Complex Algebraic Group Research Articles

Orbits of strongly solvable spherical subgroups on the flag variety

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup \(H \subset B\) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes.

The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

Quasi-hereditary algebras were introduced by E. Cline, B. Parshall and L. Scott in order to deal with highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups. These categories have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander, and used in his proof of the first Brauer–Thrall conjecture and later used systematically in his joint work with I. Reiten on stable equivalence, as well as many other applications. Recently, functor categories were used by Martinez-Villa and Solberg to study the Auslander–Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category. We can think of the Auslander–Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category $$\mathrm {Mod}(\mathcal {C} )$$ , with $$\mathcal C$$ a component of the Auslander–Reiten quiver.

Groups with the same complex group algebras as some extensions of psl(2, pn )

Abstract For a natural number n and the prime p, let L be an almost simple group with the socle PSL(2,p n ) such that p does not divide [L: PSL(2,p n )]. In this paper, we prove that L is uniquely determined by the first column of its character table. In particular, this implies that L is uniquely determined by the structure of its complex group algebra.

Vust's Theorem and higher level Schur–Weyl duality for types B, C and D

Let G be a complex linear algebraic group, g=Lie(G) its Lie algebra and e∈g a nilpotent element. Vust's Theorem says that in case of G=GL(V), the algebra EndGe(V⊗d), where Ge⊂G is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group Sd and the linear maps {1⊗(i−1)⊗e⊗1⊗(d−i)|i=1,…,d}. In this paper, we give an analogue of Vust's Theorem for G=O(V) and SP(V) when the nilpotent elements e satisfy that G⋅e‾ is normal. As an application, we study the higher Schur–Weyl duality in the sense of [4] for types B, C and D, which establishes a relationship between W-algebras and degenerate affine braid algebras.

Dimension of character varieties for $3$-manifolds

Let M be an orientable 3-manifold, compact with boundary and Γ its fundamental group. Consider a complex reductive algebraic group G. The character variety X(Γ, G) is the GIT quotient Hom(Γ, G)//G of the space of morphisms Γ → G by the natural action by conjugation of G. In the case G = SL(2, C) this space has been thoroughly studied. Following work of Thurston [Thu80], as presented by Culler-Shalen [CS83], we give a lower bound for the dimension of irreducible components of X(Γ, G) in terms of the Euler characteristic χ(M) of M , the number t of torus boundary components of M , the dimension d and the rank r of G. Indeed, under mild assumptions on an irreducible component X0 of X(Γ, G), we prove the inequality dim(X0) ≥ t · r − dχ(M).

Recognition of PSL(2,p) ×PSL(2,p) by its complex group algebra

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61–68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of [Formula: see text], the complex group algebra of [Formula: see text]. One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if [Formula: see text] is an odd prime number, then [Formula: see text] is uniquely determined by the structure of its complex group algebra.

Moduli spaces of bundles on an abelian variety

Let [Formula: see text] be a complex abelian variety and [Formula: see text] a complex reductive affine algebraic group. We describe the connected component, containing the trivial bundle, of the moduli spaces of topologically trivial principal [Formula: see text]-bundles and [Formula: see text]-Higgs bundles on [Formula: see text]. We also describe the moduli spaces of [Formula: see text]-connections and the [Formula: see text]-character variety for [Formula: see text].

A classification of equivariant principal bundles over nonsingular toric varieties

We classify holomorphic as well as algebraic torus equivariant principal [Formula: see text]-bundles over a nonsingular toric variety [Formula: see text], where [Formula: see text] is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.

Complete intersections in spherical varieties

Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G / H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete intersections are smooth varieties. We compute their arithmetic genus as well as some of their \(h^{p,0}\) numbers. The answers are given in terms of the moment polytopes and Newton–Okounkov polytopes associated to G-invariant linear systems. We also give a necessary and sufficient condition on a collection of linear systems so that the corresponding generic complete intersection is nonempty. This criterion applies to arbitrary quasi-projective varieties (i.e., not necessarily spherical homogeneous spaces). When the spherical homogeneous space under consideration is a complex torus \((\mathbb {C}^*)^n\), our results specialize to well-known results from the Newton polyhedra theory and toric varieties.

Total positivity, Schubert positivity, and geometric Satake

Let G be a simple, simply-connected complex algebraic group, and let X⊂G∨ be the centralizer of a principal nilpotent. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian GrG. Peterson furthermore connected X to the quantum cohomology rings of partial flag varieties G/P.In this paper we study three notions of positivity for X: (1) Schubert positivity arising via Peterson's work, (2) Lusztig's total positivity and (3) Mirković–Vilonen positivity obtained from the MV-cycles in GrG. The first main theorem establishes that these three notions of positivity coincide. Our second main theorem proves a parametrization of the totally nonnegative part of X, confirming a conjecture of the second author.In type A the parametrization and relationship with Schubert positivity were proved earlier by the second author. Here we tackle the general type case and also introduce a crucial new connection with the affine Grassmannian and geometric Satake correspondence.

Character varieties of free groups are Gorenstein but not always factorial

Fix a rank g free group Fg and a connected reductive complex algebraic group G. Let X(Fg,G) be the G-character variety of Fg. When the derived subgroup DG<G is simply connected we show that X(Fg,G) is factorial (which implies it is Gorenstein), and provide examples to show that when DG is not simply connected X(Fg,G) need not even be locally factorial. Despite the general failure of factoriality of these moduli spaces, using different methods, we show that X(Fg,G) is always Gorenstein.

FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.

EQUIVARIANT COMPACTIFICATIONS OF A NILPOTENT GROUP BY G/P

Let $G$ be a simple complex algebraic group, $P$ a parabolic subgroup of $G$ and $N$ the unipotent radical of $P.$ The so-called equivariant compactification of $N$ by $G/P$ is given by an action of $N$ on $G/P$ with a dense open orbit isomorphic to $N$. In this article, we investigate how many such equivariant compactifications there exist. Our result says that there is a unique equivariant compactification of $N$ by $G/P$, up to isomorphism, except $\P^n$.

Lifting free divisors

Let $\varphi:X\to S$ be a morphism between smooth complex analytic spaces, and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi$, then $f\circ \varphi=0$ defines a free divisor on $X$; this is generalized in several directions. Among applications we recover a result of Mond-van Straten, generalize a construction of Buchweitz-Conca, and show that a map $\varphi:\mathbb{C}^{n+1}\to \mathbb{C}^n$ with critical set of codimension $2$ has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi$ is the algebraic quotient $X\to S=X// G$ with $X// G$ smooth, we describe sufficient conditions for $T^1_{X/S}$ to be Cohen-Macaulay of codimension $2$. In one such case, a free divisor on $\mathbb{C}^{n+1}$ lifts under the operation of "castling" to a free divisor on $\mathbb{C}^{n(n+1)}$, partially generalizing work of Granger-Mond-Schulze on linear free divisors. We give several other examples of such representations.

A new characterization for some extensions of PSL(2, q) for some q by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p2) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p2) of order 2|PSL(2,p2)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p3. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.

Equivariant principal bundles and logarithmic connections on toric varieties

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.

Equivariant principal bundles and logarithmic connections on toric varieties

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.

A characterization of PGL (2,pn) by some irreducible complex character degrees

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2,p) |, p ? cd(G) and max(cd(G)) = p+1, then G ? PGL(2,p), SL(2, p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also, we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G ? PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely determined by the structure of its complex group algebra.

Equivariant quantum cohomology of cotangent bundle of G/P

Let G denote a complex semisimple linear algebraic group, P a parabolic subgroup of G and P=G/P. We identify the quantum multiplication by divisors in T⁎P in terms of stable basis, which is introduced in [9]. Using this and the restriction formula for stable basis [17], we show that the G×C⁎-equivariant quantum multiplication formula in T⁎P is conjugate to the formula conjectured by Braverman.

Lie–Poisson theory for direct limit Lie algebras

In the first half of this paper, we develop the fundamentals of Lie–Poisson theory for direct limits G=lim→Gn of complex algebraic groups and their Lie algebras g=lim→gn. We describe the Poisson pro- and ind-variety structures on g⁎=lim←gn⁎ and the coadjoint orbits of G, respectively. While the existence of symplectic foliations remains an open question for most infinite-dimensional Poisson manifolds, we show that for direct limit algebras, the coadjoint orbits give a weak symplectic foliation of the Poisson provariety g⁎.The second half of the paper applies our general results to the concrete setting of G=GL(∞) and g⁎=M(∞), the space of infinite-by-infinite complex matrices with arbitrary entries. We use the Poisson structure of g⁎ to construct an integrable system on M(∞) that generalizes the Gelfand–Zeitlin system on gl(n,C) to the infinite-dimensional setting. We further show that this integrable system integrates to a global action of a direct limit group on M(∞), whose generic orbits are Lagrangian ind-subvarieties of the coadjoint orbits of GL(∞) on M(∞).