AI 中文总结
研究仿射幂零考克斯特代数\(A\)在任意特征域\(k\)上的表示理论，通过研究其大交换子代数\(C\)，证明\(A\)是诺特素仿射PI代数，得出单模性质及相关结论。
AI 中文摘要
我们研究\(\tilde A_{n - 1}\)型仿射幂零考克斯特代数\(A\)在任意特征的域\(k\)上的表示理论。我们的主要定理表明，这是一个PI次数为\(n!\)的诺特素仿射PI代数。因此，简单\(A -\)模都是有限维的，并且在\(k\)的合适有限扩张上，简单模的最大维数是\(n!\)。为了实现这一点，我们研究一个作为代数有限生成的大交换子代数\(C\)，并且\(A\)作为\(C -\)模有限生成。我们表明\(C\)的相关素理想是极小素理想，并且有\(n!\)个这样的素理想，由\(\mathfrak{S}_n\)正则置换。代数\(R = C^{\mathfrak{S}_n}\)等于\(A\)的中心，并且对于每个极小素理想\(\mathfrak{p}\)，同构于\(C / \mathfrak{p}\)。我们证明环\(R\)同构于\(k[X_1,\dots,X_{n - 1}]^{\mu_n}\)，其中\(\mu_n\)是\(n\)次单位根的有限群概型，其作用使得\(X_i\)模\(n\)的次数为\(i\)。环\(R\)是科恩 - 麦考利环，并且当且仅当\(n\)为奇数或\(n = 2\)时是戈伦斯坦环。它是一个环面环，除子类群\(\mathsf{Cl}(R)\cong\mathbb{Z}/n\)，并且每个投射\(R -\)模是自由的。
英文摘要
We study the representation theory of the affine nilCoxeter algebra $A$ of type $\tilde A_{n-1}$, over a field $k$ of any characteristic. Our main theorem states that this is a Noetherian prime affine PI algebra of PI degree $n!$. As a consequence, the simple $A$-modules are all finite dimensional, and the maximum dimension of a simple module is $n!$ over a suitable finite extension of $k$. To achieve this, we investigate a large commutative subalgebra $C$ which is finitely generated as an algebra and over which $A$ is finitely generated as a module. We show that the associated primes of $C$ are minimal primes, and there are $n!$ of them, regularly permuted by $\mathfrak{S}_n$. The algebra $R=C^{\mathfrak{S}_n}$ is equal to the centre of $A$, and isomorphic to $C/\mathfrak{p}$ for each of the minimal primes $\mathfrak{p}$. We prove that the ring $R$ is isomorphic to $k[X_1,\dots,X_{n-1}]^{μ_n}$, where $μ_n$ is the finite group scheme of $n$th roots of unity, acting so that $X_i$ has degree $i$ modulo $n$. The ring $R$ is Cohen--Macaulay, and is Gorenstein if and only if $n$ is odd or $n=2$. It is a toric ring, with divisor class group $\mathsf{Cl}(R)\cong\mathbb{Z}/n$, and every projective $R$-module is free.
