Symmetric functions and Springer representations

The characters of the (total) Springer representations afford the Green functions, that can understood as generalizations of Hall–Littlewood’s $Q$-functions. In this paper, we present a purely algebraic proof that the (total) Springer representations of $GL\left(n\right)$ are $\mathrm{Ext}$-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.     

Even degree characters in principal blocks

We characterise finite groups such that for an odd prime p all the irreducible characters in its principal p-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by p unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\ne 7$ or if $M_{22}$ is not a composition factor of a group G, then the condition above is equivalent to $G/{\mathbf{O}}_{p^{\prime }}(G)$ having odd order.     

On the Baer–Lovász–Tutte construction of groups from graphs: Isomorphism types and homomorphism notions

Let $p$ be an odd prime. From a simple undirected graph $G$, through the classical procedures of Baer (1938), Tutte (1947) and Lovász (1989), there is a $p$-group $P_G$ of class 2 and exponent $p$ that is naturally associated with $G$. Our first result is to show that this construction of groups from graphs respects isomorphism types. That is, given two graphs $G$ and $H$, $G$ and $H$ are isomorphic as graphs if and only if $P_G$ and $P_H$ are isomorphic as groups. Our second contribution is a new homomorphism notion for graphs. Based on this notion, a category of graphs can be defined, and the Baer–Lovász–Tutte construction naturally leads to a functor from this category of graphs to the category of groups.     

On invariants of modular categories beyond modular data

We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the W-matrix—the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the W-matrix and the set of punctured S-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.     

Representation and factorization theorems for almost-Lp-spaces

We extend the notions of $p$-convexity and $p$-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost $p$-convex and almost $p$-concave space is order isomorphic to an almost-$L^p$-space. The class of almost-$L^p$-spaces contains, in particular, direct sums of (infinitely many) $L^p$-spaces with different norms, that are not in general $p$-convex – nor $p$-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for $p$-concave operators acting in $p$-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than $L^p$-spaces.     

The Alon–Tarsi conjecture: A perspective on the main results

The Alon–Tarsi conjecture states that if $n$ is even, then the sum of the signs of the Latin squares of order $n$ is non-zero (Alon and Tarsi, 1992). The conjecture has been proven in the cases $n=p+1$ (Drisko, 1997), and $n=p?1$ (Glynn, 2010), where $p$ is an odd prime. This paper is intended to be a concise and largely self-contained account of these results, along with streamlined, and in some cases, original proofs that should be readily accessible to a mathematician with a background in combinatorics. We also discuss the relation between the Alon–Tarsi conjecture and Rota’s basis conjecture (Huang and Rota, 1994), and present some related problems, such as Zappa’s extension of the Alon–Tarsi conjecture (Zappa, 1997), and Drisko’s proof of the extended conjecture for $n=p$ (Drisko, 1998).     

Computing discrete invariants of varieties in positive characteristic I. Ekedahl-Oort types of curves

We develop a method to compute the Ekedahl–Oort type of a curve C over a field k of characteristic p (which is the isomorphism type of the p-kernel group scheme $J[p]$, where J is the Jacobian of C). Part of our method is general, in that we introduce the new notion of a Hasse–Witt triple, which re-encodes in a useful way the information contained in the Dieudonné module of $J[p]$. For complete intersection curves we then give a simple method to compute this Hasse–Witt triple. An implementation of this method is available in Magma.     

Isomorphism problems for Hopf–Galois structures on separable field extensions

Let $L/K$ be a finite separable extension of fields whose Galois closure $E/K$ has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on $L/K$ has the form $E{[N]}^G$ for some group N of order $[L:K]$. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree $p^n$, for p an odd prime number.     

On generalized quadrangles with a point regular group of automorphisms

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group $G$ acting regularly on the point set of a generalized quadrangle of order $(u^2,u^3)$ or $(s,s)$, where $s$ is odd and $s+1$ is coprime to 3, cannot have any nonabelian minimal normal subgroups.     

Supersingular quartic surfaces

In this paper, we prove that, over an algebraically closed field of odd characteristic p ($p\ne 5$), every supersingular K3 surface is isomorphic to a quartic surface in the three dimensional projective space.     

Symmetric graphs of valency five and their basic normal quotients

A graph $\Gamma$ is symmetric or arc-transitive if its automorphism group $\mathrm{Aut}\left(\Gamma \right)$ is transitive on the arc set of the graph, and $\Gamma$ is basic if $\mathrm{Aut}\left(\Gamma \right)$ has no non-trivial normal subgroup $N$ such that the quotient graph ${\Gamma }_N$ has the same valency as $\Gamma$. In this paper, we classify symmetric basic graphs of order $2q{p}^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5|(q-1)$, the complete graph $K_6$ of order 6, the complete bipartite graph $K_{5,5}$ of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order $k{p}^n$ for some small integers $k$ and $n$ are classified.     

A large family of cospectral Cayley graphs over dicyclic groups

For a finite group G and an inverse closed subset $S\subseteq G\setminus \{e\}$, the Cayley graph $X(G,S)$ has vertex set G and two vertices $x,y\in G$ are adjacent if and only if $x{y}^{-1}\in S$. Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let $p\ge 3$ be a prime number and $T_{4p}$ be the dicyclic group of order 4p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group $T_{4p}$ with $p\ge 23$, and find several pairs of non-isomorphic and cospectral Cayley graphs for $5\le p\le 19$.     

Complex ball quotients from manifolds of K3[n]-type

We describe periods of irreducible holomorphic symplectic manifolds of $K{3}^{[n]}$-type with a non-symplectic automorphism of prime order $p\ge 3$. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective by introducing the notion of $K(T)$-generality.