The (semi-infinite) Pfaff lattice was introduced by Adler and van Moerbeke [2] to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the τ -functions of the Pfaff lattice. In this paper, we study a finite version of...

We consider the situation of a compact irreducible subvariety of a smooth compact complex variety equipped with a Kähler form preserved by a torus action. We study the image of that subvariety under the moment map of the Kähler form. 1. Main result The moment map of a Hamiltonian T -symplectic/Kähler manifold has been one of the main interests of study for mathematical as well as physical reaso...

We define a collection of polytopes associated to a semisimple group G. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirković and K. Vilonen. These cycles are a canonical basis for the intersection homology of (the closures o...

Abstract The positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real whose Plücker coordinates are non-negative. In this paper we consider image and its positroid cells under two different maps: moment map$\mu $ onto hypersimplex [ 31] amplituhedron map$\tilde{Z}$ 6]. For either map, define dissection to be collection images that disjoint cover dense sub...

The main contribution of this manuscript is a local normal form for Hamiltonian actions Poisson-Lie groups $K$ on symplectic manifold equipped with an $AN$-valued moment map, where $AN$ the dual group $K$. Our proof uses delinearization theorem Alekseev which relates classical action $\mathfrak{k}^*$-valued map to via deformation structures. We obtain our result by proving ``delinearization com...

We compute the first and second homotopy groups of a class of contact toric manifolds in terms of the images of the associated moment map.

We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.

We prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges to a holomorphic map with zero average moment map. Conversely, any such map is the limit of such a sequence. We use these results to equate the gauged Gromov-Witten invariants for sufficiently small area with the invariant part of equivariant Gromov-Witten invariants, for convex tar...

Let X be any nonsingular complex projective variety with a linear action of a complex reductive group G, and let X and X be the sets of semistable and stable points of X in the sense of Mumford’s geometric invariant theory [17]. We can choose a maximal compact subgroup K of G and an inner product on the Lie algebra k of K which is invariant under the adjoint action. Then X has a G-equivariantly...

The purpose of this note is to define a tri-moment map for certain manifolds with an Sp(1)-action. We show how this map can be used ro reduce such manifolds. The images of such maps for quaternionic flag manifolds, which are defined using the Dieudonné determinant, resemble the polytopes from the complex case.