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Bin Shu

Publications and source records attributed to Bin Shu.

At least 19 recordsLinked to original sources
arXiv2026

Towards Precision Therapy in Hepatocellular Carcinoma: A Clinical-Reasoning LLM for Risk Stratification and Treatment Guidance

Hepatocellular carcinoma (HCC) is a common malignancy and a leading cause of cancer-related mortality. Current guidelines and staging systems provide coarse categories, but often miss within-stage heterogeneity and the clinical context in electronic medical records (EMRs). We present HCC-STAR (Hepatocellular Carcinoma Staging, Treatment And pRognosis), a clinically aligned large language model that reads routine EMR narratives and jointly outputs risk score-based staging, ranked guideline-consistent treatments with evidence-based rationales, and individualized survival estimates. We curated about 30,000 HCC cases from SEER and expanded them into EMR-style narrative training data using a clinician-validated, prompt-based augmentation workflow. On this corpus, we developed a knowledge-aligned reasoning framework optimized with a step-verifiable composite reward, moving beyond text-level memorization of clinical guidelines. In a multi-center cohort of 6,668 patients from 12 hospitals in China, HCC-STAR achieved state-of-the-art performance in treatment recommendation and risk stratification compared with clinical guidelines and competitive models, including GPT-5 and Gemini-2.5 Pro. Hypothetical overall-survival analysis showed a median survival of 51 months under adherence to HCC-STAR recommendations, compared with 29 and 32 months under BCLC and CNLC. In clinician-centric evaluations, blinded hepatobiliary specialists rated HCC-STAR's reasoning and evidence-based justifications as trustworthy. The model surpassed resident and attending physicians in treatment accuracy and helped physicians make more accurate decisions faster when used as an assistant. These findings support HCC-STAR as a reliable and verifiable decision-support system for risk stratification and precision therapy in HCC.

cs.AI↗
arXiv2026

Whittaker Category and Finite W-superalgebras for Cartan Type Lie Superalgebras

Let $W(n)$ be the finite-dimensional simple Lie superalgebra of fundamental type in the Cartan type series of Kac's classification result \cite{Kac77} over an algebraically closed field of characteristic $0$. Let $\mathbf{g}$ be the graded-zero part of $W(n)$ which is isomorphic to $\mathfrak{gl}(n)$. In the first part of this paper, following the basic idea of taking the ``minimal" parabolic subalgebra $\mathsf{P}$ as a working platform in \cite{DSY} we introduce the Whittaker category $\mscrw$ for representations of $W(n)$ associated with a nilpotent element $e$ in $\mathbf{g}_0$ and with $W(n)_{-1}$. This Whittaker category turns out to be close to the classical Whittaker category McDowell and Miličić-Soergel studied in \cite{Mc} and \cite{MS}, respectively (or see \cite{Back}). We finally classify the simple objects in $\mscrw$. In the second part, we introduce the finite $W$-algebra associated with $e$, we then establish a generalized Skryabin's equivalence between the representation category of the finite $W$-superalgebra and the category $\mscrw'$ of so-called weakened Whittaker modules over $W(n)$. Here $\mscrw'$ naturally contains $\mscrw$ as a full subcategory.

math.RT↗
arXiv2026

Defining sequences for fundamental root systems and Coxeter graphs for super Weyl groups

The super Weyl group of a basic classical Lie superalgebra was introduced and studied in \cite{PS}, which turns out to play an important role for the study of representations of the basic classical Lie superalgebras and algebraic supergroups (see \cite{PS, LS}). These groups turn out to be some quotients of Coxeter groups. It is deserved to specially investigate super Weyl groups via revealing the related Coxeter systems. The purpose of this paper is twofold. One is to describe the Coxeter systems for super Weyl groups of basic classical Lie superalgebras. The other one is to introduce defining sequences which are a kind of new descriptions of fundamental root systems for classical Lie superalgebras of type $A,B,C$ and $D$. Based on defining sequences, we decide the Coxeter groups associated with those super Weyl groups via Coxeter graphs.

math.RT↗
arXiv2026

On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and its rational representation $(ρ, M)$ over an algebraically closed filed $\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_ρM$ which is a product variety $\GL_n\times M$, endowed with an enhanced cross product. In this paper, we first show that the nilpotent cone $\ucaln:=\caln(\ugg)$ of the enhanced Lie algebra $\ugg:=\Lie(\uG)$ has finite nilpotent orbits under adjoint $\uG$-action if and only if up to tensors with one-dimensional modules, $M$ is isomorphic to one of the three kinds of modules: (i) a one-dimensional module, (ii) the natural module $\bk^n$, (iii) the linear dual of $\bk^n$ when $n>2$; and $M$ is an irreducible module of dimension not bigger than $3$ when $n=2$. We then investigate the geometry of enhanced nilpotent orbits when the finiteness occurs. Our focus is on the enhanced group $\uG=\GL(V)\ltimes_ηV$ with the natural representation $(η, V)$ of $\GL(V)$, for which we give a precise classification of finite nilpotent orbits via a finite set $\scrpe$ of so-called enhanced partitions of $n=\dim V$, then give a precise description of the closures of enhanced nilpotent orbits via constructing so-called enhanced flag varieties. Finally, the $\uG$-equivariant intersection cohomology decomposition on the nilpotent cone of $\ugg$ along the closures of nilpotent orbits is established.

math.RT↗
arXiv2026

Basic quasi-reductive root data and supergroups

We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is isomorphic to $G$, and the quotient $\mathbb{G}/\mathbb{G}_{\text{ev}}$ is $G$-equivariantly isomorphic to $Y$. We prove that, if $Y$ satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain $0$; (ii) $\mathfrak{g}:=\text{Lie}(\mathbb{G})$ admits a non-degenerate even symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type.

math.RT↗
arXiv2026

On the Zassenhaus varieties of finite $W$-algebras in prime characteristic

Let $Z(\mathcal{W})$ be the center of the finite $W$-algebra $\mathcal{W}({\mathfrak{g}},e)$ associated with $\mathfrak{g}=\text{Lie}(G)$ and a nilpotent element $e\in\mathfrak{g}$ for a connected reductive algebraic group $G$ over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$ under the standard hypotheses (H1)-(H3) in [Jantzen]. In this paper, we first demonstrate that our previous results in [Shu-Zeng] on the structure and geometric properties of $Z(\mathcal{W})$ for $p>>0$ are still true under the present weakened restriction on $p$. Then we study the Zassenhaus variety $\mathscr{Z}$ of $\mathcal{W}(\mathfrak{g},e)$, which is by definition the maximal spectrum $\text{Specm}(Z(\mathcal{W}))$ of $Z({\mathcal{W}})$. On basis of the structure properties of $Z({\mathcal{W}})$, we describe $\mathscr{Z}$ via a good transverse slice $\mathcal{S}$ and show that $\mathscr{Z}$ is birationally equivalent to $\mathcal{S}$, thereby a rational affine scheme. In the special case when $e=0$, we reobtain one of the main results of [Tange] on the rationality of the Zassenhaus varieites for reductive Lie algebras in prime characteristic.

math.RT↗
arXiv2025

Lie-Cartan modules and cohomology

As a sequel to [Duan-Shu-Yao], we introduce here a category $\mathscr{LC}$ arising from the BGG category $\mathcal{O}$ defined in [Duan-Shu-Yao] for Lie algebras of polynomial vector fields. The objects of $\mathscr{LC}$ are so-called Lie-Cartan modules which admit both Lie-module structure and compatible $R$-module structure ($R$ denotes the corresponding polynomial ring). This terminology is natural, coming from affine connections in differential geometry through which the structure sheaves in topology and the vector fields in geometry are integrated for differential manifolds. In this paper, we study Lie-Cartan modules and their categorical and cohomology properties. The category $\mathscr{LC}$ is abelian, and a ``highest weight category" with depths. Notably, the set of co-standard objects in the category $\mathcal{O}$ turns out to represent the isomorphism classes of simple objects of $\mathscr{LC}$. We then establish the cohomology for this category (called the $\mathscr{uLC}$-cohomology), extending Chevalley-Eilenberg cohomology theory. Another notable result says that in the fundamental case $\mathfrak{g}= W(n)$, the extension ring $\text{Ext}^\bullet_{\mathscr{uLC}}(R,R)$ for the polynomial algebra $R$ in the $\mathscr{uLC}$-cohomology is isomorphic to the usual cohomology ring $H^\bullet(\mathfrak{gl}(n))$ of the general linear Lie algebra $\mathfrak{gl}(n)$.

math.RT↗
arXiv2025

Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories

Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_θ\in\mathfrak{g}_{\bar0}$ with $-θ$ being a minimal root of $\mathfrak{g}$. Set $U(\mathfrak{g},e)$ to be the minimal finite $W$-superalgebras associated with the pair $(\mathfrak{g},e)$. In this paper we study the highest weight theory for $U(\mathfrak{g},e)$, introduce the Verma modules and give a complete isomorphism classification of finite-dimensional irreducible modules, via the parameter set consisting of pairs of weights and levels. Those Verma modules can be further described via parabolic induction from Whittaker modules for $\mathfrak{osp}(1|2)$ or $\mathfrak{sl}(2)$ respectively, depending on the detecting parity of $\textsf{r}:=\dim\mathfrak{g}(-1)_{\bar1}$. We then introduce and investigate the BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highest weight theory, as a counterpart of the works for finite $W$-algebras by Brundan-Goodwin-Kleshchev and Losev, respectively. In comparison with the non-super case, the significant difference here lies in the situation when $\textsf{r}$ is odd, which is a completely new phenomenon. The difficulty and complicated computation arise from there.

math.RT↗
arXiv2025

Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras

Considering the general linear Lie superalgebra $\mathfrak{gl}(m|n)=\mathfrak{gl}(m|n)_{\bar{\bar 0}}\oplus \mathfrak{gl}(m|n)_{\bar{\bar 1}}$ over $\mathbb{C}$, we first formulate a super version of Vust theorem associated with a principal nilpotent element $e\in \mathfrak{gl}(m|n)_{\bar{\bar 0}}$. As an application of this theorem, we then obtain a Schur-Sergeev duality for principal finite $W$-superalgebras which is partially a super version of Brundan-Kleshchev's higher level Schur-Weyl duality established in \cite{BKl}

math.RT↗
arXiv2025

Representations of a class of infinite-dimensional primitive Lie superalgebras

In [Kac77, Section 5.4] and [Kac 98], V. G. Kac tried to raise, and finished a classification of infinite-dimensional primitive Lie superalgebras. The series $\mathbf{W}(m,n)$ with $m,n$ being positive integers are the fundamental ones. In this article, we introduce the BGG category $\mathcal{O}$ of modules over $\textbf{W}(m,n)$, and try to systematically investigate the representations of $\mathbf{W}(m,n)$ in this category, analogue of the study in [Duan-Shu-Yao2024} dealing with finite-dimensional Lie superalgebra case $\mathbf{W}(0,n)$, or analogue of the study in [Duan-Shu-Yao2020] dealing with infinite-dimensional Lie algebra case $\mathbf{W}(m,0)$. Beyond a compound of the arguments in [Duan-Shu-Yao2020} and in [Duan-Shu-Yao2024], it is nontrivial to understand irreducible modules in $\mathcal{O}$, which is the main goal of this article. We solve the question with aid of homological analysis on costandard modules along with extending Skryabin's theory on independence of operators for graded differential operator Lie algebras in [Skryabin] to the super case. After classifying irreducible modules in this category and describing their structure, we finally obtain irreducible characters. In the end, by confirming the semi-infinite character property, and applying Soergel's tilting module theory in [Soergel], we study indecomposable tilting modules in $\mathcal{O}$, obtaining their character formulas.

math.RT↗
arXiv2024

Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture

Suppose $g=g_0+g_1$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(g)$ of $g$, as a super generalization of the celebrated first Kac-Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

math.RT↗
arXiv2024

On automorphisms of affine superspaces

In this note, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field $\mathbb{F}$ of characteristic $0$, which predicts that for a homomorphism $φ$ of the polynomial superalgebra $\mathcal{R}:=\mathbb{F}[x_1,\ldots,x_m; ξ_1,\ldots,ξ_m]$ over $\mathbb{F}$, if $φ$ satisfies the super version of Jacobian condition (SJ for short), then $φ$ gives rise to an automorphism of the affine superspace $\mathbb{A}_{\mathbb{F}}^{m|n}$. We verify the conjecture if additionally, the set $\mathscr{M}$ of maximal $\mathbb{Z}_2$-homogeneous ideals of $\mathcal{R}$ is assumed to be preserved under $φ$. The statement is actually proved in any characteristic, i.e. a homomorphism $φ$ gives rise to an automorphism of $\mathbb{A}_{\mathbb{F}}^{m|n}$ if SJ is satisfied with $φ$ and the set $\mathscr{M}$ is preserved under $φ$ for an algebraically closed field $\mathbb{F}$ of any characteristic.

math.AG↗
arXiv2024

Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1}$ be a basic classical Lie superalgebra over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. Denote by $\mathcal{Z}$ the center of the universal enveloping algebra $U(\mathfrak{g})$. Then $\mathcal{Z}$ turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction $\text{Frac}(\mathcal{Z})$ is isomorphic to $\text{Frac}(\mathfrak{Z})$ for the center $\mathfrak{Z}$ of $U(\mathfrak{g}_{\bar 0})$. Consequently, both Zassenhaus varieties for $\mathfrak{g}$ and $\mathfrak{g}_{\bar 0}$ are birationally equivalent via a subalgebra $\widetilde{mathcal{Z}}\subset\mathcal{Z}$, and $\text{Spec}(\mathcal{Z})$ is rational under the standard hypotheses.

math.RT↗
arXiv2023

Duplex Hecke Algebras of type B

As a sequel to [14], in this article we first introduce a so-called duplex Hecke algebras of type B which is a Q(q)-algebra associated with the Weyl group W (B) of type B, and symmetric groups S_l for l = 0, 1, . . . ,m, satisfying some Hecke relations. This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur-Weyl duality of Levi-type, extending the duplex Hecke algebra of type A arising from the related q-Schur-Weyl duality of Levi-type. A duplex Hecke algebra of type B admits natural representations on certain tensor spaces. We then establish a Levi-type q-Schur-Weyl duality of type B, which reveals the double centralizer property between such duplex Hecke algebras and ıquantum groups studied by Bao-Wang in [1].

math.RT↗
arXiv2023

Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over complex numbers. The gradation of such a Lie superalgebra $\ggg$ naturally arises, with the zero component $\ggg_0$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\ggg_0$: the ``maximal one" $\sfp_\max$ and the ``minimal one" $\sfp_\min$. Furthermore, the parabolic BGG category arising from $\sfp_\max$, essentially turns out to be a subcategory of the one arising from $\sfp_\min$. Such a priority of $\sfp_\min$ in the sense of representation theory reduces the question to the study of the ``minimal parabolic" BGG category $\comi$ associated with $\sfp_\min$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows: (1) We classify and obtain a precise description of the blocks of $\comi$. (2) We investigate indecomposable tilting and indecomposable projective modules in $\comi$, and compute their character formulas.

math.RT↗
arXiv2023

Modular representations of strange classical Lie superalgebras and the first super Kac-Weisfeiler conjecture

Suppose $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1} is a Lie superalgebra of queer type or periplectic type over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. In this article, we initiate preliminarily to investigate modular representations of periplectic Lie superalgebras and then verify the first super Kac-Weisfeiler conjecture on the maximal dimensions of irreducible modules for $\mathfrak{g}$ proposed by the second-named author in [Shu] where the conjecture is targeted at all finite-dimensional restricted Lie superalgebras over $\bk$, and already proved to be true for basic classical Lie superalgebras and completely solvable restricted Lie superalgebras.

math.RT↗
arXiv2023

Jantzen filtration of Weyl modules for general linear supergroups

Let $G=GL(m|n)$ be a general linear supergroup over an algebraically closed field $k$ of odd characteristic $p$. In this paper we construct Jantzen filtration of Weyl modules $V(λ)$ of $G$ when $λ$ is a typical weight in the sense of Kac's definition, and consequently obtain a sum formula for their characters. By Steinberg's tensor product theorem, it is enough for us to study typical weights with aim to formulate irreducible characters. As an application, it turns out that an irreducible $G$-module $L(λ)$ can be realized as a Kac module if and only if $λ$ is $p$-typical.

math.RT↗