On deformation method in invariant theory

Let G be a complex reductive algebraic group. Fix a Borel subgroup B of G and a maximal torus T in B. Call the monoid of dominant weights Λ + and let S be a finitely generated submonoid of Λ + . V. Alexeev and M. Brion introduced a moduli scheme M S which classifies affine G-varieties X equipped with a T-equivariant isomor-where U is the unipotent radical of B. Examples of M S have been obtained by S. Jansou, P. Bravi and S. Cupit-Foutou. In this paper, we prove that M S is isomorphic to an affine space when S is the weight monoid of a spherical G-module with G of type A. Unlike the earlier examples, this includes cases where S does not satisfy the condition S Z ∩ Λ + = S.

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“…The following condition on submonoids of Λ + was considered by D. Panyushev in [25]. It also occurs in [29].…”

Section: Further Results and Notions Frommentioning

confidence: 99%

Equivariant degenerations of spherical modules for groups of type \mathsf {A}

Papadakis¹,

Steirteghem²

2012

Ann. inst. Fourier

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“…For the first step, let us denote the maximal unipotent subgroup of SL n of lower triangular matrices by U , the opposite maximal unipotent subgroup of upper triangular matrices by U o and the normalizing maximal torus by T . By Theorem 0.2 of [22], the algebra C[X×Y ] SLn is a deformation of (C[X] U ⊗C[Y ] U o ) T for affine varieties X, Y and both algebras share the same Hilbert series with respect to a common SL n -stable grading. We explicitly compute the algebra…”

Section: The Serial Casesmentioning

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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“…Note that λ(G/H) ≤λ(G/H) since BH is dense in G, and more generally Z λ(G/H) ∩ P + ≤ λ(G/H) ( [34], part 2 of the proof of Proposition 1.5). Moreover…”

Section: Definition 32 We Putλ(g/h)mentioning

confidence: 99%

“…We shall prove the crucial fact thatλ [34], Definition 1.3). In the following, x is a fixed element in O ∩ wB andẇ a representative of w in N such that x =ẇu, u ∈ U .…”

Section: We Describeλ(o) For This Purpose We Denote Bymentioning

confidence: 99%