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Oded Yacobi - One of the best experts on this subject based on the ideXlab platform.
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An analysis of the multiplicity spaces in branching of symplectic groups, Selecta Math N.S., Volume 16, Issue 4, (2010) E-mail address : skim@maths.uq.edu.au School of Mathematics and
2016Co-Authors: Oded YacobiAbstract:Abstract. Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching Algebra B. The Algebra B is a Graded Algebra whose components encode the multiplicities of irreducible representations of Sp2n−2 in irreducible representations of Sp2n. Our first theorem states that the map taking an element of Sp2n to its principal n × (n+ 1) submatrix induces an isomorphism of B to a different branching Algebra B ′. The Algebra B ′ encodes multiplicities of irreducible representations of GLn−1 in certain irreducible representations of GLn+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp2n to Sp2n−2 is canonically an irreducible module for the n-fold product of SL2. In particular, this induces a canonical decomposition of the multiplicity spaces into on
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a basis for the symplectic group branching Algebra
Journal of Algebraic Combinatorics, 2012Co-Authors: Sangjib Kim, Oded YacobiAbstract:The symplectic group branching Algebra, $\mathcal {B}$ , is a Graded Algebra whose components encode the multiplicities of irreducible representations of Sp2n?2(?) in each finite-dimensional irreducible representation of Sp2n (?). By describing on $\mathcal {B}$ an ASL structure, we construct an explicit standard monomial basis of $\mathcal {B}$ consisting of Sp2n?2(?) highest weight vectors. Moreover, $\mathcal {B}$ is known to carry a canonical action of the n-fold product SL2×?×SL2, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi Algebras we describe a deformation of $\mathrm{Spec}(\mathcal {B})$ into an explicitly described toric variety.
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a basis for the symplectic group branching Algebra
arXiv: Representation Theory, 2010Co-Authors: Sangjib Kim, Oded YacobiAbstract:The symplectic group branching Algebra, B, is a Graded Algebra whose components encode the multiplicities of irreducible representations of Sp(2n-2,C) in each irreducible representation of Sp(2n,C). By describing on B an ASL structure, we construct an explicit standard monomial basis of B consisting of Sp(2n-2,C) highest weight vectors. Moreover, B is known to carry a canonical action of the n-fold product SL(2) \times ... \times SL(2), and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi Algebras we describe a deformation of Spec(B) into an explicit toric variety.
Sangjib Kim - One of the best experts on this subject based on the ideXlab platform.
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a basis for the symplectic group branching Algebra
Journal of Algebraic Combinatorics, 2012Co-Authors: Sangjib Kim, Oded YacobiAbstract:The symplectic group branching Algebra, $\mathcal {B}$ , is a Graded Algebra whose components encode the multiplicities of irreducible representations of Sp2n?2(?) in each finite-dimensional irreducible representation of Sp2n (?). By describing on $\mathcal {B}$ an ASL structure, we construct an explicit standard monomial basis of $\mathcal {B}$ consisting of Sp2n?2(?) highest weight vectors. Moreover, $\mathcal {B}$ is known to carry a canonical action of the n-fold product SL2×?×SL2, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi Algebras we describe a deformation of $\mathrm{Spec}(\mathcal {B})$ into an explicitly described toric variety.
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a basis for the symplectic group branching Algebra
arXiv: Representation Theory, 2010Co-Authors: Sangjib Kim, Oded YacobiAbstract:The symplectic group branching Algebra, B, is a Graded Algebra whose components encode the multiplicities of irreducible representations of Sp(2n-2,C) in each irreducible representation of Sp(2n,C). By describing on B an ASL structure, we construct an explicit standard monomial basis of B consisting of Sp(2n-2,C) highest weight vectors. Moreover, B is known to carry a canonical action of the n-fold product SL(2) \times ... \times SL(2), and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi Algebras we describe a deformation of Spec(B) into an explicit toric variety.
He Ji-wei - One of the best experts on this subject based on the ideXlab platform.
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Clifford deformations of Koszul Frobenius Algebras and noncommutative quadrics
2021Co-Authors: He Ji-weiAbstract:Let $E$ be a Koszul Frobenius Algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-Graded Algebra $E(\theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. It turns out that the bounded derived category $D^b(\text{gr}_{\mathbb Z_2}E(\theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(\theta)$ is a semisimple $\mathbb Z_2$-Graded Algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius Algebra are related to the Kn\"{o}rrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Kn\"{o}rrer Periodicity Theorem without using of matrix factorizations.Comment: We added a new sectio
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Pre-resolutions of noncommutative isolated singularities
2020Co-Authors: He Ji-weiAbstract:We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version of quasi-resolutions introduced by Qin-Wang-Zhang. We prove that right quasi-resolutions for noetherian bounded below and locally finite Graded Algebra with right injective dimension 2 are always Morita equivalent. When we restrict to noncommutative quadric hypersurfaces, we prove that a noncommutative quadric hypersurface, which is a noncommutative isolated singularity, always admits a right pre-resolution. Besides, we provide a method to verify whether a noncommutative quadric hypersurface is an isolated singularity. An example of noncommutative quadric hypersurfaces with detailed computations of indecomposable maximal Cohen-Macaulay modules and right pre-resolutions is included as well
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Clifford deformations of Koszul Frobenius Algebras and noncommutative quadrics
2019Co-Authors: He Ji-weiAbstract:Let $E$ be a Koszul Frobenius Algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-Graded Algebra $E(\theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. It turns out that the bounded derived category $D^b(\text{gr}_{\mathbb Z_2}E(\theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(\theta)$ is a semisimple $\mathbb Z_2$-Graded Algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius Algebra are related to the Kn\"{o}rrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Kn\"{o}rrer Periodicity Theorem without using of matrix factorizations
Shouchuan Zhang - One of the best experts on this subject based on the ideXlab platform.
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the von neumann regular radical and jacobson radical of crossed products
arXiv: Quantum Algebra, 2003Co-Authors: Shouchuan ZhangAbstract:We construct the $H$-von Neumann regular radical for $H$-module Algebras and show that it is an $H$-radical property. We obtain that the Jacobson radical of twisted Graded Algebra is a Graded ideal. For twisted $H$-module Algebra $R$, we also show that $r_{j}(R#_\sigma H)= r_{Hj}(R)#_\sigma H$ and the Jacobson radical of $R$ is stable, when $k$ is an Algebraically closed field or there exists an Algebraic closure $F$ of $k$ such that $r_j(R \otimes F) = r_j(R) \otimes F$, where $H$ is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf Algebra over $k$. In particular, we answer two questions J.R.Fisher asked.
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the von neumann regular radical and jacobson radical of crossed products
Acta Mathematica Hungarica, 2000Co-Authors: Shouchuan ZhangAbstract:We construct the H-von Neumann regular radical for H-module Algebras and show that it is an H-radical property. We obtain that the Jacobson radical of a twisted Graded Algebra is a Graded ideal. For a twisted H-module Algebra R, we also show that rj(R#σH) = rHj(R)#σH and the Jacobson radical of R is stable, when k is an Algebraically closed field or there exists an Algebraic closure F of k such that rj(R⊗F) = rj(R) ⊗ F, where H is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf Algebra over k. In particular, we answer two questions of J. R. Fisher.
Antonio Giambruno - One of the best experts on this subject based on the ideXlab platform.
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Graded polynomial identities and codimensions computing the exponential growth
Advances in Mathematics, 2010Co-Authors: Antonio Giambruno, D La MattinaAbstract:Abstract Let G be a finite abelian group and A a G-Graded Algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the Graded polynomial identities satisfied by A. We study the asymptotic behavior of c n G ( A ) , n = 1 , 2 , … , the sequence of Graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, lim n → ∞ c n G ( A ) n exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G × Z 2 -Graded Algebra related to A.
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exponential codimension growth of pi Algebras an exact estimate
Advances in Mathematics, 1999Co-Authors: Antonio Giambruno, Mikhail ZaicevAbstract:Abstract LetAbe an associative PI-Algebra over a fieldFof characteristic zero. By studying the exponential behavior of the sequence of codimensions {cn(A)} ofA, we prove thatInv(A)=limn→∞ c n ( A ) always exists and is an integer. We also give an explicit way for computing such integer: letBbe a finite dimensionalZ2-Graded Algebra whose Grassmann envelopeG(B) satisfies the same identities ofA; thenInv(A)=Inv(G(B))=dim C(0)+dim C(1)whereC(0)+C(1)is a suitableZ2-Graded semisimple subAlgebra ofB.
