Highest-weight vectors for the adjoint action of GL<sub><i>n</i></sub>on polynomials

Tange, Rudolf

doi:10.2140/pjm.2012.258.497

Cited by 6 publications

(9 citation statements)

References 14 publications

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“…In this paper we will give, for certain weights, explicit bases for the spaces of highest weight vectors in k[gl n ] as modules over the ring k[gl n ] GLn of invariants. Our main result Theorem 2 extends Theorems 1 and 2 in [24]. The method we use here is quite different: in [24] we used evaluation at certain special nilpotent matrices, in this paper we use certain Cayley-Hamilton type identities and a morphism from the nilpotent cone to the variety of n × (n − 1) matrices which intertwines the adjoint action of the upper triangular matrices with the left regular action.…”

Section: Introductionmentioning

confidence: 64%

“…The algebra k[g] is a polynomial algebra in the matrix entry functions x ij , and we have k[g] U χ = 0 if and only if χ is dominant and in the root lattice. Furthermore, dim k [24,Prop. 1] for references and more explanation.…”

Section: Preliminariesmentioning

confidence: 99%

“…By [17,Thm. 11] (see [24,Prop. 1] for references for the case of prime characteristic) k[N ] U χ has dimension equal to dim L C (χ) 0 , where χ = [(r), λ].…”

Section: Denote the Variety Of N×m-matrices By Matmentioning

confidence: 99%

“…Assume k = C. In [24,Sect. 4] a general construction is given to produce k[gl n ] GLn -module generators for the module k[gl n ] U λ of highest weight vectors of weight λ.…”

Section: Introductionmentioning

confidence: 99%

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HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

Tange

2015

Transformation Groups

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Let G = GLn be the general linear group over an algebraically closed field k and let g = gl n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of polynomial functions on g and let k[g] G be the algebra of invariants under the conjugation action of G. For all weights χ ∈ Z n with χ 2 ≤ 0 or χ n−1 ≥ 0 we give explicit bases for the k[g] G -module k[g] U χ of highest weight vectors of weight χ. We also give bases for the vector spaces k[C] U χ where C is a nilpotent orbit closure. This extends earlier results to a much bigger class of weights. To express our semi-invariants in terms of matrix powers we prove certain Cayley-Hamilton type identities.

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Section: Introductionmentioning

confidence: 64%

Section: Preliminariesmentioning

confidence: 99%

“…By [17,Thm. 11] (see [24,Prop. 1] for references for the case of prime characteristic) k[N ] U χ has dimension equal to dim L C (χ) 0 , where χ = [(r), λ].…”

Section: Denote the Variety Of N×m-matrices By Matmentioning

confidence: 99%

“…Assume k = C. In [24,Sect. 4] a general construction is given to produce k[gl n ] GLn -module generators for the module k[gl n ] U λ of highest weight vectors of weight λ.…”

Section: Introductionmentioning

confidence: 99%

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HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

Tange

2015

Transformation Groups

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“…4. Corollary 1 to Theorem 4 proves a weaker version of the "conjecture" in [26,Sect. 4]: in the notation there, with χ = [λ, µ], the elements ϑ ψ t ((τ, id) · E χ ) · s i 1 ⊗ · · · ⊗ s it , 2 ≤ i 1 , .…”

Section: 2mentioning

confidence: 83%

Highest Weight Vectors and Transmutation

Tange

2018

Transformation Groups

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Let G = GLn be the general linear group over an algebraically closed field k, let g = gl n be its Lie algebra and let U be the subgroup of G which consists of the upper uni-triangular matrices. Let k[g] be the algebra of polynomial functions on g and let k[g] G be the algebra of invariants under the conjugation action of G. We consider the problem of giving finite homogeneous spanning sets for the k[g] G -modules of highest weight vectors for the conjugation action on k[g]. We prove a general result in arbitrary characteristic which reduces the problem to giving spanning sets for the vector spaces of highest weight vectors for the action of GLr × GLs on tuples of r × s matrices. This requires the technique called "transmutation" by R. Brylinsky which is based on an instance of Howe duality. In characteristic zero, we give for all dominant weights χ ∈ Z n finite homogeneous spanning sets for the k[g] G -modules k[g] U χ of highest weight vectors. This result was already stated by J. F. Donin, but he only gave proofs for his related results on skew representations for the symmetric group. We do the same for tuples of n × n-matrices under the diagonal conjugation action.2010 Mathematics Subject Classification. 13A50, 16W22, 20G05.

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Hermitian matrices with a bounded number of eigenvalues

Domokos

2013

Linear Algebra and its Applications

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Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n partial information on the minimal degree component of the vanishing ideal of the variety of n × n Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices. MSC: Primary: 13F20, 13A50, 14P05; Secondary: 15A15, 15A72, 20G05, 22E47

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Highest-weight vectors for the adjoint action of GL_non polynomials

Cited by 6 publications

References 14 publications

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

Highest Weight Vectors and Transmutation

Hermitian matrices with a bounded number of eigenvalues

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Highest-weight vectors for the adjoint action of GLnon polynomials

Cited by 6 publications

References 14 publications

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

Highest Weight Vectors and Transmutation

Hermitian matrices with a bounded number of eigenvalues

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Highest-weight vectors for the adjoint action of GL_non polynomials