Semisimple Representations of Quivers

Bruyn, Lieven Le

doi:10.2307/2001477

Cited by 70 publications

(108 citation statements)

References 2 publications

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“…Proof For any quiver Q, the proof of [20,Theorem 2] shows that the stratification by representation type of Rep(Q, α)// G(α) is equal to the stratification by orbit type. Since G(α) is reductive one can identify µ −1 α (λ)// G(α) with a closed subvariety of Rep(Q, α)// G(α).…”

Section: Stratifyingmentioning

confidence: 99%

Stratifications of Marsden–Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities

Martino

2007

Math. Z.

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We investigate the Poisson geometry of the Marsden-Weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver. We show that the stratification by representation type equals the stratification by symplectic leaves. The deformed symplectic quotient singularities-spectra of centres of symplectic reflection algebras associated to a wreath product-are shown to be isomorphic to reductions of a certain quiver. This establishes a method to calculate when these deformations are smooth. Furthermore, the isomorphism identifies symplectic leaves and so one can give a description of their symplectic leaves in terms of roots of the quiver.

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Section: Stratifyingmentioning

confidence: 99%

Stratifications of Marsden–Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities

Martino

2007

Math. Z.

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“…It is well-known [30] that the orbit GL(W) · x is closed if and only if x is semisimple as a representation of Q dim V , namely there exists a direct sum decomposition Since the vector space at the vertex ∞ of the representation x is just C, we may assume that the vector space at ∞ of x(0) is nonzero and those of all the other x(i) are zero. Then we must have m 0 = 1 and the vector space at ∞ of x(0) is just C. Now for each i ≥ 0, the vector space W (i) is given by the direct sum of those of x(i) among all the vertices contained in D.…”

Section: Preliminary Results From Gitmentioning

confidence: 99%

Middle convolution and Harnad duality

Yamakawa

2010

Math. Ann.

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Dettweiler-Reiter's additive description of Katz's middle convolution can be interpreted in terms of the Harnad duality. Using this interpretation together with Mumford's geometric invariant theory, we generalize the additive middle convolution to have a multi-parameter and act on systems of linear ordinary differential equations with irregular singularities. We show that the generalized operation holds basic properties of the original one, and additionally, show that Katz's algorithm involving our generalization works well in generic case.

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“…Let Q be a quiver (a directed graph allowing loops and multiple arrows) [11,19], i.e., a four-tuple (Q 0 , Q 1 , t, h), where Q 0 = {1, . .…”

Section: Generalization To Quiversmentioning

confidence: 99%

“…This can be obtained by extending the proof of Theorem 7.6, using Proposition 10.9, and replacing Theorem 6.2 by its generalization for quivers (cf. Theorem 1 in [11]). This generalization states that K[V ] G is generated by the trace-monomials associated with the oriented cycles in Q of length ≤ |m| 2 .…”

Section: Generalization To Quiversmentioning

confidence: 99%

Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley

2016

J. Amer. Math. Soc.

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We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry.In particular, we show that:(1) The categorical quotient for any finite dimensional representation V of SL m , with constant m, is explicit in characteristic zero.(2) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in the dimension of V .(3) The categorical quotient of the space of r-tuples of m×m matrices by the simultaneous conjugation action of SL m is explicit in any characteristic.(4) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in m and r in any characteristic p ∈ [2, ⌊m/2⌋].(5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory.The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

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Semisimple Representations of Quivers

Cited by 70 publications

References 2 publications

Stratifications of Marsden–Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities

Stratifications of Marsden–Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities

Middle convolution and Harnad duality

Geometric complexity theory V: Efficient algorithms for Noether normalization

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