First Fundamental Theorem for Covariants of Classical Groups

Shmelkin, D. A.

doi:10.1006/aima.2001.2037

Cited by 4 publications

(4 citation statements)

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“…We will depict these dummy edges in a grey color. In fact, the algebra of covariants is isomorphic to the algebra of invariants C[W ] U (G) for a maximal unipotent subgroup U (G) of G, see [30]. The isomorphism is given -if we choose the upper triangular matrices for U (G) -by evaluation at e 1 ∧ • • • ∧ e k for each dummy k-edge.…”

Section: The Graph Methods For (Anti-)symmetric Tensorsmentioning

confidence: 99%

Completing the Classification of Representations of SLn with Complete Intersection Invariant Ring

Braun

2020

Transformation Groups

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We present a full list of all representations of the special linear group SLn over the complex numbers with complete intersection invariant ring of homological dimension greater than or equal to two, completing the classification of Shmelkin. For this task, we combine three techniques. Firstly, the graph method for invariants of SLn developed by the author to compute invariants, covariants and explicit forms of syzygies. Secondly, a new algorithm for finding a monomial order such that a certain basis of an ideal is a Gröbner basis with respect to this order, in between usual Gröbner basis computation and computation of the Gröbner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.

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Section: The Graph Methods For (Anti-)symmetric Tensorsmentioning

confidence: 99%

Completing the Classification of Representations of SLn with Complete Intersection Invariant Ring

Braun

2020

Transformation Groups

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“…. ∧ e k ), compare [5,14,24,29] and Subsection 3.1 of Section 3. Now the Plücker relation from above becomes a graph relation in the algebra generated by these graphs.…”

Section: The Basic Techniquesmentioning

confidence: 99%

“…3], covariant graphs are similar to invariant graphs but can in addition contain looping dummy k-edges, behaving as if they correspond to additional copies of Λ k V . In fact, the algebra of covariants is isomorphic to the algebra of invariants C[W ] U(G) for a maximal unipotent subgroup U (G) of G, see [29]. The isomorphism is given -if we choose the upper triangular matrices for U (G) -by evaluation at e 1 ∧ .…”

Section: Rings Of Covariantsmentioning

confidence: 99%

Completing the classification of representations of $\mathrm{SL}_n$ with complete intersection invariant ring

Braun

2018

Preprint

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We present a full list of all representations of the special linear group SLn over the complex numbers with complete intersection invariant ring, completing the classification of Shmelkin. For this task, we combine three techniques. Firstly, the graph method for invariants of SLn developed by the author to compute invariants, covariants and explicit forms of syzygies. Secondly, a new algorithm for finding a monomial order such that a certain basis of an ideal is a Gröbner basis with respect to this order, inbetween usual Gröbner basis computation and computation of the Gröbner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.

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“…Proof. The invariants have been described by Shmelkin, see [18,Theorem 1.1]. For (ii) we define ι by its comorphism…”

Section: Point Configurations On P 1 and Translationsmentioning

confidence: 99%

Point configurations and translations

Bäker

2014

manuscripta math.

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The spaces of point configurations on the projective line up to the action of SL(2, K) and its maximal torus are canonically compactified by the Grothdieck-Knudsen and Losev-Manin moduli spaces M 0,n and L n respectively. We examine the configuration space up to the action of the maximal unipotent group G a ⊆ SL(2, K) and define an analogous compactification. For this we first assign a canonical quotient to the action of a unipotent group on a projective variety. Moreover, we show that similar to M 0,n and L n this quotient arises in a sequence of blow-ups from a product of projective spaces.

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First Fundamental Theorem for Covariants of Classical Groups

Abstract: Let U(G) be a maximal unipotent subgroup of one of the classical groups Sp(V). Let W be a direct sum of copies of V and its dual V

Cited by 4 publications

References 7 publications

Completing the Classification of Representations of SLn with Complete Intersection Invariant Ring

Completing the Classification of Representations of SLn with Complete Intersection Invariant Ring

Completing the classification of representations of $\mathrm{SL}_n$ with complete intersection invariant ring

Point configurations and translations

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