تقرير
On singularity properties of word maps and applications to probabilistic Waring type problems
| العنوان: | On singularity properties of word maps and applications to probabilistic Waring type problems |
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| المؤلفون: | Glazer, Itay, Hendel, Yotam I. |
| سنة النشر: | 2019 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Group Theory, Mathematics - Number Theory, Mathematics - Probability, 11P05, 14B05, 14B07, 14G05, 17B45, 20F69, 20G25, 20P05, 60B15 (Primary) 03C98, 11G25, 17B01, 20E18, 20G40, 22E50 (Secondary) |
| الوصف: | We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. Given a word $w$ in a free Lie algebra $\mathcal{L}_{r}$, it induces a word map $\varphi_{w}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ for every semisimple Lie algebra $\mathfrak{g}$. Given two words $w_{1}\in\mathcal{L}_{r_{1}}$ and $w_{2}\in\mathcal{L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $\varphi_{w_{1}}*\varphi_{w_{2}}:=\varphi_{w_{1}}+\varphi_{w_{2}}:\mathfrak{g}^{r_{1}+r_{2}}\rightarrow\mathfrak{g}$. We show that for any word $w\in\mathcal{L}_{r}$ of degree $d$, and any simple Lie algebra $\mathfrak{g}$ with $\varphi_{w}(\mathfrak{g}^{r})\neq0$, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking $O(d^{4})$ self-convolutions of $\varphi_{w}$. We deduce that a group word map of length $\ell$ becomes (FRS) at $(e,\ldots,e)\in G^{r}$ after $O(\ell^{4})$ self-convolutions, for any semisimple algebraic group $G$. We furthermore bound the dimensions of the jet schemes of the fibers of Lie algebra word maps, and the fibers of group word maps in the case where $G=\mathrm{SL}_{n}$. For the commutator word $\nu=[X,Y]$, we show that $\varphi_{\nu}^{*4}$ is (FRS) for any semisimple Lie algebra, obtaining applications in representation growth of compact $p$-adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form $\mathbb{Z}/p^{k}\mathbb{Z}$. This allows us to relate them to properties of some natural families of random walks on finite and compact $p$-adic groups. We explore these connections, and provide applications to $p$-adic probabilistic Waring type problems. Comment: 78 pages, a few new results are added, with improved bounds. The proof for low rank Lie algebras is simplified. Light changes in style. Comments welcome |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1912.12556 |
| رقم الأكسشن: | edsarx.1912.12556 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |