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Showing papers on "Equivariant map published in 2013"


Journal ArticleDOI
TL;DR: In this paper, a representation of the affine W-algebra of the group SU(r) on the equivariant homology space of the moduli space of U r -instantons is presented.
Abstract: We construct a representation of the affine W-algebra of ${\mathfrak{g}}{\mathfrak{l}}_{r}$ on the equivariant homology space of the moduli space of U r -instantons, and we identify the corresponding module. As a corollary, we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). Our approach uses a deformation of the universal enveloping algebra of W 1+∞, which acts on the above homology space and which specializes to $W({\mathfrak{g}}{\mathfrak{l}}_{r})$ for all r. This deformation is constructed from a limit, as n tends to ∞, of the spherical degenerate double affine Hecke algebra of GL n .

244 citations


Journal ArticleDOI
TL;DR: The notion of twisted equivariant K-theory of the Brillouin torus has been introduced in this article for topological phases of quantum systems, and it has been shown that there is a canonical twisted version of the Ktheory for systems with a lattice of translational symmetries.
Abstract: We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/ quaternionic representations as well as a corresponding tenfold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles, there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries, we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.

209 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define the virtual motive of the Hilbert scheme of n points on a smooth complex, and give a formula for the generating function for arbitrary X as a motivic exponential.
Abstract: Given a smooth complex threefold X, we define the virtual motive \([\operatorname{Hilb}^{n}(X)]_{\operatorname {vir}}\) of the Hilbert scheme of n points on X. In the case when X is Calabi–Yau, \([\operatorname{Hilb}^{n}(X)]_{\operatorname{vir}}\) gives a motivic refinement of the n-point degree zero Donaldson–Thomas invariant of X. The key example is X=ℂ3, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef–Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives \([\operatorname{Hilb}^{n} (\mathbb{C}^{3})]_{\operatorname{vir}}\) via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Gottsche’s formula for the Poincare polynomials of the Hilbert schemes of points on surfaces.

142 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the energy critical Schrodinger map problem with the 2-sphere target for equivariant initial data of homotopy index k = 1.
Abstract: We consider the energy critical Schrodinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

125 citations


Journal ArticleDOI
TL;DR: In this article, the convolution algebra in the equivariant K-theory of the Hilbert scheme of A2 was shown to be isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL∞.
Abstract: In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A2. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL∞. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over P2, virtual fundamental classes, shuffle algebras, …).

103 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduce two families of symmetric functions generalizing the factorial Schur P -and Q -functions due to Ivanov, and show that these functions represent the Schubert classes in the K-theory of torus equivariant coherent sheaves on the maximal isotropic Grassmannians of symplectic and orthogonal types.

92 citations


Journal ArticleDOI
TL;DR: In this article, the generalized Jack polynomials, depending on the Nples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, and this resolves the problem of original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model.
Abstract: Original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model did not work for beta different from one, because Nekrasov functions were not properly reproduced by Selberg-Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper we describe a complete proof of AGT in the simple case of GL(2) (N=2) Yang-Mills theory, i.e. the 4-point spherical conformal block of the Virasoro algebra.

84 citations


Posted Content
TL;DR: In this article, the authors apply supersymmetric localization to N=(2,2) gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries.
Abstract: We apply supersymmetric localization to N=(2,2) gauged linear sigma models on a hemisphere, with boundary conditions, i.e., D-branes, preserving B-type supersymmetries. We explain how to compute the hemisphere partition function for each object in the derived category of equivariant coherent sheaves, and argue that it depends only on its K theory class. The hemisphere partition function computes exactly the central charge of the D-brane, completing the well-known formula obtained by an anomaly inflow argument. We also formulate supersymmetric domain walls as D-branes in the product of two theories. In particular 4d line operators bound to a surface operator, corresponding via the AGT relation to certain defects in Toda CFT's, are constructed as domain walls. Moreover we exhibit domain walls that realize the sl(2) affine Hecke algebra.

78 citations


Posted Content
TL;DR: The authors recast equivariant stable homotopy theory in terms of point-set level categories of G-spans and nonequivariant spectra, and gave a more topologically grounded model based on Atiyah duality.
Abstract: Let G be a finite group. We give Quillen equivalent models for the category of G-spectra as categories of spectrally enriched functors from ex- plicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point-set level categories of G-spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality.

72 citations


Journal ArticleDOI
TL;DR: In this article it was shown that the Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broue-Malle-Michel are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to the character has connected center.
Abstract: We show that several character correspondences for finite reductive groups $$G$$ are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to $$G$$ has connected center. The correspondences we consider are the so-called Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broue–Malle–Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from Isaacs–Malle–Navarro for the non-abelian finite simple groups of Lie types $$^3\mathsf{D }_4,\mathsf{E }_8,\mathsf{F }_4,^2\mathsf{F }_4$$ , and $$\mathsf{G }_2$$ .

71 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define a minimal simplicial action of the group Fn on a simplicial tree T such that the stabilizer of each edge of T is the trivial subgroup of Fn.
Abstract: Given a free group Fn of finite rank n 2, a free splitting over Fn is a minimal, simplicial action of the group Fn on a simplicial tree T such that the stabilizer of each edge of T is the trivial subgroup of Fn . A free splitting is denoted Fn O T , or just T , when the group and its action are understood. Although the tree T is allowed to have vertices of valence 2, there is a unique natural cell structure on T the vertices of which are the points of valence at least 3. We say that T is a k –edge free splitting if k is the number of natural edge orbits, a number which can take on any value from 1 to 3n 3. The equivalence relation amongst free splittings is conjugacy, where two free splittings of Fn are conjugate if there exists an Fn –equivariant homeomorphism between them. See the beginning of Section 1 for the details of these definitions.

Journal ArticleDOI
Abstract: In previous work we have shown that the equivariant index of multi-centered $ \mathcal{N}=2 $ black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription — the Coulomb branch formula — for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke’s result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincare polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package “CoulombHiggs.m” implementing the Coulomb and Higgs branch formulae.

Journal ArticleDOI
TL;DR: In this paper, Yetter-Drinfeld modules over hom-bialgebras with bijective structure maps are defined and studied, and a quasi-braided pre-tensor category for them is defined.
Abstract: The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hom-bialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category $_H^H{\mathcal YD}$ of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively coquasitriangular) the first (respectively second) quasi-braided pre-tensor category $_H^H{\mathcal YD}$ contains, as a quasi-braided pre-tensor subcategory, the category of modules (respectively comodules) with bijective structure maps over H.

Journal ArticleDOI
TL;DR: In this paper, the authors define S1-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo.
Abstract: We define S1-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.

Posted Content
TL;DR: In this article, Kobayashi et al. introduced a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras.
Abstract: We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, this http URL {Transf. Groups (2012)}], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan--Holder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators [Juhl, this http URL {Progr. Math. 2009}] and its generalizations. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

Journal ArticleDOI
TL;DR: This paper proposes an observer structure with a pre-observer or internal model augmented by an equivariant innovation term that leads to autonomous error evolution and a control Lyapunov function construction is used to design the observer innovation.

Posted Content
TL;DR: In this article, the authors derived an explicit expression for the R-matrix associated with the instanton moduli space, which is the standard rational solution of the Yang-Baxter equation, which are well known in the theory of quantum integrable systems.
Abstract: A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting R-matrices are the standard rational solutions of the Yang-Baxter equation, which are well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated R-matrices, depending additionally on two equivariant parameters t_1 and t_2. In this paper we derive an explicit expression for the R-matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this R-matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the R-matrix the well known Lehn's formula for the first Chern class. We explicitly compute the first several coefficients for the power series expansion of the R-matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields.

Journal ArticleDOI
TL;DR: In this article, the authors construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A).
Abstract: We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant K-theory.

Journal ArticleDOI
TL;DR: In this paper, the Coulomb branch formula is used to compute the cohomology of the moduli space of quiver representations, which is a generalization of the Higgs branch formula.
Abstract: In previous work we have shown that the equivariant index of multi-centered N=2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -- the Coulomb branch formula -- for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincare polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package "CoulombHiggs.m" implementing the Coulomb and Higgs branch formulae.

Journal ArticleDOI
TL;DR: In this article, the authors identify the problem of equivariant vortex counting in a two-dimensional quiver gauged linear sigma model with that of computing the Gromov-Witten invariants of the GIT quotient target space determined by the quiver.
Abstract: In this paper we identify the problem of equivariant vortex counting in a $(2,2)$ supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the ${\cal I}$ and ${\cal J}$-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov-Witten theory follow just by deforming the integration contour. In particular we apply our formalism to compute Gromov-Witten invariants of the $\mathbb{C}^3/\mathbb{Z}_n$ orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on $\mathbb {C}^2$ and of $A_n$ and $D_n$ singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a new method to study Birational maps between Fano varieties based on multiplier ideal sheaves and proved equivariant Birational rigidity of four Fano threefolds acted on by the group A6.
Abstract: We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that Bir(P 3 ) has at least five non-conjugate subgroups isomorphic to A6.

Journal ArticleDOI
TL;DR: In this article, it was shown that every cubical cubical complex T(X) is cubical, and that the cubical subdivision of any locally CAT(0) cubical group is a cubical subgroup.
Abstract: For every simplicial complex X we construct a locally CAT(0) cubical complex T(X), a cellular isometric involution tau on T(X) and a map t from T(X) to X with the following properties: t is equivariant for tau; t is a homology isomorphism; the induced map from the quotient space T(X)/tau to X is a homotopy equivalence; the induced map from the tau-fixed point set in T(X) to X is a homology isomorphism. The construction is functorial in X. One corollary is an equivariant Kan-Thurston theorem: every connected proper G-CW-complex has the same equivariant homology as the classifying space for proper actions of some other group. From this we obtain extensions of a theorem of Quillen on the spectrum of a (Borel) equivariant cohomology ring and of a result of Block concerning assembly conjectures. Another corollary of our main result is that there can be no algorithm to decide whether a CAT(0) cubical group is generated by torsion. In appendices we prove some foundational results concerning cubical complexes, notably in the infinite-dimensional case. We characterize the cubical complexes for which the natural metric is complete; we establish Gromov's criterion for infinite-dimensional cubical complexes; we show that every CAT(0) cube complex is cubical; we deduce that the second cubical subdivision of any locally CAT(0) cube complex is cubical. [A version of this paper was submitted in September 2010. This is a revised version I made in April 2011 (improvements to some material in the appendices).]

Journal ArticleDOI
TL;DR: In this paper, the authors considered equivariant solutions for the Schrodinger map problem from ℝ2+1 to S2 with energy less than 4π and showed that they are global in time and scatter.
Abstract: We consider equivariant solutions for the Schrodinger map problem from ℝ2+1 to S2 with energy less than 4π and show that they are global in time and scatter.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the special case of toric Sasaki-Einstein manifolds known as $Y^{p,q}$ manifolds and derived the full perturbative part of the partition function.
Abstract: On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as $Y^{p,q}$ manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large $N$ behaviour for the case of single hypermultiplet in adjoint representation and we derive the $N^3$-behaviour in this case.

Posted Content
TL;DR: In this article, the authors studied 1-equivariant wave maps of finite energy from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere and established relaxation of such a map of arbitrary energy and degree to the unique stationary harmonic map in its degree class.
Abstract: In this paper we study 1-equivariant wave maps of finite energy from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the unit sphere in R^3 gets mapped to the north pole. Finite energy implies that spacial infinity gets mapped to either the north or south pole. In particular, each such equivariant wave map has a well-defined topological degree which is an integer. We establish relaxation of such a map of arbitrary energy and degree to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizon, Chmaj, Maliborski who observed this asymptotic behavior numerically.

Journal ArticleDOI
01 Jun 2013
TL;DR: The Packer-Raeburn Stabilization Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions as discussed by the authors.
Abstract: C � -algebras form a 2-category with � -homomorphisms or corre- spondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant rep- resentations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith. The Packer-Raeburn Stabilisation Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalise this to actions of strict 2-groupoids.

Journal ArticleDOI
TL;DR: In this article, it was shown that the integrable representations of a q-deformed and elliptic affine W_N-algebra can be derived from the spacetime BPS spectra of string-dual M-theory compactifications.
Abstract: We generalize our analysis in [arXiv:1301.1977], and show that a 5d and 6d AGT correspondence for SU(N) -- which essentially relates the relevant 5d and 6d Nekrasov instanton partition functions to the integrable representations of a q-deformed and elliptic affine W_N-algebra -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Via an appropriate defect, we also derive a "fully-ramified" version of the 5d and 6d AGT correspondence where integrable representations of a quantum and elliptic affine SU(N)-algebra at the critical level appear on the 2d side, and argue that the relevant "fully-ramified" 5d and 6d Nekrasov instanton partition functions are simultaneous eigenfunctions of commuting operators which define relativistic and elliptized integrable systems. As an offshoot, we also obtain various mathematically novel and interesting relations involving the double loop algebra of SU(N), elliptic Macdonald operators, equivariant elliptic genus of instanton moduli space, and more.

Posted Content
TL;DR: In this article, it was shown that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C u -categories over ReppGq up to natural equivalence.
Abstract: An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module structure over the tensor category ReppGq of finite-dimensional representations of G. We show that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C u -categories over ReppGq up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts.

Posted Content
TL;DR: In this article, the authors consider finite energy corotationnal wave maps with a target manifold and prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow case) or a linear scattering term (in global case), up to an error which tends to 0 in the energy space.
Abstract: We consider finite energy corotationnal wave maps with target manifold $\m S^2$. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space.

Posted Content
TL;DR: In this article, the authors studied the problem of proper biharmonic immersion in the Euclidean space of invariant biconservative hypersurfaces, and showed that there exists no proper immersion in these invariant families.
Abstract: In this paper, using the framework of equivariant differential geometry, we study proper $SO(p+1) \times SO(q+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+q+2$) and proper $SO(p+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+2$). Moreover, we show that, in these two classes of invariant families, there exists no proper biharmonic immersion.