Factorial Characters and Tokuyama's Identity for Classical Groups

International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.

The Homology Class of a Poisson Transversal

This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.

Towards a Goldberg–Shahidi pairing for classical groups

Let{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

Exact functors on perverse coherent sheaves

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $R{\rm\Gamma}_{Z}{\mathcal{F}}$ is concentrated in degree $0$ for special subvarieties $Z$ of $X$. These subvarieties $Z$ are analogs of Lagrangians in the symplectic case.

On Almost Hyper-Para-Kähler Manifolds

In this paper it is shown that a -dimensional almost symplectic manifold can be endowed with an almost paracomplex structure , , and an almost complex structure , , satisfying for , for and , if and only if the structure group of can be reduced from (or ) to . In the symplectic case such a manifold is called an almost hyper-para-Kähler manifold. Topological and metric properties of almost hyper-para-Kähler manifolds as well as integrability of are discussed. It is especially shown that the Pontrjagin classes of the eigenbundles of to the eigenvalues depend only on the symplectic structure and not on the choice of .

CAPACITIES AND DISPLACEMENT ENERGY IN CONTACT HOFER GEOMETRY

A contact version of a Laudenbach's engulfing theorem is proved. Some properties of the notions of contact displacement energy and contact Hofer–Zehnder capacities are presented and, under the condition of existence of a modified action selector on a contact manifold, we can prove some inequalities involving these invariants. These inequalities are similar to the ones obtained by Frauenfelder, Ginzburg and Schlenk, in the symplectic case.

Constructing the symplectic Evans matrix using maximally analytic individual vectors

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.