Explanipedia

Classification of nilpotent and semisimple fourvectors of a real eight-dimensional space Open

Di Bella, Emanuele, de Graaf, Willem A., Santi, Andrea · 2025

In 1981 Antonyan classified the orbits of SL$(8,\mathbb{C})$ on $\bigwedge^4 \mathbb{C}^8$. This is an example of a $θ$-group action as introduced and studied by Vinberg. The orbits of a $θ$-group are divided into three classes: nilpotent,…

The Relative Trace Formula for Galois Periods Open

Mahendraker, Siddharth · 2025

Let $E/F$ be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair $G \subset \mathrm{Res}_{E/F} G_E$, subject to the constraint that $G$ and $\mathrm{…

The Relative Trace Formula for Galois Periods Open

Mahendraker, Siddharth · 2025

Let $E/F$ be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair $G \subset \mathrm{Res}_{E/F} G_E$, subject to the constraint that $G$ and $\mathrm{…

Polynomially effective equidistribution for certain unipotent subgroups in quotients of semisimple Lie groups Open

Lin Zuo · 2025

We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of semisimple Lie groups with a polynomial error term. This provides the first infinite family of examples where effective equ…

Polynomially effective equidistribution for certain unipotent subgroups in quotients of semisimple Lie groups Open

Lin Zuo · 2025

We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of semisimple Lie groups with a polynomial error term. This provides the first infinite family of examples where effective equ…

On an infinite family of unipotent Sylvester–Kac-like matrices Open

Zhibin Du, Carlos M. da Fonseca · 2025

Classical Sylvester–Kac matrices are tridiagonal integral matrices with positive off-diagonal entries and fully integral spectra. Here, by relaxing the positivity requirement and using a lower Pascal triangle framework, we define, for each…

Minimal generation of finite simple groups of Lie type by regular unipotent elements Open

Pellegrini, M. A., Zalesski, A. E. · 2025

We prove that every finite simple group of Lie type $G$ can be generated by three regular unipotent elements. In certain cases we show that two regular unipotents are sufficient to generate $G$.

Minimal generation of finite simple groups of Lie type by regular unipotent elements Open

Pellegrini, M. A., Zalesski, A. E. · 2025

We prove that every finite simple group of Lie type $G$ can be generated by three regular unipotent elements. In certain cases we show that two regular unipotents are sufficient to generate $G$.

Characterization of the unit object in localized quantum unipotent category Open

Matsuura, Koh, Nakashima, Toshiki · 2025

For the quiver Hecke algebra $R$, let $R\hbox{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\hbox{-gmod}[w]}$ be the localization of $R\hbox{-gmod}$. Kashiwara and the second author showed the set o…

Characterization of the unit object in localized quantum unipotent category Open

Matsuura, Koh, Nakashima, Toshiki · 2025

For the quiver Hecke algebra $R$, let $R\hbox{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\hbox{-gmod}[w]}$ be the localization of $R\hbox{-gmod}$. Kashiwara and the second author showed the set o…

Noncommutative tensor triangular geometry: modules, bimodules, and unipotent Hopf algebras Open

Solberg, Øyvind, Vashaw, Kent B., Witherspoon, Sarah · 2025

We initiate a program aimed at classifying thick ideals, Balmer spectra, and submodule categories of various stable categories of bimodules and modules for finite dimensional selfinjective algebras, and at clarifying the relationship betwe…

Arithmetic monodromy in $\mathrm{Sp}(2n)$ Open

Jitendra. Bajpai, Daniele Donà, Martin Nitsche · 2025

Based on a result of Singh–Venkataramana, Bajpai–Dona–Singh–Singh gave a criterion for a discrete Zariski-dense subgroup of \mathrm{Sp}(2n,\mathbb{Z}) to be a lattice. We adapt this criterion so that it can be used in some situations that …

Noncommutative tensor triangular geometry: modules, bimodules, and unipotent Hopf algebras Open

Solberg, Øyvind, Vashaw, Kent B., Witherspoon, Sarah · 2025

We initiate a program aimed at classifying thick ideals, Balmer spectra, and submodule categories of various stable categories of bimodules and modules for finite dimensional selfinjective algebras, and at clarifying the relationship betwe…

Brauer groups of abelian varieties over fields of finite characteristic Open

Grammatica, Livia, Skorobogatov, Alexei N., Yang Yuan · 2025

We study the Brauer group of an abelian variety A over an algebraically closed field of characteristic p focusing on the p-primary torsion, the key part of which is a certain quasi-algebraic unipotent group U_A. We determine its dimension …

Brauer groups of abelian varieties over fields of finite characteristic Open

Grammatica, Livia, Skorobogatov, Alexei N., Yang Yuan · 2025

We study the Brauer group of an abelian variety A over an algebraically closed field of characteristic p focusing on the p-primary torsion, the key part of which is a certain quasi-algebraic unipotent group U_A. We determine its dimension …

AdditiveToricVarieties: A Macaulay2 package for working with additive complete toric varieties Open

Levicán, Fabián, Montero, Pedro · 2025

We introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. More precisely, we implement…

AdditiveToricVarieties: A Macaulay2 package for working with additive complete toric varieties Open

Levicán, Fabián, Montero, Pedro · 2025

We introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. More precisely, we implement…

On good A1 subgroups, Springer maps, andovergroups of distinguished unipotent elements in reductive groups Open

Michael Bate, Sören Böhm, Benjamin Martin, Gerhard Röhrle · 2025

We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript, and to A. Thomas for providing the G2 example in Example 4.13. We thank the referee for a number of comments clarifying some poin…

Maximal commutative unipotent subgroups and a characterization of affine spherical varieties Open

Andriy Regeta, Immanuel van Santen · 2025

We describe maximal commutative unipotent subgroups of the automorphism group \mathrm{Aut}(X) of an irreducible affine variety X . Furthermore, we show that a group isomorphism \mathrm{Aut}(X) \allowbreak\to \mathrm{Aut}(Y) maps unipotent …

Unicity of 𝐴₁-subgroups associated to unipotent elements in simple algebraic groups Open

Rachel Pengelly, Donna Testerman · 2025

Let be an algebraically closed field of positive characteristic and a simple algebraic group over . Under the assumption that the characteristic is a good prime for , we determine which unipotent elements , with of order , satisfy the p…

Expressing matrices in $\mathrm{SL}_{n}(F)$ as products of commutators of unipotent matrices Open

Truong Huu Dung, Vo Thi Van Anh · 2025

This paper aims to show that for two positive integers $n \ge k$, every nonscalar matrix in the special linear group of degree $n$ over a field can be written as a product of a maximum of two commutators of unipotent matrices of index $k$.…

Special unipotent representations of real classical groups: Counting and reduction Open

Dan Barbasch, Jia-Jun Ma, Binyong Sun, Chen-Bo Zhu · 2025

Let G be a real reductive group in Harish-Chandra’s class. We derive some consequences of the theory of coherent continuation representations to the counting of irreducible representations of G with a given infinitesimal character and a gi…

Crystal Structure of Localized Quantum Unipotent Coordinate Category Open

Masaki Kashiwara, Toshiki Nakashima · 2025

A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of …

Unipotent similarity for matrices over commutative domains Open

Grigore Cǎlugǎreanu · 2025

A unit u of a ring is called unipotent if u − 1 is nilpotent. We characterize the similarity of 2×2 matrices over commutative domains, realized by unipotent matrices, i.e., B = U−1AU with unipotent matrix U.

The non-Archimedean Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1 Open

Jackson S. Morrow, Paul Vojta · 2025

Let $k$ be algebraically closed field of characteristic zero, let $G$ be a commutative algebraic group over $k$ such that the linear part of $G$ is isomorphic to $\mathbb{G}_a$, and let $X$ be a closed subvariety of $G$. We show that the K…

Time change rigidity for unipotent flows Open

Elon Lindenstrauss, Daren Wei · 2025

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u^{(1)}_t$ acting on $\mathbf{G}_{1}/Γ_1$ is such a flow it satisfies exactly one of the…

Random unipotent Sylow subgroups of groups of Lie type of bounded rank Open

Saveliy V. Skresanov · 2025

In 2001 Liebeck and Pyber showed that a finite simple group of Lie type is a product of $ 25 $ carefully chosen unipotent Sylow subgroups. Later, in a series of works it was shown that $ 4 $ unipotent Sylow subgroups suffice. We prove that…

Actions of nilpotent groups on nilpotent groups Open

Michael C. Burkhart · 2025

For finite nilpotent groups $J$ and $N$ , suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$ -subgroups of $J$ that mirrors the primary deco…

A nonabelian Fourier transform for tempered unipotent representations Open

Anne‐Marie Aubert, Dan Ciubotaru, Beth Romano · 2025

We define an involution on the elliptic space of tempered unipotent representations of inner twists of a split simple $p$ -adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact …

Equivariant Deformation theory for Nilpotent slices in Symplectic lie Algebras Open

Filippo Ambrosio, Lewis Topley · 2024

The Slodowy slice is a flat Poisson deformation of its nilpotent part, and it was demonstrated by Lehn–Namikawa–Sorger that there is an interesting infinite family of nilpotent orbits in symplectic Lie algebras for which the slice is not t…