Positivity of Equivariant Schubert Classes Through Moment Map Degeneration

For a flag manifold $M=G/B$ with the canonical torus action, the $T-$equivariant cohomology is generated by equivariant Schubert classes, with one class $\tau_u$ for every element $u$ of the Weyl group $W$. These classes are determined by their restrictions to the fixed point set $M^T \simeq W$, and the restrictions are polynomials with nonnegative integer coefficients in the simple roots. The main result of this article is a positive formula for computing $\tau_u(v)$ in types A, B, and C. To obtain this formula we identify $G/B$ with a generic co-adjoint orbit and use a result of Goldin and Tolman to compute $\tau_u(v)$ in terms of the induced moment map. Our formula, given as a sum of contributions of certain maximal ascending chains from $u$ to $v$, follows from a systematic degeneration of the moment map, corresponding to degenerating the co-adjoint orbit. In type A we prove that our formula is manifestly equivalent to the formula announced by Billey in \cite{Bi}, but in type C, the two formulas are not equivalent.Comment: 20 page

Equivariant de Rham Theory and Graphs

Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.Comment: AMSLaTex, 26 pages New sections added; some terminology improve

Morse theory on graphs

Let $\Gamma$ be a finite d-valent graph and G an n-dimensional torus. An ``action'' of G on $\Gamma$ is defined by a map, $\alpha$, which assigns to each oriented edge e of $\Gamma$ a one-dimensional representation of G (or, alternatively, a weight, $\alpha_e$, in the weight lattice of G). For the assignment, $e \to \alpha_e$, to be a schematic description of a ``G-action'', these weights have to satisfy certain compatibility conditions: the GKM axioms. We attach to $(\Gamma, \alpha)$ an equivariant cohomology ring, $H_G(\Gamma)=H(\Gamma,\alpha)$. By definition this ring contains the equivariant cohomology ring of a point, \SS(\fg^*) = H_G(pt), as a subring, and in this paper we will use graphical versions of standard Morse theoretical techniques to analyze the structure of $H_G(\Gamma)$ as an \SS(\fg^*)-module.Comment: 23 pages, 1 figur

G-actions on graphs

Let G be an n-dimensional torus and $\tau$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $\tau$ a representation of G on a virtual vector space, Q(M), by \spin^{\CC}-quantization. If M is a symplectic GKM manifold we will show that several well-known theorems about this ``quantum action'' of G: for example, the convexity theorem, the Kostant multiplicity theorem and the ``quantization commutes with reduction'' theorem for circle subgroups of G, are basically just theorems about G-actions on graphs.Comment: 19 page

Combinatorial formulas for products of Thom classes

Let G be a torus of dimension n > 1 and M a compact Hamiltonian G-manifold with $M^G$ finite. A circle, $S^1$, in G is generic if $M^G = M^{S^1}$. For such a circle the moment map associated with its action on M is a perfect Morse function. Let $\{ W_p^+ ; p \in M^G\}$ be the Morse-Whitney stratification of M associated with this function, and let $\tau_p^+$ be the equivariant Thom class dual to $W_p^+$. These classes form a basis of $H_G^*(M)$ as a module over \SS(\fg^*) and, in particular, $$\tau_p^+ \tau_q^+ = \sum c_{pq}^r \tau_r^+$$ with c_{pq}^r \in \SS(\fg^*). For manifolds of GKM type we obtain a combinatorial description of these $\tau_p^+$'s and, from this description, a combinatorial formula for $c_{pq}^r$.Comment: 30 page

One-skeleta, Betti numbers and equivariant cohomology

The one-skeleton of a G-manifold M is the set of points p in M where $\dim G_p \geq \dim G -1$; and M is a GKM manifold if the dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such a manifold this one-skeleton has the structure of a ``labeled" graph, $(\Gamma, \alpha)$, and that the equivariant cohomology ring of M is isomorphic to the ``cohomology ring'' of this graph. Hence, if M is symplectic, one can show that this ring is a free module over the symmetric algebra \SS(\fg^*), with $b_{2i}(\Gamma)$ generators in dimension 2i, $b_{2i}(\Gamma)$ being the ``combinatorial'' 2i-th Betti number of $\Gamma$. In this article we show that this ``topological'' result is , in fact, a combinatorial result about graphs.Comment: Revised and added content, AMSLaTex, 51 pages, 8 figure

Cardinality of Balls in Permutation Spaces

For a right invariant distance on a permutation space $S_n$ we give a sufficient condition for the cardinality of a ball of radius $R$ to grow polynomially in $n$ for fixed $R$. For the distance $\ell_1$ we show that for an integer $k$ the cardinality of a sphere of radius $2k$ in $S_n$ (for $n \geqslant k$) is a polynomial of degree $k$ in $n$ and determine the high degree terms of this polynomial

Polynomial Assignments for Bott-Samelson manifolds

Polynomial assignments for a torus $T$-action on a smooth manifold $M$ were introduced by Ginzburg, Guillemin, and Karshon in 1999; they form a module over $\mathbb{S}(\mathfrak{t}^*)$, the algebra of polynomial functions on $\mathfrak{t}$, the Lie algebra of $T$. In this paper we describe the assignment module $\mathcal{A}_T(M)$ for a natural $T$-action on a Bott-Samelson manifold $M = BS^I$ and present a method for computing generators

A GKM description of the equivariant cohomology ring of a homogeneous space

Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ is a GKM manifold if the set of zero dimensional orbits in the orbit space $M/T$ is zero dimensional and the set of one dimensional orbits in $M/T$ is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about $M$ is encoded in this graph. In this paper we prove that every compact homogeneous space $M$ of non-zero Euler characteristic is of GKM type and show that the graph associated with $M$ encodes \emph{geometric} information about $M$ as well as topological information. For example, from this graph one can detect whether $M$ admits an invariant complex structure or an invariant almost complex structure.Comment: 19 pages, 3 figure

Modeling how windfarm geometry affects bird mortality

Birds flying across a region containing a windfarm risk death from turbine encounters. This paper describes a geometric model that helps estimate that risk and a spreadsheet that implements the model