GLn(q) 上的文字度量和自由群集代数

IF 0.9 1区数学 Q2 MATHEMATICS

Algebra & Number Theory Pub Date : 2024-10-18 DOI:10.2140/ant.2024.18.2047

Danielle Ernst-West, Doron Puder, Matan Seidel

引用次数: 0

摘要

固定一个阶数为 q 的有限域 K 和一个自由群 F 中关于 r 个发电机的字 w。GL N(K)中的 w-随机元素是通过采样 r 个独立的均匀随机元素 g1,... ,gr∈ GL N(K)并求值 w(g1,...,gr)得到的。考虑𝔼w[fix ]，即由 w 个随机元素固定的 KN 中向量的平均数。我们将证明𝔼w[fix ] 是 qN 中的有理函数。如果 w= ud，而 u 是非幂，那么极限 lim N→∞𝔼w[fix ] 只取决于 d 而不取决于 u。这项工作的一个主要特点是我们在 GL N(K) 上的字计量和自由群代数 K[F] 之间建立了联系。Cohn (1964) 和 Lewin (1969) 的一个经典结果是，K[F] 的每一个单边理想都是一个自由 K[F] 模块，具有定义明确的秩。我们证明，对于非幂级数的 w，𝔼w[fix ]= 2+ CqN+O( 1q2N)，其中 C 是包含 w- 1 但不作为基元的秩 2 右理想 I≤K[F] 的数目。在此过程中，我们证明了关于自由群集的几个新结果。例如，我们证明了如果 T 是 F 的 Cayley 图的任意有限子树，而 I≤K[F] 是一个右理想，其生成集支持在 T 上，那么 I 允许一个支持在 T 上的基。

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Word measures on GLn(q) and free group algebras

Fix a finite field $K$ of order $q$ and a word $w$ in a free group $F$ on $r$ generators. A $w$ -random element in ${GL}_{N} (K)$ is obtained by sampling $r$ independent uniformly random elements $g_{1}, \dots, g_{r} \in {GL}_{N} (K)$ and evaluating $w (g_{1}, \dots, g_{r})$ . Consider $𝔼_{w} [fix]$ , the average number of vectors in $K^{N}$ fixed by a $w$ -random element. We show that $𝔼_{w} [fix]$ is a rational function in $q^{N}$ . If $w = u^{d}$ with $u$ a nonpower, then the limit $\lim_{N \to \infty} 𝔼_{w} [fix]$ depends only on $d$ and not on $u$ . These two phenomena generalize to all stable characters of the groups ${{GL}_{N} (K)}_{N}$ .

A main feature of this work is the connection we establish between word measures on ${GL}_{N} (K)$ and the free group algebra $K [F]$ . A classical result of Cohn (1964) and Lewin (1969) is that every one-sided ideal of $K [F]$ is a free $K [F]$ -module with a well-defined rank. We show that for $w$ a nonpower, $𝔼_{w} [fix] = 2 + \frac{C}{q^{N}} + O (\frac{1}{q^{2 N}})$ , where $C$ is the number of rank-2 right ideals $I \leq K [F]$ which contain $w - 1$ but not as a basis element. We describe a full conjectural picture generalizing this result, featuring a new invariant we call the $q$ -primitivity rank of $w$ .

In the process, we prove several new results about free group algebras. For example, we show that if $T$ is any finite subtree of the Cayley graph of $F$ , and $I \leq K [F]$ is a right ideal with a generating set supported on $T$ , then $I$ admits a basis supported on $T$ . We also prove an analog of Kaplansky’s unit conjecture for certain $K [F]$ -modules.

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来源期刊

Algebra & Number Theory MATHEMATICS-

CiteScore

1.80

自引率

7.70%

发文量

审稿时长

6-12 weeks

期刊介绍： ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.