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In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type.

A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits.(i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the $1$-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-action.(ii) We axiomize the definition of a stacky GKM graph and introduce abstract stacky GKM graphs which are more general than stacky GKM graphs of honest smooth GKM stacks. From an abstract GKM graph we construct a formal smooth GKM stack.(iii) We define equivariant orbifold Gromov–Witten invariants of smooth GKM stacks, as well as formal equivariant orbifold Gromov–Witten invariants of formal smooth GKM stacks. These invariants can be computed by virtual localization and depend only the stacky GKM graph or the abstract stacky GKM graph. Formal equivariant orbifold Gromov–Witten invariants of the stacky GKM graph of a smooth GKM stack $\mathcal{X}$ are refinements of equivariant orbifold Gromov–Witten invariants of $\mathcal{X}$.

For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional torus gives rise to different interacting phases, where $P$ denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases., Comment: Stronger results. Included examples in section 6. Accepted for publication in JHEP

In this paper, motivated by Chen–Ruan’s stringy orbifold theory on almost complex orbifolds, we construct a new equivariant cohomology ring $${\mathscr {H}}^*_G(X)$$ for an equivariant almost complex pair (X, G), where X is a compact connected almost complex manifold, G is a connected compact Lie group which acts on X and preserves the almost complex structure.

Using the equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape λ / μ . Morales, Pak and Panova found two q-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape λ / μ and reverse plane partitions of shape λ / μ . When λ and μ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape λ / μ can be expressed as a determinant whose entries are related to q-analogues of the Euler numbers. The objective of this paper is to settle this conjecture.

J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas, A. Yong. Our proof then combines a saturation theorem of D. Anderson, E. Richmond, A. Yong, a reading order independence property, and E. Tardos’ algorithm for combinatorial linear programming.

We describe the torus-equivariant cohomology of weighted partial flag orbifolds $${\mathrm {w}}\Sigma $$ of type A. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as “Schubert Calculus on $${\mathrm {w}}\Sigma $$ ”. For the weighed Schubert classes in $${\mathrm {w}}\Sigma $$, we give the Chevalley’s formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley–Monk’s formula.

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group., Comment: 20 pages, one figure. Minor corrections and referee suggestions and additional references included. To appear in Geometriae Dedicata

We compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

This paper is dedicated to introducing and studying a cohomology theory for monoids enriched with the action by endomorphisms of a fixed monoid of operators. This equivariant cohomology theory extends both Whitehead’s cohomology for groups with operators and Leech’s cohomology for monoids.

This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism and can still derive the classical result of tom Dieck., Comment: 13 pages. This is the new version to cover the previous version "Equivariant unitary bordism and equivariant cohomology Chern numbers" in 2014

We study two different one-parameter generalizations of Littlewood–Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by Knutson and Tao in their work on the equivariant cohomology of the Grassmannian.

We investigate the topological nilpotence degree, in the sense of Henn-Lannes-Schwartz, of a connected Noetherian unstable algebra $R$. When $R$ is the mod $p$ cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian $p$-subgroups. By replacing centralizers of elementary abelian $p$-subgroups with components of Lannes' $T$-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn's result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn's result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac-Moody groups. In fact, our results apply much more generally, for example, we establish results for $p$-local compact groups in the sense of Broto-Levi-Oliver, for connected $H$-spaces with Noetherian mod $p$ cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson's depth conjecture in the case of a Noetherian unstable algebra of minimal depth., 42 pages; comments welcome v2; Significantly revised version, accepted for publication in Selecta Mathematica

We introduce eight “flavors” of cyclic homology of a mixed complex and study their properties. In particular, we determine their behavior with respect to Chen’s iterated integrals.

In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are one-parameter (ħ) deformations of the notion of fundamental class in their respective extraordinary cohomology theories. Examining these three classes in conjunction is justified by their relation to Okounkov’s stable envelope notion. We review formulas for the ħ-deformed classes originating from Tarasov–Varchenko weight functions, as well as their orthogonality relations. As a consequence, explicit formulas are obtained for the Littlewood-Richardson type structure constants.