Let K be a field of characteristic zero. Let V be an n -dimensional vector space over K and let S be the graded ring of polynomial functions on V …
Let K be a field of characteristic zero. Let V be an n -dimensional vector space over K and let S be the graded ring of polynomial functions on V . If G is a group of linear transformations of V , then G acts naturally as a group of automorphisms of S if we define The elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G .
Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a …
Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a reflection group to be a group generated by reflections. Chevalley (1) (and also Coxeter (2) together with Shephard and Todd (4)) has shown that a reflection group G, acting on a space of n dimensions, possesses a set of n algebraically independent (polynomial) invariants which form a polynomial basis for the set of all invariants of G.
The author studies basis invariants of finite groups generated by reflections in the real space .Bibliography: 27 titles.
The author studies basis invariants of finite groups generated by reflections in the real space .Bibliography: 27 titles.
If W is a finite subgroup of GL(V) generated by reflections, it acts in a natural way on the ring of polynomial functions on V. This chapter will be devoted …
If W is a finite subgroup of GL(V) generated by reflections, it acts in a natural way on the ring of polynomial functions on V. This chapter will be devoted to the study of this action, emphasizing the remarkable features of the subring of invariants, which turns out to be a polynomial ring on generators of certain well-determined degrees (whose product is |W|). This is a far-reaching generalization of the fundamental theorem on symmetric polynomials (the case of a symmetric group).
Let R be a commutative ring, and let V be a finitelygenerated freei?-module. Let R[V] be a polynomial ring over R associated with V. Then a finitesubgroup G of GL(V) …
Let R be a commutative ring, and let V be a finitelygenerated freei?-module. Let R[V] be a polynomial ring over R associated with V. Then a finitesubgroup G of GL(V) acts naturally on R[V]. We denote by R[V]G the ring of invariants of R[V] under the action of G. Let R=k be a fieldand suppose that \G\is a unit of k. It is known ([4],[9], [3],[8]) that k[V]G is a polynomial ring if and only if G is generated by pseudoreflectionsin GL{V). But, in the case where \G\=0 mod char{k), there are only the following results: (1) L. E. Dickson [5]; FqlTu ・・・,rB]O£cn.9) an(jFq[Tu ■-,TnfLin^ are polynomial rings, where Fq is the finitefieldof q elements. (2) M.-J. Bertin [1]; Fq[Tu ■・-, Tnfnipin-^ is a polynomial ring, where
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and …
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean space by a finite orthogonal group and hence in the theory of orbifolds.
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and …
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean space by a finite orthogonal group and hence in the theory of orbifolds.
a finite group.Let $V$ be a (finite dimen- sional) $kG$ -module, $i$ .$e.$ , a representation module of $G$ over $k$ .Then $G$ acts naturally on the quotient field $F$ …
a finite group.Let $V$ be a (finite dimen- sional) $kG$ -module, $i$ .$e.$ , a representation module of $G$ over $k$ .Then $G$ acts naturally on the quotient field $F$ of the symmetric algebra $S(V)$ of $V$ as k- automorphisms.We denote the field $F$ with this action of $G$ by $k(V)$ .An extension $L/k$ is said to be rational if $L$ is finitely generated and purely transcendental over $k$ .To simplify our notation, we say that a triple $\langle k, G, V\rangle$ has the propertyThe following problem is the classical and basic one (e.g. [11]).Does $\langle k, G, V\rangle$ have the ProPerty (R) ?It is well known that the answer to the problem is affirmative in each of the following cases:(i) $G$ is the symmetric group, $k$ is any field and $V=kG$.(ii) $G$ is an abelian group of exponent $e$ and $k$ is a field whose charac- teristic does not divide $e$ and which contains a primitive e-th root of unity.(Fisher [5], etc.)$G$ is a $P$ -group and $k$ is a field of characteristic $p$ . (Kuniyoshi [6], etc.) (iv) $k$ is a field of characteristic $0$ and $G$ is a finite group generated by reflections of a k-module $V$ (Chevalley [2]).However the problem has been kept open even in the case where $G$ is abelian and $k$ is an algebraic number field.K. Masuda proved in [7] and [8] that $\langle Q, G\rangle$ has the property (R) when $G$ is a cyclic group of order $n\leqq 7$ or $n=11$ , and reduced the problem to the one on integral representations, in case $G$ is a cyclic group of order $p$ .Recently R. G. Swan [15] showed, using the Masuda's result, that $\langle Q, G\rangle$ does not have the property(R) when $G$ is a cyclic group of order $P=47,113$, 233, $\cdots$In this paper we will refine the Masuda-Swan's method and will give some further consequences on the problem in case $G$ is abelian.
Let S(gl n ) be the symmetric algebra of the Lie algebra of the matrices of size n × n over the field C of complex numbers.For ξ ∈ gl …
Let S(gl n ) be the symmetric algebra of the Lie algebra of the matrices of size n × n over the field C of complex numbers.For ξ ∈ gl * n ∼ = gl n let F ξ (gl n ) be the Mishchenko-Fomenko subalgebra of S(gl n ) constructed by the argument shift method associated with the parameter ξ.It is known that if ξ is a semisimple regular element or nilpotent regular element then the subalgebra F ξ (gl n ) is generated by a regular sequence in S(gl n ).In this thesis we prove that in gl 3 the result is extended to all ξ ∈ gl 3 , this is, the Mishchenco-Fomenko subalgebras F ξ (gl 3 ) ⊂ S(gl 3 ) are generated by a regular sequence in S(gl 3 ), A consequence of this fact is that the irreducible modules over certain commutative subalgebras of the universal enveloping algebra U (gl 3 ) can it be lifted to irreducible modules over U (gl 3 ).Furthermore, is proved that this result is true for all elements nilpotente ξ ∈ gl 4 .The general case, which is determined when the Mishchenko-Fomenko subalgebras F ξ (gl n ) ⊂ S(gl n ), with ξ ∈ gl n , are generated by a regular sequence in S(gl n ), it is still an open problem.
We give an introduction to the McKay correspondence and its connection to quotients of $\mathbb{C}^n$ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants …
We give an introduction to the McKay correspondence and its connection to quotients of $\mathbb{C}^n$ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection groups. This paper is an extended version of E.F.'s talk with the same title delivered at the ICRA.
For a finite group, we present three algorithms to compute a generating set of invariants simultaneously to generating sets of basic equivariants,<italic>i.e.,</italic>equivariants for the irreducible representations of the group. The …
For a finite group, we present three algorithms to compute a generating set of invariants simultaneously to generating sets of basic equivariants,<italic>i.e.,</italic>equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants. Its symmetry adapted basis delivers the fundamental equivariants. Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering. Our first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The fundamental invariants can be read off the H-basis of the ideal of the orbit while the fundamental equivariants are obtained from a symmetry adapted basis of an invariant direct complement to this ideal in the polynomial ring. The second algorithm takes as input primary invariants and the output provides not only the secondary invariants but also free bases for the modules of basic equivariants. These are constructed as the components of a symmetry adapted basis of the orthogonal complement, in the polynomial ring, to the ideal generated by primary invariants. The third and main algorithm proceeds degree by degree, determining the fundamental invariants as forming a H-basis of the Hilbert ideal,<italic>i.e.,</italic>the polynomial ideal generated by the invariants of positive degree. The fundamental equivariants are simultaneously computed degree by degree as the components of a symmetry adapted basis of the orthogonal complement of the Hilbert ideal.
We consider a finite-dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G …
We consider a finite-dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic, this yields that the algebra k[V]G of coinvariants is a free module over its subalgebra generated by kG-module generators of V*. This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order [M. Sezer, Decomposing modular coinvariants, J. Algebra423 (2015), 87–92]. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and Shank [M. Sezer and R. J. Shank, Rings of invariants for modular representations of the Klein four group, Trans. Amer. Math. Soc. 368 (2016), 5655–5673].
We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use …
We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the An, Bn, and Dn case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2d stabilizes with n>2d. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms.
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We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we …
We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three dimensional indecomposable representation.
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called <italic>exponents</italic>. We give new analogues of such generating functions that accommodate orbits of reflecting …
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called <italic>exponents</italic>. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes using similar invariants we call <italic>reflexponents</italic>. Our verifications are case-by-case.
Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant …
Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant polynomial maps $V \rightarrow W$, and is a module over $k[V]^G$. We give a formula for the Noether bound $\beta(k[V,W]^G,k[V]^G)$, i.e. the minimal degree $d$ such that $k[V,W]^G$ is generated over $k[V]^G$ by elements of degree at most $d$.
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>W</m:mi> </m:mrow> </m:math> {V,W} be representations of a cyclic group G of prime order p over a field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝕜</m:mi> </m:math> {\Bbbk} …
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>W</m:mi> </m:mrow> </m:math> {V,W} be representations of a cyclic group G of prime order p over a field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝕜</m:mi> </m:math> {\Bbbk} of characteristic p . The module of covariants <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕜</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>W</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mi>G</m:mi> </m:msup> </m:mrow> </m:math> {\Bbbk[V,W]^{G}} is the set of G -equivariant polynomial maps <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>V</m:mi> <m:mo>→</m:mo> <m:mi>W</m:mi> </m:mrow> </m:math> {V\rightarrow W} , and is a module over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕜</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mi>V</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mi>G</m:mi> </m:msup> </m:mrow> </m:math> {\Bbbk[V]^{G}} . We give a formula for the Noether bound <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>𝕜</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>W</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mi>G</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>𝕜</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mi>V</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mi>G</m:mi> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})} , i.e. the minimal degree d such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕜</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>W</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mi>G</m:mi> </m:msup> </m:mrow> </m:math> {\Bbbk[V,W]^{G}} is generated over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕜</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mi>V</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mi>G</m:mi> </m:msup> </m:mrow> </m:math> {\Bbbk[V]^{G}} by elements of degree at most d .
Classical invariant theory of a complex reflection group W beautifully describes the W-invariant polynomials, the W-invariant differential forms, and the relative invariants of any W-representation. When W is a duality …
Classical invariant theory of a complex reflection group W beautifully describes the W-invariant polynomials, the W-invariant differential forms, and the relative invariants of any W-representation. When W is a duality (or well-generated) group, we give an explicit description of the isotypic component within the differential forms of the irreducible reflection representation. This resolves a conjecture of Armstrong, Rhoades, and the first author, and relates to Lie-theoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and also Deconcini, Papi, and Procesi. We establish this result by examining the space of W-invariant differential derivations; these are derivations whose coefficients are not just polynomials, but differential forms with polynomial coefficients. For every complex reflection group W, we show that the space of invariant differential derivations is finitely generated as a module over the invariant differential forms by the basic derivations together with their exterior derivatives. When W is a duality group, we show that the space of invariant differential derivations is free as a module over the exterior subalgebra of W-invariant forms generated by all but the top-degree exterior generator. (The basic invariant of highest degree is omitted.) Our arguments for duality groups do not rely on any reflection group classification.
EQUIVARIANT TENSORS ON POLAR MANIFOLDSRicardo MendesWolfgang Ziller, AdvisorThis PhD dissertation has two parts, both dealing with extension questions for equivariant tensors on a polar G-manifold M with section Σ ⊂ …
EQUIVARIANT TENSORS ON POLAR MANIFOLDSRicardo MendesWolfgang Ziller, AdvisorThis PhD dissertation has two parts, both dealing with extension questions for equivariant tensors on a polar G-manifold M with section Σ ⊂ M.Chapter 3 contains the first part, regarding the so-called smoothness conditions: If atensor defined only along Σ is equivariant under the generalized Weyl group W (Σ), then it exends to a G-equivariant tensor on M if and only if it satisfies the smoothness conditions. The main result is stated and proved in the first section, and an algorithm is also provided that calculates smoothness conditions.Chapter 4 contains the second part, which consists of a proof that every equivariantsymmetric 2-tensor defined on the section of a polar manifold extends to a symmetric 2-tensor defined on the whole manifold. This is stated in detail in the first section, withproof. The main technical result used, called the Hessian Theorem, concerns the InvariantTheory of reflection groups, and is possibly of independent interest.
Building on results of Kollár, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.
Building on results of Kollár, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.
This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic \(0\). Our most general result determines the top component …
This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic \(0\). Our most general result determines the top component in total degree, which we prove for all Shephard-Todd groups \(G(m, p, n)\) with \(m \neq p\) or \(m=1\). Our strongest result gives tight bi-degree bounds and is proven for all \(G(m, 1, n)\), which includes the Weyl groups of types \(A\) and \(B\)/\(C\). For symmetric groups (i.e. type \(A\)), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund-Remmel-Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman.Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first paper in this sequence. In this paper we use concrete constructions including Gröbner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups \(G(m, p, n)\), which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences. Mathematics Subject Classifications: 05E16 (Primary), 20F55, 05A15 (Secondary)Keywords: Coinvariant algebras, pseudo-reflection groups, Gröbner basis, Artin basis, differential forms, exterior derivatives
Suppose ${{\mathbf {R}}^*}$ is an unstable algebra over the Steenrod algebra of the form ${{\mathbf {P}}^*}(\sqrt [k]{d})$, where ${{\mathbf {P}}^*}$ is a polynomial algebra over the Steenrod algebra. If ${{\mathbf …
Suppose ${{\mathbf {R}}^*}$ is an unstable algebra over the Steenrod algebra of the form ${{\mathbf {P}}^*}(\sqrt [k]{d})$, where ${{\mathbf {P}}^*}$ is a polynomial algebra over the Steenrod algebra. If ${{\mathbf {R}}^*}$ is integrally closed then ${{\mathbf {R}}^*} = P{(V)^{{G_\mathcal {X}}}}$, where $C \leqslant GL(V)$ is generated by pseudoreflections and ${G_\mathcal {X}} = \ker \{ \mathcal {X}:G \to {\mathbf {F}}_p^*\}$ is a character of degree $k$.
On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to …
On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.
We classify isolated terminal cyclic quotient singularities in dimension three, and isolated Gorenstein terminal cyclic quotient singularities in dimension four.In addition, we give a new proof of a combinatorial lemma …
We classify isolated terminal cyclic quotient singularities in dimension three, and isolated Gorenstein terminal cyclic quotient singularities in dimension four.In addition, we give a new proof of a combinatorial lemma of G. K. White using Bernoulli functions.Let A' be a smooth algebraic variety over C, and let ux be the canonical bundle of X.For each n > 0, if T(X, to®") ^ 0, there is a natural pluricanonical map <p": X -> PT( X, o)®")*.An algebraic variety is of general type if i>n is a birational map for n sufficiently large.For a variety of general type, the pluricanonical images <t>n(X) are the most natural birational models of X to study.Canonical singularities are the singularities which may occur in the pluricanonical models of varieties of general type.In dimension 1, the pluricanonical models are smooth so there are no canonical singularities; in dimension 2 the canonical singularities coincide with the classical rational double points.One characterization of the rational double points is as quotient singularities: if G is any finite subgroup of Sl(2, C), then the quotient C2/G has a rational double point, and every rational double point is analytically isomorphic to such a quotient singularity.Reid and Shepherd-Barron [10], and independently Tai [14], have given a condition for quotient singularities to be canonical in arbitrary dimensions (although not all canonical singularities are quotient singularities in dimensions greater than two).Terminal singularities are a class of canonical singularities which play an important role in birational geometry (as evidenced by recent work of Mori [8], Reid [12], and Tsunoda [15]).In this note we study cyclic quotient singularities which are terminal.In dimension three we explicitly describe all isolated terminal cyclic quotient singularities, while in dimension four, we describe isolated terminal cyclic quotient singularities which are also Gorenstein.The description uses a combinatorial lemma due to G. K. White [17]; we have given a new proof of this lemma (Corollary 1.4 below) using Bernoulli functions.
We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a …
We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves.
We demonstrate that the linear quotient singularity for the exceptional subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal …
We demonstrate that the linear quotient singularity for the exceptional subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal p left-parenthesis 4 comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Sp}(4,\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Superscript 4 Baseline slash upper G"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}^4/G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n minus 6"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2n-6</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; for example, we show that there are 1684 projective crepant resolutions when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 6"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=6</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also prove that the resulting affine cones are <italic>not</italic> quotient singularities for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 6"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geq 6</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
On considere des varietes et des sous-varietes de Hilbert riemanniennes. On demontre qu'une sous-variete de Fredholm propre a une structure de Fredholm naturelle et on montre que les operateurs de …
On considere des varietes et des sous-varietes de Hilbert riemanniennes. On demontre qu'une sous-variete de Fredholm propre a une structure de Fredholm naturelle et on montre que les operateurs de forme sont compacts et que chaque fonction distance euclidienne fa satisfait la condition C de Palais et Smale. On etudie la geometrie des sous-varietes de Fredholm propre a fibre normal plat