Showing papers on "Symplectic group published in 2019"

Squeezing formalism and canonical transformations in cosmology

Abstract: Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameter-isations, but also generate the dynamics of the system. In the first part of this work we study the symplectic structure associated with linear canonical transformations. After reviewing salient mathematical properties of the symplectic group in a pedagogical way, we introduce the squeezing formalism, and show how any linear dynamics can be cast in terms of an invariant representation. In the second part, we apply these results to the case of cosmological perturbations, and focus on scalar field fluctuations during inflation. We show that di↵erent canonical variables select out di↵erent vacuum states, and that this leaves an ambiguity in observational predictions if initial conditions are set at a finite time in the past. We also discuss how the e↵ectiveness of the quantum-to-classical transition of cosmological perturbations depends on the set of canonical variables used to describe them. Keywords: quantum field theory on curved space, physics of the early universe, inflation

Howe Correspondence of Unipotent Characters for a Finite Symplectic/Even-orthogonal Dual Pair

Abstract: In this paper we give a complete description of the Howe correspondence of unipotent characters for a finite dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization under a mild restriction of the characteristic of the base field. That is, the conjecture by Aubert-Michel-Rouquier is confirmed.

Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group

Abstract: In this work we develop an integral representation for the partial $L$-function of a pair $\pi\times\tau$ of genuine irreducible cuspidal automorphic representations, $\pi$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $\tau$ of a certain covering group of $GL_k$, with arbitrary $m$, $n$ and $k$. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Possible applications include an analytic definition of local factors for representations of covering groups, and a Shimura type lift of representations from covering groups to general linear groups.

Canonical transformations and squeezing formalism in cosmology

Abstract: Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameterisations, but also generate the dynamics of the system. In the first part of this work we study the symplectic structure associated with linear canonical transformations. After reviewing salient mathematical properties of the symplectic group in a pedagogical way, we introduce the squeezing formalism, and show how any linear dynamics can be cast in terms of an invariant representation. In the second part, we apply these results to the case of cosmological perturbations, and focus on scalar field fluctuations during inflation. We show that different canonical variables select out different vacuum states, and that this leaves an ambiguity in observational predictions if initial conditions are set at a finite time in the past. We also discuss how the effectiveness of the quantum-to-classical transition of cosmological perturbations depends on the set of canonical variables used to describe them.

The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices

Abstract: Horn's problem, i.e., the study of the eigenvalues of the sum C " A`B of two matrices, given the spectrum of A and of B, is reexamined , comparing the case of real symmetric, complex Hermitian and self-dual quaternionic 3 ˆ 3 matrices. In particular, what can be said on the probability distribution function (PDF) of the eigenvalues of C if A and B are independently and uniformly distributed on their orbit under the action of, respectively, the orthogonal, unitary and symplectic group ? While the two latter cases (Hermitian and quaternionic) may be studied by use of explicit formulae for the relevant orbital integrals, the case of real symmetric matrices is much harder. It is also quite intriguing, since numerical experiments reveal the occurrence of singularities where the PDF of the eigenvalues diverges. Here we show that the computation of the PDF of the symmetric functions of the eigenvalues for traceless 3 ˆ 3 matrices may be carried out in terms of algebraic functions-roots of quartic polynomials-and their integrals. The computation is carried out in detail in a particular case, and reproduces the expected singular patterns. The divergences are of logarithmic or inverse power type. We also relate this PDF to the (rescaled) structure constants of zonal polynomials and introduce a zonal analogue of the Weyl SUpnq characters.

Lusztig Correspondence and Howe Correspondence for Finite Reductive Dual Pairs

Abstract: Let $(G,G')$ be a finite reductive dual pair of a symplectic group and an orthogonal group. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a subset of irreducible characters of $G'$. The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^*)^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^*$. In this paper, we prove the commutativity (up to a twist of the sign character) between these two correspondences under some restriction on the characteristic of the finite field.

Equivariant cohomology of moduli spaces of genus three curves with level two structure

Abstract: We study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked point and strata of the moduli space of Abelian differentials of genus three.

Equivariant quantum cohomology of the odd symplectic Grassmannian

Abstract: The odd symplectic Grassmannian $$\mathrm {IG}:=\mathrm {IG}(k, 2n+1)$$ parametrizes k dimensional subspaces of $${\mathbb {C}}^{2n+1}$$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $$\mathrm {IG}$$ with two orbits, and $$\mathrm {IG}$$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $$\mathrm {IG}(k, 2n+2)$$ . We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $$\mathrm {IG}$$ , i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case $$k=2$$ , and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

Crystal structure on King tableaux and semistandard oscillating tableaux

Abstract: In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(2m)$ in terms of semistandard oscillating tableaux.

Involution pipe dreams

Abstract: Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Grobner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.

Modular invariants of finite gluing groups

Abstract: 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.

On the Bruhat-Tits stratification of a quaternionic unitary Shimura variety

Abstract: In this note we study the special fiber of the Rapoport-Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat-Tits stratification and is intimately related to the Bruhat-Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.

Symplectic resolutions of character varieties

Abstract: In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that these varieties are symplectic singularities and classify which admit symplectic resolutions. The classification reduces to the semi-simple case, where we show that a resolution exists if and only if either $g=1$ and $G$ is a product of special linear groups of any rank and copies of the group $PGL_2$, or if $g=2$ and $G = (SL_2)^m$ for some $m$.

Euler-Poincaré Equation for Lie Groups with Non Null Symplectic Cohomology. Application to the Mechanics

Abstract: Let G be a symplectic group on a symplectic manifold \(\mathcal {N}\). To any momentum map \(\psi : \mathcal {N} \rightarrow \mathfrak {g}^*\), one can associate a class of symplectic cohomology cocs. It does not depend on the choice of the momentum map but only on the structure of the Lie group G. It forwards an affine left action \(\mu = a \cdot \mu ' = Ad^* (a)\, \mu ' + cocs (a)\) of G on \(\mathfrak {g}^*\).

The Langlands parameter of a simple supercuspidal representation: symplectic groups

Abstract: Let $$\pi $$ be a simple supercuspidal representation of the symplectic group $${\mathrm {Sp}}_{2l}(F)$$, over a p-adic field F. In this work, we explicitly compute the Rankin–Selberg $$\gamma $$-factor of rank-1 twists of $$\pi $$. We then completely determine the Langlands parameter of $$\pi $$, if $$p e 2$$. In the case that $$F = \mathbb {Q}_2$$, we give a conjectural description of the functorial lift of $$\pi $$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.

The Shintani double zeta functions

Abstract: In this paper, we give an explicit formula of 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.

Noncommutative coordinates for symplectic representations

Abstract: We introduce coordinates on the space of Lagrangian decorated and framed representations of the fundamental group of a surface with punctures into the symplectic group Sp(2n,R). These coordinates provide a non-commutative generalization of the parametrizations of the spaces of representations into SL(2,R) given by Thurston, Penner, and Fock-Goncharov. With these coordinates, the space of framed symplectic representations provides a geometric realization of the non-commutative cluster algebras introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of decorated maximal representations. We use this to determine the homotopy type of the space of decorated maximal representations, and its homeomorphism type when n=2.

Scattering Amplitudes & Orbits of Cognitive Representations under Subgroup of Symplectic Group Respecting the Extension of Rationals

Abstract: In this article the ideas inspired by the work of number theorist Minhyong Kim are applied to the construction of scattering amplitudes with finite cognitive precision in terms of cognitive representations and their orbits under subgroup S D of symplectic group respecting the extension of rationals defining the adele. One could pose to S D the additional condition that it leaves the value of action invariant: call this group S D,S : this would define what I have called micro-canonical ensemble (MCE). The obvious question is whether the simplest zero energy states could correspond to single orbit of S D or whether several orbits are required. For the more complex option zero energy states would be superposition of states corresponding to several orbits of S D with coefficients constructed of symplectic invariants. The following arguments lead to the conclusion that MCE and single orbit orbit option are non-realistic, and raise the question whether the orbits of S D could combine to an orbit of its Yangian analog. A generalization of the formula for scattering amplitudes in terms of n-point functions emerges and somewhat surprisingly one finds that the unitarity is an automatic consequence of state orthonormalization in zero energy ontology.

A new look at the classification of the tri-covectors of a 6-dimensional symplectic space

Abstract: Let F be a field of characteristic ≠2 and 3, let V be a F -vector space of dimension 6, and let Ω∈∧2V∗ be a non-degenerate form. A system of generators for polynomial invariant functions under the ...

Covariant projective representation of symplectic group on discrete phase space

Abstract: The phase point operator ∆(q, p) is the quantum mechanical counterpart of the classical phase point (q, p). The discrete form of ∆(q, p) was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective unitary representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.

Unit triangular factorization of the matrix symplectic group

Abstract: 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, (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.

Product of symplectic groups and its cyclic orbit code

Abstract: Constant dimension subspace codes are subsets of the finite Grassmann Variety. Orbit codes are constant dimension subspace codes that arise as the orbit of subgroup of general linear group acting o...

Linear representations of hyperelliptic mapping class groups

Abstract: Let $p:S\to S_g$ be a finite $G$-covering of a closed surface of genus $g\geq 1$ and let $B$ its branch locus. To this data, it is associated a representation of a finite index subgroup of the mapping class group $\operatorname{Mod}(S_g\smallsetminus B)$ in the centralizer of the group $G$ in the symplectic group $\operatorname{Sp}(H_1(S,{\mathbb Q}))$. They are called \emph{virtual linear representations} of the mapping class group and are related, via a conjecture of Putman and Wieland, to a question of Kirby and Ivanov on the abelianization of finite index subgroup of the mapping class group. The purpose of this paper is to study the restriction of such representations to the hyperelliptic mapping class group $\operatorname{Mod}(S_g,B)^\iota$, which is a subgroup of $\operatorname{Mod}(S_g\smallsetminus B)$ associated to a given hyperelliptic involution $\iota$ on $S_g$. We extend to hyperelliptic mapping class groups some previous results on virtual linear representations of the mapping class group. We then show that, for all $g\geq 2$, there are virtual linear representations of the hyperelliptic mapping class group with nontrivial finite orbits, associated to $G$-coverings of $(S_g,\iota)$ ramified over the locus of Weierstrass points.

Gauss-Manin connection in disguise: Genus two curves

Abstract: We describe an algebra of meromorphic functions on the Siegel domain of genus two which contains Siegel modular forms for an arithmetic index six subgroup of the symplectic group and it is closed under three canonical derivations of the Siegel domain. The main ingredients of our study are the moduli of enhanced genus two curves, Gauss-Manin connection and the modular vector fields living on such moduli spaces.

Radial wavelet transform on Iwasawa subgroups

Abstract: Let N2A2K2 be the Iwasawa decomposition of symplectic group Sp(2,R). The Iwasawa subgroup N2A2 can be realized as the tube domain T. We obtain the Hankel transform on Hardy spaces via the radial wa...

Secret Link Between Pure Math & Physics Uncovered

Abstract: Number theorist Minhyong Kim has speculated about very interesting general connection between number theory and physics. The reading of a popular article about Kim's work revealed that number theoretic vision about physics provided by TGD has led to a very similar ideas and suggests a concrete realization of Kim's ideas. In the following I briefly summarize what I call identification problem. The identification of points of algebraic surface with coordinates, which are rational or in extension of rationals, is in question. In TGD framework the imbedding space coordinates for points of space-time surface belonging to the extension of rationals defining the adelic physics in question are common to reals and all extensions of p-adics induced by the extension. These points define what I call cognitive representation, whose construction means solving of the identification problem. Cognitive representation defines discretized coordinates for a point of "world of classical worlds" (WCW) taking the role of the space of spaces in Kim's approach. The symmetries of this space are proposed by Kim to help to solve the identification problem. The maximal isometries of WCW necessary for the existence of its Kahler geometry provide symmetries identifiable as symplectic symmetries. The discrete subgroup respecting extension of rationals acts as symmetries of cognitive representations of space-time surfaces in WCW, and one can identify symplectic invariants characterizing the space-time surfaces at the orbits of the symplectic group.

Codes, designs and graphs obtained from some projective symplectic group

Abstract: After the classification of finite simple groups, there is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in enumerating and characterizing a structure which either has a particular group acting on it or which has some degree of symmetry from a group action. It has been shown that there exists interplay between finite simple groups and codes. In this thesis we construct and enumerate binary linear codes for the projective symplectic group S8(2) from the permutation representations of degree 120, 136, 255, 2295, 5355, 5440 and 11475. We find that the support of codewords of a given weight in a code hold a combinatorial design, or represent points of a projective space PG(2m− 1, q), or represent the rows of the adjacency matrix of a graph or equivalently are the incidence vectors of the blocks of a design. Through coding theory, the interplay between the combinatorial objects is enhanced and the internal structures of the group characterized.

Automorphic Forms and Hecke Operators

Abstract: We first introduce the Hecke ring of a \(\mathbb {Z}\)-group G and discuss it basic properties (local-global structure, compatibility with isogenies, criterion for commutativity…). An elementary description of the Hecke rings of classical groups is given. Then, we recall the notion of a square integrable automorphic form for G, and that of a discrete automorphic representation of G. When G is the symplectic group Sp2g, we explain how the theory of Siegel modular forms fits into this picture. We also show how the p-neighbor problem for even unimodular lattices in rank n may be viewed as a question about automorphic representations for the orthogonal \(\mathbb {Z}\)-group On.