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.     

Pseudo Frobenius numbers

For a prime $p$, we call a positive integer $n$ a Frobenius $p$-number if there exists a finite group with exactly $n$ subgroups of order $p^a$ for some $a\ge 0$. Extending previous results on Sylow’s theorem, we prove in this paper that every Frobenius $p$-number $n\equiv 1(mod{p}^2)$ is a Sylow $p$-number, i. e., the number of Sylow $p$-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order $3^a$ for any $a\ge 0$.     

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 $5\le p\le 19$.     

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.     

On the Hamming distances of repeated-root constacyclic codes of length 4ps

Let $p$ be an odd prime, $s$, $m$ be positive integers, $\gamma ,\lambda$ be nonzero elements of the finite field ${\mathbb{F}}_{p^m}$ such that ${\gamma }^{p^s}=\lambda$. In this paper, we show that, for any positive integer $\eta$, the Hamming distances of all repeated-root $\lambda$-constacyclic codes of length $\eta p^s$ can be determined by those of certain simple-root $\gamma$-constacyclic codes of length $\eta$. Using this result, Hamming distances of all constacyclic codes of length $4p^s$ are obtained. As an application, we identify all MDS $\lambda$-constacyclic codes of length $4p^s$.     

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.     

A note on complex conference matrices

Let ${\mathbb{F}}_q$ be the Galois field of order $q=p^m$, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ϰ of ${\mathbb{F}}_q^{⁎}$ and for any $b\in {\mathbb{F}}_q^{⁎}$ we have$\sum _{a\in {\mathbb{F}}_q^{⁎}}\varkappa (a)\overline{\varkappa (a+b)}=-1$. Whenever q is odd and ϰ is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that $C^{⁎}C=(n-1)I$. Paley used finite fields with odd orders $q=p^m$, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders $q+1$ whenever $q\equiv 1(\mathrm{mod}4)$ and real skew-symmetric conference matrices of orders $q+1$ whenever $q\equiv -1(\mathrm{mod}4)$. In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order $q+1$ whenever $q\ge 4$ is any prime power as well as a complex skew-symmetric conference matrix of order $q+1$ whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ϰ and that one constructed with ${\varkappa }^{p^k}$ for any integer $k=1,...m-1$ are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.     

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.     

Antimagic orientations of disconnected even regular graphs

A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\dots ,m\}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u\in V\left(D\right)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz et al. (2010) raised the question: Does every graph admit an antimagic orientation? It had been proved that every 2$d$-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider 2$d$-regular graphs with more than two odd components. We show that every 2$d$-regular graph with $k(3\le k\le 5d+4)$ odd components has an antimagic orientation. And we show that each 2$d$-regular graph with $k(k\ge 5d+5)$ odd components admits an antimagic orientation if each odd component has at least $2x_0+5$ vertices with $x_0=?\frac{k?(5d+4)}{2d?2}?$.     

Distribution of the reduced quadratic irrationals arising from the odd continued fraction expansion

This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface $\Gamma ?\mathbb{H}$, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.     

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).     

Convergence of Picard’s iteration using projection algorithm for noncyclic contractions

Given $A$ and$B$ two nonempty subsets of a metric space, a mapping $T:A\cup B\to A\cup B$ is noncyclic provided that $T\left(A\right)?A$ and $T\left(B\right)?B$. A point $(p,q)\in A\times B$ is called a best proximity pair for the noncyclic mapping $T$ if $p=Tp,q=Tq$ and $d(p,q)=\mathrm{dist}(A,B)$. In this article, we survey the convergence of Picard’s iteration to a best proximity pair for noncyclic contractions using a projection algorithm in uniformly convex Banach spaces, where the initial point is in the proximal set of $A$. We also provide some sufficient conditions to ensure the existence of a common best proximity pairs for a pair of noncyclic mappings.