Results 1  10
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33
Quantum cohomology of minuscule homogeneous spaces
 Transform. Groups
"... Abstract. We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q = 1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this inv ..."
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Cited by 32 (14 self)
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Abstract. We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q = 1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce VafaIntriligator type formulas for the GromovWitten invariants.
Lifting smooth curves over invariants for representations of compact Lie groups
 II, J. Lie Theory
"... Abstract. Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space. ..."
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Cited by 19 (13 self)
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Abstract. Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space.
Classical invariant theory for finite reflection groups, Transform. Groups 2
, 1997
"... Abstract. We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). As an application of the results we prove a generalization of Chevalley’s restriction theorem for the cla ..."
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Cited by 11 (0 self)
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Abstract. We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). As an application of the results we prove a generalization of Chevalley’s restriction theorem for the classical Lie algebras. In the interesting case when the group is of Coxeter type Dn (n ≥ 4) we use higher polarization operators introduced by Wallach. The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables. We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of type F4.
Normal forms and tensor ranks of pure states of four qubits
"... Abstract. We examine the SLOCC classification of the nonnormalized pure states of four qubits obtained by F. Verstraete et al. in [31]. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of nonno ..."
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Cited by 7 (0 self)
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Abstract. We examine the SLOCC classification of the nonnormalized pure states of four qubits obtained by F. Verstraete et al. in [31]. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of nonnormalized pure states under SLOCC transformations of determinant 1 and qubit permutations. As a byproduct, we produce a new set of generators for the invariants of the Weyl group of type F4. We complete the determination of the tensor ranks of fourqubit pure states initiated by J.L. Brylinski [3]. As a result we obtain a simple algorithm for computing these ranks. We obtain also a very simple classification of states of rank ≤ 3. 1.
CHOOSING ROOTS OF POLYNOMIALS WITH SYMMETRIES SMOOTHLY
, 2006
"... The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not C 1,α for any α> 0. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refine ..."
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Cited by 5 (5 self)
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The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not C 1,α for any α> 0. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a C 3n curve of hyperbolic polynomials of degree n admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptions in that statement.
The conjugate dimension of algebraic numbers
 Quart. J. Math
"... We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Qdimension of the space spanned by its conjugates. For all but seven nonnegative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The ..."
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Cited by 4 (0 self)
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We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Qdimension of the space spanned by its conjugates. For all but seven nonnegative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GLn(Q); this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when Q is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension Q(ω) of Q. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number.
Frobenius manifolds associated to Coxeter groups of type E7
 and E8, arXiv:0910.5453
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THE CHOW RINGS OF THE ALGEBRAIC GROUPS E6 AND E7
"... Abstract. We determine the Chow rings of the complex algebraic groups E6 and E7, giving generators and relations in terms of Schubert classes of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SOn, Spin n, G2, and F4. ..."
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Abstract. We determine the Chow rings of the complex algebraic groups E6 and E7, giving generators and relations in terms of Schubert classes of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SOn, Spin n, G2, and F4.
PARTIAL NORMALIZATIONS OF COXETER ARRANGEMENTS AND DISCRIMINANTS
, 2012
"... We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin’s Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also desc ..."
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We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin’s Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also describe an independent approach to these structures via duality of maximal Cohen–Macaulay fractional ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter group. Finally, we show that our partial normalizations give rise to new free divisors.