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Daniel S. Sage - One of the best experts on this subject based on the ideXlab platform.
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c © 2005 Heldermann Verlag Quantum Racah coefficients and Subrepresentation semirings
2014Co-Authors: Daniel S. Sage, Communicated K. SchmüdgenAbstract:Abstract. Let G be a group and A a G-algebra. The Subrepresentation semiring of A is the set of Subrepresentations of A endowed with operations induced by the algebra operations. The introduction of these semirings was motivated by a problem in material science. Typically, physical properties of composite materials are strongly dependent on microstructure. However, in ex-ceptional situations, exact relations exist which are microstructure-independent. Grabovsky has constructed an abstract theory of exact relations, reducing the search for exact relations to a purely algebraic problem involving the product of SU(2)-Subrepresentations in certain endomorphism algebras. We have shown that the structure of the associated semirings can be described explicitly in terms of Racah coefficients. In this paper, we prove an analogous relationship between Racah coefficients for the quantum algebra Ŭq(sl2) and semirings for endomor-phism algebras of representations of Ŭq(sl2). We generalize the construction of Subrepresentation semirings to the Hopf algebra setting. For Ŭq(sl2), we compute these semirings for the endomorphism algebra of an arbitrary complex finite-dimensional representation. When the representation is irreducible, we show that the Subrepresentation semiring can be described explicitly in terms of the vanishing of q-Racah coefficients. We further show that q-Racah coefficients can be defined entirely in terms of the multiplication of Subrepresentations. 1
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QUANTUM RACAH COEFFICIENTS AND Subrepresentation SEMIRINGS
2010Co-Authors: Daniel S. SageAbstract:Abstract. Let G be a group and A a G-algebra. The Subrepresentation semiring of A is the set of Subrepresentations of A endowed with operations induced by the algebra operations. The introduction of these semirings was motivated by a problem in material science. Typically, physical properties of composite materials are strongly dependent on microstructure. However, in exceptional situations, exact relations exist which are microstructureindependent. Grabovsky has constructed an abstract theory of exact relations, reducing the search for exact relations to a purely algebraic problem involving the product of SU(2)-Subrepresentations in certain endomorphism algebras. We have shown that the structure of the associated semirings can be described explicitly in terms of Racah coefficients. In this paper, we prove an analogous relationship between Racah coefficients for the quantum algebra Ŭq(sl2) and semirings for endomorphism algebras of representations of Ŭq(sl2). We generalize the construction of Subrepresentation semirings to the Hopf algebra setting. For Ŭq(sl2), we compute these semirings for the endomorphism algebra of an arbitrary complex finite-dimensional representation. When the representation is irreducible, we show that the Subrepresentation semiring can be described explicitly in terms of the vanishing of q-Racah coefficients. We further show that q-Racah coefficients can be defined entirely in terms of the multiplication of Subrepresentations. 1
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Subrepresentation semirings and an analog of 6j-symbols
Journal of Mathematical Physics, 2008Co-Authors: Namhee Kwon, Daniel S. SageAbstract:Let V be a complex representation of the compact group G. The Subrepresentation semiring associated to V is the set of Subrepresentations of the algebra of linear endomorphisms of V with operations induced by the matrix operations. The study of these semirings has been motivated by recent advances in materials science, in which the search for microstructure-independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group SO(3). In this case, the structure constants for Subrepresentation semirings can be described explicitly in terms of the 6j-symbols familiar from the quantum theory of angular momentum. In this paper, we investigate Subrepresentation semirings for the class of quasisimply reducible groups defined by Mackey [“Multiplicity free representations of finite groups,” Pac. J. Math. 8, 503 (1958)]. We introduce a new class of symbols called twisted 6j-symbols for these groups, and we explicitly calculate the structure constan...
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Racah coefficients, Subrepresentation semirings, and composite materials
Advances in Applied Mathematics, 2005Co-Authors: Daniel S. SageAbstract:Typically, physical properties of composite materials are strongly dependent on microstructure. However, in exceptional situations, exact relations exist which are microstructure-independent. Grabovsky has constructed an abstract theory of exact relations, reducing the search for exact relations to a purely algebraic problem involving the multiplication of SO(3)-Subrepresentations in certain endomorphism algebras. This motivates us to introduce Subrepresentation semirings, algebraic structures which formalize Subrepresentation multiplication. We study the ideals and subsemirings of these semirings, relating them to properties of the underlying G-algebra and proving classification theorems in the case of endomorphism algebras of representations. For SU(2), we compute these semirings for general V. When V is irreducible, we describe the semiring structure explicitly in terms of the vanishing of Racah coefficients, coefficients familiar from the quantum theory of angular momentum. In fact, we show that Racah coefficients can be defined entirely in terms of Subrepresentation multiplication.
Claus Günther Schmidt - One of the best experts on this subject based on the ideXlab platform.
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The critical numbers of Rankin–Selberg convolutions of cohomological representations
Journal of Number Theory, 2017Co-Authors: Claus Günther SchmidtAbstract:Abstract We study the critical numbers of the Rankin–Selberg convolution of arbitrary pairs of cohomological cuspidal automorphic representations and we parametrize these critical numbers by certain 1-dimensional Subrepresentations attached to the corresponding pair of finite dimensional representations of the related general linear groups.
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The Critical Numbers of Rankin-Selberg Convolutions of Cohomological Representations
arXiv: Number Theory, 2016Co-Authors: Claus Günther SchmidtAbstract:We study the critical numbers of the Rankin-Selberg convolution of arbitrary pairs of cohomological cuspidal automorphic representations and we parametrize these critical numbers by certain 1-dimensional Subrepresentations attached to the corresponding pair of finite dimensional representations of the related general linear groups.
Wang Jian-yong - One of the best experts on this subject based on the ideXlab platform.
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The Subrepresentation Theorem of(l~(β_1)_(β_2)~* and the Non-local β_2-convexity of l~(β_1)(0
Advances in Mathematics, 2011Co-Authors: Wang Jian-yongAbstract:In this paper,we obtain the Subrepresentation theorem of theβ_2-conjugatecone of l~β_1 for 0β_1β_2≤1,and prove the non-localβ_2-convexity of l~β_1.
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The Subrepresentation Theorem of the Conjugate Cone of l~p(X)(0
Advances in Mathematics, 2010Co-Authors: Wang Jian-yongAbstract:For a Banach space X,it is well known that the dual of l~1(X) can be represented as l~∞(X~*).l~p(X) is not locally convex if 0p1,but it is locally p-convex,its conjugate cone[l~p(X)]_p~* is large enough to separate its points.This paper explores the representation problem of the conjugate cone of l~p(X)(0p1),and obtains the Subrepresentation Theorem [l~o(X)]_p~*■l~∞(X_p~*) for every Banach space X.When X=R or C,the Subrepresentation theorem has the simplified version[l~p(R)]_p~*■m+×m~+and[l~p(C)]_p~*■mM_p~+(T) respectively.
Wang Jianyong - One of the best experts on this subject based on the ideXlab platform.
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the Subrepresentation theorem of l β_1 _ β_2 and the non local β_2 convexity of l β_1 0 β_1 β_2 1
Advances in Mathematics, 2011Co-Authors: Wang JianyongAbstract:In this paper,we obtain the Subrepresentation theorem of theβ_2-conjugatecone of l~β_1 for 0β_1β_2≤1,and prove the non-localβ_2-convexity of l~β_1.
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the Subrepresentation theorem of the conjugate cone of l p x 0 p 1
Advances in Mathematics, 2010Co-Authors: Wang JianyongAbstract:For a Banach space X,it is well known that the dual of l~1(X) can be represented as l~∞(X~*).l~p(X) is not locally convex if 0p1,but it is locally p-convex,its conjugate cone[l~p(X)]_p~* is large enough to separate its points.This paper explores the representation problem of the conjugate cone of l~p(X)(0p1),and obtains the Subrepresentation Theorem [l~o(X)]_p~*■l~∞(X_p~*) for every Banach space X.When X=R or C,the Subrepresentation theorem has the simplified version[l~p(R)]_p~*■m+×m~+and[l~p(C)]_p~*■mM_p~+(T) respectively.
Avner Segal - One of the best experts on this subject based on the ideXlab platform.
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The degenerate principal series representations of exceptional groups of type E_6 over p-adic fields
Israel Journal of Mathematics, 2020Co-Authors: Hezi Halawi, Avner SegalAbstract:In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type E _6. Furthermore, we calculate the maximal semi-simple Subrepresentation and quotient of these representations.
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The degenerate principal series representations of exceptional E_6 over p-adic fields
Israel Journal of Mathematics, 2020Co-Authors: Hezi Halawi, Avner SegalAbstract:In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type E _6. Furthermore, we calculate the maximal semi-simple Subrepresentation and quotient of these representations.
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The Degenerate Principal Series Representations of Exceptional Groups of Type $E_7$ over $p$-adic Fields.
arXiv: Representation Theory, 2019Co-Authors: Hezi Halawi, Avner SegalAbstract:In this paper, we study the degenerate principal series of a split, simply-connected, simple p-adic group of type $E_7$. We determine the points of reducibility and the maximal semi-simple Subrepresentation at each point.
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The Structure of Degenerate Principal Series Representations of Exceptional Groups of Type $E_6$ over $p$-adic Fields
arXiv: Representation Theory, 2018Co-Authors: Hezi Halawi, Avner SegalAbstract:In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type $E_6$. Furthermore, we calculate the maximal semi-simple Subrepresentation and quotient of these representations.
