General Linear Supergroup Research Articles

The Center of Dist (GL(m|n)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.

Cohomological finite-generation for finite supergroup schemes

In this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain “universal extension classes” for the general linear supergroup. Some of these classes restrict to the universal extension classes for the general linear group exhibited by Friedlander and Suslin, while others arise from purely super phenomena. We then use these extension classes to show that the cohomology ring of a finite supergroup scheme—equivalently, of a finite-dimensional cocommutative Hopf superalgebra—over a field is a finitely-generated algebra. Implications for the rational cohomology of the general linear supergroup are also discussed.

Primitive vectors in induced supermodules for general linear supergroups

The purpose of the paper is to derive formulas that describe the structure of the induced supermodule HG0(λ) for the general linear supergroup G=GL(m|n) over an algebraically closed field K of characteristic p≠2. Using these formulas we determine primitive Gev=GL(m)×GL(n)-vectors in HG0(λ). We conclude with remarks related to the linkage principle in positive characteristic.

Deligne’s category \underline{𝑅𝑒}𝑝(𝐺𝐿_{𝛿}) and representations of general linear supergroups

We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric Schur polynomials, and give a method for decomposing their tensor products. Along the way, we describe indecomposable objects in Re _ p ⁡ ( G L δ ) \underline {\operatorname {Re}}\!\operatorname {p}(GL_\delta ) and explain how to decompose their tensor products.

Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup

We prove that blocks of the general linear supergroup are Morita equivalent to a limiting version of Khovanov's diagram algebra. We deduce that blocks of the general linear supergroup are Koszul.

Pseudocompact Algebras and Highest Weight Categories

We develop a new approach to highest weight categories \(\cal{C}\) with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For \(\cal{C}\) admitting a Chevalley duality, we define and investigate tilting modules and Ringel duals of the corresponding pseudocompact algebras. Finally, we illustrate all these concepts on an explicit example of the general linear supergroup GL(1|1).

Generators of supersymmetric polynomials in positive characteristic

In Kantor and Trishin (1997) [3], Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie superalgebra gl ( m | n ) and a related algebra A s of what they called pseudosymmetric polynomials over an algebraically closed field K of characteristic zero. The algebra A s was investigated earlier by Stembridge (1985) who in [9] called the elements of A s supersymmetric polynomials and determined generators of A s . The case of positive characteristic p of the ground field K has been recently investigated by La Scala and Zubkov (in press) in [6]. We extend their work and give a complete description of generators of polynomial invariants of the adjoint action of the general linear supergroup GL ( m | n ) and generators of A s .

Donkin–Koppinen Filtration for General Linear Supergroups

We consider a generalization of Donkin–Koppinen filtrations for coordinate superalgebras of general linear supergroups. More precisely, if G = GL(m|n) is a general linear supergroup of (super)degree (m|n), then its coordinate superalgebra K[G] is a natural G × G-supermodule. For every finitely generated ideal \(\Gamma\subseteq \Lambda\times\Lambda\), the largest subsupermodule OΓ(K[G]) of K[G], which has all composition factors of the form L(λ) ⊗ L(μ) where (λ, μ) ∈ Γ, has a decreasing filtration \(O_{\Gamma}(K[G])=V_0\supseteq V_1\supseteq\ldots\) such that ∩ t ≥ 0Vt = 0 and \(V_t/V_{t+1}\simeq V_-(\lambda_t)^*\otimes H_-^0(\lambda_t)\) for each t ≥ 0. Here \(H_-^0(\lambda)\) is a costandard G-supermodule, and V − (λ) is a standard G-supermodule, both of highest weight λ ∈ Λ (see Zubkov, Algebra Log 45(3): 257–299, 2006). We deduce the existence of such a filtration from more general facts about standard and costandard filtrations in certain highest weight categories which will be proved in Section 5. Until now, analogous results were known only for highest weight categories with finite sets of weights. We believe that the reader will find the results of Section 5 interesting on its own. Finally, we apply our main result to describe invariants of (co)adjoint action of G.

Highest Weight Categories Arising from Khovanov's Diagram Algebra I: Cellularity

This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions. In this article we prove that H(n,m) is a cellular symmetric algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent articles, we relate these algebras to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively.

COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

In this paper we consider the problem of describing the costandard modules ∇(λ) of a Schur superalgebra S(m|n,r) over a base field K of arbitrary characteristic. Precisely, if G = GL(m|n) is a general linear supergroup and Dist (G) its distribution superalgebra we compute the images of the Kostant ℤ-form under the epimorphism Dist (G) → S(m|n,r). Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain an isomorphism ∇(λ) ≈ ∇(λ+|0) ⊗ ∇(0|λ-) assuming that λ = (λ+|λ-) and λm = 0. If char (K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ)p where ∇(μ) is a costandard module of the ordinary Schur algebra S(n,r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux.

SUPERMETRICS ON SUPERMANIFOLDS

By virtue of the well-known theorem, a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold X. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthogonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.

Some properties of general linear supergroups and of Schur superalgebras

It is shown that the category of rational supermodules over a general linear supergroup is a highest weight category. More exactly, we construct superanalogs of the theory of modules with good filtration and of the dual theory of modules with Weyl’s. Using these, we show that indecomposable injective supermodules have good filtration of a certain kind.

Dual canonical bases for the quantum general linear supergroup

Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup O q ( SL m | n ) . We apply the dual canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case n = 1 , it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of Borel–Weil theorem.

Spherical functions on homogeneous superspaces

Homogeneous superspaces arising from the general linear supergroup are studied within a Hopf algebraic framework. Spherical functions on homogeneous superspaces are introduced, and the structures of the superalgebras of the spherical functions on classes of homogeneous superspaces are described explicitly.

Quantum superalgebra representations on cohomology groups of non-commutative bundles

Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic sub-supergroup to the category of locally finite modules of the quantum general linear supergroup. The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles. We explicitly compute the cohomology groups, which are given in terms of well understood modules over the quantized universal enveloping algebra of the general linear superalgebra.

The general linear supergroup and its Hopf superalgebra of regular functions

The structure of the Hopf superalgebra B of regular functions on the general linear supergroup is developed, and applied to study the representation theory of the supergroup. It is shown that the general linear supergroup can be reconstructed from B in a way reminiscent of the Tannaka–Krein theory. A Borel–Weil type realization is also obtained for the irreducible integrable representations. As a side result on the structure of B , Schur superalgebras are introduced and are shown to be semi-simple over the complex field, with the simple ideals determined explicitly.

Structure and Representations of the Quantum General Linear Supergroup

The structure and representations of the quantum general linear supergroup GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq of its representative functions. Gq is factorized into \(G_q^\pi G_q^{\bar \pi }\), and a Peter–Weyl basis is constructed for each factor. Parabolic induction for the quantum supergroup is developed. The underlying geometry of induced representations is discussed, and an analog of Frobenius reciprocity is obtained. A quantum Borel–Weil theorem is proven for the irreducible covariant and contravariant tensorial representations, and explicit realizations are given for classes of irredducible tensorial representatins in terms of sections of quantum super vector bundles over quantum projective superspaces.

On representations of super coalgebras

The general structure of the representation theory of a Z2-graded coalgebra is discussed. The result contains the structure of Fourier analysis on compact supergroups and quantizations thereof as a special case. The general linear supergroups serve as an explicit illustration and the simplest example is carried out in detail.

The quantum general linear supergroup and braid statistics

We generalize to braid statistics some of Manin's work (1989) on the quantum deformation of the general linear supergroup. The key ingredient in this construction is the introduction of a non-standard transposition map Psi ( Psi 2 not=1) which is defined in terms of a generalized permutation operator Pmu (pmu 2 not=1). The dual space (enveloping algebra) with Pmu -statistics is defined. Consideration of coactions on quantum spaces clarifies the resulting structure.

Canonical differential calculus on quantum general linear groups and supergroups

We specify a set of relations between non-commuting matrix elements and their differentials, defined in terms of an R-matrix satisfying the braid relation, which are uniquely determined by the requirements of consistency with the relations between non-commuting coordinates and their differentials. We also give a necessary condition for the existence of a matrix inverse (antipode) in the form of an additional equation to be satisfied by the R-matrix.