this conjecture as a \(p\)-adic analytic phenomenon whose natural base is a local, not a global field. Mochizuki's insight (from global to \(p\)-adic fields) was indeed very powerful culminating in the proof of a more general result and (GC2) over fields which are finitely generated over \({\mathbb Q}\) can be deduced as a special case. To begin with, Mochizuki formulated and proved the \(p\)-adic analogue of the Grothendieck conjecture.
Theorem: For any smooth algebraic variety \(S\) and any hyperbolic curve \(X\) (both) over a sub-\(p\)-adic field \(K\), the natural maps
\[
\Hom^{\text{dom}}_K(S,X)\to \Hom^{\text{open}}_{\text{Gal}(K)}(\pi_1(S),\pi_1(X)) \to \Hom^{\text{open}}_{\text{Gal}(K)}(\pi_1^{(p)}(S),\pi_1^{(p)}(X))
\]
are bijections. Here \(\Hom_K^{\text{dom}}\) denotes the ``set of all dominant \(K\)-morphisms''; \(\Hom^{\text{open}}_{\text{Gal}(K)}\) denotes the ``set of all equivalent classes of open homomorphisms which are compatible with the projections to \(\text{Gal}(K)\); and \(\pi_1^{(p)}(V)\) is the natural pro-\(p\) analogue of \(\pi_1(V)\).
This theorem may be regarded as an analogue of the uniformization theorem of hyperbolic Riemann surfaces. This theorem resolves (GC2) in a fairly strong form.
As a corollary, the birational version of the Grothendieck conjecture was proved.
Corollary: For (regular) function fields \(L\) and \(M\) of arbitrary dimension over a field of constants \(K\) which is sub-\(p\)-adic, the natural map
\[
\Hom_K(M,L)\to \Hom^{\text{open}}_{\text{Gal}(K)}(\text{Gal}(L),\text{Gal}(M))
\]
is bijective. Here \(\Hom_K\) denotes the ``set of ring homomorphisms over \(K\)''; and \(\Hom^{\text{open}}_{\text{Gal}(K)}\) denotes the ``set of equivalence classes of open homomorphisms which are compatible with the projections to \(\text{Gal}(K)\).
A sketch of proof of the above theorem is presented. Let \(K\) be a finite extension of \({\mathbb Q}_p\), and let \(X\) and \(S\) be proper, non-hyperelliptic hyperbolic curves. The most essential problem was to reconstruct \(X\) from \(\pi_1^{(p)}(X)\to \text{Gal}(K)\) group-theoretically, and Mochizuki did this using \(p\)-adic analysis. Grothendieck's conjecture; hyperbolic algebraic curves; étale fundamental groups; arithmetic fundamental groups; anabelian conjectures; \(p\)-adic Grothendieck conjecture; birational Grothendieck conjecture Nakamura, H.; Tamagawa, A.; Mochizuki, S.: The Grothendieck conjecture on the fundamental groups of algebraic curves. Sugaku expositions 14, 31-53 (2001) Coverings of curves, fundamental group, Arithmetic ground fields for curves, Homotopy theory and fundamental groups in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Curves over finite and local fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields The Grothendieck conjecture on the fundamental groups of algebraic curves | 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 obtain the quantum cohomology ring of the flag manifold of type \(G_{2}\). The corresponding quantum Schubert polynomials are explicitly computed. The authors utilize the moment graph of such manifold to calculate all the curve neighborhoods of Schubert classes. Curve neighborhoods method is used to write down Chevalley formula for class multiplication. quantum cohomology; Schubert polynomial; \(G_2\) flag manifold; Chevalley formulas; moment graph Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum Schubert polynomials for the \(G_2\) flag 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 In this article, we prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the formula with the \(K\)-theoretic divided difference operators. Our formula specializes to the one obtained by \textit{W. Y. C. Chen} et al. [Eur. J. Comb. 25, No. 8, 1181--1196 (2004; Zbl 1055.05149)] for the (double) skew Schur polynomials. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; set-valued tableaux; 321-avoiding permutations Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices A tableau formula of double Grothendieck polynomials for 321-avoiding permutations | 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 affine Lie group, \(B\) a Borel subgroup and \(X= G/B\) the corresponding affine flag manifold which is decomposed into the union of affine Schubert varieties \(X_w = \overline{BwB/B}\). \(X\) is an infinite-dimensional (not quasi-compact) scheme over \(\mathbb C\) and its \(B\)-orbits are parametrized by the elements of the Weyl group \(W\). Each \(B\) orbit is a locally closed subscheme with finite codimension and is isomorphic to the scheme \(\mathbb A^\infty = \text{Spec} (\mathbb C(x_1,x_2,\dots,))\). The \(K\)-group is decomposed into the product \(K_B(X) \cong \prod_{w\in W} K_B(pt)[\mathcal{O}_{X^w}]\). The group \(K_B(pt)\) is isomorphic to the group ring \(\mathbb Z[P]\) of the weight lattice \(P\) of a maximal torus of \(B\). Similar to the finite-dimensional case there is a homomorphism \(\mathbb Z[P] \otimes \mathbb Z[P] \cong K_B(pt) \otimes K_B(pt) \to K_B(X)\) which factors through the equivariant Atiyah-Hirzebruch homomorphism \(\mathbb Z[P] \bigotimes_{\mathbb Z[P]^W} \mathbb Z[P] \to K_B(X)\), where \(\mathbb Z[P]^W\) is the ring of invariants with respect to the action of the Weyl group \(W\).
In the affine case, this morphism in injective but not surjective, not all \([{\mathcal O}_{X^w}]\) are in the image of this morphism. Nevertheless, after localization by a generator \(\delta\) of null roots, taking tensor product with the subring \(\mathbb Q[\delta]\) we get the main result of the paper: Theorem 4.4: all the elements \([{\mathcal O}_{X^w}]\) are in \(\mathbb Q[\delta] \bigotimes_{\mathbb Z[e^{\pm\delta}]} K_B(X)\). The authors call the elements of \(R \bigotimes_{\mathbb Z[e^{\pm\delta}]} \mathbb Z[P] \bigotimes_{\mathbb Z[P]^W} \mathbb Z[P]\) the affine Grothendieck polynomials. \(K\)-theory; affine flag manifold; Grothendieck polynomial Kashiwara, Masaki; Shimozono, Mark, Equivariant \(K\)-theory of affine flag manifolds and affine Grothendieck polynomials, Duke Math. J., 148, 3, 501-538, (2009) Equivariant \(K\)-theory, Homogeneous spaces and generalizations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties Equivariant \(K\)-theory of affine flag manifolds and affine Grothendieck 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 study matrix elements of a change of basis between two different bases of representations of the quantum algebra \({\mathcal{U}}_q(\mathfrak{su}(1,1))\). The two bases, which are multivariate versions of Al-Salam-Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman's multivariate Askey-Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey-Wilson polynomials are solutions of a multivariate bispectral \(q\)-difference problem. Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Lie algebras of linear algebraic groups, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Estimation in multivariate analysis, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) A quantum algebra approach to multivariate Askey-Wilson 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 One perspective on the motive of an algebraic variety is to consider it as a collection of realizations, e.g. its Betti-, Hodge- and \(\ell\)-adic-realization. Equipped with the world of motives and étale cohomology is the formalism of Grothendieck's six functors for étale and \(\ell\)-adic sheaves. In his two groundbreaking œuvres [The Grothendieck six operations and the vanishing cycles formalism in the motivic world. I. Astérisque 314. Paris: Société Mathématique de France. (2007; Zbl 1146.14001)] and [Astérisque 315. Paris: Société Mathématique de France. (2007; Zbl 1153.14001)] the author has developed Grothendieck's six operations and the formalism of vanishing cycles in the stable motivic homotopy category of schemes. This leads to the natural question how this formalism behaves with respect to realizations in the world of stable motivic homotopy. The present paper provides a complete treatment of the case of Betti realization.
For a variety over a field of characteristic zero with an embedding into the field of complex numbers, the Betti realization is obtained by taking singular cohomology of the associated analytic space. The assignment of forming the associated analytic space can be extended to a functor from presheaves over smooth varieties to presheaves over smooth analytic spaces. Moreover it is possible to extend this functor to the stable motivic homotopy category. For this purpose, the author provides a thorough treatment of the homotopy category of presheaves over smooth analytic spaces in the first section. The Betti realization is then defined as the left derived functor of forming associated analytic spaces from the stable motivic homotopy category over a variety to the derived category of sheaves of abelian groups on the corresponding analytic objects. The main result states that Betti realization is compatible with Grothendieck's six operations and the nearby cycles functors. This is proven by constructing canonical isomorphisms between the respective composition of functors. This construction is given by providing general criteria for the compatibility of functors.
Despite the necessary level of abstract formalism the article is nicely written and provides detailed explanations with many interesting remarks at the sidelines. (The only obstacle for reading it is that the author assumes familiarity to a certain extent with the above cited fundamental articles and its notations and definitions. But this is just another impetus to study the work of the author.) motives; Betti realizations; Grothendieck's six operations; vanishing cycles formalism; model categories Ayoub, Joseph, Note sur les opérations de Grothendieck et la réalisation de Betti, J. Inst. Math. Jussieu, 9, 2, 225-263, (2010) Classical real and complex (co)homology in algebraic geometry, Motivic cohomology; motivic homotopy theory, Nonabelian homotopical algebra, Étale and other Grothendieck topologies and (co)homologies, Grothendieck topologies and Grothendieck topoi, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Resolutions; derived functors (category-theoretic aspects) Notes on Grothendieck's operations and Betti realizations | 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 smooth complete complex algebraic variety. A connective \(K\)-theory \(K(X,\beta)\) of \(X\) is, roughly speaking, an interpolation of the cohomology theory and the \(K\)-theory of \(X\), in the sense that \(K(X,0)=H^*(X)\), the ordinary singular cohomology ring of \(X\), and \(K(X,1)=K(X)\), the ordinary Grothendieck ring of \(X\).
To say it with a slogan, the paper under review is concerned with the quantum equivariant connective \(K\)-theory, \(qh^*_n:=qh^*(X;\beta)\), of the Grassmann variety \(X:=G(n,N)\), parametrizing \(n\)-dimensional subspaces of \(\mathbb{C}^N\).
The ring \(qh^*_n\) can be seen as a multiparameter deformation of the classical cohomology ring of \(X\). The involved deformation parameters \((t_1,\ldots,t_n)\), \(q\) and \(\beta\) play different roles. The first are the equivariant parameters related with the action of the algebraic torus \(\mathbb{T}:=(\mathbb{C}^*)^n\), induced by the diagonal action on \(\mathbb{P}^{N-1}\), \(q\) is the quantum deformation parameter and \(\beta\) is a parameter that connects the generalized (i.e. quantum, equivariant) \(K\)-theory of \(X\) (for \(\beta=1\)) to the quantum, equivariant cohomology ring \(QH^*_{\mathbb{T}}(X)\) (for \(\beta=0\)).
The main result of this paper is without doubt the description of the ring \(qh^*_n\). Its impact is described in another main result, named Theorem 1.1. in the introduction, where three different specializations of \(qh^*_n\), obtained by setting to zero some of the deformation parameters, are considered. It is so shown that \(qh^*_n\) generalizes all the presentations known so far, relying on one hand on the classical Schubert Calculus, ruled by Giambelli's and Pieri's formula and, on the other, on important work appeared along the last couple of decades, due to \textit{D. Peterson} [``Quantum Cohomology of \(G/P\)'', Lecture Notes, M.I.T. (1997)], \textit{B. Kostant} and \textit{S. Kumar} [J. Differ. Geom. 32, No. 2, 549--603 (1990; Zbl 0731.55005)] and more recently to \textit{A. S. Buch} and \textit{L. C. Mihalcea} [Duke Math. J. 156, No. 3, 501--538 (2011; Zbl 1213.14103)].
In particular, Theorem 1.1. shows that i) setting \(\beta=0\) one recovers the presentation due to \textit{L. C. Mihalcea} [Adv. Math. 203, No. 1, 1--33 (2006; Zbl 1100.14045)] of the quantum cohomology ring \(QH^*_{\mathbb{T}}(X)\); ii) setting \(\beta=1\) and \((t_1,\ldots, t_N)=(0,\ldots,0)\), one obtains Buch's quantum \(K\) theory \(KQ(X)\) and that iii) for \(\beta=-1\), \(q=0\) and \(t_j\) equal to certain expressions involving generators of the character ring of \({\mathfrak gl}(N)\), recovers \(K_{\mathbb{T}}(X)\), the equivariant \(K\)-functor. The proof of iii) above is certainly the most intriguing, as it involves a generalization of the celebrated Goresky-Kottwitz-MacPherson theory [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] and the localized Schubert classes are identified with certan polynomials related to the Bethe ansatz of quantum integrable models, which is another part of the story told in this impressive article.
As a matter of fact, the topic faced in the paper is so wide and important that it is hard to bound it within the narrow borders of conventional subject classifications. Indeed, the ring \(qh^*_n\), the main character of the paper, allows the authors to dig up a breath taking relationship between Schubert Calculus and certain quantum integrable systems that in statistical mechanics are known as \textsl{asymmetric six-vertex model}, invented to describe the physics of anti-ferroelectric materials.
This very well written paper is rather long and dense but the authors put a special effort not to loose the readers by clearly segmenting it in sections, with the aim to provide pre-requisites with graduality. Although combinatorial tools are inspired by the Yang Baxter equations as well as the six vertex models in statistical mechanics, a preliminary knowledge of the latter is not necessary to follow the mathematical content of the paper, which candidates itself to be a must for all mathematicians interested in the cohomological theories of homogeneous varieties. quantum cohomology; quantum \(K\)-theory; enumerative combinatorics; exactly solvable models; Bethe ansatz; Yang Baxter equations; statistical mechanics V. Gorbounov and C. Korff. ''Quantum integrability and generalised quantum Schubert calculus''. Adv. Math. 313 (2017), pp. 282--356.DOI. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Bordism and cobordism theories and formal group laws in algebraic topology, Symmetric functions and generalizations, Exactly solvable models; Bethe ansatz, Equivariant \(K\)-theory Quantum integrability and generalised quantum 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 An efficient review of Gromov-Witten invariant and quantum cohomology is given. Let \((M,\omega)\) be a symplectic manifold, \(J\) an \(\omega\)-tamed almost complex structure, and \(\Sigma\) be a Riemann surface with the complex structure \(j\). Then \(f:\Sigma\to M\) is called a pseudoholomorphic curve if \(J\circ df= df\circ j\). Gromov-Witten invariants count the number of pseudoholomorphic curves and are used to distinguish symplectic manifolds that are diffeomorphic but not equivalent under symplectic deformations [\textit{Y. Ruan}, J. Differ. Geom. 39, No. 1, 215--227 (1994; Zbl 0810.53021)]. Gromov-Witten invariants are certain correlation functions in the topological sigma model and its coupling to gravity [\textit{E. Witten}, Commun. Math. Phys. 118, No. 3, 411--449 (1988; Zbl 0674.58047); Surv. Differ. Geom., Suppl. J. Differ. Geom. 1, 243--310 (1991; Zbl 0757.53049)]. This argument leads to the definition of the three point function and the quantum product \(*_w\) on \(H^*(M)\). \(*_w\) is associative [\textit{Y. Ruan} and \textit{G. Tian}, J. Differ. Geom. 42, No. 2, 259--367 (1995; Zbl 0860.58005); Invent. Math. 130, No. 3, 455--516 (1997; Zbl 0904.58066)].
The algebra obtained by the cohomology group and the quantum product is denoted by \(H^*_Q(M)\) and called the quantum cohomology ring. Three-dimensional minimal models in the same birational class have isomorphic quantum cohomology rings [\textit{A.-M. Li} and \textit{Y. Ruan}, Invent. Math. 145, No. 1, 151--218 (2001; Zbl 1062.53073), hereafter refered to as [1]].
Let \({\mathcal M}_{g,k}\) be the moduli space of Riemann surfaces of genus \(g\) with \(k\) marked points, \({\mathcal M}^M_A\) the moduli space of pseudoholomorphic maps \(f\) such that \(f_*(\Sigma)= A\), \(A\in H_2(M,\mathbb{Z})\). Intuitively, the Gromov-Witten invariant is defined as the integral of a polynomial of \(K\in H^*(\overline{{\mathcal M}}_{g,k})\) and \(\alpha_i\in H^*(M)\), \(i= 1,\dots, k\) on \(\overline{{\mathcal M}}^M_A\), the compactification of \({\mathcal M}^M_A\). But \(\overline{{\mathcal M}}^M_A\) is not smooth, the author uses the virtual neighborhood method, to give a rigorous definition of the Gromov-Witten invariant \(\Phi^M_{(A,g)}(K; \alpha_1,\dots, \alpha_k)\) [\textit{Y. Ruan}, Turk. J. Math. 23, No. 1, 161--231 (1999; Zbl 0967.53055)]. \(\overline{{\mathcal M}}^M_A\) is imbedded in a finite-dimensional open manifold \(U\) and a vector bundle \(E\) over \(U\) with a section \(s: U\to E\) such that the zero locus \(s^{-1}(0)\) is \(\overline{{\mathcal M}}^M_A\). The Gromov-Witten invariant is described as the integral of the integrand used in the formal definition of the Gromov-Witten invariant multiplied by the Thom form of \(E\to U\). This definition does not depend on the choice of \(U\) and the Thom form, and is invariant under symplectic deformation (section 1). The author states that Gromov-Witten invariants can be calculated only in the genus 0 cases.
In section 1.2, examples of Gromov-Witten invariants are given. The quantum product is defined in section 1.3 by using the three-point function
\[
\Phi^M_w(\alpha_1,\alpha_2,\alpha_3)= \sum_A\, \sum^\infty_{k=1} {1\over k!}\Phi^M_{(A,0)}(\alpha_1, \alpha_2,\alpha_3,\underbrace{w,\dots,w}_{k\text{ copies}}) q^A.
\]
Here \(w\in H^*(M)\) and \(q\) is a symbol satisfying \(q^Aq^B= q^{A+B}\). The (big) quantum product is defined via
\[
\langle\alpha*_w \alpha_2,\alpha^*\rangle= \Phi^M_w(\alpha_1,\alpha_2,\alpha_3).
\]
When \(w= 0\), \(*_0= *\) is called the small quantum product. As an example, \(H^*_Q(\mathbb{P}^n)= \mathbb{Z}[x]/\langle x^{n+1}- q\rangle\) is stated.
In Section 2, after dealing with semistable degeneration and the symplectic normal sum [\textit{R. E. Gompf}, Ann. Math (2) 142, No. 3, 527--595 (1995; Zbl 0849.53027)], the relative Gromov-Witten invariant \(\Phi^{(M,Z)}_{(A,g)}\) of the pair \((M, Z)\), \(Z\) a codimension 2 almost complex submanifold of \(M\), is defined. If there is a Hamiltonian action \(H: U\to \mathbb{R}\), \(U\) an open set of \(M\), such that \(N= H^{-1}(0)\) divide \(M\) into two components, then we have symplectic cuts \(M^{\pm}\), each containing the symplectic quotient \(Z= N/S^1\) as a symplectic submanifold of codimension 2.
Let \(\pi: M\to M^+\cup M^-\) be the degneration map, \([A]= A + \ker\pi_*\), and let
\[
\alpha^+_i|_Z= \alpha^-_i|_Z,\quad \alpha^{\pm}_i\in H^*(M^{\pm}),\quad \alpha_i= \pi^*(\alpha^+_i\cup_Z \alpha^-_i)\in H^*(M).
\]
Then the degeneration formula (in the case \(K= 1\), [1], it is the quantum counter part of the Meyer-Vietoris long exact sequence in the ordinary cohomology)
\[
\sum_{B\in [A]} \Phi^M_{(B,g)}(\alpha_1,\dots, \alpha_k)= \sum_C \Phi_C,
\]
is stated in section 2.4. Here each \(C\) is a graph representing the topological type of a Riemann surface. Examples of the explicit forms of \(\Phi_C\) are given.
Fibrations (holomorphic) embeddings are not morphisms of quantum cohomology. But studies of surgeries on Calabi-Yau manifolds suggest that certain extended birational transformations are morphisms of quantum cohomology. This is explained in section 3. Here, only even cohomology groups are considered to avoid introducing superstructure. To state results and conjectures, Mori's program [\textit{S. Mori}, Ann. Math. (2) 116, 133--176 (1982; Zbl 0557.14021)] is reviewed in section 3.2. Then results and conjectures related to the birational geometry and quantum cohomology are corrected in section 3.3. Gromov-Witten invariant; quantum cohomology; birational geometry; pseudo-holomorphic curve; symplectic deformation; minimal model; Calabi-Yau manifold Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Applications of global analysis to structures on manifolds, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Gromov-Witten invariants and quantum cohomology | 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 family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes, our formulas generalize the classical hook-length formula and the Littlewood formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its \(q\)-analogs, which were studied in previous papers of the series.
For Part III see [the authors, Algebr. Comb. 2, No. 5, 815--861 (2019; Zbl 1425.05158)]. hook-length formulas; Littlewood formula; Naruse hook-length formula Combinatorial aspects of representation theory, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds Hook formulas for skew shapes. IV: Increasing tableaux and factorial Grothendieck 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 cohomology ring of a complete flag variety has a basis of Schubert classes labelled by elements of the symmetric group, in which the structure constants can be described combinatorially. In a polynomial presentation of the cohomology ring, this basis corresponds to Schubert polynomials.
Similar statements are known for the equivariant cohomology ring, and for the so-called ``quantum'' deformation of the ordinary (non-equivariant) cohomology. The relevant polynomials are double Schubert polynomials and quantum Schubert polynomials, respectively. It the latter case, the structure constants encode genus \(0\) Gromov-Witten invariants of the flag variety.
In this paper, it is shown the the quantum double Schubert polynomials introduced by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, represent equivariant quantum Schubert classes of the complete flag variety. Furthermore, the result is generalized to partial flag varieties and parabolic quantum double Schubert polynomials. This affords in particular a combinatorial description of equivariant Gromov-Witten invariants.
The discussion of the maximally extended results is somewhat difficult to follow. The main result has also been obtained independently by \textit{D. Anderson} and \textit{L. Chen} [Adv. Math. 254, 300--330 (2014; Zbl 1287.14025)]. Lam, T.; Shimozono, M.: Quantum double Schubert polynomials represent Schubert classes Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum double Schubert polynomials represent 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 [For the entire collection see Zbl 0657.00005.]
Mathematical questions connected with the problem of computing the Polyakov partition function and its fermionic analogue are discussed in the frames of quantum string theory. The results, partly joint with those by A. A. Bejlinson and A. S. Shvarts, rely upon calculating the Mumford form and Polyakov measure, as well as upon a study of the superfuchsian uniformization, Schottky's uniformization, and superautomorphic functions. superstrings; supersymmetry; Polyakov partition function; quantum string theory; Mumford form; Polyakov measure; superautomorphic functions Yu. I. Manin.Quantum strings and algebraic curves, Berkeley ICM talk (1986). Families, moduli of curves (algebraic), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Strong interaction, including quantum chromodynamics Quantum strings and algebraic curves | 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 0742.00081.]
The authors state that the goal of this paper is to place the Alexander polynomial of knot theory into the context of topological quantum field theory. The following illustrates how this is done. Let SC be the category whose objects are tuples of (oriented) surfaces and whose morphisms are (oriented) cobordisms. For a commutative ring \(R\) let \(PM_ R\) be the category whose objects are graded \(R\)-modules of finite type and whose morphisms are equivalence classes of graded \(R\)-linear maps, where a map \(f\) is equivalent to \(\lambda f\) for units \(\lambda\) of \(R\). Let \(Q:\text{SC}\to PM_ R\) be a functor with certain properties. For a given closed oriented 3-manifold \(M\) and a given homology class \(\zeta\in H_ 2(M)\), represent \(\zeta\) by an embedded surface \(F\) and let \(M'\) be \(M\) cut along \(F\). Considering \(F\times 0\), \(F\times 1\) as objects in SC and \(M'\) as a morphism of SC, the authors define the polynomial \(\lambda_ Q(M,\zeta)(t)\) as the Lefschetz polynomial of \(Q(M')\) and show that this polynomial only depends on \(M\) and \(\zeta\) up to multiplication by \(ut^ m\), where \(u\) is a unit of \(R\) and \(m\) ranges over certain integers. In particular, if \(M\) is the result of longitudinal surgery on a knot \(K\) in \(S^ 3\) then the Seifert surface of \(K\) determines a \(\zeta\in H_ 2(M)\). For a certain functor \(Q\) the authors show that in this case \(\lambda_ Q(M,\zeta)(t)\) coincides up to multiplication by \(\pm t^ k\) with the (un-normalized) Alexander polynomial of \(K\). graded \(R\)-modules; graded \(R\)-linear maps; Alexander polynomial; topological quantum field theory; surfaces; cobordisms; oriented 3- manifold; homology class; embedded surface; Lefschetz polynomial; longitudinal surgery on a knot; Seifert surface Frohman, C., Nicas, A.: The Alexander Polynomial via Topological Quantum Field Theory. Differential Geometry, Global Analysis, and Topology, Canadian Mathematical Society Conference Proceedings, Volume 12, American Mathematical Society, Providence, Rhode Island, pp. 27-40 (1992) Topology of general 3-manifolds, Knots and links in the 3-sphere, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Algebraic moduli problems, moduli of vector bundles The Alexander polynomial via topological quantum field 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 This chapter is devoted to a discussion of Gromov-Witten-Welschinger (GWW) classes and their applications. In particular, Horava's definition of quantum cohomology of real algebraic varieties is revisited by using GWW classes and is introduced as a differential graded operad. In light of this definition, we speculate about mirror symmetry for real varieties. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Real algebraic and real-analytic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Towards quantum cohomology of real 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 Orbifold Hurwitz numbers are the weighted count of connected genus \(g\) covers of the projective line with specific ramification behavior over \(0\) and \(\infty\) and simple ramification elsewhere, if any.
These numbers appear in many different contexts with many different disguises, such as the intersection theory of moduli space of curves, low dimensional topology, topological recursion and matrix models, etc.
Among many such instances, the authors use the reincarnation of Hurwitz numbers in the context of semi-infinite wedge formalism, [\textit{A. Alexandrov} et al., J. High Energy Phys. 2016, No. 5, Paper No. 124, 31 p. (2016; Zbl 1388.81016)].
In this interpretation, the authors verify the conjectures of \textit{N. Do} and \textit{M. Karev}, [J. Math. Sci., New York 226, No. 5, 568--587 (2017; Zbl 1388.14097)], and \textit{N. Do} and \textit{D. Manescu}, [Commun. Number Theory Phys. 8, No. 4, 677--701 (2014; Zbl 1366.14034)].
More precisely, they show that the dependence of various versions of Hurwitz numbers on the ramification parameters is polynomial. The proof is combinatorial and obtained by defining analogues of \(\mathcal{A}\) operators of \textit{A. Okounkov} and \textit{R. Pandharipande}, [Ann. Math. (2) 163, No. 2, 561--605 (2006; Zbl 1105.14077)], in the context of semi-infinite wedge formalism.
It must be noted that their result in the context of ordinary orbifold Hurwitz numbers exists already in literature, see [\textit{N. Do} et al., Math. Res. Lett. 23, No. 5, 1281--1327 (2016; Zbl 1371.14061)]. enumerative geometry; spectral curves; Hurwitz numbers; dessins d'enfants Enumerative problems (combinatorial problems) in algebraic geometry, Dessins d'enfants theory, Symmetric functions and generalizations Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants | 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 first gives a sketch of Grothendieck's dramatic life. After some interesting remarks on the role of generalization and specialization in mathematical heuristics, he then illustrates Grothendieck's work by means of the monodromy theorem: Grothendieck's highly original and general approach to the fundamental group makes it possible to prove a difficult and deep result by means of simple linear algebra. Grothendieck; monodromy theorem F. Oort, Some questions in algebraic geometry. History of mathematics in the 20th century, History of algebraic geometry Grothendieck: the new algebraic 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 The authors describe how the theory of ordinary and modular characters may be localized at the prime ideals of certain commutative rings acting on the representation ring of a finite group over a field. In addition, this localized character theory, and a Lefschetz-Riemann-Roch theorem, are applied to study the Galois module structure of the cohomology of the structure sheaves of semi-stable curves over rings of algebraic integers. modular characters; representation rings; finite groups; localized character theory; Lefschetz-Riemann-Roch theorem; Galois modules; sheaves; semi-stable curves Chinburg, Ted; Erez, Boas; Pappas, Georgios; Taylor, Martin: Localizations of Grothendieck groups and Galois structure, Contemp. math. 224, 47-63 (1999) Ordinary representations and characters, Frobenius induction, Burnside and representation rings, Modular representations and characters, Relations of \(K\)-theory with cohomology theories, Riemann-Roch theorems, Arithmetic ground fields for curves, de Rham cohomology and algebraic geometry, Integral representations related to algebraic numbers; Galois module structure of rings of integers Localizations of Grothendieck groups and Galois 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 We propose a new characterization of integral zeros of binary Krawtchouk polynomials in terms of the range criterion for entangled states in quantum information theory. Krawtchouk polynomial; positive partial transpose; entangled state; range criterion Quantum coherence, entanglement, quantum correlations, Linear equations (linear algebraic aspects), Topological properties in algebraic geometry, General theory of \(C^*\)-algebras On characterizing integral zeros of krawtchouk polynomials by quantum entanglement | 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 quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of \(G/B\). We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph.
We define path Schubert polynomials, which are quantum cohomology analogs of skew Schubert polynomials recently introduced by \textit{C. Lenart} and \textit{F. Sottile} [Proc. Am. Math. Soc. 131, 3319--3328 (2003; Zbl 1033.05097)]. They are given by sums over paths in the quantum Bruhat graph of type \(A\). The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes. quantum Bruhat graph; Bruhat order; quantum cohomology; Schubert classes; path Schubert polynomials Postnikov, A., Quantum Bruhat graph and Schubert polynomials. Proc. Amer. Math. Soc., 133 (2005), 699--709. Symmetric functions and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum Bruhat graph 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 a new perspective on the \(K\)-theory of exact categories via the notion of a \textit{CGW-category}. CGW-categories are a generalization of exact categories that admit a Quillen \(Q\)-construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of dévissage and localization we define \textit{ACGW-categories}, an analogous generalization of abelian categories for which we prove theorems akin to dévissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying dévissage and localization allows us to calculate a filtration on the \(K\)-theory of schemes of finite type. As an application of this theory we construct a comparison map showing that the two authors' definitions of the Grothendieck spectrum of varieties are equivalent. Algebraic K-Theory, Dévissage, Localization, Grothendieck ring of varieties, motives Dévissage and localization for the Grothendieck spectrum of 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 Given a linear differential equation in vector form \(dY/dt=AY\), with \(A\) an \(n\times n\) matrix over \(K=\mathbb{Q}(t)\), the Grothendieck-Katz conjecture [\textit{N. Katz}, Bull. Soc. Math. Fr. 110, 203--239 (1982; Zbl 0504.12022)] states that: (GK) If for almost all primes \(p\), the reduction of the equation mod \(p\) has a fundamental matrix of solutions, with coefficients from the separable closure \(\mathbb{F}_p(t)^{\text{sep}}\) of \(\mathbb{F}_p(t)\), then the original equation has a fundamental matrix of solutions with coefficients from \(\mathbb{Q}(t)^{\text{alg}}\).
Given a smooth and irreducible variety \(X\) over an algebraically closed field of any characteristic and an involutive distribution \(\mathcal{F}\) (an algebraic vector subbundle of the tangent bundle \(T(X)\) which is closed under Lie brackets), \((X,\mathcal{F})\) is said to be algebraically integrable if there is a rational map \(f: X\rightarrow Y\) to a variety \(Y\) such that \(\ker df=\mathcal{F}|_{U}\), for a nonempty Zariski open subset \(U\) of \(X\). In [\textit{T. Ekedahl, N. I. Sheperd-Barron} and \textit{R. L. Taylor}, A conjecture on the existence of compact leaves of algebraic foliations, preprint (1999)] the following conjecture (considered a kind of nonlinear version of (GK)), was stated and pointed out to imply (GK) over fields of zero characteristic: (EST) \((X,\mathcal{F})\) is algebraically integrable if and only if for almost all primes \(p\), the reduction \((X_p,\mathcal{F}_p) \mod p\) of \((X,\mathcal{F})\) has ``p-curvature'' zero.
The paper under review, influenced by \textit{Z. Chatzidakis} and \textit{E. Hrushovski} [Ill. J. Math. 47, No. 3, 593--618 (2003; Zbl 1025.03025)] proves that (EST) is equivalent to [(GK) and (D)] over a field \(K\) of characteristic zero, being: (D) If \(r(x)\) is a complete type over \(K\) and the reduction mod \(p\) has ``\(p\)-curvature \(0\)'' for almost all primes ``\(p\)'', then \(r\) is nonorthogonal to the constants. The conclusions of (EST) and (D) are ``geometric'' statements in differential algebra concerning nontrivial maps into the constants, possibly after a base change in (EST) and without a base change in (D). The gap between (EST) and (D) is explained by a ``differential Galois group''. Grothendieck-Katz conjecture; p-curvature; algebraically integrable Differential algebra, Vector bundles on curves and their moduli Differential algebra and generalizations of Grothendieck's conjecture on the arithmetic of linear differential equations | 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 Given a homology theory, one can attempt to turn an additive homology class into a natural transformation. The authors suggest the following approach to a ``categorification'' of additive invariants. Consider a cospan of categories \(S: \mathcal C_s\to \mathcal B\) and \(T: \mathcal C_t\to \mathcal B\) where \(\mathcal B\) is a category with coproducts \(\sqcup\), \((\mathcal C_s, \sqcup)\) is a strict monoidal category with \(S\) a strict monoidal functor, and \(T\) is just a functor. (Here the notation \(\mathcal C_s\) and \(\mathcal C_t\) indicate source and target, respectively.) Choose an object \(X\) in \(\mathcal C_t\). Given two triples \((V,X,h)\) and \((V', X, h')\), \(V, V'\in \mathcal C_s\), \(h\in \hom_{\mathcal B}(S(V), T(X))\), \(h'\in \hom_{\mathcal B}(S(V'), T(X))\), we define them to be isomorphic if there exists an isomorphism \(\phi: V \to V'\) such that \(h=h'\circ S(\phi)\). The set of isomorphic classes can be turned into a monoid by the operation of disjoint union. Denote the associated Grothendieck group by \(K(S,T)(X)\), this is a covariant functor with respect to \(X\).
Consider a functor \(H: \mathcal B \to Ab\) where \(Ab\) is the category of abelian groups. Let \(\alpha\) be an additive invariant with values in \(H\), i.e. \(\alpha(V)\) is an object of \(H(\mathcal C_s(V))\) where \(V \in \mathcal C_s\). Given two triples \((V, X, h)\) and \((V', X, h')\) as above, we say that an isomorphism \(\phi: V \to V'\) is an \(\alpha\)-isomorphism if \(S(\phi)_*(\alpha(V))=\alpha(V')\). Now, given a class \([(V,X,h)]_{\alpha}\) of \(\alpha\)-isomorphisms, we turn the set of these classes to a monoid (using the additivity of \(\alpha\)). The corresponding Grothendieck group is denoted by \(K_{\alpha}(S,T)(X)\).
Theorem (the categorification of an additive invariant) Let \(H:\mathcal B \to Ab\) be an additive functor on \(\mathcal B\) and \(T'=H\circ T\). Then an additive invariant \(\alpha\) as above induces a natural transformation on \(\mathcal C_t\):
\[
\tau_{\alpha}: K_{\alpha}(S,T)(-) \to T'(-), \quad \tau_{\alpha}([V,X,h]):=h_*(\alpha(V)).
\]
As an example, let \(\mathcal B\) be the category \(TOP\) of topological spaces, \(T\) the identity functor id, and \(C\) the category of smooth closed manifolds with forgetful functor \(S: C\to \mathcal B\). Then we can construct additive invariants \(\alpha\) as follows. Let \([M]\) be the fundamental class of a smooth closed manifold \(M\). Given a characteristic class \(c: \text{Vect}(-) \to H^*(-;R)\), put \( \alpha([M])=c(TM)\cap [M]\in H_*(M;R). \) This gives us a natural transformation \(\tau_{\alpha}: K_{\alpha}(S, \text{id})(-) \to H_*(-;R)\). Now the oriented bordism group \(\Omega_*^{SO}(X)\) is isomorphic to \(K_{\alpha}(S, \text{id})(X)/\sim\) where \(\sim\) is the bordism relation. Furthermore, \(\tau_{\alpha}\) passes through \(\Omega_*^{SO}\) if \(c\) is a stable characteristic class. categorification; characteristic class; cospan; homology class; bordism; Steenrod's problem Generalized (extraordinary) homology and cohomology theories in algebraic topology, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, General theory of categories and functors Grothendieck groups and a categorification of additive invariants | 0 |
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