Torus Orbits Research Articles

The HHMP Decomposition of the Permutohedron and Degenerations of Torus Orbits in Flag Varieties

Abstract Let $Z\subset \operatorname{Fl}(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside $\operatorname{Fl}(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.

One-skeleton posets of Bruhat interval polytopes

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these polytopes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee–Masuda, and of Lee–Masuda–Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth.

Non-exactness of toric Poisson structures

We prove that a Poisson structure on a projective toric variety which is invariant by the torus action and whose symplectic leaves are the torus orbits is not exact. This is deduced from a geometric criterion for non-exactness of Poisson structures with a finite number of symplectic leaves.

Torus orbit closures in flag varieties and retractions on Weyl groups

A finite Coxeter group W has a natural metric d and if [Formula: see text] is a subset of W, then for each [Formula: see text], there is [Formula: see text] such that [Formula: see text]. Such q is not unique in general but if [Formula: see text] is a Coxeter matroid, then it is unique, and we define a retraction [Formula: see text] so that [Formula: see text]. The T-fixed point set [Formula: see text] of a T-orbit closure Y in a flag variety G/B is a Coxeter matroid, where G is a semi-simple algebraic group, B is a Borel subgroup, and T is a maximal torus of G contained in B. We define a retraction [Formula: see text] geometrically, where W is the Weyl group of [Formula: see text], and show that [Formula: see text]. We introduce another retraction [Formula: see text] algebraically for an arbitrary subset [Formula: see text] of W when W is a Weyl group of classical Lie type, and show that [Formula: see text] when [Formula: see text] is a Coxeter matroid.

Cohomology of generalized Dold spaces

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e., the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where X,S are arbitrary topological spaces. The resulting space P(S,X) will be called a generalized Dold space. When S and X are CW complexes satisfying certain natural requirements, we obtain a CW-structure on P(S,X). Under certain further hypotheses, we determine the mod 2 cohomology groups of P(S,X). We determine the Z2-cohomology algebra when X is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over P(S,X) associated to σ-conjugate complex bundles over X when the S is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds.

Toric invariant theory for maximum likelihood estimation in log-linear models

We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.

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.

Non-vanishing of class group L-functions for number fields with a small regulator

Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.

Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group

We provide a functorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. Ours is an adaptation of the moment graph approach of Goresky, Kottwitz, and MacPherson. We first define a generalized moment graph, a combinatorial object determined by the structure of a finite collection of low-dimensional torus orbits. We then show how an algorithm of Braden and MacPherson computes the stalks of the equivariant intersection cohomology complex, producing “sheaves” on the generalized moment graph that functorially encode the equivariant intersection cohomology. Because the Braden-MacPherson algorithm uses only basic commutative algebra, this approach greatly simplifies the a priori very difficult task of computing intersection cohomology stalks. This paper extends previous work of Springer, who had computed the (ordinary) intersection cohomology Betti numbers of the Borel orbit closures in the wonderful compactification.

Generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety G/B of type An−1 is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety Xw(w∈Sn) and study its closure Yw. We identify the maximal cone in the fan of Yw corresponding to a fixed point uB(u≤w), associate a graph Γw(u) to each u≤w, and show that Yw is smooth at uB if and only if Γw(u) is a forest. We also introduce a polynomial Aw(t) for each w, which agrees with the Eulerian polynomial when w is the longest element of Sn, and show that the Poincaré polynomial of Yw agrees with Aw(t2) when Yw is smooth.

Arithmetic of double torus quotients and the distribution of periodic torus orbits

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates and homogeneous dynamics.

The Smooth Torus Orbit Closures in the Grassmannians

It is known that for the natural algebraic torus actions on the Grassmannians, the closures of torus orbits are toric varieties, and that these toric varieties are smooth if and only if the corresponding matroid polytopes are simple. We prove that simple matroid polytopes are products of simplices and smooth torus orbit closures in the Grassmannians are products of complex projective spaces. Moreover, it turns out that the smooth torus orbit closures are uniquely determined by the corresponding simple matroid polytopes.

Generic Torus Orbit Closures in Flag Bott Manifolds

In this article the generic torus orbit closure in a flag Bott manifold is shown to be a non-singular toric variety, and its fan structure is explicitly calculated.

Joint equidistribution of CM points

We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes; assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all whose components have the same fundamental discriminant; assuming the CM fields are split at two fixed primes and have no exceptional zero. The joinings theorem of Einsiedler and Lindenstrauss applies to the toral orbits arising in these results. Yet it falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates. The main novel contribution is a method to exclude intermediate measures for toral periods. The crux is a geometric expansion of the cross-correlation between the periodic measure on a torus orbit and a Hecke correspondence, expressing it as a short shifted convolution sum. The latter is bounded from above generalizing the method of Shiu and Nair to polynomials in two variables on smooth domains.

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

Torus orbits on homogeneous varieties and Kac polynomials of quivers

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a new way to compute them. We also discuss connections with the Gel’fand–MacPherson correspondence.

Tropical Skeletons

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of X as the subset of the associated Berkovich space X an which collects all Shilov boundary points in the fibers of the Kajiwara–Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When X is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm–Katz.

Poisson deleting derivations algorithm and Poisson spectrum

ABSTRACTCauchon [5] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.

Complex instabilities near codimension-2 bifurcation and torus breakdown in a Ćuk converter with an inductive impedance load

Power switching converters exhibit a wide range of nonlinear behavior, such as bifurcation and chaos. Complex instabilities near the codimension-2 bifurcation point are shown in some nonlinear higher order systems. In this study, codimension-2 bifurcation and complex instabilities in a single inner current loop controlled Cuk converter with an inductive impedance load are analyzed by studying the Floquet multipliers and switching modes of the system. The period1 orbit loses its stability by period-doubling and subcritical Neimark-Sacker bifurcations. The period2 orbit enters a two-loop torus orbit via border collision and Neimark-Sacker bifurcations. The dynamic behavior of the system near the codimension-2 bifurcation points is clearly studied. The mechanism of the coexisting phenomenon from the subcritical Neimark-Sacker bifurcation is explained. Finally, chaos from torus breakdown is detected in this higher order switching converter with an inductive impedance load and its phase shift characteristics to different initial conditions are investigated.

Remarks on Lagrangian intersections in toric manifolds

I will consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. The remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. I will show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits. This is joint work with Leonardo Macarini.