环半群的仿射Schur--Weyl理论与多项式表示
Affine Schur--Weyl theory for loop Semigroups and polynomial representations
- Tongji University（同济大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对Laurent多项式环半群发展仿射Schur-Weyl理论，证明相关代数同态的满射性、双中心化子性质，确定仿射Schur代数中心，分类其有限维不可约多项式表示，得到不可约特征的显式公式。
AI中文摘要：
经典Schur–Weyl对偶在一般线性群与对称群的表示理论中具有基础性作用。本文针对Laurent多项式环半群发展仿射Schur–Weyl理论。设$\hat G_K(n)$为域$K$上的环群$GL_n(K((t)))$中的Laurent多项式环半群，$\hat S_K(n,r)$为仿射Schur代数。$\hat G_K(n)$在仿射张量空间$\Omega_K^{\otimes r}$上的自然作用诱导出代数同态$K\hat G_K(n)\rightarrow \hat S_K(n,r).$。我们证明，对满足$|K|>r$的任意域$K$，该同态是满射。此外，对所有$n\geq2$，我们在唯一分解整环上建立仿射Schur代数的双中心化子性质。对$n\geq2$及满足$|K|>r$的任意域$K$，这给出$\hat G_K(n)$对应的中心化子性质。相反，对$n=1$和$r\geq2$，对应的自然同态在任意域上都不是满射。对$n\geq 2$，我们利用双中心化子定理确定仿射Schur代数的中心；当$K$是满足$|K|>r$的域时，证明$\hat\zeta_{r,K}$将$K\hat G_K(n)$的中心映到该中心上。在任意特征的代数闭域$K$上，我们证明每个有限维不可约$\hat S_K(n,r)$-模都是赋值模的张量积。此外，我们得到显式特征公式，将这些不可约特征表示为不可约经典Schur代数模的特征的乘积。最后，我们研究代数闭域$K$上$\hat G_K(n)$的多项式表示，并对其有限维不可约多项式表示进行分类。对满足$q>r$的有限域$\mathbb F_q$，这给出定义特征情形下的分类。
英文摘要：
Classical Schur--Weyl duality plays a fundamental role in the representation theory of general linear and symmetric groups. In this paper, we develop an affine Schur--Weyl theory for the Laurent polynomial loop semigroup. Let $\hat G_K(n)$ denote the Laurent polynomial loop semigroup inside the loop group $GL_n(K((t)))$ over a field $K$, and let $\hat S_K(n,r)$ be the affine Schur algebra. The natural action of $\hat G_K(n)$ on the affine tensor space $Ω_K^{\otimes r}$ induces an algebra homomorphism $K\hat G_K(n)\rightarrow \hat S_K(n,r).$ We prove that this homomorphism is surjective for every field $K$ satisfying $|K|>r$. Furthermore, we establish the double centralizer property for affine Schur algebras over unique factorization domains for all $n\geq2$. For $n\geq2$ and any field $K$ with $|K|>r$, this yields the corresponding centralizer property for $\hat G_K(n)$. In contrast, for $n=1$ and $r\geq2$, the corresponding natural homomorphism is not surjective over any field. For $n\geq 2$, we use the double centralizer theorem to determine the center of the affine Schur algebra and, when $K$ is a field with $|K|>r$, show that $\hatζ_{r,K}$ maps the center of $K\hat G_K(n)$ onto it. Over an algebraically closed field $K$ of arbitrary characteristic, we prove that every finite-dimensional irreducible $\hat S_K(n,r)$-module is a tensor product of evaluation modules. Moreover, we obtain explicit character formulas expressing these irreducible characters as products of characters of irreducible classical Schur algebra modules. Finally, we study polynomial representations of $\hat G_K(n)$ over algebraically closed fields $K$, and classify its finite-dimensional irreducible polynomial representations. For finite fields $\mathbb F_q$ with $q>r$, this yields a classification in the defining-characteristic setting.
