Irreducibility of induced modules for general linear supergroups

In this note we determine when is an induced module H^0_G(\lambda), corresponding to a dominant integral highest weight \lambda of 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(\lambda). This extends the result of Kac who proved that, for ground fields of characteristic zero, V(\lambda) is irreducible if and only if \lambda is typical

Supersymmetric elements in divided powers algebras

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 $m\leq 2, n=2$

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 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(\lambda). We conclude with remarks related to the linkage principle in positive characteristic

Routh's theorem for simplices

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.Comment: 15 pages, 8 figure

Blocks for general linear supergroup GL(m∣n)GL(m|n)

We prove the linkage principle and describe blocks of the general linear supergroups $GL(m|n)$ over the ground field $K$ of characteristic $p\neq 2$

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 $GL(1|1)$.Comment: 43 page

Central elements in the distribution algebra of a general linear supergroup and supersymmetric elements

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 $T_r$

A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroups

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, 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 $\phi^*(x_{ij})$ 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 $\Gamma$ be a finitely generated ideal $\Gamma$ of $X(T)^+$ and $O_{\Gamma}(K[G])$ be the largest $\Gamma$-subsupermodule of $K[G]$ having simple composition factors of highest weights $\lambda\in \Gamma$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_{\Gamma}(K[G])$

Symmetrizers for Schur superalgebras

For the Schur superalgebra $S=S(m|n,r)$ over a ground field $K$ of characteristic zero, we define symmetrizers $T^{\lambda}[i:j]$ of the ordered pairs of tableaux $T_i, T_j$ of the shape $\lambda$ and show that the $K$-span $A_{\lambda,K}$ of all symmetrizers $T^{\lambda}[i:j]$ has a basis consisting of $T^{\lambda}[i:j]$ for $T_i,T_j$ semistandard. The $S$-superbimodule $A_{\lambda,K}$ is identified as %$\Delta(\lambda)^*\otimes_K \nabla(\lambda)$, where $\Delta(\lambda)^*$ is the dual of the standard supermodule %and $\nabla(\lambda)$ is the costandard supermodule of the highest weight $\lambda$. $D_{\lambda}\otimes_K D^o_{\lambda}$, where $D_\lambda$ and $D^o_\lambda$ are left and right irreducible $S$-supermodules of the highest weight $\lambda$. We define modified symmetrizers $T^{\lambda}\{i:j\}$ and show that their $\mathbb{Z}$-span form a $\mathbb{Z}$-form $A_{\lambda,\mathbb{Z}}$ of $A_{\lambda, \mathbb{Q}}$. We show that every modified symmetrizer $T^\lambda\{i:j\}$ is a $\mathbb{Z}$-linear combination of symmetrizers $T^\lambda\{i:j\}$ for $T_i, T_j$ semistandard. Using modular reduction to a field $K$ of characteristic $p>2$, we obtain that $A_{\lambda,K}$ has a basis consisting of modified symmetrizers $T^\lambda\{i:j\}$ for $T_i, T_j$ semistandard

A note on the geometry of figurate numbers

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