Symplectic Group Research Articles

Symplectic Radon Transform and the Metaplectic Representation.

We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform; the inverse Radon transform thus appears as a “demarginalization process” for the Wigner distribution.

Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable distillation protocols that use bilocal Clifford operations followed by a single round of communication. We introduce tools to enumerate and optimise over all protocols for up to n=5 (not necessarily equal) Bell-diagonal states using a commodity desktop computer. Furthermore, by exploiting the symmetries of the input states, we find all protocols for up to n=8 copies of a Werner state. For the latter case, we present circuits that achieve the highest fidelity with perfect operations and no decoherence. These circuits have modest depth and number of two-qubit gates. Our results are based on a correspondence between distillation protocols and double cosets of the symplectic group, and improve on previously known protocols.

Quasienergy operators and generalized squeezed states for systems of trapped ions

Collective many-body dynamics for time-dependent quantum Hamiltonians is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, along with the total energy. Moreover, we also infer the dispersions for the coordinate and momentum, together with the asymmetry and the flatness parameter for the distribution. The system interaction with laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is derived. The optical system is modelled as a HO (trapped ion) that undergoes interaction with an external laser field and we use it to engineer a squeezed state of the electromagnetic (EM) field. We consider coherent and squeezed states associated to both ion dynamics and to the EM field. The approach used enables one to build CS in a compact and smart manner by use of the group theory.

Poisson Noether's Problem and Poisson rationality

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces.

Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

We study the combinatorial structure of the irreducible characters of the classical groups GLn(C), SO2n+1(C), Sp2n(C), SO2n(C) and the “non-classical” odd symplectic group Sp2n+1(C), finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of Sp2n(C) and SO2n+1(C) and between characters of SO2n(C) and SO2n+1(C). We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of irreducible characters, can be viewed as branching rules. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains (Duke Math. J., 2001). Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.

Periods and (χ,b)-factors of cuspidal automorphic forms of metaplectic groups

We give constraints on the existence of (χ,b)-factors in the global A-parameter of a genuine cuspidal automorphic representation σ of a metaplectic group in terms of the invariant, lowest occurrence index, of theta lifts to odd orthogonal groups. We also give a refined result that relates the invariant, first occurrence index, to non-vanishing of period integrals of residues of Eisenstein series associated to the cuspidal datum χ⊗σ. This complements our previous results for symplectic groups.

Web of Seiberg-like dualities for 3D N=2 quivers

We propose a universal manipulation to obtain Seiberg-like dualities of 3d $\mathcal{N}=2$ general quiver gauge theories with unitary, symplectic and orthogonal gauge groups coupled to fundamental and bifundamental matter fields. We illustrate this with several examples of linear, circular and star-shaped quiver gauge theories. We examine the local operators in the theories by computing supersymmetric indices and also find precise matching for the proposed dualities as strong evidence. We also generalize the dualities in the presence of a boundary on which the theories obey $\mathcal{N}=(0,2)$ chiral half-BPS boundary conditions.

Explicit strong boundedness for higher rank symplectic groups

This paper gives an explicit argument to show strong boundedness for Sp2n(R) for R a ring of S-algebraic integers or a semi-local ring thereby giving a quantitative version of the abstract result in the paper [15]. The results presented further generalize older results regarding strong boundedness by Kedra, Libman and Martin [6] and Morris [9] from SLn to Sp2n. The results also completely solve the question of the asymptotic of strong boundedness for Sp2n(R) for R semi-local case with an argument that immediately generalizes to all other split Chevalley groups.

Splittings of global Mackey functors and regularity of equivariant Euler classes

We establish natural splittings for the values of global Mackey functors at orthogonal, unitary and symplectic groups. In particular, the restriction homomorphisms between the orthogonal, unitary and symplectic groups of adjacent dimensions are naturally split epimorphisms. The interest in the splitting comes from equivariant stable homotopy theory. The equivariant stable homotopy groups of every global spectrum form a global Mackey functor, so the splittings imply that certain long exact homotopy group sequences separate into short exact sequences. For the real and complex global Thom spectra $\mathbf{MO}$ and $\mathbf{MU}$, the splittings imply the regularity of various Euler classes related to the tautological representations of $O(n)$ and $U(n)$.

A Combinatorial Translation Principle and Diagram Combinatorics for the Symplectic Group

Let k be an algebraically closed field of characteristic p > 2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups.

Zero-form and one-form symmetries of the ABJ and related theories

The zero-form and one-form global symmetries of the Aharony-Bergman-Jafferis (ABJ) and related theories, with at least mathcal{N} = 6 supersymmetry in three dimensions, are examined in detail. Starting from well-known dualities between theories with orthogonal and symplectic gauge groups and those with unitary gauge groups, we gauge their one-form symmetries or their subgroups and obtain new dualities. One side of the latter involves theories with special orthogonal and symplectic gauge groups, and the other side involves theories with unitary gauge groups; there is a discrete quotient on one or both sides of the duality. We study the refined superconformal indices of such theories and map the symmetries across the dualities, with particular attention to their discrete part. As a generalisation, we also find a new duality between a circular quiver with a discrete quotient of alternating special orthogonal and symplectic gauge groups and a three-dimensional mathcal{N} = 4 circular (Kronheimer-Nakajima) quiver with unitary gauge groups, whose Higgs or Coulomb branch describes an instanton on a singular orbifold.

Symplectic mapping class groups of K3 surfaces and Seiberg–Witten invariants

The purpose of this note is to prove that the symplectic mapping class groups of many K3 surfaces are infinitely generated. Our proof makes no use of any Floer-theoretic machinery but instead follows the approach of Kronheimer and uses invariants derived from the Seiberg–Witten equations.

On symplectic stabilisations and mapping classes

We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For n ⩾ 2 $n \geqslant 2$ , consider a configuration of Lagrangian S n $S^n$ s in a Weinstein domain M 2 n $M^{2n}$ . If it is analogous, in some sense that we make precise, to a configuration of exact Lagrangian S 1 $S^1$ s on a surface Σ $\Sigma$ , we show that any relation between Dehn twists in the S n $S^n$ s must also hold between the S 1 $S^1$ s. Such analogous pairs of configurations include plumbings of T * S 1 $T^\ast S^1$ s and T * S n $T^\ast S^n$ s with the same plumbing graph, and vanishing cycles for a two-variable singularity and for its stabilisation. We give a number of corollaries for subgroups of symplectic mapping class groups.

Bounded reduction of orthogonal matrices over polynomial rings

We prove that a matrix from the split orthogonal group over a polynomial ring with coefficients in a small-dimensional ring can be reduced to a smaller matrix by a bounded number of elementary orthogonal transformations. The bound is given explicitly. This result is an effective version of the early stabilisation of the orthogonal K1 functor proven by Suslin and Kopeiko in [39]. Since the similar effective results for special linear and symplectic groups are obtained by Vaserstein in [44], the present paper closes the problem for split classical groups.

A COMBINATORIAL TRANSLATION PRINCIPLE AND DIAGRAM COMBINATORICS FOR THE GENERAL LINEAR GROUP

Let k be an algebraically closed field of characteristic p > 0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GLn-modules to the category of modules for the walled Brauer algebra. As a corollary, we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra.

Auto-correlation functions of Sato–Tate distributions and identities of symplectic characters

The Sato–Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this paper, we explicitly compute the auto-correlation functions of Sato–Tate distributions for genus 2 curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups Sp(2m) for all m∈Z≥1, which have interest in their own rights.

Normality of Closures of Orthogonal Nilpotent Symmetric Orbits

We study closures of conjugacy classes in the symmetric matrices of the orthogonal group and we determine which one are normal varieties. In contrast to the result for the symplectic group where all classes have normal closure, there is only a relatively small portion of classes with normal closure. We perform a combinatorial computation on top of the same methods used by Kraft-Procesi and Ohta.

Conway subgroup symmetric compactifications redux

We extend the investigation in [1] of special toroidal compactifications of heterotic string theory for which the half-BPS states provide representations of subgroups of the Conway group. We also explore dual descriptions of these theories and find that they are all linked to either F-theory or type IIA string theory on K3 surfaces with symplectic automorphism groups that are the same Conway subgroups as those of the heterotic dual. The matching with type IIA K3 dual theories includes both the matching of symmetry groups and a comparison between the Narain lattice on the heterotic side and the cohomology lattice on the type IIA side. We present twelve examples where we can identify a type IIA dual K3 orbifold theory as the dual description of the heterotic theory. In addition, we include a supplementary Mathematica package that performs most of the computations required for these comparisons.

The Shintani double zeta functions

Abstract In this paper, we give an explicit formula for the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functions in addition to Shintani’s functional equations. Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur–Selberg trace formula of the symplectic group. Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet L-functions.

Integral cohomology of quotients via toric geometry

We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to degenerate at the second page. As an application, we compute the Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.