Quotient complete intersections of affine spaces by finite linear groups

Nakajima, Haruhisa

doi:10.1017/s0027763000021334

Cited by 8 publications

(5 citation statements)

References 20 publications

(58 reference statements)

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“…(2), /, (l^i^n + 1) are known in [6], [37], [39]. The last assertion was shown in [27], [28]. The last assertion was shown in [27], [28].…”

Section: 3) Lemma Let H Be a Normal Subgroup Of G Such That φ (H) mentioning

confidence: 83%

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Representations of a reductive algebraic group whose algebras of invariants are complete intersections.

1986

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…(2), /, (l^i^n + 1) are known in [6], [37], [39]. The last assertion was shown in [27], [28]. The last assertion was shown in [27], [28].…”

Section: 3) Lemma Let H Be a Normal Subgroup Of G Such That φ (H) mentioning

confidence: 83%

“…This case has already been treated in [28]. In order to show this, without loss of generality, we may assume that φ (G) does not contain nontrivial pseudo-reflections and both φ/s are irreducible.…”

Section: 3) Lemma Let H Be a Normal Subgroup Of G Such That φ (H) mentioning

confidence: 99%

Representations of a reductive algebraic group whose algebras of invariants are complete intersections.

1986

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Before a complete (but highly nontrivial) classification of CI-groups was given by Gordeev [15] and Nakajima [22][23][24], a concise necessary condition on G for it to be a CI-group was given by Kac and Watanabe. They showed in [18, Theorem C] that if G is a CI-group then necessarily every parabolic subgroup of G is generated by either pseudo-reflections or bireflections.…”

Section: Remark 12 By An Easy Induction (Seementioning

confidence: 99%

“…Since the finite subgroups of Sp 2 (C) = SL 2 (C) are well-known to be CI-groups, it remains to understand which symplectic reflection groups of rank 4 are CI-groups. In theory, one could use the classification of Gordeev and Nakajima [15,[22][23][24] for this, but this appears to be very difficult to do in practice.…”

Section: Complete Intersectionsmentioning

confidence: 99%

On parabolic subgroups of symplectic reflection groups

2023

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Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups. Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

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“…When A is a commutative polynomial ring over C, Question 0.3 was answered by Gordeev [22] and Nakajima [38,39] independently. A very important tool in the classification of groups G, such that k[x 1 , · · · , x n ] G is a complete intersection, is a result of Kac and Watanabe [27] and Gordeev [21] independently; they prove that if G is a finite subgroup of GL n (k) and k[x 1 , · · · , x n ] G is a complete intersection, then G is generated by bireflections (i.e.…”

Section: Introductionmentioning

confidence: 99%