Invariants of quivers under the action of classical groups

Lopatin, Artem

doi:10.1016/j.jalgebra.2008.11.018

Cited by 26 publications

(27 citation statements)

References 31 publications

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“…The involution allows us to define a nondegenerate bilinear form <, > on a representation V of Q. We call the pair (V, <, >) orthogonal representation (respectively symplectic) A similar result has been obtained by Lopatin in [14] in a different setting, using ideas from [15]. The strategy of the proofs is the following.…”

Section: Introductionmentioning

confidence: 81%

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Semi-invariants of Symmetric Quivers of Finite Type

Aragona

2012

Algebr Represent Theor

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ABSTRACT. Let (Q, σ) be a symmetric quiver, where Q = (Q0, Q1) is a finite quiver without oriented cycles and σ is a contravariant involution on Q0 ⊔ Q1. The involution allows us to define a nondegenerate bilinear form <, > on a representation V of Q. We shall call the representation orthogonal if <, > is symmetric and symplectic if <, > is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c V and, in the case when matrix defining c V is skew-symmetric, by the Pfaffians pf V .

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Section: Introductionmentioning

confidence: 81%

“…. Hence ϕ Sp x,α = φ Sp x,α | SpSI(Q,α) and so for every representation Z of dimension vector α of (Q, σ) we have (14) ϕ Sp…”

Section: Semi-invariants Of Symmetric Quivers Of Finite Typementioning

confidence: 99%

Semi-invariants of Symmetric Quivers of Finite Type

Aragona

2012

Algebr Represent Theor

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“…Orthogonal and symplectic representations of symmetric quivers studied by Derksen and Weyman in [2] as well as mixed representations of quivers are partial cases of supermixed representations. More details can be found in Section 2.1 of [7]. Section 6 is dedicated to the algebra of invariants I(Q, n, ı) of mixed representations of a quiver Q.…”

Section: 2mentioning

confidence: 99%

Free relations for matrix invariants in the modular case

Lopatin¹

2012

Journal of Pure and Applied Algebra

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A classical linear group G < GL(n) acts on d-tuples of n × n matrices by simultaneous conjugation. Working over an infinite field of characteristic different from two we establish that the ideal of free relations, i.e. relations valid for matrices of any order, between generators for matrix O(n)-and Sp(n)-invariants is zero. We also prove similar result for invariants of mixed representations of quivers.These results can be considered as a generalization of the characteristic isomorphism ch : S → J between the graded ring S = ⊗ ∞ d=0 S d , where S d is the character group of the symmetric group S d , and the inverse limit J with respect to n of rings of symmetric polynomials in n variables.As a consequence, we complete the description of relations between generators for O(n)-invariants as well as the description of relations for invariants of mixed representations of quivers. We also obtain an independent proof of the result that the ideal of free relations for GL(n)invariants is zero, which was proved by Donkin in [Math. Proc. Cambridge Philos.

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“…Proof. This follows from [Lop,Theorem 1], but we will give a simpler proof using classical invariant theory in the case when the characteristic of K is 0. For the definitions of full polarization and restitution, we refer to [Pro,§3.2.2].…”

Section: Fundamental Invariantsmentioning

confidence: 99%

Symmetric quivers, invariant theory, and saturation theorems for the classical groups

Sam

2012

Advances in Mathematics

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Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights λ 1 , . . . , λ r such that the tensor product V N λ 1 ⊗· · ·⊗V N λ r contains nonzero G-invariants for some N ≥ 1, we show that the tensor product V 2λ 1 ⊗ · · · ⊗ V 2λ r also contains nonzero G-invariants. This extends results of Kapovich-Millson and Belkale-Kumar and complements similar results for the general linear group due to Knutson-Tao and Derksen-Weyman. Our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations.

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Invariants of quivers under the action of classical groups

Cited by 26 publications

References 31 publications

Semi-invariants of Symmetric Quivers of Finite Type

Semi-invariants of Symmetric Quivers of Finite Type

Free relations for matrix invariants in the modular case

Symmetric quivers, invariant theory, and saturation theorems for the classical groups

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