arXiv subjects

Nariel Monteiro

Publications and source records attributed to Nariel Monteiro.

5 recordsLinked to original sources

arXiv2024

The conjugation representation of $\operatorname{GL}_{2}$ and $\operatorname{SL}_{2}$ over finite local rings

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for $G=\operatorname{SL}_{2}(\mathbb{Z}/p^{r})$, where $r\geq1$ and $p\geq5$ is a prime, any irreducible representation of $G$ that is trivial on the centre of $G$ is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to $p=2$ or $3$. We answer the Hain--Tiep question in the affirmative and also prove analogous statements for $\operatorname{SL}_{2}$ and $\operatorname{GL}_{2}$ over any finite local principal ideal ring with residue field of odd characteristic.

math.RT↗

arXiv2024

The Defining Characteristic Case of the Representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over Principal Ideal Local Rings

Let $W_{r}(\mathbb{F}_{q})$ be the ring of Witt vectors of length $r$ with residue field $\mathbb{F}_{q}$ of characteristic $p$. In this paper, we study the defining characteristic case of the representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over the principal ideal local rings $W_{r}(\mathbb{F}_{q})$ and $\mathbb{F}_{q}[t]/t^{r}$. Let ${\mathbf{G}}$ be either $\mathrm{GL}_{n}$ or $\mathrm{SL}_{n}$ and $F$ a perfect field of characteristic $p$, we prove that for most $p$ the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not stably equivalent of Morita type. Thus, the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not isomorphic in the defining characteristic case.

math.RT↗

arXiv2024

The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups

Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^Δ(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $k\otimes_\mathbb{Z} T^Δ(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^Δ(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^Δ(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $k\otimes_\mathbb{Z} T^Δ(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras.

math.RT↗

arXiv2023

The stable representations of $\mathrm{GL}_{N}$ over finite local principal ideal rings

Let $\mathcal{O}$ be a discrete valuation ring with maximal ideal $\mathfrak{p}$ and with finite residue field $\mathbb{F}_{q}$, the field with $q$ elements where $q$ is a power of a prime $p$. For $r \ge 1$, we write $\mathcal{O}_r$ for the reduction of $\mathcal{O}$ modulo the ideal $\mathfrak{p}^r$. An irreducible ordinary representation of the finite group $\mathrm{GL}_{N}(\mathcal{O}_{r})$ is called stable if its restriction to the principal congruence kernel $K^l=1+\mathfrak{p}^{l}\mathrm{M}_{N}(\mathcal{O}_r)$, where $l=\lceil \frac{r}{2} \rceil$, consists of irreducible representations whose stabilizers modulo $K^{l'}$, where $l'=r-l$, are centralizers of certain matrices in $\mathfrak{g}_{l'}=\mathrm{M}_{N}(\mathcal{O}_{l'})$, called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group $\mathrm{GL}_{N}(\mathcal{O}_{r})$ for $N \ge 2$.

math.RT↗

arXiv2020

The $\ell$-modular representation of reductive groups over finite local rings of length two

Let $\mathcal{O}_2$ and $\mathcal{O}'_2$ be two distinct finite local rings of length two with residue field of characteristic $p$. Let $\mathbb{G}(\mathcal{O}_2)$ and $\mathbb{G}(\mathcal{O}'_2)$, be the group of points of any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G} \times \mathbb{F}_q$. We prove that there exists an isomorphism of group algebra $K[\mathbb{G}(\mathcal{O}_2)] \cong K[\mathbb{G}(\mathcal{O}'_2)]$, where $K$ is a sufficiently large field of characteristic different from $p$.

math.RT↗