Symplectic Group Research Articles

Symmetric Dellac configurations and symplectic/orthogonal flag varieties

The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag varieties of types A and C. The definitions are given in terms of linear algebra: they are based on the quiver Grassmannian realization of the degenerate flag varieties and odd symplectic and odd and even orthogonal groups. We study basic properties of our varieties; in particular, we construct cellular decomposition in all the three cases above (as well as in the case of even symplectic group). We also compute the Poincaré polynomials in terms of certain statistics on the set of symmetric Dellac configurations.

Theta functions and a presentation of [formula omitted

We investigate a Curtis–Tits style group presentation based on the Dynkin diagram Cg, in which the short root generators have order two while the long root generators have order four. We prove that this describes a finite group with an almost extraspecial normal subgroup of order 22g+2 and quotient isomorphic to the symplectic group Sp(2g,2). Such an extension has to be non-split if g⩾3, and was proved to exist in papers of Bolt, Room and Wall from 1961/2 and Griess from 1973. Our presentation proves that it is a double cover of a finite quotient of Sp(2g,Z). We investigate a 2g dimensional complex representation on a suitable space of theta functions, and produce some consequences for the signatures of 4-manifolds described as surface bundles over surfaces. In particular, we prove that if the monodromy is contained in the theta subgroup Spq(2g,Z)⩽Sp(2g,Z) then the signature of the 4-manifold is divisible by eight.

A Transcendence Basis in the Differential Field of Invariants of Pseudo-Galilean Group

Let G be a subgroup in the group of all invertible linear transformations of a finite-dimensional real space X. One of the problems in differential geometry is that of finding easily verified necessary and sufficient conditions for G-equivalence of paths in X. In solving this problem, we use methods of the theory of differential invariants which give descriptions of transcendence bases of differential fields of G-invariant differential rational functions. Having explicit forms of transcendence bases, we can obtain efficient criteria for G-equivalence of paths with respect to actions of the special linear, orthogonal, pseudoorthogonal, and symplectic groups. We present a description of one finite transcendence basis in the differential field of differential rational functions invariant with respect to the action of the pseudo-Galilean group ΓO. Based on this, we establish necessary and sufficient conditions of ΓO-equivalence of paths.

Viterbo’s transfer morphism for symplectomorphisms

We construct an analogue of Viterbo’s transfer morphism for Floer homology of an automorphism of a Liouville domain. As an application we prove that the Dehn–Seidel twist along any Lagrangian sphere in a Liouville domain of dimension [Formula: see text] has infinite order in the symplectic mapping class group.

String structures associated to indefinite Lie groups

String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group O(n) to the indefinite-signature orthogonal group O(p,q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simpler, constructions hold for other noncompact Lie groups such as the unitary group U(p,q) and the symplectic group Sp(p,q). This extension provides a starting point for an abundance of constructions in (higher) geometry and applications in physics.

Universality of Toda equation in mathcal{N}=2 superconformal field theories

We show that extremal correlators in all Lagrangian mathcal{N}=2 superconformal field theories with a simple gauge group, when suitably defined the {mathbb{S}}^4 , are governed by the same universal Toda system of equations, which dictates the structure of extremal correlators to all orders in the perturbation series. A key point is the construction of a convenient orthogonal basis for the chiral ring, by arranging towers of operators in order of increasing dimension, which has the property that the associated two-point functions satisfy decoupled Toda chain equations. We explicitly verify this in all known SCFTs based on SU(N) gauge groups as well as in superconformal QCD based on orthogonal and symplectic groups. As a by-product, we find a surprising non-renormalization property for the mathcal{N}=2 SU(N) SCFT with one hypermultiplet in the rank-2 symmetric representation and one hypermultiplet in the rank-2 antisymmetric representation, where the two-loop terms of a large class of supersymmetric observables identically vanish.

USp(2Nc) SQCD3 with antisymmetric: dualities and symmetry enhancements

We study various aspects of the 4d/3d reduction of mathcal{N} = 1 dualities involving USp(2Nc) gauge theories with 2Nf fundamentals and one antisymmetric. We discuss the non-trivial role played by the monopole superpotentials in the reduction and obtain new 3d dualities for models with both symplectic and unitary gauge groups. For Nf = 4 we observe interesting webs of dualities and symmetry enhancements, recovering and extending some results already appeared in the literature.

Traces and Supertraces on Symplectic Reflection Algebras

The symplectic reflection algebra H1,ν (G) has a T(G)-dimensional space of traces, and if it is regarded as a superalgebra with a natural parity, then it has an S(G)-dimensional space of supertraces. The values of T(G) and S(G) depend on the symplectic reflection group G and are independent of the parameter ν. We present values of T(G) and S(G) for the groups generated by the root systems and for the groups G = Γ ≀ SN, where Γ is a finite subgroup of Sp(2,ℂ).

On the tangent Lie group of a symplectic Lie group

Motivated by the recent work of Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185–195, 2018) on the Riemannian geometry of tangent Lie groups, we prove that the tangent Lie group $${ TG}$$ of a symplectic Lie group $$(G,\omega )$$ admits the structure of a symplectic Lie group. On $${ TG}$$, we construct a left invariant symplectic form $${\widetilde{\omega }}$$ which is induced from $$\omega $$ using complete and vertical lifts of left invariant vector fields on G. The aforementioned construction can be viewed as the symplectic analogue of the left invariant Riemannian metrics on the tangent Lie groups that were constructed in Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185–195, 2018). One immediate upshot of our construction is that by taking iterated tangent bundles of a non-abelian symplectic Lie group, one obtains a convenient means of generating non-abelian symplectic Lie groups of arbitrarily high dimension.

The diagram category of framed tangles and invariants of quantized symplectic group

In this paper, we present a categorical version of the first and second fundamental theoremsof the invariant theory for the quantized symplectic groups. Our methods depend onthe theory of braided strict monoidal categories which are pivotal, more explicitly, thediagram category of framed tangles.

Regular characters of classical groups over complete discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic, and let G be a symplectic or special orthogonal group scheme over o. For any ℓ∈N let Gℓ denote the ℓ-th principal congruence subgroup of G(o). An irreducible character of the group G(o) is said to be regular if it is trivial on a subgroup Gℓ+1 for some ℓ, and if its restriction to Gℓ/Gℓ+1≃Lie(G)(k) consists of characters of minimal G(kalg)-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over o, and construct and enumerate all regular characters of G(o), when the characteristic of k is greater than two. As a result, we compute the regular part of their representation zeta function.

The Gelfand–Graev representation of GSp(4,𝔽q)

In this article, we demonstrate a technique for calculating the irreducible representations of the endomorphism algebra of the Gelfand–Graev representations of the general symplectic group where q is an odd prime power.

-ADIC -FUNCTIONS FOR ORDINARY FAMILIES ON SYMPLECTIC GROUPS

We construct the$p$-adic standard$L$-functions for ordinary families of Hecke eigensystems of the symplectic group$\operatorname{Sp}(2n)_{/\mathbb{Q}}$using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group$\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to$p$-adically interpolate the restrictions of the Siegel Eisenstein series to$\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

Symplectic harmonicity and generalized coeffective cohomologies

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by Tsai, Tseng and Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms $$\omega _t$$ such that the dimensions of these symplectic cohomology groups vary with respect to t. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifolds, and concrete examples with special cohomological properties are obtained on an 8-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.

A Method for Determining the Affine Equivalence of Boolean Functions

Determining the affine equivalence of Boolean functions has significant applications in circuit and cryptography. Previous methods for determining this require a large amount of computation when Boolean functions are bent functions or when the truth table is sparse. This paper presents a new method to determine the affine equivalence based on matrix algebra. By transforming Boolean function to the corresponding matrix representation, we first propose and prove the congruent standard form of Boolean function. It lays the foundation for the determination of equivalence because affine Boolean functions must have the same standard form. Then we find the generators of orthogonal matrix group and symplectic matrix group, which greatly reduce the search space. The computation complexity of our method is o(2(r 2/2+n * (n-r) )), where n is the number of bit operations, and r is the rank of the matrix, which is the product of Boolean-1 matrix of the test Boolean function and its transposition. The experimental results show that our method is useful when the test Boolean function is no more than 7 bits and r is greater than 2.

Jordan blocks of cuspidal representations of symplectic groups

Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters for $G$, compatible with the local Langlands correspondence. The main tool consists in analysing the intertwining Hecke algebra of a good cover, in the sense of Bushnell--Kutzko, for parabolic induction from a cuspidal representation of $G\times\mathrm{GL}_n$, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine its (ir)reducibility; a criterion of Moeglin then relates this to Langlands parameters.

Descending congruences of theta lifts on GSp4

We study the question of when a congruence between two theta lifts on GSp4/Q descends to a congruence on modular forms on GL2 over a quadratic field. In order to accomplish that, we use the theory of the local theta correspondence between similitude orthogonal groups and the similitude symplectic group GSp4, together with a classification for the degeneration modulo a prime of conductors for the L-parameters of irreducible admissible representations of GSp4 over a non-archimedean local field. We explain that this is unlikely to be used in conjunction with existing results on congruences for GSp4/Q to deduce a theory of congruences over imaginary quadratic fields. On the other hand, we prove a result which does give some such congruence results by twisting.

Homogeneous Einstein–Randers metrics on symplectic groups

In this paper, we prove that Lie group Sp(n)(n≥3) admits at least one family of invariant Einstein–Randers metrics, which are not naturally reductive. We also give the classification of these metrics under isometries and determine their group of isometries. Moreover, we prove that if homogeneous Randers space (M,F) with navigation data (h,W) is Berwald, then (M,F) is naturally reductive if and only if the Riemannian metric (M,h) is naturally reductive.

Cohomology of symplectic groups and Meyer’s signature theorem

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z} \to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to 1$ the pullback of the universal cover of $\mathsf{ Sp}(2g,\mathbb{R})$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g, \mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf {Sp}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}/2\to\tilde{{\mathfrak{H}}}\to\mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$. It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\widetilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.

Arthur's multiplicity formula for certain inner forms of special orthogonal and symplectic groups

Let \mathbf G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which \mathbf G has discrete series and outside of which \mathbf G is quasi-split. We prove Arthur’s multiplicity formula for automorphic representations of \mathbf G having algebraic regular infinitesimal character at all places in S .