arXiv subjects
Publications and source records attributed to Frantisek Marko.
We consider the periplectic supergroup ${\bf P} (n)$ over a ground field $\Bbbk$ of characteristic $p>2$. We show that there are four blocks of ${\bf P} (n)$ of simple supermodules $L^{\epsilon}(\lambda)$ corresponding to dominant weights $\lambda$ of even and odd lengths, and the even and odd parity $\epsilon$ of their highest weight vector.
Let $G=GL(m|n)$ be a general linear supergroup and $G_{ev}$ be its even subsupergroup isomorphic to $GL(m)\times GL(n)$. In this paper we use the explicit description of $G_{ev}$-primitive vectors in the costandard supermodule $\nabla(λ)$, the largest polynomial $G$-subsupermodule of the induced supermodule $H^0_G(λ)$, for $(m|n)$-hook partition $λ$, and a properties of certain morphisms $ψ_k$ to derive results related to the odd linkage for $G$ over a field $F$ of characteristic different from $2$.
For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $ϕ: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $ϕ^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $Γ$ be a finitely generated ideal $Γ$ of $X(T)^+$ and $O_Γ(K[G])$ be the largest $Γ$-subsupermodule of $K[G]$ having simple composition factors of highest weights $λ\in Γ$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_Γ(K[G])$.
The paper contains results that characterize the Donkin-Koppinen filtration of the coordinate superalgebra $K[G]$ of the general linear supergroup $G=GL(m|n)$ by its subsupermodules $C_Γ=O_Γ(K[G])$. Here, the supermodule $C_Γ$ is the largest subsupermodule of $K[G]$ whose composition factors are irreducible supermodules of highest weight $λ$, where $λ$ belongs to a finitely-generated ideal $Γ$ of the poset $X(T)^+$ of dominant weights of $G$. A decomposition of $G$ as a product of subsuperschemes $U^-\times G_{ev}\times U^+$ induces a superalgebra isomorphism $ϕ^* : K[U^-]\otimes K[G_{ev}]\otimes K[U^+]\simeq K[G]$. We show that $C_Γ=ϕ^*(K[U^-]\otimes M_Γ\otimes K[U^+])$, where $M_Γ=O_Γ(K[G_{ev}])$. Using the basis of the module $M_Γ$, given by generalized bideterminants, we describe a basis of $C_Γ$. Since each $C_Γ$ is a subsupercoalgebra of $K[G]$, its dual $C_Γ^*=S_Γ$ is a (pseudocompact) superalgebra, called the generalized Schur superalgebra. There is a natural superalgebra morphism $π_Γ:Dist(G)\to S_Γ$ such that the image of the distribution algebra $Dist(G)$ is dense in $S_Γ$. For the ideal $X(T)^+_{l}$, of all weights of fixed length $l$, the generators of the kernel of $π_{X(T)^+_{l}}$ are described.
For the Schur superalgebra $S=S(m|n,r)$ over a ground field $K$ of characteristic zero, we define symmetrizers $T^λ[i:j]$ of the ordered pairs of tableaux $T_i, T_j$ of the shape $λ$ and show that the $K$-span $A_{λ,K}$ of all symmetrizers $T^λ[i:j]$ has a basis consisting of $T^λ[i:j]$ for $T_i,T_j$ semistandard. The $S$-superbimodule $A_{λ,K}$ is identified as %$Δ(λ)^*\otimes_K \nabla(λ)$, where $Δ(λ)^*$ is the dual of the standard supermodule %and $\nabla(λ)$ is the costandard supermodule of the highest weight $λ$. $D_λ\otimes_K D^o_λ$, where $D_λ$ and $D^o_λ$ are left and right irreducible $S$-supermodules of the highest weight $λ$. We define modified symmetrizers $T^λ\{i:j\}$ and show that their $\mathbb{Z}$-span form a $\mathbb{Z}$-form $A_{λ,\mathbb{Z}}$ of $A_{λ, \mathbb{Q}}$. We show that every modified symmetrizer $T^λ\{i:j\}$ is a $\mathbb{Z}$-linear combination of symmetrizers $T^λ\{i:j\}$ for $T_i, T_j$ semistandard. Using modular reduction to a field $K$ of characteristic $p>2$, we obtain that $A_{λ,K}$ has a basis consisting of modified symmetrizers $T^λ\{i:j\}$ for $T_i, T_j$ semistandard.
We give a short proof of the formula $n^p=\sum_{\ell=0}^{p-1} (-1)^{\ell} c_{p,\ell} F^{p-\ell}_n$, where $F^{p-\ell}_n$ is the figurate number and $c_{p,\ell}$ is the number of $(p-\ell)$-dimensional facets of $p$-dimensional simplices obtained by cutting the $p$-dimensional cube.
In this paper we investigate the image of the center $Z$ of the distribution algebra $Dist(GL(m|n))$ of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism $h:Z \to Dist(T)$ obtained by the restriction of the natural map $Dist(GL(m|n))\to Dist(T)$. We define supersymmetric elements in $Dist(T)$ and show that each image $h(c)$ for $c\in Z$ is supersymmetric. The central part of the paper is devoted to a description of a minimal set of generators of the algebra of supersymmetric elements over Frobenius kernels $T_r$.
Description of adjoint invariants of general Linear Lie superalgebras $\mathfrak{gl}(m|n)$ by Kantor and Trishin is given in terms of supersymmetric polynomials. Later, generators of invariants of the adjoint action of the general linear supergroup $GL(m|n)$ and generators of supersymmetric polynomials were determined over fields of positive characteristic. In this paper, we introduce the concept of supersymmetric elements in the divided powers algebra $Div[x_1, \ldots, x_m,y_1, \ldots, y_n]$, and give a characterization of supersymmetric elements via a system of linear equations. Then we determine generators of supersymmetric elements for divided powers algebras in the cases when $n=0$, $n=1$, and $m\leq 2, n=2$.
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 $H^0_G(λ)$, corresponding to a dominant weight $λ$ of $G$, can be represented as $H^0_{G_{ev}}(λ)\otimes Λ(Y)$, 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 $Λ(Y)$ is the exterior algebra of $Y$. For a dominant weight $λ$ of $G$, we construct explicit $G_{ev}$-primitive vectors in $H^0_G(λ)$. Related to this, we give explicit formulas for $G_{ev}$-primitive vectors of the supermodules $H^0_{G_{ev}}(λ)\otimes \otimes^k Y$. Finally, we describe a basis of $G_{ev}$-primitive vectors in the largest polynomial subsupermodule $\nabla(λ)$ of $H^0_G(λ)$ (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 $H^0_G(λ)$.
The purpose of the paper is to derive linkage principle for modular representations of ortho-symplectic supergroups. We follow the approach of Doty and investigate in detail the representation theory of the orthosymplectic group $OSP(2|1)$ and that of its Frobenius thickening. Using the description of flags and adjacent Borel supersubgroups we derive first the strong linkage for the Frobenius thickening $G_rT$ of the orthosymplectic supergroup $G$ of type $SpO(2m|2n+1)$ and $SpO(2m|2n)$. Based on this, we derive the linkage principle for orthosymplectic supergroup $SpO(2m|2n+1)$ and $SpO(2m|2n)$.
We investigate questions related to the minimal degree of invariants of finitely generated diagonalizable groups. These questions were raised in connection to security of a public key cryptosystem based on invariants of diagonalizable groups. We derive results for minimal degrees of invariants of finite groups, abelian groups and algebraic groups. For algebraic groups we relate the minimal degree of the group to the minimal degrees of its tori. Finally, we investigate invariants of certain supergroups that are superanalogs of tori. It is interesting to note that a basis of these invariants is not given by monomials.
We develop a public key cryptosystem based on invariants of diagonalizable groups and investigate properties of such cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on Euclidean algorithm).
We prove the linkage principle and describe blocks of the general linear supergroups $GL(m|n)$ over the ground field $K$ of characteristic $p\neq 2$.
It is shown in our earlier paper that, using only tools of elementary geometry, the classical Routh's theorem for triangles can be fully extended to tetrahedra. In this article we first give another proof of Routh's theorem for tetrahedra where methods of elementary geometry are combined with the inclusion-exclusion principle. Then we generalize this approach to $(n-1)-$ dimensional simplices. A comparison with the formula obtained using vector analysis yields an interesting algebraic identity.
The purpose of this paper is to investigate central elements in distribution algebras $Dist(G)$ of general linear supergroups $G=GL(m|n)$. As an application, we compute explicitly the center of $Dist(GL(1|1))$ and its image under Harish-Chandra homomorphism.
The purpose of the paper is to derive formulas that describe the structure of the induced supermodule H^0_G(\la) for the general linear supergroup G=GL(m|n) over an algebraically closed field K of characteristic p\neq 2. Using these formulas we determine primitive G_{ev}=GL(m)\times GL(n)-vectors in H^0_G(λ). We conclude with remarks related to the linkage principle in positive characteristic.
In this note we determine when is an induced module H^0_G(λ), corresponding to a dominant integral highest weight λof the general linear supergroup G=GL(m|n) irreducible. Using the contravariant duality given by the supertrace we obtain a characterization of irreducibility of Weyl modules V(λ). This extends the result of Kac who proved that, for ground fields of characteristic zero, V(λ) is irreducible if and only if λis typical.
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 $GL(1|1)$.