arXiv subjects

Alexander Stasinski

Publications and source records attributed to Alexander Stasinski.

At least 19 recordsLinked to original sources

arXiv2025

On Fillmore's theorem over integrally closed domains

A well-known theorem of Fillmore says that if $A\in\operatorname{M}_{n}(K)$ is a non-scalar matrix over a field $K$ and $\gamma_{1},\dots,\gamma_{n}\in K$ are such that $\gamma_{1}+\dots+\gamma_{n}=\operatorname{Tr}(A)$, then $A$ is $K$-similar to a matrix with diagonal $(\gamma_{1},\dots,\gamma_{n})$. Building on work of Borobia, Tan extended this by proving that if $R$ is a unique factorisation domain with field of fractions $K$ and $A\in\operatorname{M}_{n}(R)$ is non-scalar, then $A$ is $K$-similar to a matrix in $\operatorname{M}_{n}(R)$ with diagonal $(\gamma_{1},\dots,\gamma_{n})$. We note that Tan's argument actually works when $R$ is any integrally closed domain and show that the result cannot be generalised further by giving an example of a matrix over a non-integrally closed domain for which the result fails. Moreover, Tan gave a necessary condition for $A\in\operatorname{M}_{n}(R)$ to be $R$-similar to a matrix with diagonal $(\gamma_{1},\dots,\gamma_{n})$. We show that when $R$ is a PID and $n\geq3$, Tan's condition is also sufficient.

math.RA↗

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↗

arXiv2023

The algebraisation of higher level Deligne--Lusztig representations II: odd levels

In this paper we study higher level Deligne--Lusztig representations of reductive groups over discrete valuation rings, with finite residue field $\mathbb{F}_q$. In previous work we proved that, at even levels, these geometrically constructed representations are isomorphic to certain algebraically constructed representations (referred to as the algebraisation theorem at even levels). In this paper we work with an arbitrary level $>1$. Our main result is (1) the algebraisation theorem at all levels $>1$ (with the sign being explicitly determined for $q\geq7$). As consequences, we obtain (2) the regular semisimplicity of orbits of generic higher level Deligne--Lusztig representations, and the dimension formula; in the course of the proof, we give (3) an induction formula of higher level Deligne--Lusztig representations, and a new proof of the character formula at regular semisimple elements.

math.RT↗

arXiv2022

Rationality of twist representation zeta functions of compact $p$-adic analytic groups

We prove that for any twist rigid compact $p$-adic analytic group $G$, its twist representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its twist representation zeta function is rational in $p^{-s}$. To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass. Second part of arXiv:2007.10694.

math.GR↗

arXiv2022

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

For a principal ideal domain $A$, the Latimer--MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\operatorname{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$. We prove that when $A[x]/(f(x))$ is maximal (i.e., integrally closed, i.e., a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

math.RA↗

arXiv2020

Representations of $\mathrm{SL}_{n}$ over finite local rings of length two

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$ and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any integer $n$ such that $p$ divides $n$, the groups $\mathrm{SL}_{n}(\mathbb{F}_{q}[t]/t^{2})$ and $\mathrm{SL}_{n}(W_{2}(\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$.

math.RT↗

arXiv2020

Rationality of representation zeta functions of compact $p$-adic analytic groups

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its representation zeta function is rational in $p^{-s}$. These results were proved by Jaikin-Zapirain for $p>2$ or for $G$ uniform and pro-$2$, respectively. We give a new proof which avoids the Kirillov orbit method and works for all $p$. First part of arXiv:2007.10694, second part uploaded as a separate paper.

math.GR↗

arXiv2020

A uniform proof of the finiteness of the class group of a global field

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of rings of integers of global fields.

math.AC↗

arXiv2019

Representations of reductive groups over finite local rings of length two

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G}\times\mathbb{F}_{q}$, the groups $\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})$ and $\mathbb{G}(W_{2}(\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$. Equivalently, there exists an isomorphism of group algebras $\mathbb{C}[\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})]\cong\mathbb{C}[\mathbb{G}(W_{2}(\mathbb{F}_{q}))]$.

math.RT↗

arXiv2018

Representation growth of compact linear groups

We study the representation growth of simple compact Lie groups and of $\mathrm{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\mathrm{GL}_n(\mathcal{O})$. This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is $r/κ$, where $r$ is the rank and $κ$ the number of positive roots. We then show that the twist zeta function of $\mathrm{GL}_n(\mathcal{O})$ exists and has the same abscissa of convergence as the zeta function of $\mathrm{SL}_n(\mathcal{O})$, provided $n$ does not divide $\text{char}\,{\mathcal{O}}$. We compute the twist zeta function of $\mathrm{GL}_2(\mathcal{O})$ when the residue characteristic $p$ of $\mathcal{O}$ is odd, and approximate the zeta function when $p=2$ to deduce that the abscissa is $1$. Finally, we construct a large part of the representations of $\mathrm{SL}_2(\mathbb{F}_q[[t]])$, $q$ even, and deduce that its abscissa lies in the interval $[1,\,5/2]$.

math.RT↗

arXiv2017

Representations of $\mathrm{GL}_N$ over finite local principal ideal rings - an overview

We give a survey of the representation theory of $\mathrm{GL}_N$ over finite local principal ideal rings via Clifford theory, with an emphasis on the construction of regular representations. We review results of Shintani and Hill, and the generalisation of Takase. We then summarise the main features, with some details but without proofs, of the recent constructions of regular representations due to Krakovski--Onn--Singla and Stasinski--Stevens, respectively.

math.RT↗

arXiv2017

Commutators of trace zero matrices over principal ideal rings

We prove that for every trace zero matrix $A$ over a principal ideal ring $R$, there exist trace zero matrices $X,Y$ over $R$ such that $XY-YX=A$. Moreover, we show that $X$ can be taken to be regular mod every maximal ideal of $R$. This strengthens our earlier result that $A$ is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one. Shalev has conjectured an analogous statement for group commutators in $\mathrm{SL}_{n}$ over $p$-adic integers. We prove Shalev's conjecture for $n=2$.

math.RA↗

arXiv2016

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

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.

math.RT↗

arXiv2016

Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces

We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating $\mathfrak{p}$-adic integrals associated to certain rank varieties of matrices of linear forms.

math.GR↗

arXiv2016

The algebraisation of higher Deligne--Lusztig representations

In this paper we study higher Deligne--Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations coincide with certain induced representations in the generic case; this gives a solution to a problem raised by Lusztig. In particular, we determine the dimensions of these representations. As an immediate application we verify a conjecture of Letellier for $\mathrm{GL}_2$ and $\mathrm{GL}_3$.

math.RT↗

arXiv2014

Reductive group schemes, the Greenberg functor, and associated algebraic groups

Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group $G:=(\mathcal{F}\mathbf{G})(k)$ over $k$, such that $G\cong\mathbf{G}(A)$. We prove that if $\mathbf{G}$ is a reductive group scheme over $A$, and $\mathbf{T}$ is a maximal torus of $\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for smooth affine schemes with reduced fibre over $A$, and that the Greenberg functor preserves certain normaliser group schemes over $A$. Moreover, we prove that if $\mathbf{G}$ is reductive and $\mathbf{P}$ is a parabolic subgroup of $\mathbf{G}$, then $P$ is a self-normalising subgroup of $G$, and if $\mathbf{B}$ and $\mathbf{B}'$ are two Borel subgroups of $\mathbf{G}$, then the corresponding subgroups $B$ and $B'$ are conjugate in $G$.

math.AG↗

arXiv2014

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and p-adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class-2-nilpotent groups obtained from three infinite families of group schemes, generalising the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type B. This allows us to establish a formula for the joint distribution of three functions, or 'statistics', on such Weyl groups. Finally, we compare our explicit formulae to p-adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.

math.GR↗

arXiv2013

A new statistic on the hyperoctahedral groups

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank. For several descent classes we prove the conjectural formula. For this we construct suitable "supporting sets" for the relevant generating functions. We prove cancellations on the complements of these supporting sets using suitably defined sign reversing involutions.

math.CO↗