• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.933-943
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• 2020

Let M be a Fano manifold, and H^🟉(M; ℂ) be the quantum cohomology ring of M with the quantum product 🟉. For 𝜎 ∈ H^🟉(M; ℂ), denote by [𝜎] the quantum multiplication operator 𝜎🟉 on H^🟉(M; ℂ). It was conjectured several years ago [7,8] and has been proved for many Fano manifolds [1,2,10,14], including our cases, that the operator [c₁(M)] has a real valued eigenvalue 𝛿₀ which is maximal among eigenvalues of [c₁(M)]. Galkin's lower bound conjecture [6] states that for a Fano manifold M, 𝛿₀ ≥ dim M + 1, and the equality holds if and only if M is the projective space ℙⁿ. In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.525-535
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• 2017

In this paper, we consider two kinds of derivatives for the shape operator of a real hypersurface in a $K{\ddot{a}}hler$ manifold which are named the Lie derivative and the covariant derivative with respect to the k-th generalized Tanaka-Webster connection ${\hat{\nabla}}^{(k)}$. The purpose of this paper is to study Hopf hypersurfaces in complex Grassmannians of rank two, whose Lie derivative of the shape operator coincides with the covariant derivative of it with respect to ${\hat{\nabla}}^{(k)}$ either in direction of any vector field or in direction of Reeb vector field.

• Honam Mathematical Journal
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• v.32 no.2
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• pp.271-281
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• 2010

Inspired by the reduction formulae between intersection numbers on Grassmannians obtained by Griffiths-Harris and the factorization theorem of Littlewood-Richardson coefficients by King, Tollu and Toumazet, eight reduction formulae has been discovered by the author and others. In this paper, we prove r = 1 reduction formula by constructing a bijective map between suitable sets of Littlewood-Richardson tableaux.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.257-266
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• 2023

Let G_k,n(ℍ) for 2 ≤ k < n denote the Grassmann manifold of k-dimensional vector subspaces of ℍⁿ. In this paper, we determine the evaluation subgroups of the Plüker embedding $G{_{2,n}}({\mathbb{H}}){\hookrightarrow}{\mathbb{H}}P^{N-1}$, where $N\;=\;({n \atop 2}\)$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.423-434
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• 2012

In this paper we give a characterization of real hypersurfaces of type (A) in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which is a tube over a totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$, in terms of two commuting conditions related to the normal Jacobi operator and the shape operator.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.485-494
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• 2008

There are two well known reduction formulae for structural constants of the cohomology ring of Grassmannians, i.e., Littlewood-Richardson coefficients. Two reduction formulae are a conjugate pair in the sense that indexing partitions of one formula are conjugate to those of the other formula. A nice bijective proof of the first reduction formula is given in the authors' previous paper while a (combinatorial) proof for the second reduction formula in the paper depends on the identity between Littlewood-Richardson coefficients of conjugate shape. In this article, a direct bijective proof for the second reduction formula for Littlewood-Richardson coefficients is given. Our proof is independent of any previously known results (or bijections) on tableaux theory and supplements the arguments on bijective proofs of reduction formulae in the authors' previous paper.