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A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2 n < 250 , specifically, the question remains open only 7 values of n. [ABSTRACT FROM AUTHOR]

In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses. [ABSTRACT FROM AUTHOR]

In this paper, we study the Majorana representations of symplectic and orthogonal groups over GF ⁢ ( 2 ) \mathrm{GF}(2) . We show that such a group admits a Majorana representation only if it can be embedded into the Monster group as a 2 ⁢ A 2A -generated subgroup. [ABSTRACT FROM AUTHOR]

The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices (PDMs) due to Heisenberg's uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix (CM) produces a PDM that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a CM of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state. [ABSTRACT FROM AUTHOR]

For reductive groups G over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations π of G (A) to irreducible cohomological automorphic representations of H (A) for the quasi-split inner form H of G , and other inner forms as well. We show the existence of nontrivial weak global cohomological liftings in many cases, in particular for the case where G is anisotropic at the archimedean places. A priori, for these weak liftings we do not give a description of the precise nature of the corresponding local liftings at the ramified places, nor do we characterize the image of the lifting. For inner forms of the group H = GSp (4) however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level. [ABSTRACT FROM AUTHOR]

An automorphism u of a vector space is called unipotent of index 2 whenever (u − id) 2 = 0. Let b be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space V over a field F of characteristic different from 2. Here, we characterize the elements of the isometry group of b that are the product of two unipotent isometries of index 2. In particular, if b is skewsymmetric and nondegenerate we prove that an element of the symplectic group of b is the product of two unipotent isometries of index 2 if and only if it has no Jordan cell of odd size for the eigenvalue −1. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index 2 (and no fewer in general). For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article [7]. [ABSTRACT FROM AUTHOR]

We show that the natural bi-invariant cone structure on the linear symplectic group Sp (2 n) is globally hyperbolic in the positively elliptic region Sp ell + (2 n). This answers a question by Abbondandolo, Benedetti and Polterovich and implies a formula for a bi-invariant Lorentzian distance function defined by these authors for elements in this region. Moreover we give a characterization of the positively elliptic region and of − id ∈ Sp (2 n) in terms of the causality of this cone structure. [ABSTRACT FROM AUTHOR]

Let G be a symplectic or orthogonal group defined over a finite field with odd characteristic and let D ≤ G be a Sylow 2-subgroup. In this paper, we classify the essential 2-subgroups and determine the essential 2-rank of the Frobenius category ℱD(G). Together with the results of An-Dietrich and Cao–An–Zeng, this completes the work of essential subgroups and essential ranks of classical groups. [ABSTRACT FROM AUTHOR]

We study commutative Schur rings over the symplectic groups Sp (n , 2) that contain the class C of symplectic transvections. We find the possible partitions of C determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp (n , 2). [ABSTRACT FROM AUTHOR]

In this paper, we focus on Oliver's p -group conjecture. We prove that Oliver's p -group conjecture holds for Sylow p -subgroups of symplectic groups and orthogonal groups. [ABSTRACT FROM AUTHOR]

Let F be a non-archimedean local field of characteristic zero and (π , V) a depth zero, irreducible, supercuspidal representation of GSp (4 , F). We calculate the dimensions of the spaces of Klingen-invariant vectors in V of level p n for all n ≥ 0. [ABSTRACT FROM AUTHOR]

Let s be an n -dimensional symplectic form over an arbitrary field with characteristic not 2, with n > 2. The simplicity of the group Sp (s) / { ± id } and the existence of a non-trivial involution in Sp (s) yield that every element of Sp (s) is a product of involutions. Extending and improving recent results of Awa, de La Cruz, Ellers and Villa with the help of a completely new method, we prove that if the underlying field is infinite, every element of Sp (s) is the product of four involutions if n is a multiple of 4, and of five involutions otherwise. The first part of this result is shown to be optimal for all multiples of 4 and all fields, and is shown to fail for the fields with three elements and for n = 4. Whether the second part of the result is optimal remains an open question. Finite fields will be tackled in a subsequent article. [ABSTRACT FROM AUTHOR]

In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez, Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters, following Kaletha, Minguez, Shin, and White. [ABSTRACT FROM AUTHOR]

We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat-Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters. [ABSTRACT FROM AUTHOR]

We consider morphisms of smooth projective varieties over . We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections. [ABSTRACT FROM AUTHOR]

In this paper we give a complete and explicit description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization. That is, the conjecture by Aubert-Michel-Rouquier is confirmed. [ABSTRACT FROM AUTHOR]

We consider the extended metaplectic representation of the semi-direct product of the Heisenberg group and the symplectic group (the Jacobi group). We give explicit formulas for the Berezin symbols and for the complex Weyl symbols of the corresponding representation operators. Then we deduce formulas for the symbols of the representation operators in the classical Weyl calculus. As an application, we find the classical Weyl symbol of the exponential of an operator whose Weyl symbol is a polynomial on ℝ2n of degree ≤2, recovering a result of L. Hörmander. [ABSTRACT FROM AUTHOR]

We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group S p (2 N) and the orthogonal ensembles S O (2 N) and O − (2 N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L -functions with symplectic and orthogonal symmetry. [ABSTRACT FROM AUTHOR]

For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of ladder representations for split odd special orthogonal groups and symplectic groups over a non-archimedean local field of characteristic zero. This is a natural class in the admissible dual, which contains both strongly positive discrete series representations and irreducible representations with irreducible |$A$| -parameters. We compute Jacquet modules and the Aubert duals of ladder representations, and we establish a formula for describing ladder representations in terms of linear combinations of standard modules. [ABSTRACT FROM AUTHOR]

In this paper, we determine all simple block-transitive 3-designs on 65 points with block stabilizer of order greater than 2, admitting the 2-transitive action of the group S z (8). As a result, we find 23 3-designs on 65 points that have S z (8) as an automorphism group. Also we construct four 3-(3 6 , 1 8 , λ 1 ) designs for λ 1 ∈ { 7 6 8 , 5 1 8 4 , 6 9 1 2 , 9 2 1 6 } and two 3-(3 6 , 1 7 , λ 2 ) designs for λ 2 ∈ { 2 8 8 0 , 8 6 4 0 }. All of these designs have S 6 (2) as the full automorphism group. Some of these designs are flag and anti-flag-transitive. [ABSTRACT FROM AUTHOR]

We study Artin–Schreier varieties with natural actions of finite orthogonal groups. We compute the characters of the cohomology groups of these varieties as representations of the groups. Our main ingredient is Shinoda's character formulae on Weil representations for symplectic groups. We give trace formulae for projective hypersurfaces with actions of finite orthogonal groups generalizing Fermat hypersurfaces. [ABSTRACT FROM AUTHOR]

In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety X, with values in the space of Picard Lie algebroid over X. As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition. [ABSTRACT FROM AUTHOR]

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$ , producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture. [ABSTRACT FROM AUTHOR]

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest to vary the orthogonal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. In this work we study the existence of generic lifts in the resulting theta tower. In the classical case, there are two orthogonal groups that may support a generic lift of an irreducible cuspidal automorphic representation of a symplectic group. We show that in general the Whittaker range consists of r + 1 groups for the lift from the r -fold cover of a symplectic group. We also give a period criterion for the genericity of the lift at each step of the tower. [ABSTRACT FROM AUTHOR]

In this paper, we prove a formula, realizing certain residual Eisenstein series on symplectic groups as regularized kernel integrals. These Eisenstein series, as well as the kernel integrals, are attached to Speh representations. This forms an initial step to a new type of a regularized Siegel-Weil formula that we propose. This new formula bears the same relation to the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan, as does the regularized Siegel-Weil formula to the doubling integrals of Piatetski-Shapiro and Rallis. [ABSTRACT FROM AUTHOR]

We construct new three-family N = 1 supersymmetric Pati–Salam models from intersecting D6-branes with original gauge group U (4) C × USp (2) L × U (2) R on a Type IIA T 6 / (Z 2 × Z 2) orientifold. We find that replacing a U (2) with a USp (2) group severely restricts the number of three-generation supersymmetric models such that there are only five inequivalent models. Exchanging the left and right sectors, we obtain five dual models with gauge group U (4) C × U (2) L × USp (2) R . These ten models have different gauge coupling relations at string scale. The highest wrapping number is 4, and one of the models contains no filler O6-planes. Moreover, we discuss in detail the particle spectra, the composite particles through strong coupling dynamics, and the exotic particle decouplings. Also, we study how to realize the string-scale gauge coupling relations in some models. [ABSTRACT FROM AUTHOR]

We prove that a word hyperbolic group whose Gromov boundary properly contains a 2‐sphere cannot admit a projective Anosov representation into Sp2m(C)$\mathsf {Sp}_{2m}(\mathbb {C})$, m∈N$m\in \mathbb {N}$. We also prove that a word hyperbolic group that admits a projective Anosov representation into Sp2m(R)$\mathsf {Sp}_{2m}(\mathbb {R})$ is virtually a free group or virtually a surface group, a result established independently by Dey–Greenberg–Riestenberg. [ABSTRACT FROM AUTHOR]

Let G be a finite group, cd(G) the set of all irreducible character degrees of G, and p(G) the set of all prime divisors of integers in cd(G). For a prime p and a positive integer n, let np denote the p-part of n. The degree prime-power graph of G is a graph whose vertex set is V(G) = {...| p ∈ p(G)}, where ... = max {np | n ∈ cd(G)}, and there is an edge between distinct numbers x, y ∈ V(G) if xy divides some integer in cd(G). The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group PS p4(p) can be uniquely identified by its order and degree prime-power graph. [ABSTRACT FROM AUTHOR]

We study the relation between the set of oriented Z/d-homology 3-spheres and the level-d mapping class groups, the kernels of the canonical maps from the mapping class group of an oriented surface to the symplectic group with coefficients in Z/dZ. We formulate a criterion to decide whenever a Z/d-homology 3-sphere can be constructed from a Heegaard splitting with gluing map an element of the level-d mapping class group. Then, we give a tool to construct invariants of Z/d-homology 3-spheres from families of trivial 2-cocycles on the level-d mapping class groups. We apply this tool to find all the invariants of Z/p-homology 3-spheres constructed from families of 2-cocycles on the abelianization of the level-p mapping class group with p prime and to disprove the conjectured extension of the Casson invariant modulo a prime p to rational homology 3-spheres due to B. Perron. [ABSTRACT FROM AUTHOR]

For noncuspidal irreducible admissible representations of GSp(4, k) over a local non‐Archimedean field k, we determine the exceptional poles of the spinor L‐factor attached to anisotropic Bessel models by Piatetski–Shapiro. This completes the classification of spinor L‐factors for GSp(4, k). [ABSTRACT FROM AUTHOR]

In this paper, we give an algorithm to determine all local 퐴-packets containing a given irreducible representation of a split special odd orthogonal group or a symplectic group over a 푝-adic field. Especially, we can determine whether a given irreducible representation is of Arthur type or not. [ABSTRACT FROM AUTHOR]

Let (G , Ω) be a symplectic Lie group, i.e., a Lie group endowed with a left invariant symplectic form. If g is the Lie algebra of G then we call (g , ω = Ω (e)) a symplectic Lie algebra. The product • on g defined by 3 ω (x • y , z) = ω ([ x , y ] , z) + ω ([ x , z ] , y) extends to a left invariant connection ∇ on G which is torsion free and symplectic ( ∇ Ω = 0) . When ∇ has vanishing curvature, we call (G , Ω) a flat symplectic Lie group and (g , ω) a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to ω. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection ∇ of a flat symplectic Lie group is always complete. We prove that the flat symplectic double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of flat symplectic double extension of flat symplectic Lie algebras starting from {0}. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension ≤ 6 . [ABSTRACT FROM AUTHOR]

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group W E (R , I) is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol V R , I : U m 3 (R , I) / E 3 (R , I) → W E (R , I). We establish injectivity of this symbol if R is a non-singular affine algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective. [ABSTRACT FROM AUTHOR]

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is 960 and if such a group has order 960, then it is isomorphic to the Mathieu group M20$M_{20}$. In this paper, we are interested in projective K3 surfaces admitting a faithful symplectic action of the group M20$M_{20}$. We show that there are infinitely many K3 surfaces with this action and we describe them and their projective models, giving some explicit examples. [ABSTRACT FROM AUTHOR]

We use a method developed by Björklund and Gorodnik to show a central limit theorem (as $ T $ tends to $ \infty $) for the counting functions $ \# \left(\Lambda \cap \Omega_T \right) $ where $ \Lambda $ ranges over the space $ {Y_{2d}} $ of symplectic lattices in $ {\bf R}^{2d} $ ($ d \geqslant 4 $). Here $ \left\lbrace \Omega_T \right\rbrace_T $ is a certain family of bounded domains in $ {\bf R}^{2d} $ that can be tessellated by means of the action of a diagonal semigroup contained in $ \mathrm{Sp} (2d, {\bf R}) $. In the process we obtain new $ L^p $ bounds on a certain height function on $ {Y_{2d}} $ originally introduced by Schmidt. [ABSTRACT FROM AUTHOR]

We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed φ-congruence subgroups, are obtained by reducing homomorphisms φ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasiunipotent case, we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve y² = x³ - 1728 defined by the commutator subgroup of the modular group. [ABSTRACT FROM AUTHOR]

We present a different symplectic point of view in the definition of weighted modulation spaces M m p , q (R d) and weighted Wiener amalgam spaces W (F L m 1 p , L m 2 q) (R d). All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the τ -Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions μ (A) (f ⊗ g ¯) , where μ (A) is the metaplectic operator and A is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13] , the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications. [ABSTRACT FROM AUTHOR]

The Torelli group of a genus g$g$ oriented surface Σg$\Sigma _g$ is the subgroup Ig$\mathcal {I}_g$ of the mapping class group Mod(Σg)${\rm Mod}(\Sigma _g)$ consisting of all mapping classes that act trivially on H1(Σg,Z)${\rm H}_1(\Sigma _g, \mathbb {Z})$. The quotient group Mod(Σg)/Ig${\rm Mod}(\Sigma _g) / \mathcal {I}_g$ is isomorphic to the symplectic group Sp(2g,Z)${\rm Sp}(2g, \mathbb {Z})$. The cohomological dimension of the group Ig$\mathcal {I}_g$ equals to 3g−5$3g-5$. The main goal of the present paper is to compute the top homology group of the Torelli group in the case g=3$g = 3$ as Sp(6,Z)${\rm Sp}(6, \mathbb {Z})$‐module. We prove an isomorphism H4(I3,Z)≅IndS3⋉SL(2,Z)×3Sp(6,Z)Z,$$\begin{equation*} \hspace*{4pc}{\rm H}_4(\mathcal {I}_3, \mathbb {Z}) \cong {\rm Ind}^{{\rm Sp}(6, \mathbb {Z})}_{S_3 \ltimes {\rm SL}(2, \mathbb {Z})^{\times 3}} \mathcal {Z}, \end{equation*}$$where Z$\mathcal {Z}$ is the quotient of Z3$\mathbb {Z}^3$ by its diagonal subgroup Z$\mathbb {Z}$ with the natural action of the permutation group S3$S_3$ (the action of SL(2,Z)×3${\rm SL}(2, \mathbb {Z})^{\times 3}$ is trivial). We also construct an explicit set of generators and relations for the group H4(I3,Z)${\rm H}_4(\mathcal {I}_3, \mathbb {Z})$. [ABSTRACT FROM AUTHOR]

Let ω$\omega$ be a Kähler form on the real 4‐torus T4$T^4$. Suppose that ω$\omega$ satisfies an irrationality condition that can be achieved by an arbitrarily small perturbation of ω$\omega$. This note shows that the smoothly trivial symplectic mapping class group of the one‐point symplectic blowup of (T4,ω)$(T^4,\omega)$ is infinitely generated. [ABSTRACT FROM AUTHOR]

We introduce the Q (s) -index ind Γ for a symplectic orthogonal group Q (s) and Q (s) invariant subset Γ of R 2 n and prove that ind S 2 n − 1 = n. Using this fact, we study multiple rotating periodic orbits of Hamiltonian systems. For an orthogonal matrix Q , a Q -rotating periodic solution z (t) has the form z (t + T) = Q z (t) for all t ∈ R and some constant T > 0. According to the structure of Q , it can be periodic, anti-periodic, subharmonic, or just a quasi-periodic one. Under a non-resonant condition, we prove that on each energy surface near the equilibrium, the Hamiltonian system admits at least n Q -rotating periodic orbits, which can be regarded as a Lyapunov type theorem on rotating periodic orbits. [ABSTRACT FROM AUTHOR]

We compute the Lens space index for 4d supersymmetric gauge theories involving symplectic gauge groups. This index can distinguish between different gauge groups from a given algebra and it matches across theories related by supersymmetric dualities. We provide explicit calculations for N = 4 SYM and for classes of N = 2 and N = 1 Lagrangian quivers related by S-duality. In these cases the index matches across the S-dual phases, while models in different S-duality orbits have a different Lens index. We provide analogous computations for a 4d N = 1 toric quiver gauge theory corresponding to a ℤ7 orbifold of ℂ3. This SU(n)7 gauge theory becomes interesting in the case of n = 2 because it is conformally dual to other two models, with symplectic and unitary gauge groups, bifundamentals and antisymmetric tensors. We explicitly check this triality at the level of the Lens space index. [ABSTRACT FROM AUTHOR]

Let G be a finite group and c d (G) denote the set of irreducible complex character degrees of G. Suppose that H is a projective conformal symplectic group PCS p 4 (q) for some prime power q. The main result of this paper is to prove that if c d (G) = c d (H) , then G / Z (G) is isomorphic to H. This confirms a recent generalization of Huppert's conjecture for the almost simple groups of Lie type. [ABSTRACT FROM AUTHOR]

In this paper we continue with our investigation of principal and antiprincipal solutions at infinity solutions of a dynamic symplectic system. The paper is a continuation of part I appeared in Differential Equations and Applications in 2022, where we have presenteded a theory of genera of conjoined bases for symplectic dynamic systems on time scales and its connections with principal solutions at infinity and antiprincipal solutions at infinity for these systems together with some basic properties of this new concept on time scales. Here we provide a characterization of all principal solutions of dynamic symplectic system at infinity in the given genus in terms of the initial conditions and a fixed principal solution at infinity from this genus. Further, we provide a characterization of all antiprincipal solutions of dynamic symplectic system at infinity in the given genus in terms of the initial conditions and a fixed principal solution at infinity from this genus. We also establish mutual limit properties of principal and antiprincipal solutions at infinity. [ABSTRACT FROM AUTHOR]

Let |${\operatorname {GL}}_c^{(m)}$| be the covering group of |${\operatorname {GL}}_c$|⁠ , obtained by restriction from the |$m$| -fold central extension of Matsumoto of the symplectic group. We introduce a new family of Rankin–Selberg integrals for representations of |${\operatorname {GL}}_c^{(m)}\times {\operatorname {GL}}_k^{(m)}$|⁠. The construction is based on certain assumptions, which we prove here for |$k=1$|⁠. Using the integrals, we define local |$\gamma $| -, |$L$| -, and |$\epsilon $| -factors. Globally, our construction is strong in the sense that the integrals are truly Eulerian. This enables us to define the completed |$L$| -function for cuspidal representations and prove its standard functional equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss an important discourse, as most literatures highlighted the dimension of partial flag in a certain manner, according to a fixed definition that does not include all possible cases. In the present paper the existence of two different cases of flags in the partial flag varieties of the groups SO(n,c) and SP(2n,c) is explained wholly. In addition to this we will proceed by calculating the dimensions for all the given cases after pertaining the signature of these flags that are studied. [ABSTRACT FROM AUTHOR]

We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for Mp2 (A), we give a new approach to prove the result that for an automorphic cuspidal representation of GL2(A) of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central L-values. [ABSTRACT FROM AUTHOR]

For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. This article mainly describes the endomorphism semigroups of both the types of extra-special $p$-groups and computes their cardinalities as polynomials in $p$ for each $n$. Firstly a new way of representing the extra-special $p$-group of exponent $p^2$ is given. Using the representations, explicit formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types are found. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.