The Experts below are selected from a list of 30 Experts worldwide ranked by ideXlab platform
Daisuke Sagaki - One of the best experts on this subject based on the ideXlab platform.
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LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞
Transformation Groups, 2020Co-Authors: Satoshi Naito, Daisuke SagakiAbstract:Let λ ∈ P ^+ be a level-zero dominant integral weight, and w the Coset Representative of minimal length for a Coset in W / W _λ, where W _λ is the stabilizer of λ in a finite Weyl group W . In this paper, we give a module K w − λ $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial E _wλ( q, t ) multiplied by a certain explicit finite product of rational functions of q of the form (1 − q ^− r )^−1 for a positive integer r . This module K w − λ $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module V w − λ $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules V z − λ $$ {V}_z^{-}\left(\uplambda \right) $$ for all those Coset Representatives z of minimal length for Cosets in W/W _λ such that z > w in the Bruhat order < on W .
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Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t = \infty$
arXiv: Quantum Algebra, 2018Co-Authors: Satoshi Naito, Daisuke SagakiAbstract:Let $\lambda \in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary Coset Representative of minimal length for the Cosets in $W/W_{\lambda}$, where $W_{\lambda}$ is the stabilizer of $\lambda$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(\lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mathbb{K}_{w}(\lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(\lambda)$ by the sum of the submodules $V_{z}^{-}(\lambda)$ for all those Coset Representatives $z$ of minimal length for $W/W_{\lambda}$ such that $z > w$ in the Bruhat order $
Sundar B. Rajan - One of the best experts on this subject based on the ideXlab platform.
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Block-coded modulation using two-level group codes over generalized quaternion groups
IEEE Transactions on Information Theory, 1999Co-Authors: T.v. Selvakumaran, Sundar B. RajanAbstract:A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2/sup m/), m/spl ges/3, are constructed using a binary code and a code over Z(2/sup m-1/), the ring of integers modulo 2/sup m-1/ as component codes and a mapping f from Z/sub 2//spl times/Z(2/sup m-1/)to Q(2/sup m/). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2/sup m/). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2/sup m/), it is shown that the Euclidean space codes obtained from the group codes over Q(2/sup m/) have Euclidean distance profiles which are independent of the Coset Representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2/sup m/) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2/sup m/) are automorphic Euclidean-distance equivalent.
Satoshi Naito - One of the best experts on this subject based on the ideXlab platform.
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LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞
Transformation Groups, 2020Co-Authors: Satoshi Naito, Daisuke SagakiAbstract:Let λ ∈ P ^+ be a level-zero dominant integral weight, and w the Coset Representative of minimal length for a Coset in W / W _λ, where W _λ is the stabilizer of λ in a finite Weyl group W . In this paper, we give a module K w − λ $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial E _wλ( q, t ) multiplied by a certain explicit finite product of rational functions of q of the form (1 − q ^− r )^−1 for a positive integer r . This module K w − λ $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module V w − λ $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules V z − λ $$ {V}_z^{-}\left(\uplambda \right) $$ for all those Coset Representatives z of minimal length for Cosets in W/W _λ such that z > w in the Bruhat order < on W .
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Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t = \infty$
arXiv: Quantum Algebra, 2018Co-Authors: Satoshi Naito, Daisuke SagakiAbstract:Let $\lambda \in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary Coset Representative of minimal length for the Cosets in $W/W_{\lambda}$, where $W_{\lambda}$ is the stabilizer of $\lambda$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(\lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mathbb{K}_{w}(\lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(\lambda)$ by the sum of the submodules $V_{z}^{-}(\lambda)$ for all those Coset Representatives $z$ of minimal length for $W/W_{\lambda}$ such that $z > w$ in the Bruhat order $
T.v. Selvakumaran - One of the best experts on this subject based on the ideXlab platform.
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Block-coded modulation using two-level group codes over generalized quaternion groups
IEEE Transactions on Information Theory, 1999Co-Authors: T.v. Selvakumaran, Sundar B. RajanAbstract:A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2/sup m/), m/spl ges/3, are constructed using a binary code and a code over Z(2/sup m-1/), the ring of integers modulo 2/sup m-1/ as component codes and a mapping f from Z/sub 2//spl times/Z(2/sup m-1/)to Q(2/sup m/). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2/sup m/). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2/sup m/), it is shown that the Euclidean space codes obtained from the group codes over Q(2/sup m/) have Euclidean distance profiles which are independent of the Coset Representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2/sup m/) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2/sup m/) are automorphic Euclidean-distance equivalent.
E J Beggs - One of the best experts on this subject based on the ideXlab platform.
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further results on Coset Representative categories
arXiv: Quantum Algebra, 2007Co-Authors: M M Alshomrani, E J BeggsAbstract:This paper is devoted to further results on the nontrivially associated categories C and D, which are constructed from a choice of Coset Representatives for a subgroup of a finite group. We look at the construction of integrals in the algebras A and D in the categories. These integrals are used to construct abstract projection operators to show that general objects in D can be split into a sum of simple objects. The braided Hopf algebra D is shown to be braided cocommutative, but not braided commutative. Extensions of the categories and their connections with conjugations and inner products are discussed.
