Periods and nonvanishing of central <Emphasis Type="Italic">L</Emphasis>-values for GL(2<Emphasis Type="Italic">n</Emphasis>)

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idèle class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ? η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula.We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.     

Properties of element orders in covers for L<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>) and U<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

We show that if a finite simple group G, isomorphic to PSL_n(q) or PSU_n(q) where either n ≠ 4 or q is prime or even, acts on a vector space over a field of the defining characteristic of G; then the corresponding semidirect product contains an element whose order is distinct from every element order of G. We infer that the group PSL_n(q), n ≠ 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.     

<Emphasis Type="Italic">n</Emphasis>-Tuple algebras of associative type

We study properties of n-tuple algebras of associative type. We show that the nilpotency of an n-tuple algebra of associative type is determined by the nilpotency of each element. In addition, we characterize the nilpotency of an n-tuple algebra of associative type in terms of the trace function. In the final part of the paper, we show that a homogeneously semisimple n-tuple algebra of associative type is the direct sum of two-sided ideals each of which is a homogeneously simple n-tuple algebra of associative type.     

On the centers of quantum groups of <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-type

Let g be the finite dimensional simple Lie algebra of type An, and let U? = U _q (g,Λ) and U = U _q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U? for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U? = U _q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U? = U _q (g,Λ) and U = U _q (g,Q).     

The Center of <Emphasis Type="Italic">Dist</Emphasis> (<Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">m</Emphasis>|<Emphasis Type="Italic">n</Emphasis>)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

Stable equivalence classes of self-injective algebras of tree class <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A classification up to stable equivalence of representation-finite self-injective algebras with associated Dynkin diagram D _n is suggested. In each stable equivalence class, a representative is chosen, and the representatives are partitioned into five families of algebras, which are described in terms of quivers with relations.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

Note edge-colourings of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n,n</Emphasis></Subscript> with no long two-coloured cycles

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K _n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K _n,n with n colours having all two-coloured cycles of length ≤ 2q ², for integers n in a set of density 1 ? 3/(q ? 1). One consequence is that L(n, K _n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n ? 4.     

The characterization problem for designs with the parameters of AG<Subscript>d</Subscript>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We start a new characterization of the geometric 2-design AG _d (n,q) among all simple 2-designs with the same parameters by handling the cases d ∈ {1,2,3,n — 2}. For d ≠ 1, our characterization is in terms of line sizes, and for d = 1 in terms of the number of affine hyperplanes. We also show that the number of non-isomorphic resolvable designs with the parameters of AG₁(n,q) grows exponentially with linear growth of n.     

A new functor from <Emphasis Type="Italic">D</Emphasis><Subscript>5</Subscript>-Mod to <Emphasis Type="Italic">E</Emphasis><Subscript>6</Subscript>-Mod

We find a new representation of the simple Lie algebra of type E ₆ on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D ₅-Mod to E ₆-Mod. A condition for the functor to map a finite-dimensional irreducible D ₅-module to an infinite-dimensional irreducible E ₆-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.     

Limit theorems for inverse process <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> of Hawkes process

We consider linear Hawkes process N _t and its inverse process T _n . The limit theorems for N _t are well known and studied by many authors. In this paper, we study the limit theorems for T _n . In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for T _n . The main tool of the proof is based on immigration-birth representation and the observations on the relation between N _t and T _n .     

On <Emphasis Type="Italic">n</Emphasis>-cubic Pyramid Algebras

In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid algebras can be characterized by n-cuboids, and prove that they are periodic. So these algebras are almost Koszul and (n?1)-translation algebras. We also recover Iyama’s cone construction for n-Auslander absolutely n-complete algebras using n-cubic pyramid algebras and the theory of n-translation algebras.     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(²G₂(q)), where q = 3²ⁿ⁺¹ for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to ² G ₂(q). We infer that if G is a finite group satisfying |G| = |² G ₂(q)| and Γ(G) = Γ (² G ₂(q)) then G ? = ² G ₂(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.