Structure of Classical Groups

Goodman, Roe; Wallach, Nolan R.

doi:10.1007/978-0-387-79852-3_2

Cited by 31 publications

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“…In this paper, we take a different approach, studying all the invariants, and treating the problem using the methods of invariant theory [16][17][18], which considers the ring of polynomials that are invariant under the action of a group. Polynomial invariants also are the relevant objects for physics applications, since an effective Lagrangian is written as a polynomial in the basic variables which describe the theory.…”

Section: Jhep10(2009)094mentioning

confidence: 99%

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Algebraic structure of lepton and quark flavor invariants andCPviolation

Jenkins

Manohar

2009

J. High Energy Phys.

132

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Lepton and quark flavor invariants are studied, both in the Standard Model with a dimension five Majorana neutrino mass operator, and in the seesaw model. The ring of invariants in the lepton sector is highly non-trivial, with non-linear relations among the basic invariants. The invariants are classified for the Standard Model with two and three generations, and for the seesaw model with two generations, and the Hilbert series is computed. The seesaw model with three generations proved computationally too difficult for a complete solution. We give an invariant definition of the CP -violating angle θ in the electroweak sector.

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Section: Jhep10(2009)094mentioning

confidence: 99%

“…To study the invariants, it is useful to introduce several mathematical results from invariant theory [16][17][18]. The general problem is the following: one has a set of variables x 1 , .…”

Section: Jhep10(2009)094mentioning

confidence: 99%

Algebraic structure of lepton and quark flavor invariants andCPviolation

Jenkins

Manohar

2009

J. High Energy Phys.

132

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“…Essentially by Schur's lemma (cf. [14], Section 3.3) each W i is then an H-representation space and the action of G × H on V is given by…”

Section: Commuting Group Actionsmentioning

confidence: 99%

Geometry and arithmetic of Maschke’s Calabi–Yau three-fold

Bini

Geemen

2011

Communications in Number Theory and Physics

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Maschke's Calabi-Yau three-fold is the double cover of projective three space branched along Maschke's octic surface. This surface is defined by the lowest degree invariant of a certain finite group acting on a four-dimensional (4D) vector space. Using this group, we show that the middle Betti cohomology group of the three-fold decomposes into the direct sum of 150 2D Hodge substructures. We exhibit 1D families of rational curves on the three-fold and verify that the associated Abel-Jacobi map is non-trivial. By counting the number of points over finite fields, we determine the rank of the Néron-Severi group of Maschke's surface and the Galois representation on the transcendental lattice of some of its quotients.We also formulate precise conjectures on the modularity of the Galois representations associated to Maschke's three-fold (these have now been proven by M. Schütt) and to a genus 33 curve, which parametrizes rational curves in the three-fold.The Hodge structure on the middle dimensional Betti cohomology group of a Calabi-Yau (CY) three-fold carries important information on the moduli and the one-dimensional (1D) algebraic cycles on the three-fold. However, if the three-fold is easy to define, say by one equation in a (weighted) projective space, the dimension h 3 of this vector space tends to be large. For example, a smooth quintic three-fold in P 4 has h 3 = 204 and a double octic, i.e., a double cover of P 3 branched along a smooth surface of degree 8, has h 3 = 300. Using automorphisms of the three-folds, one can decompose the cohomology into subrepresentations, which give rise to Hodge substructures. In this paper, we consider a double octic with a particularly large automorphism group G of order 16 ·

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“…This means that (G : V ) is one of the cases (1) to (10) in the table of Appendix A. We may assume here that G = GC, C ∼ = C * .…”

Section: Application To D-modulesmentioning

confidence: 99%

“…As observed above the category mod θ (U ) has a nice combinatorial description, which would give, when G is simply connected, a classification of the regular holonomic modules on V whose characteristic variety is contained inC(V ). (SO(n) × C * : C n ) 2 (s + 1)(s + n/2) yes yes (2) (GL(n) : S 2 C n ) n n i=1 (s + (i + 1)/2) yes yes (3) (GL(n) : 2 C n ), n even n/2 n/2 i=1 (s + 2i − 1) yes yes (4) (GL(n) × SL(n) : M n (C)) n n i=1 (s + i) yes yes (5) (Sp(n) × GL(2) : M 2n,2 (C)) 2 (s + 1)(s + 2n) yes no (6) (SO(7) × C * : spin = C 8 ) 2 (s + 1)(s + 4) yes no (7) (SO(9) × C * : spin = C 16 ) 2 (s + 1)(s + 8) no no (8) (G 2 × C * : C 7 ) 2 (s + 1)(s + 7/2) yes no (9) (E 6 × C * : C 27 ) 3 (s + 1)(s + 5)(s + 9) no yes (10) (GL(4) × Sp(2) : M 4 (C)) 4 (s + 1)(s + 2)(s + 3)(s + 4) yes yes (3') (GL(n) : 2 C n ), n odd --yes no (4') (GL(n) × SL(m) : M n,m (C)), n = m --yes no (11) (Sp(n) × GL(1) : C 2n ) --yes no (12) (Sp(n) × GL(3) : M 2n,3 (C)) --no no (10') (GL(n) × Sp(2) : M n,4 (C)), n = 4 --yes no (13) (SO(10) × C * : 1 2 spin = C 16 ) --yes no…”

Section: Regular Holonomic Modulesmentioning

confidence: 99%

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Levasseur

2008

International Mathematics Research Notices

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Abstract. Let (G : V ) be a finite dimensional representation of a connected reductive complex Lie group (G : V ). Denote by G the derived subgroup ofG and assume that the categorical quotient V / /G is one dimensional, i.e. C[V ] G = C[f ] for a non constant polynomial f . In this situation there exists a homomorphism rad : D(V ) G → A 1 (C), the radial component map, where A 1 (C) is the first Weyl algebra. We show that the image of rad is isomorphic to the spherical subalgebra of a rational Cherednik algebra whose multiplicity function is defined by the roots of the Bernstein-Sato polynomial of f . In the case where (G : V ) is also multiplicity free we describe the kernel of rad and prove a Howe duality result between representations of G occuring in C[V ] and lowest weight modules over the Lie algebra generated by f and the "dual" differential operator ∆ ∈ S(V ); this extends results of H. Rubenthaler obtained when (G : V ) is a parabolic prehomogeneous vector space. If (G : V ) satisfies a Capelli type condition, some applications are given to holonomic and equivariant D-modules on V . These applications are related to results proved by M. Muro or P. Nang in special cases of the representation (G : V ).

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Structure of Classical Groups

Cited by 31 publications

References 0 publications

Algebraic structure of lepton and quark flavor invariants andCPviolation

Algebraic structure of lepton and quark flavor invariants andCPviolation

Geometry and arithmetic of Maschke’s Calabi–Yau three-fold

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

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