GKM Theory Research Articles

Modular law through GKM theory

Modular law through GKM theory

Generalized Splines on Graphs with Two Labels and Polynomial Splines on Cycles

Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley-Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet.
 This paper focuses on the theory of generalized splines. A generalized spline on a graph $G$ with each edge labeled by an ideal in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.

The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory

Явно описан модуль вторых когомологий регулярного полупростого многообразия Хессенберга в терминах образующих и соотношений при помощи ГКМ-теории. Введенное Тымочко действие позволяет определить структуру модуля над группой перестановок $\mathfrak {S}_n$ на когомологиях регулярного полупростого многообразия Хессенберга. В качестве приложения найденного явного описания вторых когомологий доказывается явная формула, описывающая вторые когомологии как $\mathfrak {S}_n$-модуль. Данная формула не совпадает с известной формулой Чоу или Чо-Хон-Ли, но они эквивалентны. Также обсуждается обобщение этой формулы на старшие градуировки.

The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory

The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory

Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold

To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a flag Bott tower and a generalized Bott tower. It is known that a generalized Bott tower is a toric manifold. However a flag Bott tower is not toric in general but we show that it is a GKM manifold, and we also show that for a given generalized Bott tower we can find the associated flag Bott tower so that the closure of a generic torus orbit in the latter is a blow-up of the former along certain invariant submanifolds. We use GKM theory together with toric geometric arguments.

GKM theory for orbifold stratified spaces and application to singular toric varieties

We study the GKM theory for a equivariant stratified space having orbifold structures in its successive quotients. Then, we introduce the notion of an almost simple polytope, as well as a divisive toric variety generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.

GKM theory and Hamiltonian non-Kähler actions in dimension 6

Using the classification of 6-dimensional manifolds by Wall, Jupp and Žubr, we observe that the diffeomorphism type of simply-connected, compact 6-dimensional integer GKM T2-manifolds is encoded in their GKM graph. As an application, we show that the 6-dimensional manifolds on which Tolman and Woodward constructed Hamiltonian, non-Kähler T2-actions with finite fixed point set are diffeomorphic to Eschenburg's twisted flag manifold SU(3)//T2. In particular, they admit a noninvariant Kähler structure.

Manifolds of Isospectral Matrices and Hessenberg Varieties

Abstract We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.

Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.

Localization in equivariant operational K-theory and the Chang–Skjelbred property

We establish a localization theorem of Borel–Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne (Doc Math 20:357–399, 2015). Inspired by the work of Chang–Skjelbred and Goresky–Kottwitz–MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet–Kimura’s technique of cohomological descent for equivariant envelopes. As an application, we extend Uma’s description of the equivariant K-theory of smooth compactifications of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.

GKM theory for p-compact groups

This work studies the flag varieties of p-compact groups, principally through torus-equivariant cohomology, extending methods and tools of classical Schubert calculus and moment graph theory from the setting of real reflection groups to the broader context of complex reflection groups. In particular we give, for the infinite family of p-compact flag varieties corresponding to the complex reflection groups G(r,1,n), a generalized GKM characterization (following Goresky–Kottwitz–MacPherson [8]) of the torus-equivariant cohomology, building an explicit additive basis and showing its relationship with the polynomial or Borel presentation via the localization map.

Rational smoothness, cellular decompositions and GKM theory

We introduce the notion of Q-filtrable varieties: projective varieties with a torus action and a finite number of fixed points, such that the cells of the associated Bialynicki-Birula decomposition are all rationally smooth. Our main results develop GKM theory in this setting. We also supply a method for building nice combinatorial bases on the equivariant cohomology of any Q-filtrable GKM variety. Applications to the theory of group embeddings are provided.

The T-equivariant integral cohomology ring of F4/T

We determine the $T$-equivariant integral cohomology of $F_4/T$ combinatorially by the GKM theory, where T is a maximal torus of the exceptional Lie group $F_4$ and acts on $F_4/T$ by the left multiplication.

Non-abelian GKM theory

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.

A Divided Difference Operator

We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.

Balanced Fiber Bundles and GKM Theory

Let T be a torus and B a compact T-manifold. Goresky et al. show in [3] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H-T*(B) as a subring of H-T*(B-T). In this paper, we prove an analog of this result for T-equivariant fiber bundles: we show that if M is a T-manifold and pi:M -> B a fiber bundle for which pi intertwines the two T-actions, there is a simple combinatorial description of H-T*(M) as a subring of H-T*(pi(-1)(B-T)). Using this result, we obtain fiber bundle analogs of results of Guillemin et al. [4] on GKM theory for homogeneous spaces.

GKM theory for torus actions with nonisolated fixed Points

Let M2d be a compact symplectic manifold and T a compact n-dimensional torus. A Hamiltonian action, τ, of T on M is a GKM action if, for every p ∈ MT, the isotropy representation of T on TpM has pairwise linearly independent weights. For such an action, the image of the set of zero- and one-dimensional orbits under the projection onto M/T is aregular d-valent graph; Goresky, Kottwitz, and MacPherson have proved that the equivariant cohomology of M can be computed from the combinatorics of this graph. In this paper we define a “GKM action with nonisolated fixed points” to be an action, τ, of T on M with the property that for every connected component, F of MT and p ∈ F, the isotropy representation of T on the normal space to F at p has pairwise linearly independent weights. For such an action, we show that all components of MT are diffeomorphic and prove an analogue of the theorem above.