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Bin Shu

Publications and source records attributed to Bin Shu.

44 records · Page 3Linked to original sources
arXiv2014

Finite $W$-superalgebras and the dimensional lower bounds for the representations of basic Lie superalgebras

In this paper we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\mathfrak{g}_\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type $A$. Under the assumption of this conjecture, we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable. Such lower bounds, as a super-version of Kac-Weisfeiler conjecture, were formulated by Wang-Zhao in \cite{WZ} for the modular representations of a basic Lie superalgebra ${\ggg}_{\bbk}$ over an algebraically closed field $\bbk$ of positive characteristic $p$.

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arXiv2014

Finite W-superalgebras for basic classical Lie superalgebras

We consider the finite W-superalgebras for a basic classical Lie superalgebra g associated with an even nilpotent element in g both over the field of complex numbers field and and over a filed of positive characteristic. We present the PBW theorem for these finite W-superalgebrfas. Then we formulate a conjecture about the minimal dimensional representations of of complex finite W-superalgebras, and demonstrate it with some examples. Under the assumption that the conjecture holds, we finally show that the lower bound of dimensions predicted in the super version of Kac-Weisfeiler conjecture formulated and proved by Wang-Zhao in [40] for the modular representations of the basic classical Lie superalgebra with any p-characters can be reached.

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arXiv2014

A criterion for irreducibility of parabolic baby Verma modules of reductive Lie algebras

Let $G$ be a connected, reductive algebraic group over an algebraically closed field $k$ of prime characteristic $p$ and $\mathfrak{g}=Lie(G)$. In this paper, we study representations of $\mathfrak{g}$ with a $p$-character $χ$ of standard Levi form. When $\mathfrak{g}$ is of type $A_n, B_n, C_n$ or $D_n$, a sufficient condition for the irreducibility of standard parabolic baby Verma $\mathfrak{g}$-modules is obtained. This partially answers a question raised by Friedlander and Parshall in [Friedlander E. M. and Parshall B. J., Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395]. Moreover, as an application, in the special case that $\mathfrak{g}$ is of type $A_n$ or $B_n$, and $χ$ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen in [Jantzen J. C., Subregular nilpotent representations of $sl_n$ and $so_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999), 223-257].

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arXiv2012

On $\bbf_q$-rational structure of nilpotent orbits

Let $G$ be a simple algebraic group and $\ggg=\Lie(G)$ over $k=\bar\bbf_q$ where $q$ is a power of the prime characteristic of $k$, and $F$ a Frobenius morphism on $G$ which can be defined naturally on $\ggg$. In this paper, we investigate the relation between $F$-stable restricted modules of $\ggg$ and closed conical subvarieties defined over $\bbf_q$ in the null cone $\cn(\ggg)$ of $\ggg$. Furthermore, we clearly investigate the $\bbf_q$-rational structure for all nilpotent orbits in $\ggg$ under the adjoint action of $G$ when the characteristic of $k$ is good for $G$ and bigger than 3. The arguments are also valid when $\ggg'$ is classical simple Lie algebra in the sense of \cite{Sel} and $G'$ is the adjoint group of $\ggg'$.

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arXiv2012

Weyl groups for non-classical restricted Lie algebras and the Chevalley restriction theorem

Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.

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arXiv2007

Filtrations in Modular Representations of Reductive Lie Algebras

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with $p$-character $χ$ of standard Levi-form associated with an index subset $I$ of simple roots. With aid of support variety theory we prove a theorem that a $U_χ(\ggg)$-module is projective if and only if it is a strong "tilting" module, i.e. admitting both $\cz_Q$- and $\cz^{w^I}_Q$-filtrations (to see Theorem \ref{THMFORINV}). Then by analogy of the arguments in \cite{AK} for $G_1T$-modules, we construct so-called Andersen-Kaneda filtrations associated with each projective $\ggg$-module of $p$-character $χ$, and finally obtain sum formulas from those filtrations.

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arXiv2007

Modular representations of the ortho-symplectic supergroups

A Chevalley type integral basis for the ortho-symplectic Lie superalgebra is constructed. The simple modules of the ortho-symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified, where a key combinatorial ingredient comes from the Mullineux conjecture on modular representations of the symmetric group. A Steinberg tensor product theorem for the ortho-symplectic supergroup is also obtained.

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