组合范畴表示的Gröbner方法
Gröbner methods for representations of combinatorial categories
摘要：
给定一个组合性质的范畴C，我们研究以下基本问题：C的组合行为如何影响C的代数表示行为？我们证明了两个一般结果。第一个给出了一个组合的标准表示的C承认理论的Gröbner基地。由此，我们得到一个标准noetherianity的表示。第二类给出了C的Hilbert表示级数的一般“合理性”结果的组合判据。这个准则与形式语言的理论相联系，并充分利用了语言生成函数的结果，如转移矩阵方法和Chomsky-Schützenberger定理。我们的工作是出于最近的工作在文献中的各种具体类别的表示。我们的一般准则恢复了许多已经被特别方法证明的关于这些范畴的结果，并且经常产生更清晰的证明和更有力的陈述。举例来说：我们给出了一个新的，更强大的，证明FI-模（最初由Church-Ellenberg-Farb引入）和一族自然推广是Noether的;我们从有限域的一般表示论的研究中给出了Lannes-Schwartz artinian猜想的推广的简单证明;我们显著改进了Snowden引入的与Segre嵌入合冲有关的$Δ$ -模理论;建立了正特征扭交换代数的基本性质。
英文摘要：
Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a combinatorial criterion for representations of C to admit a theory of Gröbner bases. From this, we obtain a criterion for noetherianity of representations. The second gives a combinatorial criterion for a general "rationality" result for Hilbert series of representations of C. This criterion connects to the theory of formal languages, and makes essential use of results on the generating functions of languages, such as the transfer-matrix method and the Chomsky-Schützenberger theorem. Our work is motivated by recent work in the literature on representations of various specific categories. Our general criteria recover many of the results on these categories that had been proved by ad hoc means, and often yield cleaner proofs and stronger statements. For example: we give a new, more robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb), and a family of natural generalizations, are noetherian; we give an easy proof of a generalization of the Lannes-Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of $Δ$-modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.
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