Maximal Torus Research Articles

Explicit generators of some pro-$p$ groups via Bruhat-Tits theory

Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p-subgroups thanks to parametrizations of a suitable maximal torus and of corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of a suitable root system.

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

We use the correspondence between extremal Sasaki structures and weighted extremal Kähler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors in [1], and the theory of weighted K-stability (introduced by Lahdili in [39]) in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus and smooth equivariant test configurations with reduced central fibre is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins–Székelyhidi in [18]), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kähler cone metric found in [18] from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal S1-bundle over an admissible ruled manifold in the sense of [4], expressed in terms of the positivity of a single polynomial of one variable over a given interval.

Invariable generation of finite classical groups

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate Sn is bounded away from zero by an absolute constant for all n. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate Sn tends to zero as n→∞. In this paper, we prove an analogous result for the finite classical groups. More precisely, let Gr(q) be a finite classical group of rank r over Fq. We show that for q large enough, the probability that four randomly selected elements invariably generate Gr(q) is bounded away from zero by an absolute constant for all r, and for three elements the probability tends to zero as q→∞ and r→∞. We use the fact that most elements in Gr(q) are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups.

Initial degenerations of Grassmannians

We construct closed immersions from initial degenerations of $${{\,\mathrm{Gr}\,}}_{0}(d,n)$$ —the open cell in the Grassmannian $${{\,\mathrm{Gr}\,}}(d,n)$$ given by the nonvanishing of all Plucker coordinates—to limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms when (d, n) equals (2, n), (3, 6) and (3, 7). As an application we prove $${{\,\mathrm{Gr}\,}}_0(3,7)$$ is schon, and the Chow quotient of $${{\,\mathrm{Gr}\,}}(3,7)$$ by the maximal torus in $$ {\text {PGL}}(7)$$ is the log canonical compactification of the moduli space of 7 points in $${\mathbb {P}}^2$$ in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev.

On the 𝔸1-Euler Characteristic of the Variety of Maximal Tori in a Reductive Group

AbstractWe show that for a reductive group $G$ over a field $k$ the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an invertible element of the Grothendieck–Witt ring ${\textrm{GW}}(k)$, settling the weak form of a conjecture by Fabien Morel. As an application we obtain a generalized splitting principle that allows one to reduce the structure group of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.

Torus quotients of Schubert varieties in the Grassmannian $$G_{2, n}$$

Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular points. Further, we have computed the exact number of singular points of the GIT quotient of $G_{2,n}.$

Incidence monoids: automorphisms and complexity

The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity \(\le 1\). Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.

Hodge-Deligne polynomials of character varieties of free abelian groups

Abstract Let F F be a finite group and X X be a complex quasi-projective F F -variety. For r ∈ N r\in {\mathbb{N}} , we consider the mixed Hodge-Deligne polynomials of quotients X r / F {X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F , where F F acts diagonally, and compute them for certain classes of varieties X X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X X is the maximal torus of an affine reductive group G G , and F F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E E -polynomials of (the distinguished component of) G G -character varieties of free abelian groups. In the cases G = G L ( n , C ) G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and S L ( n , C ) SL\left(n,{\mathbb{C}}\hspace{-0.1em}) , we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

The gamma filtrations for the spin groups

Let $G$ be a compact Lie group and $T$ its maximal torus. In this paper, we try to compute $gr_{\gamma}^*(G/T)$ the graded ring associated with the gamma filtration of the complex $K$-theory $K^0(G/T)$ for $G=Spin(n)$. In particular, we give a counterexample for a conjecture by Karpenko when $G=Spin(17)$.

Embeddings of maximal tori in groups of type F4

We classify maximal tori in groups of type $F_4$ over a local or global field of characteristic different from $2$ and $3$. We prove a local-global principle for embeddings of maximal tori in groups of type $F_4$.

R-groups for unitary principal series of Spin groups

We study the unitary principal series of the split group Spinm(F), where F is a p-adic field. Let χ˜ be a unitary character of a maximal F-split torus T˜ of GSpinm(F), and let χ be its restriction to T=T˜∩Spinm(F). The R-groups Rχ˜ and Rχ of the corresponding principal series representations fit in the exact sequence 0→Rχ˜→Rχ→Rχ/Rχ˜→0. We give a complete answer to the question of splitting of this exact sequence. We also prove that the multiplicity is one when irreducible constituents in unitary principal series are restricted from GSpin(F) to Spin(F). Further, based on Keys' result, we prove Arthur's conjecture on R-groups for unitary principal series of Spin.

On the Torus quotients of Schubert varieties

In this paper, we consider the GIT quotients of Schubert varieties for the action of a maximal torus. We describe the minuscule Schubert varieties for which the semistable locus is contained in the smooth locus. As a consequence, we study the smoothness of torus quotients of Schubert varieties in the Grassmannian. We also prove that the torus quotient of any Schubert variety in the homogeneous space [Formula: see text] is projectively normal with respect to the line bundle [Formula: see text] and the quotient space is a projective space, where the line bundle [Formula: see text] and the parabolic subgroup [Formula: see text] of [Formula: see text] are associated to the highest root [Formula: see text].

Cohomological representations of parahoric subgroups

We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric.

Torus Action on Quaternionic Projective Plane and Related Spaces

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number $$\frac{1}{2}\dim X-\dim T$$ is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $${\mathbb {H}}P^2/T^3\cong S^5$$ and $$S^6/T^2\cong S^4$$ , for the homogeneous spaces $${\mathbb {H}}P^2={{\,\mathrm{Sp}\,}}(3)/({{\,\mathrm{Sp}\,}}(2)\times {{\,\mathrm{Sp}\,}}(1))$$ and $$S^6=G_2/{{\,\mathrm{SU}\,}}(3)$$ . Here, the maximal tori of the corresponding Lie groups $${{\,\mathrm{Sp}\,}}(3)$$ and $$G_2$$ act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of $$T^3$$ . This class generalizes $${\mathbb {H}}P^2$$ . We prove that their orbit spaces are homeomorphic to $$S^5$$ as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold.

Torus Quotients of Schubert Varieties

In this paper, we consider the GIT quotients of Schubert varieties for the action of a maximal torus. We describe the minuscule Schubert varieties for which the semistable locus is contained in the smooth locus. As a consequence, we study the smoothness of torus quotients of Schubert varieties in the Grassmannian. We also prove that the torus quotient of any Schubert variety in the homogeneous space $SL(n, \mathbb C)/P$ is projectively normal with respect to the line bundle $\mathcal L_{\alpha_0}$ and the quotient space is a projective space, where the line bundle $\mathcal L_{\alpha_0}$ and the parabolic subgroup $P$ of $SL (n, \mathbb C)$ are associated to the highest root $\alpha_0$.

Closures of locally divergent orbits of maximal tori and values of homogeneous forms

AbstractLet ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $ \overline{T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e. $\overline{T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

Nilpotent Elements and Reductive Subgroups Over a Local Field

Let ${\mathcal {K}}$ be a local field – i.e. the field of fractions of a complete DVR ${\mathcal {A}}$ whose residue field k has characteristic p > 0 – and let G be a connected, absolutely simple algebraic ${\mathcal {K}}$ -group G which splits over an unramified extension of ${\mathcal {K}}$ . We study the rational nilpotent orbits of G– i.e. the orbits of $G({\mathcal {K}})$ in the nilpotent elements of $\text {Lie}(G)({\mathcal {K}})$ – under the assumption p > 2h − 2 where h is the Coxeter number of G. A reductive group M over ${\mathcal {K}}$ is unramified if there is a reductive model ${{\mathscr{M}}}$ over ${\mathcal {A}}$ for which $M = {{\mathscr{M}}}_{{\mathcal {K}}}$ . Our main result shows for any nilpotent element X1 ∈Lie(G) that there is an unramified, reductive ${\mathcal {K}}$ -subgroup M which contains a maximal torus of G and for which X1 ∈Lie(M) is geometrically distinguished. The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G.

Bott–Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

Let G be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous G-spaces G/Q, we construct a finite atlas $${{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)$$ on G/Q, called the Bott–Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on G/Q. We also show that the standard Poisson structure $$\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$$ on G/Q is presented, in each of the coordinate charts of $${{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)$$ , as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making $$(G/Q, \pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}, {{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q))$$ into a Poisson–Ore variety. In addition, all coordinate functions in the Bott–Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure $$\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$$ . Examples of G/Q include G itself, G/T, G/B, and G/N, where $$T \subset G$$ is a maximal torus, $$B \subset G$$ a Borel subgroup, and N the uniradical of B.

Basic Cohomology of Canonical Holomorphic Foliations on Complex Moment-Angle Manifolds

Abstract We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

Characters and group invariant polynomials of (super)fields: road to \u201cLagrangian\u201d

The dynamics of the subatomic fundamental particles, represented by quantum fields, and their interactions are determined uniquely by the assigned transformation properties, i.e., the quantum numbers associated with the underlying symmetry of the model under consideration. These fields constitute a finite number of group invariant operators which are assembled to build a polynomial, known as the Lagrangian of that particular model. The order of the polynomial is determined by the mass dimension. In this paper, we have introduced an automated {texttt {Mathematica}}^{tiny textregistered } package, GrIP, that computes the complete set of operators that form a basis at each such order for a model containing any number of fields transforming under connected compact groups. The spacetime symmetry is restricted to the Lorentz group. The first part of the paper is dedicated to formulating the algorithm of GrIP. In this context, the detailed and explicit construction of the characters of different representations corresponding to connected compact groups and respective Haar measures have been discussed in terms of the coordinates of their respective maximal torus. In the second part, we have documented the user manual of GrIP that captures the generic features of the main program and guides to prepare the input file. We have attached a sub-program CHaar to compute characters and Haar measures for SU(N), SO(2N), SO(2N+1), Sp(2N). This program works very efficiently to find out the higher mass (non-supersymmetric) and canonical (supersymmetric) dimensional operators relevant to the effective field theory (EFT). We have demonstrated the working principles with two examples: the standard model (SM) and the minimal supersymmetric standard model (MSSM). We have further highlighted important features of GrIP, e.g., identification of effective operators leading to specific rare processes linked with the violation of baryon and lepton numbers, using several beyond standard model (BSM) scenarios. We have also tabulated a complete set of dimension-6 operators for each such model. Some of the operators possess rich flavour structures which are discussed in detail. This work paves the way towards BSM-EFT.