Narrow Results

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 λ_m = 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.

In this paper, we shall describe costandard modules $\nabla (\lambda )$ of restricted highest weight $\lambda$ for Schur superalgebra $S(2|2)$ over an algebraically closed field $K$ of positive characteristic $p>2$. Additionally, for a restricted highest weight $\lambda$, we determine all composition factors of the costandard module $\nabla (\lambda )$; in particular we compute the decomposition numbers in the process of the modular reduction of a simple module with a restricted highest weight.

In this paper, we investigate a kind of double centralizer property for general linear supergroups. For the super space $V={𝕂}^{m|n}$ over an algebraically closed field $𝕂$ whose characteristic is not equal to $2$, we consider its ${\mathbb{Z}}_2$-homogeneous one-dimensional extension $\underline{V}=V\oplus 𝕂v$, and the natural action of the supergroup $\tilde{G}:=\text{GL}(V)\times G_m$ on $\underline{V}$. Then we have the tensor product supermodule (${\underline{V}}^{\otimes r}$, ${\rho }_r$) of $\tilde{G}$. We present a kind of generalized Schur–Sergeev duality which is said that the Schur superalgebras $S^{\prime }(m|,n,r)$ of $\tilde{G}$ and the so-called weak degenerate double Hecke algebra ${\underline{{\mathscr{H}}}}_r$ are double centralizers. The weak degenerate double Hecke algebra is an infinite-dimensional algebra, which has a natural representation on the tensor product space. This notion comes from [B. Shu, Y. Xue and Y. Yao, On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras, preprint (2013), arXiv:2005.13152 [Math. RT]], with a little modification.