• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.855-865
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• 2012

Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1125-1142
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• 2011

In this paper, we define some new sequence spaces of infinite matrices regarded as operators on $l_2$ by using algebraic properties of such the matrices under the Schur product multiplication. Some of their basic properties as well as duality and preduality are discussed.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.1-28
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• 2003

The Iwahori-Hecke algebra $H_{k}$ ( $q^2$) of type A acts on the k-fold tensor product space of the natural representation of the quantum superalgebra (equation omitted)$_{q}$(gl(m, n)). We show the Hecke algebra $H_{k}$ ( $q^2$) and the quantum superalgebra (equation omitted)$_{q}$(gl(m n)) have commuting actions on the tensor product space, and determine the centralizer of each other. Using this result together with Gyoja's q-analogue of the Young symmetrizers, we construct highest weight vectors of irreducible summands of the tensor product space.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1003-1021
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• 2015

A matrix completion problem has been exploited amply because of its abundant applications and the analysis of contractions enables us to have insight into structure and space of operators. In this article, we focus on a specific completion problem related to Hankel partial contractions. We provide concrete necessary and sufficient conditions for the existence of completion of Hankel partial contractions for both extremal and non-extremal types with lower dimensional matrices. Moreover, we give a negative answer for the conjecture presented in [8]. For our results, we use several tools such as the Nested Determinants Test (or Choleski's Algorithm), the Moore-Penrose inverse, the Schur product techniques, and a congruence of two positive semi-definite matrices; all these suggest an algorithmic approach to solve the contractive completion problem for general Hankel matrices of size $n{\times}n$ in both types.

• Journal of the Korea Society of Computer and Information
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• v.11 no.3
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• pp.141-149
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• 2006

In this paper, we introduce an improved algorithm for computing matrix triple product that commonly arises in primal-dual optimization method. In computing $P=AHA^{t}$, we devise a single pass algorithm that exploits the block diagonal structure of the matrix H. This one-phase scheme requires fewer floating point operations and roughly half the memory of the generic two-phase algorithm, where the product is computed in two steps, computing first $Q=HA^{t}$ and then P=AQ. The one-phase scheme achieved speed-up of 2.04 on Intel Itanium II platform over the two-phase scheme. Based on memory latency and modeled cache miss rates, the performance improvement was evaluated through performance modeling. Our research has impact on performance tuning study of complex sparse matrix operations, while most of the previous work focused on performance tuning of basic operations.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.679-710
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• 2021

We study the tensor product algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m), where P_k(x) is the partition algebra defined by Jones and Martin. We discuss the centralizer of this algebra and corresponding Schur-Weyl dualities and also index the inequivalent irreducible representations of the algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m) and compute their dimensions in the semisimple case. In addition, we describe the Bratteli diagrams and branching rules. Along with that, we have also constructed the RS correspondence for the tensor product of m-partition algebras which gives the bijection between the set of tensor product of m-partition diagram of P_k(n₁) ⊗ P_k(n₂) ⊗ ⋯ ⊗ P_k(n_m) and the pairs of m-vacillating tableaux of shape [λ] ∈ Γ_k^m, Γ_k^m = {[λ] = (λ₁, λ₂, …, λ_m)|λ_i ∈ Γ_k, i ∈ {1, 2, …, m}} where Γ_k = {λ_i ⊢ t|0 ≤ t ≤ k}. Also, we provide proof of the identity $(n_1n_2{\cdots}n_m)^k={\sum}_{[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ f^[λ]m_k^[λ] where m_k^[λ] is the multiplicity of the irreducible representation of $S{_{n_1}}{\times}S{_{n_2}}{\times}....{\times}S{_{n_m}}$ module indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$, where f^[λ] is the degree of the corresponding representation indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ and ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}=\{[{\lambda}]=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_m){\mid}{\lambda}_i{\in}{\Lambda}^k_{n_i},i{\in}\{1,2,{\ldots},m\}\}$ where ${\Lambda}^k_{n_i}=\{{\mu}=({\mu}_1,{\mu}_2,{\ldots},{\mu}_t){\vdash}n_i{\mid}n_i-{\mu}_1{\leq}k\}$.