TY - JOUR

T1 - Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

AU - Ikeda, Takeshi

N1 - Funding Information:
Firstly, I would like to thank H. Naruse for his keen interest in the present work and plenty of helpful comments. I also thank K. Takasaki, Y. Kodama, T. Tanisaki, and M. Kaneda for valuable discussions. During the preparation of this paper I also benefited from conversations with M. Ishikawa, S. Kakei, M. Katori, and T. Maeno. I also thank H. Nakajima and the referee who encouraged me to prove a Giambelli-type formula (6.4) which was not included in the first version of this paper. Lastly, but not least, I am grateful to H.-F. Yamada for showing me the importance of Schur’s Q-functions. This research was partially supported by Grant-in-Aids for Young Scientists (B) (No. 17740101) from Japan Society of the Promotion of Science.

PY - 2007/10/20

Y1 - 2007/10/20

N2 - Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ = (λ1, ..., λk) with λ1 ≤ n there is a Schubert variety X (λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T-equivariant cohomology of LGn and the T-equivariant fundamental class σ (λ) of X (λ). The main result of the present paper is an explicit formula for the restriction of the class σ (λ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LGn. As another consequence of the main result, we obtained a presentation of the ring HT* (LGn).

AB - Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ = (λ1, ..., λk) with λ1 ≤ n there is a Schubert variety X (λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T-equivariant cohomology of LGn and the T-equivariant fundamental class σ (λ) of X (λ). The main result of the present paper is an explicit formula for the restriction of the class σ (λ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LGn. As another consequence of the main result, we obtained a presentation of the ring HT* (LGn).

KW - Equivariant cohomology

KW - Factorial Q-functions

KW - Lagrangian Grassmannian

KW - Schubert classes

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U2 - 10.1016/j.aim.2007.04.008

DO - 10.1016/j.aim.2007.04.008

M3 - Article

AN - SCOPUS:34447511175

SN - 0001-8708

VL - 215

SP - 1

EP - 23

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -