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It is well-known that the symplectic Lanczos method is an efficient tool for computing a few eigenvalues of large and sparse Hamiltonian matrices. A variety of block Krylov subspace methods were introduced by Lopez and Simoncini to compute an approximation of exp(M)V for a given large square Hamiltonian matrix M and a tall and skinny matrix V that preserves the geometric property of V. For the same purpose, in this paper, we have proposed a new method based on a global version of the symplectic Lanczos algorithm, called the global J-Lanczos method (GJ-Lanczos). To the best of our knowledge, this is probably the first adaptation of the symplectic Lanczos method in the global case. Numerical examples are given to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

We consider the residues at the poles in the half plane R e (s) ≥ 0 of Eisenstein series, on symplectic groups, or their double covers, induced from Speh representations. We show that for each such pole, there is a unique maximal nilpotent orbit, attached to Fourier coefficients admitted by the corresponding residual representation. We find this orbit in each case. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on |$\operatorname{Ext}$| -group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones. [ABSTRACT FROM AUTHOR]

Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover, we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe–Gan for quasi-split even orthogonal groups. [ABSTRACT FROM AUTHOR]

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if A is a quasi-hereditary algebra with a simple preserving duality and T is a faithful tilting A-module, then A has the double centralizer property with respect to T. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module T over A for which A = EndEndA(T)(T). As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra SKsy(m,n) and the Brauer algebra Bn(−2m) on the space of dual partially harmonic tensors under certain condition. [ABSTRACT FROM AUTHOR]

In this paper we establish a connection between two important concepts from the matrix analysis, which have fundamental applications in the oscillation theory of differential equations. These are the traditional Lidskii angles for symplectic matrices and the recently introduced comparative index for a pair of Lagrangian planes. We show that the comparative index can be calculated by a specific argument function of symplectic orthogonal matrices, which are constructed from the Lagrangian planes. The proof is based on a topological property of the symplectic group and on the Sturmian separation theorem for completely controllable linear Hamiltonian systems. We apply the main result in order to present elegant proofs of certain important properties of the comparative index. [ABSTRACT FROM AUTHOR]

We prove strong multiplicity one results for Siegel eigenforms of degree two for the symplectic group Sp4(ℤ). [ABSTRACT FROM AUTHOR]

We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups. [ABSTRACT FROM AUTHOR]

Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2. [ABSTRACT FROM AUTHOR]

We propose quivers for Coulomb branch constructions of universal implosions for orthogonal and symplectic groups, extending the work on special unitary groups in [1]. The quivers are unitary-orthosymplectic as opposed to the purely unitary quivers in the A-type case. Where possible we check our proposals using Hilbert series techniques. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and α a nontrivial character of G. The McKay graph M(G,α) has the irreducible characters of G as vertices, with an edge from χ1 to χ2 if χ2 is a constituent of α χ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G = An, we prove a conjecture made in another work by the authors: there is an absolute constant C such that diam M(G,α) ≤ C log |G|/log α (1) for all nontrivial irreducible characters α of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a product χ1χ2 ⋅⋅⋅ χl of irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents. [ABSTRACT FROM AUTHOR]

Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2. [ABSTRACT FROM AUTHOR]

We propose dualities of N = (0, 2) supersymmetric boundary conditions for 3d N = 2 gauge theories with orthogonal and symplectic gauge groups. We show that the boundary 't Hooft anomalies and half-indices perfectly match for each pair of the proposed dual boundary conditions. [ABSTRACT FROM AUTHOR]

We compute the first two symplectic quadratic K-theory groups of the integers, or equivalently, the first two stable homology groups of the group of symplectic integral matrices preserving the standard quadratic refinement. The main novelty in our calculation lies in its method, which is based on high-dimensional manifold theory. [ABSTRACT FROM AUTHOR]

We classify the commutator preserving bijections of the unipotent radical U p (2 n , F) of the Borel subgroup of the classical symplectic group of rank at least 2 over a field F such that 6F=F. Every such a bijection is shown to be the composition of a standard automorphism of U p (2 n , F) and a central map. The latter is the identity modulo the centre of U p (2 n , F). [ABSTRACT FROM AUTHOR]

Let M be a random matrix chosen according to Haar measure from the unitary group U(n,C). Diaconis and Shahshahani proved that the traces of M,M2, . . . ,Mk converge in distribution to independent normal variables as n → ∞, and Johansson proved that the rate of convergence is superexponential in n. We prove a finite field analogue of these results. Fixing a prime power q = pr, we choose a matrix M uniformly from the finite unitary group U(n,q)⊆ GL(n,q2) and show that the traces of {Mi}1≤i≤k,p|i converge to independent uniform variables in Fq2 as n → \∞. Moreover we show the rate of convergence is exponential in n2. We also consider the closely related problem of the rate at which characteristic polynomial of M equidistributes in 'short intervals' of Fq2[T]. Analogous results are also proved for the general linear, special linear, symplectic and orthogonal groups over a finite field. In the two latter families we restrict to odd characteristic. The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial. [ABSTRACT FROM AUTHOR]

In [5] the authors showed that the Liouville 1-form lying on the cotangent bundle is derived from physical potential and is related to the symplectomorphism through the ux homomorphism. On the other hand, in [7, 8], A. Weinstein constructed a chart from the group of symplectic diffeomorphisms isotopic to the identity by using Lagrangian sub-manifolds geometry and from which he derived a closed 1-form called the Weinstein 1-form. In this paper, we establish a relation between the Liouville 1-form and the Weinstein 1-form through an explicit formula from which we derive a new characterization of symplectomorphism and a new formula of the ux homomorphism. [ABSTRACT FROM AUTHOR]

Our computations show that there is a total of 40 pairs of degree six coprime polynomials f , g where f (x) = (x − 1) 6 , g is a product of cyclotomic polynomials, g (0) = 1 and f , g form a primitive pair. The aim of this article is to determine whether the corresponding 40 symplectic hypergeometric groups with a maximally unipotent monodromy follow the same dichotomy between arithmeticity and thinness that holds for the 14 symplectic hypergeometric groups corresponding to the pairs of degree four polynomials f , g where f (x) = (x − 1) 4 and g is as described above. As a result we prove that at least 18 of these 40 groups are arithmetic in Sp 6. In addition, we extend our search to all degree six symplectic hypergeometric groups and find that there is a total of 458 pairs of polynomials (up to scalar shifts) corresponding to such groups. For 211 of them, the absolute values of the leading coefficients of the difference polynomials f − g are at most 2 and the arithmeticity of the corresponding groups follows from Singh and Venkataramana, while the arithmeticity of one more hypergeometric group follows from Detinko, Flannery and Hulpke. In this article, we show the arithmeticity of 163 of the remaining 246 hypergeometric groups. [ABSTRACT FROM AUTHOR]

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and let k be a nonzero nature number. Suppose that nonabelian group G is a central extension as follows where G′ ≅ ℤpk, and ζG/G′ is a direct factor of G/G′.Then G is a central product of an extraspecial pk-group E and ζG. Let ∣E∣ = p(2n+1)k and ∣ζG∣ = p(m+1)k. Suppose that the exponents of E and ζG are pk+l and pk+r, respectively, where 0 ≤ l, r ≤ k. Let AutG′G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G′, let AutG/ζG,ζGG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG, and let AutG/ζG,ζG/G′G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G′. Then (i) The group extension 1 → AutG′G → Aut G → Aut G′ → 1 is split. (ii) AutG′G/AutG/ζG,ζGG ≅ G1 × G2, where , H is an extraspecial pk -group of order p(2n−1)k and . In particular, if and only if l = k and r = 0; if and only if l ≤ r; if and only if r = k; if and only if r = 0. (iii) AutG′G/AutG/ζG,ζG/G′G ≅ G1 × G3, where G1 is defined in (ii); . In particular, if and only if r = k; if and only if r = 0. (iv) . If m = 0, then ; If m > 0, then , and . [ABSTRACT FROM AUTHOR]

We construct explicit local systems on the affine line in characteristic p>2, whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for all n⩾2, and others whose geometric monodromy groups are the special unitary groups SUn(q) for all odd n⩾3, and q any power of p, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one‐parameter families of exponential sums of a very simple, that is, easy to remember, form. We also exhibit hypergeometric sheaves on Gm, whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for any n⩾2, and others whose geometric monodromy groups are the finite general unitary groups GUn(q) for any odd n⩾3. [ABSTRACT FROM AUTHOR]

The curvature tensor of a symplectic connection, as well as its covariant derivatives, satisfy certain identities that hold on any manifold of dimension less than or equal to a fixed n. In this paper, we prove certain results regarding these curvature identities. Our main result describes, for any fixed dimension and any even number p of indices, the first space (provided we have filtered the identities by a homogeneity condition) of p -covariant curvature identities. To this end, we use recent results on the theory of natural operations on Fedosov manifolds. These results allow us to apply the invariant theory of the symplectic group, with a method that is analogous to that used in Riemannian or Kähler geometry. [ABSTRACT FROM AUTHOR]

We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators. [ABSTRACT FROM AUTHOR]

We construct (n + 1)-variable Hida families of Klingen Eisenstein series on Sp(2n + 2) for n-variable Hida families on Sp(2n), and relate their images under the Siegel operator to p-adic L-functions. We also carry out some preliminary calculations of the non-degenerate Fourier coefficients of the constructed Klingen Eisenstein families. [ABSTRACT FROM AUTHOR]

We investigate the structure constants of the center H n of the group algebra Z S p n (q) over a finite field. The reflection length on the group G L 2 n (q) induces a filtration on the algebras H n. We prove that the structure constants of the associated graded algebra S n are independent of n. As tool in the proof we consider the embedding S p m ↪ S p n (q) and determine the behavior of the centralizers and intersection of centralizers under the embedding S p m ↪ S p n (q). We determine explicit formulas for rate of the growth of the centralizer of an element U ∈ S p m (q) in terms of dimension of the fixed space of U. [ABSTRACT FROM AUTHOR]

In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced in [CFGK19]. The second identity is a generalization of the descent construction studied in [GRS11]. [ABSTRACT FROM AUTHOR]

Discrete orthogonality relations for Hall–Littlewood polynomials are employed so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group |$SU(n;\mathbb{C})$|⁠). By passing to Macdonald's hyperoctahedral Hall–Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group |$Sp (n;\mathbb{H})$|⁠). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to |$SU(3;\mathbb{C})$| and |$Sp (2;\mathbb{H})$|⁠), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals. [ABSTRACT FROM AUTHOR]

We consider the subgroups in a symplectic or orthogonal group over a finite field of odd characteristic such that for some odd prime . We obtain a refinement of the well-known Aschbacher Theorem on subgroups of classical groups for this case. [ABSTRACT FROM AUTHOR]

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of K3TnU-type varieties. [ABSTRACT FROM AUTHOR]

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a t-symplectic Lie groupoid; the "t" is motivated by the fact that each target fiber of a t-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid G ⇉ M, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on AG (the associated Lie algebroid) and t-symplectic Lie groupoid structures on G ⇉ M. In addition, we also introduce the notion of a symplectic Lie group bundle (SLGB) which is a special case of both a t-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored. [ABSTRACT FROM AUTHOR]

In this paper, using a criterion given by J. Brough and B. Späth recently, we verify the inductive blockwise Alperin weight condition for the simple groups PSp 2 n (q) and any odd prime ℓ not dividing q under some assumptions concerning the decomposition matrices. [ABSTRACT FROM AUTHOR]

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow p -subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic p -group acting on an elementary abelian p -group. [ABSTRACT FROM AUTHOR]

Let R be a unital subring of a commutative ring S, which is a free R-module of rank m. In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups GL(n, S) and E(n, S) in G = GL(mn,R), and showed that they are equal and coincide with the set {g ∈ G:E(n, S)g ≤ GL(n, S)} = Aut(S=R) x GL(n, S). Moreover, for any proper ideal A of R, NG (E(n, S) E(mn,R,A)), = ... (NGL(mn,R/A)) (E(n,S/SA))). In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups EO(n, S), SO(n, S),O(n, S) and GO(n, S) in G are equal and coincide with the set {g ∈ G: EO(n, S)g ≤ GO(n, S)} = Aut(S/R) x GO(n, S). Similarly, the ones of subgroups Ep(n, S), Sp(n, S), and GSp(n, S) are equal and coincide with the set {g ∈ G:Ep(n, S)g ≤ GSp(n, S)} = Aut(S/R) x GSp(n, S). Moreover, for any proper ideal A of R, NG(EO(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (EO(n,S/SA))) and NG(Ep(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (Ep(n,S/SA))). When R = S, we obtain the known results of N. A. Vavilov and V. A. Petrov. [ABSTRACT FROM AUTHOR]

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers. [ABSTRACT FROM AUTHOR]

Recently, a strong exponential character bound was established in [Acta Math. 221 (2018), pp. 1-57] for all elements g ∈ GF of a finite reductive group GF which satisfy the condition that the centralizer CG(g) is contained in a (G,F)-split Levi subgroup M of G and that G is defined over a field of good characteristic. In this paper, assuming a weak version of Lusztig's conjecture relating irreducible characters and characteristic functions of character sheaves holds, we considerably generalize this result by removing the condition that M is split. This assumption is known to hold whenever Z(G) is connected or when G is a special linear or symplectic group and G is defined over a sufficiently large finite field. [ABSTRACT FROM AUTHOR]

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution σ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson–Thomas theory of quivers in which the quiver representations have orthogonal or symplectic structure groups. The associated invariants are called orientifold Donaldson–Thomas invariants. We prove the orientifold analogue of the integrality conjecture for σ -symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of orientifold Donaldson–Thomas invariants and the freeness of the CoHM of a σ -symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type A ~ 1 . We also verify the geometric conjecture in a number of examples. Finally, we construct explicit Poincaré–Birkhoff–Witt-type bases of the CoHM of finite type quivers. [ABSTRACT FROM AUTHOR]

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, exhibiting several examples and in particular constructing a fibration with s = 1 and irreducible singular fibre with 4 nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group S p (2 g , Z). [ABSTRACT FROM AUTHOR]

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain local Yang–Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks. We show that, in the important case of an electrical network on the standard graph introduced in [Curtis E B et al 1998 Linear Algebr. Appl. 283 115–50], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed in [Berenstein A et al 1996 Adv. Math. 122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory 9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra. [ABSTRACT FROM AUTHOR]

We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of 3-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp 6 (2). [ABSTRACT FROM AUTHOR]

A new class of representations of the Brauer algebra that centralizes the action of orthogonal and symplectic groups in tensor spaces is found. These representations make it possible to apply the technique of building primitive orthogonal idempotents of the Brauer algebra to the construction of integer spin Behrends–Fronsdal type projectors of an arbitrary type of symmetries. [ABSTRACT FROM AUTHOR]

In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. This structure-preserving factorization of the symplectic matrices immediately reveals two well-known features that (i) the determinant of any symplectic matrix is one and (ii) the matrix symplectic group is path connected, as well as a new feature that (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the matrix symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problems with symplectic constraints under certain circumstances. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the influence of A5on PSp(4,2) and PSp(4,3) and find the triangle group isomorphic to smallest simple group A5 using the Character table of Projective Symplectic groups. Also discuss some interesting results by use of character tables of the given groups. [ABSTRACT FROM AUTHOR]

We propose two conjectures on the tensor product of Weil representations with irreducible representations in finite symplectic and unitary groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider the set of r -symbols in a full generality. We construct Hall-Littlewood functions and Kostka functions associated to those r -symbols. We also discuss a multi-parameter version of those functions. We show that there exists a general algorithm of computing the multi-parameter Kostka functions. As an application, we show that the generalized Green functions of symplectic groups can be described combinatorially in terms of our (one-parameter) Kostka functions. [ABSTRACT FROM AUTHOR]

We consider a family of birational maps φk in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family φk using Poisson geometry tools, namely the properties of the restrictions of the maps φk and their fourth iterate φ(4)k to the symplectic leaves of an appropriate Poisson manifold (R4+,P). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product SL(2,Z)⋉R2. The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for φk characterized by the parameter values k = 1, k = 2 and k ≥ 3. [ABSTRACT FROM AUTHOR]

In this paper, we study the minimal symplectic elliptic surfaces E (k) with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction. [ABSTRACT FROM AUTHOR]

Purpose: Cantilever-based nanostructures have attracted widespread attention in nanoengineering. As key components of cantilever-based nanosensors, some cantilever nanoplates are exposed to magnetic fields in working conditions. This paper performs the buckling analysis of magnetically affected cantilever nanoplates resting on elastic foundations. The aim of the presented research is to discuss the effects of magnetic fields on the buckling characteristics of cantilever nanoplates. Methods: Based on Eringen's nonlocal elasticity theory, the exact buckling solutions of the cantilever nanoplates are obtained by using symplectic approach incorporating a superposition technique. To resolve the difficulties in dealing with the complex boundary value problems of cantilever nanoplates, the buckling problem is divided into two sub-problems which can be analytically solved by the symplectic approach. Exact solutions of the buckled cantilever nanoplate are derived by a superposition of the obtained solutions for sub-problems. Results: Comparisons are presented to show the accuracy and stability of the proposed method. Parametric studies of the in-plane magnetic field and elastic foundation on the buckling characteristics of cantilever nanoplate are also given. Conclusions: Highly accurate critical buckling loads and buckling mode are derived analytically. This study has shown that applying magnetic field is an effective way to control the critical buckling loads and buckling mode shapes of cantilever nanoplates. [ABSTRACT FROM AUTHOR]