arXiv subjects

Ryuji Tanimoto

Publications and source records attributed to Ryuji Tanimoto.

6 recordsLinked to original sources

arXiv2025

Exponential matrices, $\mathbb{G}_a$-actions on projective spaces and modular representations of elementary abelian $p$-groups

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{G}_a$ denote the additive group of $k$. Let $n \geq 1$ and let ${\rm Mat}(n, k[T])^E$ denote the set of all exponential matrices of ${\rm Mat}(n, k[T])$. Let $\mathbb{E}_{\geq 0}(n, k)$ denote the set of all group homomorphisms from $(\mathbb{Z}/p\mathbb{Z})^r$ to ${\rm GL}(n, k)$, where $r$ ranges over all non-negative integers. In the first, we show that there exists a one-to-one correspondence between the set ${\rm Mat}(n, k[T])^E$ and the set of all $\mathbb{G}_a$-actions on $\mathbb{P}^{n - 1}$. In the second, we show that there exists a one-to-one correspondence between $\mathbb{E}_{\geq 0}(n, k)$ and the set ${\rm Mat}(n, k[T])^E \times \mathbb{Z}_{\geq 0}$.

math.RT↗

arXiv2025

Birational equivalence of exponential matrices

In this article, we consider birational equivalence of exponential matrices. In characteristic zero, we give a birational classification of exponential matrices of size $n$-by-$n$ $(n \geq 2)$, which consists of two types. And in positive characteristic, we give birational classifications of exponential matrices of sizes two-by-two and three-by-three, respectively.

math.AG↗

arXiv2025

Homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ in positive characteristic

Let $k$ be an algebraically closed field of positive characteistic $p$ and let ${\rm SL}(n, k)$ denote the special linear algebraic group of degree $n$ over $k$. In this paper, we describe homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$. As by-products of this description, we give a classification of homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ and describe the indecomposable decompositions of homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$.

math.RT↗

arXiv2024

Triangular involutions of the four-dimensional polynomial ring in characteristic two

We are concerned with polynomial involutions in characteristic two. In this note, we look for involutions among triangular automorphisms of the four-dimensional polynomial ring in characteristic two and obtain three types of such involutions.

math.AC↗

arXiv2023

Representations of $\mathbb{G}_a \rtimes \mathbb{G}_m$ into ${\rm SL}(3, k)$ in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic $p$. In this article, we classify representations of $\mathbb{G}_a \rtimes \mathbb{G}_m$ into ${\rm SL}(3, k)$, and thereby we classify fundamental representations of $\mathbb{G}_a$ into ${\rm SL}(3, k)$.

math.RT↗

arXiv2018

Exponential matrices

In this article, we introduce a notion of an exponential matrix, which is a polynomial matrix with exponential properties, and a notion of an equivalence relation of two exponential matrices, and then we initiate to study classifying exponential matrices in positive characteristic, up to equivalence. We classify exponential matrices of Heisenberg groups in positive characteristic, up to equivalence. We also classify exponential matrices of size four-by-four in positive characteristic, up to equivalence. From these classifications, we obtain a classification of modular representations of elementary abelian $p$-groups into Heisenberg groups, up to equivalence, and a classification of four-dimensional modular representations of elementary abelian $p$-groups, up to equivalence.

math.RT↗