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## Stokes matrices for the quantum cohomologies of Grassmannians, Int

Venue: | Math. Res. Not |

Citations: | 8 - 1 self |

### Citations

748 |
Symmetric Functions and
- Macdonald
- 1995
(Show Context)
Citation Context ... sλ(x) = det(hλi−i+j(x))1≤i,j≤n be the Shur function, where hi(x) is the complete symmetric function (the sum of all monomials of degree i). For generalities on symmetric functions, see, for example, =-=[10]-=-. Define integers cλµν’s by sµ(x)sν(x) = ∑ λ cλµνsλ(x) (4.2) and the skew Shur function sλ/µ(x) by sλ/µ(x) = ∑ ν cλµνsν(x). (4.3) Then sλ/µ(x) = det ( hλi−µj−i+j(x) ) 1≤i,j≤n. (4.4) Lemma 4.1. Let µ, ... |

338 |
Geometry of 2-D topological field theories, in: Integrable Systems and Quantum Groups (Montecatini Terme
- Dubrovin
- 1993
(Show Context)
Citation Context ...} N−1 α=0 is the flat coordinate of the Frobenius manifold. The circle denotes the product on the tangent bundle and U, V are certain operators acting on sections of the tangent bundles. See Dubrovin =-=[3]-=- for details. Note that z loc. cit. is 1/� in this paper. (1) is an ordinary differential equation on P1 with a regular singularity at infinity and an irregular singularity at the origin, and (2) give... |

335 |
Symmetric functions and Hall polynomials. Oxford Mathematical Monographs
- Macdonald
- 1995
(Show Context)
Citation Context ...n), let sλ(x) = det(hλi−i+j(x))1≤i,j≤n be the Shur function, where hi(x) is the complete symmetric function (the sum of all monomials of degree i). For generalities on symmetric functions, see, e.g., =-=[10]-=-. Define integers cλ µν’s by sµ(x)sν(x) = ∑ c λ µνsλ(x) 9 λand the skew Shur function sλ/µ(x) by sλ/µ(x) = ∑ c λ µνsν(x). Then sλ/µ(x) = det(hλi−µj−i+j(x))1≤i,j≤n. Lemma 4.1. Let µ, ν and λ be partit... |

271 | D-branes and mirror symmetry
- Hori, Iqbal, et al.
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Citation Context ...spaces in C n . The proof consists of explicit computations on both sides of (3). The computation on the left hand side relies on the following two results: The first is a conjecture of Hori and Vafa =-=[8]-=-, proved by Bertram, Ciocan-Fontanine and Kim [2], describing the solution of (1) for the Grassmannian Gr(r, n) in terms of that of the product of projective spaces (P n−1 ) r . The second is the Stok... |

159 |
Coherent sheaves on Pn and problems in linear algebra (Russian) Funktsional. Anal. i Prilozhen
- Beilinson
- 1978
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Citation Context ...Sij = ( n − 1 + j − i j − i ) (2.10) up to the braid group action. Here, ( nr ) is the binomial coefficient. Since (OPn−1(i))n−1i=0 is an exceptional collection generating D b cohPn−1 by Beı̆linson =-=[1]-=- and ( n − 1 + j − i j − i ) = ∑ k (−1)k dimExtk ( OPn−1(i),OPn−1(j) ) , (2.11) Conjecture 1.1 holds for projective spaces.sat Pennsylvania State U niversity on Septem ber 16, 2016 http://im rn.oxford... |

127 | Equivariant Gromov Witten invariants,
- Givental
- 1996
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Citation Context ...nifold). In the case of the projective space P n−1 , semisimplicity of the quantum cohomology is well-known. The solution to (1), (2) has an integral representation by Givental: Theorem 2.2 (Givental =-=[6]-=-). Let W(x1, . . .,xn−1) = x1 + · · · + xn−1 + x1 · · ·xn−1 be a function on (C×) n−1 depending on a parameter t ∈ C and choose a basis {Γi} n i=1 of the space of flat sections of the relative homolog... |

119 |
Coherent sheaves on P n and problems in linear algebra
- Beilinson
- 1978
(Show Context)
Citation Context ... given by ( ) n − 1 + j − i Sij = j − i up to the braid group action. Here, ( ) n is the binomial coefficient. r Since (OPn−1(i))n−1 i=0 is an exceptional collection generating DbcohPn−1 by Beilinson =-=[1]-=- and ( ) n − 1 + j − i = j − i ∑ (−1) k dim Ext k (OPn−1(i), OPn−1(j)), the Conjecture 1.1 holds for projective spaces. k 6Now let us move on to the Grassmannian case. Let Gr(r, n) be the Grassmannia... |

102 |
On the derived category of coherent sheaves on some homogeneous spaces,
- Kapranov
- 1988
(Show Context)
Citation Context ... these two results, we can compute the Stokes matrix for the quantum cohomology of the Grassmannian. On the right hand side, we have an exceptional collection generating D b coh(Gr(r, n)) by Kapranov =-=[9]-=-. It consists of equivariant vector bundles on Gr(r, n) and Ext-groups between them can be computed by the Borel-Weil theory. Both of the above computations can be carried out for any r and n, and Con... |

75 | Painlevé transcendents in two-dimensional topological field theory
- Dubrovin
(Show Context)
Citation Context ...egory is an exceptional collection if each Ei is exceptional and Ext k (Ei, Ej) = 0 for any i > j and any k. 2To our knowledge, Conjecture 1.1 was previously known to hold only for projective spaces =-=[5]-=-, [7]. The main result in this paper is: Theorem 1.3. Conjecture 1.1 holds for the Grassmannian Gr(r, n) of rdimensional subspaces in C n . The proof consists of explicit computations on both sides of... |

55 | Geometry and analytic theory of Frobenius manifolds”, Talk at ICM’98
- Dubrovin
(Show Context)
Citation Context ... the choice of a semisimple point because of the isomonodromicity. The following conjecture, originally due to Kontsevich, developed by Zaslow [12], and formulated into the following form by Dubrovin =-=[4]-=-, reveals a striking connection between the Gromov-Witten invariants and the derived category of coherent sheaves: Conjecture 1.1. The quantum cohomology of a smooth projective variety X is semisimple... |

25 | Stokes matrices and monodromy of the quantum cohomology of projective spaces
- Guzzetti
- 1999
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Citation Context ... is an exceptional collection if each Ei is exceptional and Ext k (Ei, Ej) = 0 for any i > j and any k. 2To our knowledge, Conjecture 1.1 was previously known to hold only for projective spaces [5], =-=[7]-=-. The main result in this paper is: Theorem 1.3. Conjecture 1.1 holds for the Grassmannian Gr(r, n) of rdimensional subspaces in C n . The proof consists of explicit computations on both sides of (3).... |

17 | Solitons and helices: the search for a math-physics bridge
- Zaslow
- 1996
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Citation Context ...ween infinity and the origin. These data do not depend on the choice of a semisimple point because of the isomonodromicity. The following conjecture, originally due to Kontsevich, developed by Zaslow =-=[12]-=-, and formulated into the following form by Dubrovin [4], reveals a striking connection between the Gromov-Witten invariants and the derived category of coherent sheaves. Conjecture 1.1. The quantum c... |

12 |
transcendents in two-dimensional topological field theory, The Painlevé property
- Painlevé
- 1999
(Show Context)
Citation Context ...d category is an exceptional collection if each Ei is exceptional and Extk(Ei,Ej) = 0 for any i > j and any k. To our knowledge, Conjecture 1.1 was previously known to hold only for projective spaces =-=[5, 7]-=-. The main result in this paper is as follows.sat Pennsylvania State U niversity on Septem ber 16, 2016 http://im rn.oxfordjournals.org/ D ow nloaded from Stokes Matrices for Grassmannians 2077 Theore... |

6 | Two proofs of a conjecture of Hori and Vafa
- Bertram, Ciocan-Fontanine, et al.
(Show Context)
Citation Context ...mputations on both sides of (3). The computation on the left hand side relies on the following two results: The first is a conjecture of Hori and Vafa [8], proved by Bertram, Ciocan-Fontanine and Kim =-=[2]-=-, describing the solution of (1) for the Grassmannian Gr(r, n) in terms of that of the product of projective spaces (P n−1 ) r . The second is the Stokes matrix for the quantum cohomology of projectiv... |

5 |
Invariant of the hypergeometric group associated to the quantum cohomology of the projective
- Tanabé
(Show Context)
Citation Context ...e Stokes matrix is given by Γi,right = n∑ j=1 Γj,leftSji. The Stokes matrix for the quantum cohomology of P n−1 has been computed by Dubrovin [5] for n ≤ 3 and by Guzzetti [7] for general n. See also =-=[11]-=-. Theorem 2.3 (Dubrovin, Guzzetti). The Stokes matrix S for the quantum cohomology of the projective space Pn−1 is given by ( ) n − 1 + j − i Sij = j − i up to the braid group action. Here, ( ) n is t... |

2 |
Stokesmatrices andmonodromy of the quantumcohomology of projective spaces
- Guzzetti
- 1999
(Show Context)
Citation Context ...d category is an exceptional collection if each Ei is exceptional and Extk(Ei,Ej) = 0 for any i > j and any k. To our knowledge, Conjecture 1.1 was previously known to hold only for projective spaces =-=[5, 7]-=-. The main result in this paper is as follows.sat Pennsylvania State U niversity on Septem ber 16, 2016 http://im rn.oxfordjournals.org/ D ow nloaded from Stokes Matrices for Grassmannians 2077 Theore... |

1 |
Two proofs of a conjecture of Hori and Vafa,Duke
- Bertram, Ciocan-Fontanine, et al.
(Show Context)
Citation Context ...utations on both sides of (1.3). The computation on the left-hand side relies on the following two results. The first is a conjecture of Hori and Vafa [8], proved by Bertram, Ciocan-Fontanine and Kim =-=[2]-=-, describing the solution of (1.1) for the Grassmannian Gr(r, n) in terms of that of the product of projective spaces (Pn−1)r. The second is the Stokes matrix for the quantum cohomology of projective ... |