arXiv subjects

Geoffrey Janssens

Publications and source records attributed to Geoffrey Janssens.

At least 19 recordsLinked to original sources

arXiv2026

Additive categorification of the monoidal $Λ$-invariant

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$, Kashiwara-Kim-Oh-Park proved that the Hernandez-Leclerc categories form a monoidal categorification of their Grothendieck rings. Furthermore, these rings are $Λ$-cluster algebras, meaning they are equipped with a compatible Poisson structure, constructed via the $Λ$-invariant. Under certain natural conditions, where $U_q'(\mathfrak{g})$ is of untwisted simply-laced type, we provide an additive interpretation of the $Λ$-invariant within the framework of Higgs categories. More precisely, there is an ice quiver with potential associated with these cluster algebras, and a key ingredient of our work consists in proving that its relative Ginzburg algebra is proper. More generally, if the relative Ginzburg algebra associated with an arbitrary ice quiver with potential is proper, we prove that the corresponding cluster algebra admits the structure of a $Λ$-cluster algebra defined in terms of negative extensions in the Higgs category. Moreover, we provide a homological formula to compute the corresponding tropical and $F$-invariants introduced by Cao.

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arXiv2026

Center-preserving irreducible representations of finite groups

Given finite groups $H \leq G$, a representation $σ$ of $G$ is called center-preserving on $H$ if the only elements of $H$ that become central under $σ$ are those that were already central in $G$. We prove that if $H$ has a faithful irreducible representation $ρ$, then at least one of the irreducible components of the induction $\operatorname{Ind}_H^G(ρ)$ is center-preserving on $H$. In consequence, $H$ has a faithful irreducible representation if and only if every finite group $G$ containing $H$ as a subgroup has an irreducible representation whose restriction to $H$ is faithful, and which is center-preserving on $H$. In addition, we give examples illustrating the sharpness of the statement, and discuss the connection with projective representations.

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arXiv2026

Representing in Low Rank I: conjugacy, topological and homological aspects

In this series of papers, we investigate properties of a finite group which are determined by its low degree irreducible representations over a number field $F$, i.e. its representations on matrix rings $\operatorname{M}_n(D)$ with $n \leq 2$. In particular we focus on representations on $\operatorname{M}_2(D)$ where $D$ is a division algebra having an order $\mathcal{O}$ such that $\mathcal{O}$ has finitely many units, i.e. such that $\operatorname{SL}_2(\mathcal{O})$ has arithmetic rank $1$. In this first part, the focus is on two aspects. One aspect concerns characterisations of such representing spaces in terms of Serre's homological goodness property, small virtual cohomological dimension and higher Kleinian-type embeddings. As an application, we obtain several characterisations of the finite groups $G$ whose irreducible representations are of the mentioned form. In particular, such groups $G$ are precisely those such that $\mathcal{U}(R G)$, with $R$ the ring of integers of $F$, can be constructed from groups which virtually map onto a non-abelian free group. Along the way we investigate the latter property for congruence subgroups of higher modular groups and its implications for the congruence kernel. This is used to obtain new information on the congruence kernel of the unit group of a group ring. The second aspect concerns the conjugacy classes of the images of finite subgroups of $\mathcal{U}(R G)$ under the irreducible representations of $G$. More precisely, we initiate the study of a blockwise variant of the Zassenhaus conjectures and the subgroup isomorphism problem. Moreover, we contribute to them for the low rank representations above.

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arXiv2026

On set-theoretic solutions of pentagon equation and positive basis Hopf algebras

We investigate the connection between bijective, not necessarily finite, set-theoretic solutions of the pentagon equation and Hopf algebras. Firstly, we prove that finite solutions correspond to Hopf algebras with the positive basis property. As a corollary we generalise Lu-Yan-Zhu classification to arbitrary characteristic $0$ fields $k$. Secondly, we study the general problem of when a Hopf algebra has a basis yielding a set-theoretic solution. Finally, we classify all (co)commutative bijective solutions. This result requires to obtain a description of all bases of a group algebra $k[G]$ yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings $A \rtimes N$ of $G$ with $A$ a finite abelian group.

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arXiv2025

Simultaneous ping-pong for finite subgroups of reductive groups

Let $Γ$ be a Zariski-dense subgroup of a reductive group $\mathbf{G}$ defined over a field $F$. Given a finite collection of finite subgroups $H_i$ ($i \in I$) of $\mathbf{G}(F)$ avoiding the center, we establish a criterion to ensure that the set of elements of $Γ$ that form a free product with every $H_i$ (the so-called simultaneous ping-pong partners for $H_i$) is both Zariski- and profinitely dense in $Γ$. This criterion applies namely to direct products $\mathbf{G}$ of inner $\mathbb{R}$-forms of $\operatorname{(P)GL}_n$, and gives a positive answer to this particular case of a question asked by Bekka, Cowling and de la Harpe. For torsion elements, a complication arises due to the fact that a finite cyclic group can split into a direct product. When $\mathbf{G}$ is the multiplicative group of a semisimple algebra, we also give a more explicit method to obtain free products between two given finite subgroups, via first-order deformations. In the second half, we investigate the case where $\mathbf{G}$ is the multiplicative group of the group algebra $FG$ of a finite group $G$, and $Γ$ is the group of units of an order in $FG$. In this regard, we prove that the set of bicylic units that play ping-pong with a given shifted bicyclic unit, is Zariski- and profinitely dense, addressing a long-standing belief in the field of group rings. This result is deduced from the criterion above, combined with sharp existence results for well-behaved irreducible representations of $G$ that are center-preserving on a given subgroup.

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arXiv2025

A unit theorem for products of groups with several ends and Applications

In his $1994$ survey, Kleinert defined formally and formulated the problem to obtain unit theorems for unit groups of orders in a semisimple algebra $A$. If $A$ is a group algebra $FG$, it boils down to classifying all finite groups $G$ such that the unit groups of most orders in $FG$ belong to a prescribed class $\mathcal{G}$ of infinite groups. We solve this problem for $\mathcal{G}$ consisting of the groups which are virtually a direct product of groups whose Cayley graph has more than one end. Subsequently, we obtain two types of applications. A first type being about the existence of torsion-free normal complements and a second about obtaining short and uniform proofs of some of the main results in [13,18,14,15].

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arXiv2025

A basis and Schur-Weyl duality for the loop Hecke algebra

The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a basis. We use higher linear rewriting theory to show linear independence and the combinatorics of Dyck paths to compute the cardinality of the basis. This yields a conjecture of Damiani-Martin-Rowel. We also give a representation theoretic interpretation of the loop Hecke algebra in terms of (non-semisimple) Schur-Weyl duality involving the negative half of quantum $\mathfrak{gl}_{1|1}$.

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arXiv2025

On integral decomposition of unipotent elements in integral group rings

Jespers and Sun conjectured that if a finite group $G$ has the property ND, i.e. for every nilpotent element $n$ in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in \mathbb{Z}G$, then at most one of the simple components of the group algebra $\mathbb{Q} G$ has reduced degree bigger than $1$. With the exception of one very special series of groups we are able to answer their conjecture, showing that it is true - up to exactly one exception. To do so we first describe groups with the so-called SN property which was introduced by Liu and Passman in their investigation of the Multiplicative Jordan Decomposition for integral group rings. We then study further objects connected to the property ND. This concerns on one hand a certain section of the unit group of $\mathbb{Z} G$ which measures how far $G$ is from having ND and about which Jespers and Sun posed two further questions which we answer. On the other hand we introduce two properties which appeared naturally in these investigations: one is purely representation-theoretic, while the other can be regarded as indicating that it might be hard to decide ND. Among others we show these two notions are equivalent for groups with SN.

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arXiv2024

Units of twisted group rings and their correlations to classical group rings

This paper is centered around the classical problem of extracting properties of a finite group $G$ from the ring isomorphism class of its integral group ring $\mathbb{Z} G$. This problem is considered via describing the unit group $\mathcal{U}( \mathbb{Z} G)$ generically for a finite group. Since the $`90s$ several well known generic constructions of units are known to generate a subgroup of finite index in $\mathcal{U}(\mathbb{Z } G)$ if $\mathbb{Q} G$ does not have so-called exceptional simple epimorphic images, e.g. $M_2 (\mathbb{Q})$. However it remained a major open problem to find a {\it generic} construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation $\mathcal{U}(\mathbb{Z} G)^{ab}$. To obtain all this, we investigate the group ring $R Γ$ of an extension $Γ$ of some normal subgroup $N$ by a group $G$, over a domain $R$. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of $Γ$ over the fraction field $F$ of $R$ in terms of various twisted group rings of $G$ over finite extensions of $F$. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of $\mathcal{U}( RΓ)$ via twisted group rings.

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arXiv2021

Abelianization and fixed point properties of units in integral group rings

Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property $(\operatorname{T})$, both in terms of the finite group $G$ and in terms of the simple components of the semisimple algebra $\mathbb{Q}G$. Furthermore, it is shown that for $\mathcal{U}( \mathbb{Z} G)$ this property is equivalent to the weaker property $\operatorname{FAb}$ (i.e. every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property $\operatorname{FA}$, denoted $\operatorname{HFA}$. More precisely, it is described when all subgroups of finite index in $\mathcal{U} (\mathbb{Z} G)$ have both finite abelianization and are not a non-trivial amalgamated product. A crucial step for this is a reduction to arithmetic groups $\operatorname{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is an order in a finite dimensional semisimple $\mathbb{Q}$-algebra $D$, and finite groups $G$ which have the so-called cut property. For such groups $G$ we describe the simple epimorphic images of $\mathbb{Q} G$. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups $\operatorname{E}_n(D)$ of $\operatorname{SL}_n(D)$. These groups are well understood except in the degenerate case of lower rank, i.e.\ for $\operatorname{SL}_2(\mathcal{O})$ with $\mathcal{O}$ an order in a division algebra $D$ with a finite number of units. In this setting we determine Serre's property \FA for $\operatorname{E}_2(\mathcal{O})$ and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its $\mathbb{Z}$-rank.

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arXiv2020

Glider Representation Rings with a view on distinguishing groups

Let $G$ be a finite group. In the first part of the paper we develop further the foundations of the youngly introduced glider representation theory. Glider representations encompass filtered modules over filtered rings and as such carry much information of $G$. Therefore the main focus is on the glider representation ring $R_d(\widetilde{G})$, which is shown to be realisable as a concrete subring of the split Grothendieck ring of the monoidal category $\text{glid}_d(G)$ of (Noetherian) glider $\mathbb{C}$-representations of (length $d$) of $G$. In the second part we investigate a Wedderburn-Malcev type decomposition of the (infinite-dimensional) $\mathbb{Q}$-algebra $\mathbb{Q}(\widetilde{G}) := \mathbb{Q} \otimes_{\mathbb{Z}}R_1(\widetilde{G})$. The main theorem obtains a $\mathbb{Q}[G^{ab}]$-module decomposition of $\mathbb{Q}(\widetilde{G})$ relating it in a precise way to $\mathbb{C}$-representation theory of subnormal subgroups in $G$. Under certain vanishing assumptions, which are proven to hold for nilpotent groups (of class $2$), the second main theorem completely describes a $\mathbb{Q}[G^{ab}]$-algebra decomposition. We end with pointing out applications on distinguishing isocategorical groups.

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arXiv2020

A dichotomy for integral group rings via higher modular groups as amalgamated products

We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case $G$ is a finite group satisfying some mild conditions. Crucial in the proof is the construction of amalgamated decompositions of the elementary group $\operatorname{E}_2(\mathcal{O})$, where $\mathcal{O}$ is an order in a rational division algebra. A major step is to introduce subgroups $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ inside the so-called higher modular groups $\operatorname{SL}_+(Γ_n(\mathbb{Z}))$, which are discrete subgroups of certain $2 \times 2$ matrix groups with entries in a Clifford algebra. The groups $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ mimic the elementary groups in linear groups over rings. We prove that $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ has in general a non-trivial decomposition as a free product with amalgamated subgroup $\operatorname{E}_2(Γ_{n-1}(\mathbb{Z}))$. From this we obtain that also the higher modular groups do have a very clearly structured amalgam decompositions in low dimensions.

math.GR↗

arXiv2020

Codimension Growth of Lie algebras with a generalized action

Let $L$ be a finite dimensional Lie $F$-algebra endowed with a generalized action by an associative algebra $H$. We investigate the exponential growth rate of the sequence of $H$-graded codimensions $c_n^H(L)$ of $L$ which is a measure for the number of non-polynomial $H$-identities of $L$. More precisely, we construct the first example of an $S$-graded Lie algebra having a non-integer, even irrational, exponential growth rate $\lim_{n\rightarrow \infty} \sqrt[n]{c_n^{S}(L)}$. Hereby $S$ is a semigroup and an exact value is given. On the other hand, returning to general $H$, if $L$ is semisimple and also semisimple for the $H$-action we prove the analog of Amitsur's conjecture (i.e. $\lim_{n\rightarrow \infty} \sqrt[n]{c_n^{H}(L)} \in \mathbb{Z}$). Moreover if $H=FS$ is a semigroup algebra the semisimplicity on $L$ can be dropped which is in strong contract to the associative setting.

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arXiv2017

Semigroup graded algebras and graded PI-exponent

Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty} \sqrt[n]{c_n^S(A)}$. For group gradings this is always an integer. Recently in [20] the first example of an algebra with a non-integer growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large non-integers. An explicit formula is given. Surprisingly, this class consists of an infinite family of algebras simple as an $S$-graded algebra. This is in strong contrast to the group graded case for which the growth rate of such algebras always equals $\dim (A)$. In light of the previous, we also handle the problem of classification of all $S$-graded simple algebras, which is of independent interest. We achieve this goal for an important class of semigroups that is crucial for a solution of the general problem.

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arXiv2016

The Polynomial Part of the Codimension Growth of Affine PI Algebras

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $Θ(c n^t d^n)$, where $d$ is the well known (PI) exponent of $W$. In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemer's theory. In particular, we show that in case $W$ is a basic algebra, then $t = \frac{d-q}{2} + s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus proving a conjecture of Giambruno.

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arXiv2016

Free products in the unit group of the integral group ring of a finite group

Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and free monoids in the unit group $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$. For a nilpotent group $G$ with a non-central element $g$ of order $p$, explicit generic constructions are given of two periodic units $b_1$ and $b_2$ in $\mathcal{U}(\mathbb{Z}G)$ such that $\langle b_1 , b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb{Z}_p \star \mathbb{Z}_{p}$, a free product of two cyclic groups of prime order. Moreover, if $G$ is nilpotent of class $2$ and $g$ has order $p^n$, then also concrete generators for free products $\mathbb{Z}_{p^k} \star \mathbb{Z}_{p^m}$ are constructed (with $1\leq k,m\leq n $). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gon{ç}alves-Passman. Further, for an arbitrary finite group $G$ we give generic constructions of free monoids in $\mathcal{U}(\mathbb{Z}G)$ that generate an infinite solvable subgroup.

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arXiv2016

A reduction theorem for $τ$-rigid modules

We prove a theorem which gives a bijection between the support $τ$-tilting modules over a given finite-dimensional algebra $A$ and the support $τ$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are $τ$-tilting finite wild blocks with more than one simple module. We then go on to classify all support $τ$-tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most $32$ different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field $k$, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all $τ$-rigid modules over (not necessarily symmetric) string algebras.

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arXiv2014

$\mathbb ZS_n$-modules and polynomial identities with integer coefficients

We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of $\mathbb ZS_n$-submodules in the $\mathbb ZS_n$-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding $\mathbb ZS_n$-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper triangular $2\times 2$ matrices and infinitely generated Grassmann algebras over unitary rings. In addition, we calculate the factors of series of $\mathbb ZS_n$-submodules for these algebras. Also we establish relations between codimensions of rings and codimensions of algebras and show that the analog of Amitsur's conjecture holds in all torsion-free rings, and all torsion-free rings with 1 satisfy the analog of Regev's conjecture.

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