Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2209.13634 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915208432189440 |
|---|---|
| author | Maazouz, Yassine EL Lerario, Antonio |
| author_facet | Maazouz, Yassine EL Lerario, Antonio |
| contents | We prove that if $p>d$ there is a unique gaussian distribution (in the sense of Evans) on the space $\mathbb{Q}_p[x_1, \ldots, x_n]_{(d)}$ which is invariant under the action of $\mathrm{GL}(n, \mathbb{Z}_p)$ by change of variables. This gives the nonarchimedean counterpart of Kostlan's Theorem on the classification of orthogonally (respectively unitarily) invariant gaussian measures on the space $\mathbb{R}[x_1, \ldots, x_n]_{(d)}$ (respectively $\mathbb{C}[x_1, \ldots, x_n]_{(d)}$). More generally, if $V$ is an $n$--dimensional vector space over a nonarchimedean local field $K$ with ring of integers $R$, and if $λ$ is a partition of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_λ(V)$ under the action of the group $\mathrm{GL}(n,R)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_13634 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials Maazouz, Yassine EL Lerario, Antonio Number Theory Algebraic Geometry Probability 20G25, 12J25, 28C10, 51E24 We prove that if $p>d$ there is a unique gaussian distribution (in the sense of Evans) on the space $\mathbb{Q}_p[x_1, \ldots, x_n]_{(d)}$ which is invariant under the action of $\mathrm{GL}(n, \mathbb{Z}_p)$ by change of variables. This gives the nonarchimedean counterpart of Kostlan's Theorem on the classification of orthogonally (respectively unitarily) invariant gaussian measures on the space $\mathbb{R}[x_1, \ldots, x_n]_{(d)}$ (respectively $\mathbb{C}[x_1, \ldots, x_n]_{(d)}$). More generally, if $V$ is an $n$--dimensional vector space over a nonarchimedean local field $K$ with ring of integers $R$, and if $λ$ is a partition of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_λ(V)$ under the action of the group $\mathrm{GL}(n,R)$. |
| title | $\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials |
| topic | Number Theory Algebraic Geometry Probability 20G25, 12J25, 28C10, 51E24 |
| url | https://arxiv.org/abs/2209.13634 |