On a certain generator system of the ring of invariants of a finite reflection group

Saito, Kyoji; Yano, Tamaki; Sekiguchi, Jiro

doi:10.1080/00927878008822464

Cited by 101 publications

(131 citation statements)

References 12 publications

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“…This was introduced by Saito, Sekiguchi and Yano in [124] using the classi cation of Coxeter groups for all of them but E 7 and E 8 . The general proof of ateness was obtained in [123].…”

Section: Appendix Hmentioning

confidence: 99%

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Geometry of 2D topological field theories

Dubrovin

1996

Lecture Notes in Mathematics

999

1,692

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“…This was introduced by Saito, Sekiguchi and Yano in [124] using the classi cation of Coxeter groups for all of them but E 7 and E 8 . The general proof of ateness was obtained in [123].…”

Section: Appendix Hmentioning

confidence: 99%

“…I will construct explicitly the at coordinates for the metric (cf. [39,124]). Let us consider the function p = p( ) inverse to the polynomial = (p).…”

Section: Appendix Hmentioning

confidence: 99%

Geometry of 2D topological field theories

Dubrovin

1996

Lecture Notes in Mathematics

999

1,692

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“…Since d t _ x <d t the operator 9/9/i: R-+R is uniquely determined up to constant. Saito, Sekiguchi and Yano [15,16] proved that there exists a basic set Sfi for W, which they call a /Zα£ basic set, such that (8/3/,)(J Γ J) e M t (R). If G c GL(V) is a Shephard group, it follows from (1.6) that df_, < df and thus the operator 9/3/i: R-> R is again uniquely determined up to constant.…”

Section: δ(T L9mentioning

confidence: 99%

“…-,/J be a basic set for W and write J = J(/j, -,/j). Saito, Sekiguchi and Yano [15,16,24] have used the matrix J Γ J. We may choose coordinates x u , x x so that /i = Σ x?…”

Section: δ(T L9mentioning

confidence: 99%

Discriminants in the invariant theory of reflection groups

Orlik¹,

Solomon²

1988

Nagoya Mathematical Journal

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Let V be a complex vector space of dimension l and let G ⊂ GL(V) be a finite reflection group. Let S be the C-algebra of polynomial functions on V with its usual G-module structure (gf)(v) = f{g-1v). Let R be the subalgebra of G-invariant polynomials. By Chevalley’s theorem there exists a set ℬ = {f1, …, fl} of homogeneous polynomials such that R = C[f1, …, fl]. We call ℬ a set of basic invariants or a basic set for G. The degrees di = deg fi are uniquely determined by G. We agree to number them so that d1 ≤ … ≤ di. The map τ: V/G → C1 defined byis a bijection. Each reflection in G fixes some hyperplane in V.

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“…This shows that B ′ | s0=1 is a discriminant matrix for the discriminant in the versal deformation f (x) = x n + s 1 x n−1 + · · · + s n of x n = 0 over S = C n s1,...,sn . The description of the entries of the discriminant matrix in terms of the derivatives of the elementary symmetric functions s i with respect to the roots r k is precisely the form of the discriminant matrix given by Arnol'd in [3,4], and its relation to the Bezout form can be found, at least implicitly, in [45]. The form ( * ) of the entries of the discriminant matrix generalizes both to simple hypersurface singularities and to the simple elliptic surface singularities, in that it uses the action of the associated Coxeter or Weyl group and the fact that the discriminant is precisely the image of the union of the reflection hyperplanes under the orbit map.…”

Section: The Classical Discriminant Of a Polynomialmentioning

confidence: 99%

Low-dimensional singularities with free divisors as discriminants

2008

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Abstract. We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C 5 . It is shown that such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C 5 , namely the simple elliptic singularity of type A 4 , we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j-invariant is the cone over the dual variety of an elliptic curve. IntroductionOne of the remarkable results in complex singularity theory is that the discriminant in the versal deformation of any isolated complete intersection singularity is a free divisor , a highly singular hypersurface in the ambient smooth base that admits though a compact representation as determinant of any discriminant matrix expressing a basis of the liftable vector fields in terms of the coordinate vector fields; see [34] or Section 2 below.Variants on this result show the freeness of the discriminant in the base of a versal deformation in a number of further cases, for example space-curve singularities (van Straten [47]), functions on space curves (Goryunov [24], ), or representation varieties of quivers ). J. Damon gives inThe authors were partly supported by the DFG Schwerpunkt "Global Methods in Complex Geometry". The first author was also partly supported by NSERC grant 3-642-114-80 and wishes to thank his alma mater, the University of Hannover, as well as S.-O. Buchweitz-Klingsöhr and D. Klingsöhr for their hospitality during the preparation of this work. [16] a broad survey of why and how free divisors appear frequently in the theory of discriminants and bifurcations.Here we present further versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes yet again a free divisor.While we consider the question in more generality to deduce sufficient criteria, showing along the way why such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm, the new explicit instances found pertain to• finite flat maps, thus the case of relative dimension zero, where we characterize freeness of the discriminant completely and show, for example, that the discriminant in the Hilbert scheme or Douady space of points on a smooth complex surface is a free divisor, • reduced curve singularities, where we recover not only the result on spacecurve singularities due to D. van Straten [47], but extend it to include all reduced, smoothable, and unobstructed Gorenstein curve singularities, and • smoothable Gorenstein surface singularities with reflexive conormal module, thus including all Gorenstein surface singularities in C 5 .This lis...

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On a certain generator system of the ring of invariants of a finite reflection group

Cited by 101 publications

References 12 publications

Geometry of 2D topological field theories

Geometry of 2D topological field theories

Discriminants in the invariant theory of reflection groups

Low-dimensional singularities with free divisors as discriminants

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