Jason Semeraro

Loughborough University | Lough · Department of Mathematical Sciences

Let $\mathcal{F}$ be a saturated fusion system on $S$ and $\mathcal{E}$ a normal subsystem of $\mathcal{F}$ on $T$. We appeal to a result of Gross to give a construction of the $\mathcal{F}$-centraliser $C_\mathcal{F}(\mathcal{E})$ which does not rely on Aschbacher's theory of normal maps. We also interpret the $S$-centraliser $C_S(\mathcal{E})$ as...

We define a `tree of fusion systems' and give a sufficient condition for its completion to be saturated. We apply this result to enlarge an arbitrary fusion system by extending the automorphism groups of certain of its subgroups.

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...

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 $7$-bl...

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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....

In this short 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 $\mathcal{F}$ is (algebraically) connected, that is every element of $S$ is...

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 `hi...

In 1993, Broué, 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 X is a space which is a homotopy-theoretic p-local analogue of a compact Lie group. A connected p-compact group X is determined by its root...

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 a large number of cases. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\ell$-compact...

For a prime p p , we describe a protocol for handling a specific type of fusion system on a p 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 autom...

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 s...

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 automiz...

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 i...

In 1999 Brou\'{e}, Malle and Michel introduced the concept of a ``spets'' which is a mysterious object with a non-real Weyl group. In algebraic topology, a $p$-compact group $X$ is a space which is a homotopy-theoretic $p$-local analogue of a compact Lie group. Such a space is determined by the isomorphism type of its Weyl group $W_X$, which is a f...

We complete the determination of saturated fusion systems on maximal class 3 3 -groups of rank two.

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 e...

We complete the determination of saturated fusion systems on maximal class 3-groups of rank two.

To each pair consisting of a saturated fusion system over a p-group together with a compatible family of Külshammer-Puig cohomology classes, one can count weights in a hypothetical block algebra arising from these data. When the pair arises from a genuine block of a finite group algebra in characteristic p, the number of conjugacy classes of weight...

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...

For S a Sylow p-subgroup of the group \({\text {G}}_2(p)\) for p odd, up to isomorphism of fusion systems, we determine all saturated fusion systems \(\mathcal {F}\) on S with \(O_p(\mathcal {F})=1\). For \(p \ne 7\), all such fusion systems are realized by finite groups whereas for \(p=7\) there are 29 saturated fusion systems of which 27 are exot...

To each supersimple 2-(n,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M13 which is constructed from P3. We show that Sp2m(2) and 22m. Sp2m(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one...

The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples....

In 1987, John Horton Conway constructed a subset M13 of permutations on a set of size 13 for which the subset fixing any given point is isomorphic to the Mathieu group M12. The construction has fascinated mathematicians for the past thirty years, and remains remarkable in its mathematical isolation. It is based on a “moving-counter puzzle” on the p...

For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exoti...

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.

In 1997, John Conway constructed a $6$-fold transitive subset $M_{13}$ of permutations on a set of size $13$ for which the subset fixing any given point was isomorphic to the Mathieu group $M_{12}$. The construction was via a "moving-counter puzzle" on the projective plane ${\rm PG}(2,3)$. We discuss consequences and generalisations of Conway's con...

To a set $\mathcal {B} $ of 4-subsets of a set $\Omega $ of size $n,$ we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Lloyd's ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the...

A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid $M_{13}$. It turns out that several infinite families o...

Glauberman's -theorem and analogous statements for odd primes show that, for any prime p and any finite group G with Sylow p-subgroup S, the centre of is determined by the fusion system . Building on these results we show a statement that seems a priori more general: For any normal subgroup H of G with , the centralizer is expressed in terms of the...

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically...

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically...

For $r \ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x \notin R$ with the property that for each subset $A \subseteq R$ of size $r-1$, the set $A \cup \{x\}$ is a hyperedge. We show that for $r \geq 3$, the number of hyperedges...

Glauberman's Z *-theorem and analogous statements for odd primes show that, for any prime p and any finite group G with Sylow p-subgroup S, the centre of G is determined by the fusion system F S (G). Building on these results we show a statement that can be considered as a generalization: For any normal subgroup N of G, the centralizer C S (N) is e...

Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$- subgroup $S$, the centre of $G$ is determined by the fusion system $\mathcal{F}_S(G)$. Building on these results we show a statement that can be considered as a generalization: For any normal subgroup $N$ of $G$,...

To each simple $2-(n,4,\lambda)$ design $\mathcal{D}$ with the property that $\mathcal{D}$ is ``pliable'' (any two lines intersect in at most two points), one associates a `puzzle group'. This generalises a construction of the group $M_{12}$ from $\mathbb{P}_3$ due to Conway. We introduce a new infinite family of puzzle groups isomorphic with $\ope...

To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `puzzle group' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's "15 puzzle". It is shown that puzzle groups may be regarded as objects inside an objective partial group (in the sense of Ch...