scholarly journals Schur Times Schubert via the Fomin-Kirillov Algebra

10.37236/3659 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Karola Mészáros ◽  
Greta Panova ◽  
Alexander Postnikov
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Flag Manifold ◽  
Combinatorial Proof ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
Small Quantum ◽  
Special Cases ◽  
Expansion Coefficients ◽  
Complex Flag

We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_{\lambda}$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $\lambda$, including hooks and the $2\times 2$ box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box at the $(2,2)$ corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.

scholarly journals A Murgnahan-Nakayama rule for Schubert polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Andrew Morrison
Keyword(s):  
Quantum Cohomology ◽  
Chern Character ◽  
Intersection Theory ◽  
Flag Manifolds ◽  
Nous Obtenons ◽  
Schubert Polynomial ◽  
Power Sum ◽  
Schubert Polynomials ◽  
Small Quantum ◽  
International Audience

International audience We expose a rule for multiplying a general Schubert polynomial with a power sum polynomial in $k$ variables. A signed sum over cyclic permutations replaces the signed sum over rim hooks in the classical Murgnahan-Nakayama rule. In the intersection theory of flag manifolds this computes all intersections of Schubert cycles with tautological classes coming from the Chern character. We also discuss extensions of this rule to small quantum cohomology. Nous écrivons une formule pour multiplier les polynômes de Schubert avec les sommes de Newton. Une somme signée de permutations cycliques remplace la somme signée de rubans dans la formule classique de Murgnahan-Nakayama. Nous obtenons donc des relations dans l’anneau de Chow de la variété de drapeaux. Nous discutons également des extensions de cette formule en cohomologie quantique.

scholarly journals A Characterization of the Quantum Cohomology Ring of G/B and Applications

2008 ◽  
Vol 60 (4) ◽  
pp. 875-891
Author(s):  
Augustin-Liviu Mare
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Main Idea ◽  
Flag Manifold ◽  
Schubert Calculus ◽  
Generators And Relations ◽  
Cup Product ◽  
Generalized Flag Manifold ◽  
Standard Monomials ◽  
Small Quantum

AbstractWe observe that the small quantum product of the generalized flag manifold G/B is a product operation ★ on H*(G/B) ⊗ ℝ[q1, . . . , ql] uniquely determined by the facts that it is a deformation of the cup product on H*(G/B); it is commutative, associative, and graded with respect to deg(qi ) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring (H*(G/B)⊗ℝ[q1, . . . , ql], ★) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for G/B: the quantumChevalley formula of D. Peterson (see also Fulton andWoodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for G = SL(n, ℂ). The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum -module of G/B one can decode all information about the quantum cohomology of this space.

scholarly journals Residual categories for (co)adjoint Grassmannians in classical types

2021 ◽  
Vol 157 (6) ◽  
pp. 1172-1206
Author(s):  
Alexander Kuznetsov ◽  
Maxim Smirnov
Keyword(s):  
Automorphism Group ◽  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Algebraic Groups ◽  
Fano Variety ◽  
Flag Varieties ◽  
Picard Number ◽  
Coherent Sheaves ◽  
Small Quantum ◽  
Exceptional Collection

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$ , that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$ ; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

scholarly journals THREE-POINT FUNCTIONS ON THE SPHERE OF CALABI-YAU d-FOLDS

1996 ◽  
Vol 11 (02) ◽  
pp. 229-252 ◽  
Author(s):  
KATSUYUKI SUGIYAMA
Keyword(s):  
Mirror Symmetry ◽  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Chern Classes ◽  
World Sheet ◽  
The World ◽  
Expansion Coefficients ◽  
Topological Theories ◽  
Homology Cycle ◽  
Two Parameter

Using mirror symmetry in Calabi-Yau manifolds M, we study three-point functions of A(M) model operators on the genus 0 Riemann surface in cases of one-parameter families of d-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of d-folds embedded in a weighted projective space Pd+1 [2,2,2,...,2,2,1,1] (2 (d + 1)). These three-point functions [Formula: see text] are expanded by indeterminates [Formula: see text] associated with a set of Kähler coordinates {tl}, and their expansion coefficients count the number of maps with a definite degree which map each of the three-points 0, 1 and ∞ on the world sheet on some homology cycle of M associated with a cohomology element. From these analyses, we can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with Kähler forms of M. For that reason, the charge conservation of operators turns out to be a classical one. Furthermore, because their first Chern classes c1 vanish, their topological selection rules do not depend on the degree of maps (in particular, a nilpotent property of operators [Formula: see text] is satisfied). Then these fusion couplings {κl} are represented as some series adding up all degrees of maps.

scholarly journals ON THE QUANTUM COHOMOLOGY OF SOME FANO THREEFOLDS AND A CONJECTURE OF DUBROVIN

2005 ◽  
Vol 16 (08) ◽  
pp. 823-839 ◽  
Author(s):  
GIANNI CIOLLI
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Exceptional Set ◽  
Fano Threefolds ◽  
Cohomology Rings ◽  
Small Quantum ◽  
Fano Threefold

In the present paper the small quantum cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from ℙ3or the quadric Q3is explicitely computed. Because of systematic usage of the associativity property of quantum product only a very small and enumerative subset of Gromov–Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of quantum cohomology is proven by checking the computed quantum cohomology rings and by showing that a smooth Fano threefold X with b3(X) = 0 admits a complete exceptional set of the appropriate length.

scholarly journals The small quantum cohomology of the Cayley Grassmannian

2020 ◽  
Vol 31 (03) ◽  
pp. 2050019
Author(s):  
Vladimiro Benedetti ◽  
Laurent Manivel
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Derived Category ◽  
Multiplication Table ◽  
Maximal Length ◽  
Small Quantum ◽  
Exceptional Collection ◽  
Schubert Classes

We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov–Witten invariants in the multiplication table of the Schubert classes are nonnegative and deduce Golyshev’s conjecture [Formula: see text] holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin’s conjecture, an exceptional collection of maximal length in the derived category.

scholarly journals Augmented Rook Boards and General Product Formulas

10.37236/809 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Brian K. Miceli ◽  
Jeffrey Remmel
Keyword(s):  
Product Formula ◽  
Combinatorial Proof ◽  
Rook Theory ◽  
Special Cases ◽  
Ferrers Board ◽  
General Product ◽  
Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

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