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Jason Semeraro

Publications and source records attributed to Jason Semeraro.

At least 19 recordsLinked to original sources
arXiv2026

Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups

Suppose that $p$ is a prime and $X$ is a finite group with a strongly $p$-embedded subgroup, for example a rank $1$ group of Lie type in characteristic $p$. Let $m$ denote the $p$-rank of $X$ and assume that $m \ge 2$. We say that a faithful $\FF_p X$-module is $k$-active if some element of order $p$ in $X$ acts with exactly $k$ non-trivial Jordan blocks. In this paper, we determine the non-trivial composition factors of the faithful $\FF_pX$-modules which are $k$-active for some $k\leq m$.

math.GR↗
arXiv2025

On character tables for fusion systems

A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a conjecture of the second author which equates the determinant of $X \overline{X}$ (the square of the volume of this lattice) with the product of the orders of $S$-centralisers of fully $\mathcal{F}$-centralised $\mathcal{F}$-class representatives. This statement is exactly column orthogonality for the character table of $S$ when $\mathcal{F}=\mathcal{F}_S(S)$. We prove the conjecture when $\mathcal{F}=\mathcal{F}_S(G)$ is realised by some finite group $G$ with Sylow $p$-subgroup $S$, and for all simple fusion systems when $|S| \le p^4$. We also put forward a potential strategy for the general case, which would exploit properties of the characteristic idempotent of $\mathcal{F}$.

math.RT↗
arXiv2025

Partial character tables for $\mathbb{Z}_\ell$-spetses

Let ${\mathbb{G}}$ be a simply connected ${\mathbb{Z}}_\ell$-spets, let $q$ be a prime power, prime to $\ell$ and let $S$ be the underlying Sylow $\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$. Secondly, when $\ell > 2$ is a very good prime for ${\mathbb{G}}$, the Weyl group $W$ of ${\mathbb{G}}$ has order coprime with $\ell$, and $q\equiv1\pmod\ell$ we introduce a formula for the values of characters in the principal block of ${\mathbb{G}}(q)$ which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a version of block orthogonality. In both cases we formulate and provide evidence for several conjectures concerning the proposed values.

math.RT↗
arXiv2025

Weight conjectures for fusion systems on an extraspecial group

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here we verify some of these conjectures for fusion systems on an extraspecial group of order $p^3$, which contain among them the Ruiz-Viruel exotic fusion systems at the prime $7$. As a byproduct we verify Robinson's ordinary weight conjecture for principal $p$-blocks of almost simple groups $G$ realizing such (nonconstrained) fusion systems.

math.RT↗
arXiv2025

Fusion systems related to polynomial representations of $\mathrm{SL}_2(q)$

Let $q$ be a power of a fixed prime $p$. We classify up to isomorphism all simple saturated fusion systems on a certain class of $p$-groups constructed from the polynomial representations of $\mathrm{SL}_2(q)$, which includes the Sylow $p$-subgroups of $\mathrm{GL}_3(q)$ and $\mathrm{Sp}_4(q)$ as special cases. The resulting list includes all Clelland--Parker fusion systems, a simple exotic fusion system discovered by Henke--Shpectorov, and a new infinite family of exotic examples.

math.GR↗
arXiv2024

On $e$-local structures for $\mathbb{Z}_\ell$-spetses

Let $q$ be a prime power, $\ell$ a prime not dividing $q$, and $e$ the order of $q$ modulo $\ell$. We show that the geometric realisation of the nerve of the transporter category of $e$-split Levi subgroups of a finite reductive group $G$ over $\mathbb{F}_q$ is homotopy equivalent to the classifying space $BG$ up to $\ell$-completion. We suggest a generalisation of this equivalence to the setting of $\mathbb{Z}_\ell$-reflection cosets and establish a related fact involving the associated orbit spaces. We also establish a Dade-like formula for unipotent characters of $\mathbb{Z}_\ell$-spetses inspired by a question of Brou\'e.

math.GR↗
arXiv2024

Weight conjectures for Parker--Semeraro fusion systems

We prove that the Parker--Semeraro systems satisfy six of the nine Kessar--Linckelmann--Lynd--Semeraro weight conjectures for saturated fusion systems. As a by-product we obtain that Robinson's ordinary weight conjecture holds for the principal $3$-block of Aut$(G_2(3))$, the principal $5$-blocks of $HN$, $BM$, Aut$(HN)$, $Ly$, the principal $7$-block of $M$, and the principal $p$-blocks of $G_2(p)$ for $p\geq 3 $.

math.RT↗
arXiv2024

Spectra of subrings of cohomology generated by characteristic classes for fusion systems

If $\mathcal{F}$ is a saturated fusion system on a finite $p$-group $S$, we define the Chern subring $Ch(\mathcal{F})$ of $\mathcal{F}$ to be the subring of the mod-$p$ cohomology $H^*(S)$ of $S$ generated by the Chern classes of $\mathcal{F}$-stable representations of $S$. We show that $Ch(\mathcal{F})$ is contained in $H^*(\mathcal{F})$ and apply a result of Green and the first author to describe its spectrum in terms of a certain category of elementary abelian subgroups of $S$. We obtain similar results for various related subrings, including those generated by characteristic classes of $\mathcal{F}$-stable $S$-sets.

math.GR↗
arXiv2023

Weights for compact connected Lie groups

Let $\ell$ be a prime. If ${\mathbf G} $ is a compact connected Lie group, or a connected reductive algebraic group in characteristic different from $\ell$, and $\ell$ is a good prime for ${\mathbf G}$, we show that the number of weights of the $\ell$-fusion system of ${\mathbf G}$ is equal to the number of irreducible characters of its Weyl group. The proof relies on the classification of $\ell$-stubborn subgroups in compact Lie groups.

math.RT↗
arXiv2022

Weights for $\ell$-local compact groups

In this note, we initiate the study of $\mathcal{F}$-weights for an $\ell$-local compact group $\mathcal{F}$ over a discrete $\ell$-toral group $S$ with discrete torus $T$. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when $T$ is the unique maximal abelian subgroup of $S$ up to $\mathcal{F}$-conjugacy and every element of $S$ is $\mathcal{F}$-fused into $T$, the number of weights of $\mathcal{F}$ is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of $\mathcal{F}$ with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when $\mathcal{F}$ is simple and $|S:T| =\ell$. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.

math.GR↗
arXiv2022

Tur\'{a}n numbers and switching

Using a switching operation on tournaments we obtain some new lower bounds on the Tur\'{a}n number of the $r$-graph on $r+1$ vertices with $3$ edges. For $r=4$, extremal examples were constructed using Paley tournaments in previous work. We show that these examples are unique (in a particular sense) using Fourier analysis. A $3$-tournament is a `higher order' version of a tournament given by an alternating function on triples of distinct vertices in a vertex set. We show that $3$-tournaments also enjoy a switching operation and use this to give a formula for the size of a switching class in terms of level permutations, generalising a result of Babai--Cameron.

math.CO↗
arXiv2021

The principal block of a $\mathbb{Z}_\ell$-spets and Yokonuma type algebras

We formulate conjectures concerning the dimension of the principal block of a ${\mathbb Z}_\ell$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain situations. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\ell$-compact groups which may be of independent interest.

math.RT↗
arXiv2020

Weight conjectures for $\ell$-compact groups and spetses

Fundamental conjectures in modular representation theory of finite groups, more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight Conjecture, can be expressed in terms of fusion systems. We use fusion systems to connect the modular representation theory of finite groups of Lie type to the theory of $\ell$-compact groups. Under some mild conditions we prove that the fusion systems associated to homotopy fixed points of $\ell$-compact groups satisfy an equation which for finite groups of Lie type is equivalent to Alperin's Weight Conjecture. For finite reductive groups, Robinson's Ordinary Weight Conjecture is closely related to Lusztig's Jordan decomposition of characters and the corresponding results for Brauer $\ell$-blocks. Motivated by this, we define the principal block of a spets attached to a spetsial ${\mathbb Z}_\ell$-reflection group, using the fusion system related to it via $\ell$-compact groups, and formulate an analogue of Robinson's conjecture for this block. We prove this formulation for an infinite family of cases as well as for some groups of exceptional type. Our results not only provide further strong evidence for the validity of the weight conjectures, but also point toward some yet unknown structural explanation for them purely in the framework of fusion systems.

math.RT↗
arXiv2020

Algorithms for fusion systems with applications to $p$-groups of small order

For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the $p$-group. This allows a computational check of saturation. These procedures have been implemented using MAGMA. We describe a program to search for saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F})=1$ and $O^p(\mathcal{F})=\mathcal{F}$. Employing these computational methods we determine all such fusion system on groups of order $p^n$ where $(p,n) \in \{(3,4),(3,5),(3,6),(3,7),(5,4),(5,5),(5,6),(7,4),(7,5)\}$. This gives the first complete picture of which groups can support saturated fusion systems on small $p$-groups of odd order.

math.GR↗
arXiv2019

A $2$-compact group as a spets

In 1993, Brou\'{e}, Malle and Michel initiated the study of spetses on the Greek island bearing the same name. These are mysterious objects attached to non-real Weyl groups. In algebraic topology, a $p$-compact group $\mathbf{X}$ is a space which is a homotopy-theoretic $p$-local analogue of a compact Lie group. A connected $p$-compact group $\mathbf{X}$ is determined by its root datum which in turn determines its Weyl group $W_\mathbf{X}$. In this article we give strong numerical evidence for a connection between these two objects by considering the case when $\mathbf{X}$ is the exotic $2$-compact group DI$(4)$ constructed by Dwyer--Wilkerson and $W_\mathbf{X}$ is the complex reflection group $G_{24} \cong$ GL$_3(2) \times C_2$. Inspired by results in Deligne--Lusztig theory for classical groups, if $q$ is an odd prime power we propose a set Irr$(\mathbf{X}(q))$ of `ordinary irreducible characters' associated to the space $\mathbf{X}(q)$ of homotopy fixed points under the unstable Adams operation $\psi^q$. Notably Irr$(\mathbf{X}(q))$ includes the set of unipotent characters associated to $G_{24}$ constructed by Brou\'{e}, Malle and Michel from the Hecke algebra of $G_{24}$ using the theory of spetses. By regarding $\mathbf{X}(q)$ as the classifying space of a Benson--Solomon fusion system Sol$(q)$ we formulate and prove an analogue of Robinson's ordinary weight conjecture that the number of characters of defect $d$ in Irr$(\mathbf{X}(q))$ can be counted locally.

math.RT↗
arXiv2018

Weight conjectures for fusion systems

Many of the conjectures of current interest in the representation theory of finite groups in characteristic $p$ are local-to-global statements, in that they predict consequences for the representations of a finite group $G$ given data about the representations of the $p$-local subgroups of $G$. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of K\"ulshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples $(S,\mathcal{F},\alpha)$ consisting of a saturated fusion system $\mathcal{F} $ on a finite $p$-group $S$ and a compatible family $\alpha$.

math.RT↗
arXiv2017

Weights in a Benson-Solomon block

To each pair consisting of a saturated fusion system over a $p$-group together with a compatible family of K\"ulshammer-Puig cohomology classes, one can count weights in a hypothetical block algebra arising from these data. When the pair arises from a bonafide block of a finite group algebra in characteristic $p$, the number of conjugacy classes of weights is supposed to be the number of simple modules in the block. We show that there is unique such pair associated with each Benson-Solomon exotic fusion system, and that the number of weights in a hypothetical Benson-Solomon block is $12$, independently of the field of definition. This is carried out in part by listing explicitly up to conjugacy all centric radical subgroups and their outer automorphism groups in these systems.

math.GR↗