Representation theory of general linear supergroups in characteristic 2

We introduce two families of diagrammatic monoidal supercategories. The first family, depending on an associative superalgebra, generalizes the oriented Brauer category. The second, depending on an involutive superalgebra, generalizes the unoriented Brauer category. These two families of supercategories admit natural superfunctors to supercategories of supermodules over general linear supergroups and supergroups preserving superhermitian forms, respectively. We show that these superfunctors are full when the superalgebra is a central real division superalgebra. As a consequence, we obtain first fundamental theorems of invariant theory for all real forms of the general linear, orthosymplectic, periplectic, and isomeric supergroups. We also deduce equivalences between monoidal supercategories of tensor supermodules over the real forms of a complex supergroup.

The general linear supergroup and its Hopf superalgebra of regular functions

The work of our group falls within the area of Cyber Security, which is one of Qatar's Research Grand Challenges. We are working on designing a new public key cryptosystem that can improve the security of communication networks. The most widely used cryptosystem at present (like RSA) are based on the difficulty of factorization of numbers that are constructed as product of two large primes. The security of such systems was put in doubt since these systems can be attacked with a help of quantum computers. We are working on a new cryptosystem that is based on different (noncommutative) structures, like algebraic groups and supergroups. Our system is based on the difficulty of computing invariants of actions of such groups. We have designed a system that uses invariants of (super)tori of general linear (super)groups. Effectively, we are building a trapdoor function enabling us to find a suitable invariant of high degree and do the encoding of the message quickly and efficiently but which provides an attacker with computationally very expensive and difficult task to find an invariant of that high degree. As with every cryptosystem, the possibility of its break have to be scrutinized very carefully and the system has to be investigated independently by other researchers. We have established theoretical results about minimal degrees of invariants of a torus that are informing possible selection of parameters of our system. We continue getting more general theoretical results and we are working towards an implementation and testing of this new cryptosystem. A second part of our work is an extension from the classical case of algebraic groups to the case of algebraic supergroups. We are concentrating especially on general linear supergroups. We have described the center of the distribution superalgebras of general linear groups GL(m|n) using the concept of an integral in the sense of Haboush and computed explicitly all generators of invariants of the adjoint action of the group GL(1|1) on its distribution algebra. The center of the distribution algebra is related via the Harish-Chandra map to infinitesimal characters. Understanding of these characters and blocks would lead us to the description of the linkage principle, that is of composition factors of induced modules. Finding and proving linkage principle for supergroups over the field of positive characteristics is one of our main interests. This extends classical results from the representation theory that are giving scientists, mathematicians and physicists, a tool to find a theoretical model where the fundamental rules of symmetries of the space-continuum are realized. Better theoretical background could lead to better understanding of the experimental data and predictions confirming or contradicting our current understanding of the universe. As happened many times in the past, finding the right point of view and developing new language can often lead to different level of understanding. Therefore we value the theoretical part of our work the same way as the practical work related to the cryptosystems.

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.

We prove that the Brauer algebra, for all parameters for which it is quasi-hereditary, is Ringel dual to a category of representations of the orthosymplectic super group. As a consequence we obtain new and algebraic proofs for some results on the fundamental theorems of invariant theory for this super group over the complex numbers and also extend them to some cases in positive characteristic. Our methods also apply to the walled Brauer algebra in which case we obtain a duality with the general linear super group, with similar applications.

We prove Razmyslov’s theorem on trace identities for M k , l {M_{k,\,l}} using the invariant theory of pl ⁡ ( k , l ) \operatorname {pl} (k,\,l) .

Solutions of the graded Yang-Baxter equations are constructed which are invariant relative to the general linear and orthosymplectic supergroups. The Hamiltonians and other higher integrals (the transfer matrix) of spin systems on a finite lattice connected with the solutions found are diagonalized. A generalization of the Yang-Baxter equation to the case of the δ-commutative, G-graded Zamolodchikov algebra is presented.

Canonical differential calculus on quantum general linear groups and supergroups

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.

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.

Let $$G=GL(m|n)$$ be the general linear supergroup over an algebraically closed field K of characteristic zero, and let $$G_{ev}=GL(m)\times GL(n)$$ be its even subsupergroup. The induced supermodule $$H^0_G(\lambda )$$, corresponding to a dominant weight $$\lambda $$ of G, can be represented as $$H^0_{G_{ev}}(\lambda )\otimes \Lambda (Y)$$, where $$Y=V_m^*\otimes V_n$$ is a tensor product of the dual of the natural GL(m)-module $$V_m$$ and the natural GL(n)-module $$V_n$$, and $$\Lambda (Y)$$ is the exterior algebra of Y. For a dominant weight $$\lambda $$ of G, we construct explicit $$G_{ev}$$-primitive vectors in $$H^0_G(\lambda )$$. Related to this, we give explicit formulas for $$G_{ev}$$-primitive vectors of the supermodules $$H^0_{G_{ev}}(\lambda )\otimes \otimes ^k Y$$. Finally, we describe a basis of $$G_{ev}$$-primitive vectors in the largest polynomial subsupermodule $$\nabla (\lambda )$$ of $$H^0_G(\lambda )$$ (and therefore in the costandard supermodule of the corresponding Schur superalgebra S(m|n)). This yields a description of a basis of $$G_{ev}$$-primitive vectors in arbitrary induced supermodule $$H^0_G(\lambda )$$.

This is a survey of some recent results relating Khovanov's arc algebra to category O for Grassmannians, the general linear supergroup, and the walled Brauer algebra. The exposition emphasizes an extension of Young's orthogonal form for level two cyclotomic Hecke algebras.

Cohomological finite-generation for finite supergroup schemes

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.

Irreducibility of induced supermodules for general linear supergroups