The paper is motivated by an analogy between cluster algebras and Kac–Moody algebras: both theories share the same classification of finite type objects by familiar Cartan–Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac–Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

Let $\Omega=\{1,2,\dotsc,n\}$ where $n \ge 2$. The shape of an ordered set partition $P=(P_1,\dotsc, P_k)$ of $\Omega$ is the integer partition $\lambda=(\lambda_1,\dotsc,\lambda_k)$ defined by $\lambda_i = |P_i|$. Let G be a group of permutations acting on $\Omega$. For a fixed partition $\lambda$ of n, we say that G is $\lambda$-transitive if G has only one orbit when acting on partitions P of shape $\lambda$. A corresponding definition can also be given when G is just a set. For example, if $\lambda=(n-t,1,\dotsc,1)$, then a $\lambda$-transitive group is the same as a t-transitive permutation group, and if $\lambda=(n-t,t)$, then we recover the t-homogeneous permutation groups.

We use the character theory of the symmetric group Sn to establish some structural results regarding $\lambda$-transitive groups and sets. In particular, we are able to generalize a celebrated result of Livingstone and Wagner [Math. Z. 90 (1965) 393–403] about t-homogeneous groups. We survey the relevant examples coming from groups. While it is known that a finite group of permutations can be at most 5-transitive unless it contains the alternating group, we show that it is possible to construct a nontrivial t-transitive set of permutations for each positive integer t. We also show how these ideas lead to a combinatorial basis for the Bose–Mesner algebra of the association scheme of the symmetric group and a design system attached to this association scheme.

Our main aim in this paper is to show that there is a partition of the reals into finitely many classes with ‘many’ forbidden distances, in the following sense: for each positive real x, there is a natural number n such that no two points in the same class are at distance $x/n$. In fact, more generally, given any infinite set $\{ c_n:n<\omega\}$ of positive rationals, there is a partition of the reals into three classes such that for each positive real x, there is some n such that no two points in the same class are at distance $c_nx$. This result is motivated by some questions in partition regularity.

We consider a variation of the list colouring problem in which the lists are required to be sets of consecutive integers, and the colours assigned to adjacent vertices must differ by at least a fixed integer $s$. We introduce and investigate a new parameter $\tau(G)$ of a graph $G$, called the consecutive choosability ratio and defined to be the ratio of the required list size to the separation $s$ in the limit as $s\to\infty$.

We show that the above limit exists and that, for finite graphs $G,\ \tau(G)$ is rational and is a refinement of the chromatic number $\chi(G)$. We provide general bounds on $\tau(G)$, and determine its value for various classes of graphs including bipartite graphs, circuits, wheels and balanced complete multipartite graphs. Finally, we explore relationships between $\tau(G)$ and the circular chromatic number $\chi_{\mathrm{c}}(G)$.

We prove that the permutation representation of the finite orthogonal group $\Omega^{\varepsilon}(n,q)$, where $\varepsilon=+$ or $-$, on the set of anisotropic lines is multiplicity-free, if q is a power of 2 and $n\ge 6$ is even. This result is established by giving a description of orbitals of this action. The rank of this action is $(q^2+2q)/2$ if $\varepsilon=+$ and $n=6$, and $(q^2+2q+2)/2$ otherwise. Moreover, we compute the subdegrees of the orbitals of $\Omega^{\varepsilon}(n,q)$.

In this paper a formula is derived for the number of conjugacy classes of cyclic subgroups of finite order in those arithmetic Fuchsian groups which are of minimal covolume in their commensurability class. The formula is entirely in terms of the number theoretic data defining the commensurability class of the arithmetic group so that, in particular, any two such groups of minimal covolume in the class, will be isomorphic.

An RC-group is a unital group G with a distinguished compression base with respect to which G satisfies the Rickart projection and general comparison properties. We prove that a monotone $\sigma$-complete RC-group is a union of subgroups each of which is a lattice-ordered Dedekind $\sigma$-complete RC-group.

We consider a class of reductive linear groups defined in terms of weighted oriented graphs of a special sort that we call signed quivers. Each of these yields a symmetric quiver, that is, a quiver endowed with an involutive anti-automorphism together with signs for the vertices and arrows fixed by the involution. The orbits of the groups can be described in terms of the indecomposable symmetric representations of symmetric quivers. We provide a general description of the indecomposable symmetric representations and prove that their dimensions in the finite and tame cases constitute root systems corresponding to certain symmetrizable generalized Cartan matrices.

Here we develop estimates for Fourier coefficients of Siegel cusp forms. First we consider the case of Siegel modular forms for the full modular group $\Gamma_g$ having small weight (that is, weight k = g + 1). Afterwards the case of a certain subgroup $\Gamma_{g,0}(N)$ of $\Gamma_g$ is considered.

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree $d>0$ will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type $G_2$, type $F_4$, or simply connected of type $E_6$. We extend the list of cases where the answer is ‘yes’ to all groups of type $G_2$ and some nonsplit groups of type $F_4$ and $E_6$. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.

We prove that if G is a non-trivial finite group, then the integral group ring $\mathbb{Z} G$ possesses a superdecomposable pure-injective module.

We consider bihomogeneous polynomials on complex Euclidean spaces that are positive outside the origin and obtain effective estimates on certain modifications needed to turn them into squares of norms of vector-valued polynomials on complex Euclidean space. The corresponding results for hypersurfaces in complex Euclidean spaces are also proved. The results can be considered as Hermitian analogues of Hilbert's seventeenth problem on representing a positive definite quadratic form on $\mathbb{R}^n$ as a sum of squares of rational functions. They can also be regarded as effective estimates on the power of a Hermitian line bundle required for isometric projective embedding. Further applications are discussed.

A certain sequence of weight $1/2$ modular forms arises in the theory of Borcherds products for modular forms for $\textrm{SL}_{2}(\Z)$. Zagier proved a family of identities between the coefficients of these weight $1/2$ forms and a similar sequence of weight $3/2$ modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilbert modular forms.

For a real set $A$ consider the semigroup ${\mathcal S}(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A\subseteq(0,1)$ is open and non-empty, then ${\mathcal S}(A)$ is easily seen to contain all sufficiently large real numbers, and we let $G(A):=\sup\{u\in{\mathbb R}: u\notin{\mathcal S}(A)\}$. Thus $G(A)$ is the smallest number with the property that any $u>G(A)$ is representable as indicated above.

We show that if the measure of $A$ is large, then $G(A)$ is small; more precisely, writing for brevity $\alpha:= {\rm mes\,} A$, we have

\[ G(A) \le \begin{cases} (1-\alpha)\lfloor 1/\alpha\rfloor &\text{if $0<\alpha\le 0.1$}, \\ (1-\alpha+\alpha\{1/\alpha\})\lfloor 1/\alpha\rfloor &\text{if $0.1 \le \alpha \le 0.5$}, \\ 2(1-\alpha) &\text{if $0.5 \le \alpha \le 1$}. \\ \end{cases} \]

Indeed, the first and the last of these three estimates are the best possible, attained for $A=(1-\alpha,1)$ and $A=(1-\alpha,1)\setminus\{2(1-\alpha)\}$, respectively; the second is close to the best possible and can be improved by $\alpha\{1/\alpha\}\lfloor 1/\alpha\rfloor\le\{1/\alpha\}$ at most.

The problem studied is a continuous analogue of the linear Diophantine problem of Frobenius (in its extremal settings due to Erdös and Graham), also known as the ‘postage stamp problem’ or the ‘coin exchange problem’.

We prove, correct and extend several results of an earlier paper of ours (using and recalling several of our later papers) about the derived functors of projective limit in abelian categories. In particular we prove that if C is an abelian category satisfying the Grothendieck axioms AB3 and AB4* and having a set of generators then the first derived functor of projective limit vanishes on so-called Mittag-Leffler sequences in C. The recent examples given by Deligne and Neeman show that the condition that the category has a set of generators is necessary. The condition AB4* is also necessary, and indeed we give for each integer $m \geq 1$ an example of a Grothendieck category Cm and a Mittag-Leffler sequence in Cm for which the derived functors of its projective limit vanish in all positive degrees except m. This leads to a systematic study of derived functors of infinite products in Grothendieck categories. Several explicit examples of the applications of these functors are also studied.

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.

Every field $K$ admits proper projective extensions, that is, Galois extensions where the Galois group is a non-trivial projective group, unless $K$ is separably closed or $K$ is a pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products.

We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt's conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non-procyclic projective group as Galois group over ${\mathbb Q}$ produces counterexamples to the Leopoldt conjecture.

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including SL$(n,\mathbb{Z})$ for $n\geq 3$. We also give a way of thinking about Property (QFA) by breaking it down into statements about particular classes of trees.

The Skolem–Mahler–Lech theorem states that if $f(n)$ is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that $f(m)$ is equal to 0 is the union of a finite number of arithmetic progressions in $m\ge 0$ and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y, and $\sigma$ is an automorphism of Y, then the set of m such that $\sigma^m({\bf q})$ lies in X is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem.

In 1989 Happel asked the question whether, for a finite-dimensional algebra $A$ over an algebraically closed field $k,\ \text{\rm gl.dim\,} A< \infty$ if and only if $\text{\rm hch.dim\,} A < \infty$. Here, the Hochschild cohomology dimension of $A$ is given by $\text{\rm hch.dim\,} A := \inf \{ n \in \mathbb{N}_0 \mid \dim \textit{HH}^i(A)=0 \text{ for }i > n \}$. Recently Buchweitz, Green, Madsen and Solberg gave a negative answer to Happel's question. They found a family of pathological algebras $A_q$ for which $\text{\rm gl.dim\,} A_q = \infty$ but $\text{\rm hch.dim\,} A_q=2$. These algebras are pathological in many aspects. However, their Hochschild homology behaviors are not pathological any more; indeed one has $\text{\rm hh.dim\,} A_q = \infty=\text{\rm gl.dim\,} A_q$. Here, the Hochschild homology dimension of $A$ is given by $\text{\rm hh.dim\,} A := \inf \{ n \in \mathbb{N}_0 \mid \dim \textit{HH}_i(A)=0 \text{ for } i > n\}$. This suggests posing a seemingly more reasonable conjecture by replacing the Hochschild cohomology dimension in Happel's question with the Hochschild homology dimension: $\text{\rm gl.dim\,} A < \infty$ if and only if $\text{\rm hh.dim\,} A < \infty$ if and only if $\text{\rm hh.dim\,} A = 0$. The conjecture holds for commutative algebras and monomial algebras. In the case where $A$ is a truncated quiver algebra, these conditions are equivalent to the condition that the quiver of $A$ has no oriented cycles. Moreover, an algorithm for computing the Hochschild homology of any monomial algebra is provided. Thus the cyclic homology of any monomial algebra can be read off when the underlying field is characteristic 0.