The first open case of the Brown–Erdős–Sós conjecture is equivalent to the following: for every c > 0, there is a threshold n0 such that if a quasigroup has order n ⩾ n0, then for every subset S of triples of the form (a, b, ab) with |S| ⩾ cn2, there is a seven-element subset of the quasigroup which spans at least four triples of S. In this paper we prove the conjecture for finite groups.

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualized via two case studies, arising from our recent work: entanglement invariants for characterizing the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.

By using row convex tableaux, we study the section rings of Bott–Samelson varieties of type A. We obtain flat deformations and standard monomial type bases of the section rings. In a separate section, we investigate a three-dimensional Bott–Samelson variety in detail and compute its Hilbert polynomial and toric degenerations.

Let $Q$ be a finite quiver without oriented cycles, and let $k$ be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Weyl group $W_{Q}$ and the cofinite additive quotient closed subcategories of the category of finite dimensional right modules over $kQ$. We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to $Q$, which are also indexed by elements of $W_{Q}$.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Gamma $ be a compact tropical curve (or metric graph) of genus  $g$ . Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on  $\Gamma $ . We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an ‘integral’ version of this result which is of independent interest. As an application, we provide a‘geometric proof’ of (a dual version of) Kirchhoff’s celebrated matrix–tree theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma $ gives rise to a canonical polyhedral decomposition of the $g$ -dimensional real torus $\mathrm{Pic}^g(\Gamma )$ into parallelotopes $C_T$ , one for each spanning tree $T$ of $G$ , and the dual Kirchhoff theorem becomes the statement that the volume of $\mathrm{Pic}^g(\Gamma )$ is the sum of the volumes of the cells in the decomposition.

In this paper, we first prove that for $g\in \{3,4\}$, there are infinitely many 3-geodesic transitive but not 3-arc transitive graphs of girth $g$ with arbitrarily large diameter and valency. Then we classify the family of 3-geodesic transitive but not 3-arc transitive graphs of valency 3 and those of valency 4 and girth 4.

We develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the $q$-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with $q$-TASEP ($q$-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of $q$-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our $q$-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.

The problem of finding a nontrivial factor of a polynomial $f(x)$ over a finite field ${\mathbb{F}}_q$ has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let $n$ be the degree. If $(n-1)$ has a‘large’ $r$ -smooth divisor $s$ , then we find a nontrivial factor of $f(x)$ in deterministic $\mbox{poly}(n^r,\log q)$ time, assuming GRH and that $s=\Omega (\sqrt{n/2^r})$ . Thus, for $r=O(1)$ our algorithm is polynomial time. Further, for $r=\Omega (\log \log n)$ there are infinitely many prime degrees $n$ for which our algorithm is applicable and better than the best known, assuming GRH. Our methods build on the algebraic-combinatorial framework of $m$ -schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the $m$ -scheme on $n$ points, implicitly appearing in our factoring algorithm, has an exceptional structure, leading us to the improved time complexity. Our structure theorem proves the existence of small intersection numbers in any association scheme that has many relations, and roughly equal valencies and indistinguishing numbers.

A graph $\Gamma $ is $G$-symmetric if $\Gamma $ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Gamma $, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\Gamma )$, namely when $V(\Gamma )$ admits a nontrivial $G$-invariant partition ${\mathcal{B}}$, the quotient graph $\Gamma _{\mathcal{B}}$ of $\Gamma $ with respect to ${\mathcal{B}}$ is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for $\Gamma _{{\mathcal{B}}}$ to be $(G, 2)$-arc transitive (regardless of whether $\Gamma $ is $(G, 2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of ${\mathcal{B}}$ and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v, k$ and two other parameters with respect to $(\Gamma , {\mathcal{B}})$ together with a certain 2-point transitive block design induced by $(\Gamma , {\mathcal{B}})$. We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for $\Gamma _{{\mathcal{B}}}$ to be $(G, 2)$-arc transitive.

Let $d$ and $n$ be positive integers such that $n\geq d+ 1$ and ${\tau }_{1} , \ldots , {\tau }_{n} $ integers such that ${\tau }_{1} \lt \cdots \lt {\tau }_{n} $. Let ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )\subset { \mathbb{R} }^{d} $ denote the cyclic polytope of dimension $d$ with $n$ vertices $({\tau }_{1} , { \tau }_{1}^{2} , \ldots , { \tau }_{1}^{d} ),\ldots , ({\tau }_{n} , { \tau }_{n}^{2} , \ldots , { \tau }_{n}^{d} )$. We are interested in finding the smallest integer ${\gamma }_{d} $ such that if ${\tau }_{i+ 1} - {\tau }_{i} \geq {\gamma }_{d} $ for $1\leq i\lt n$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is normal. One of the known results is ${\gamma }_{d} \leq d(d+ 1)$. In the present paper a new inequality ${\gamma }_{d} \leq {d}^{2} - 1$ is proved. Moreover, it is shown that if $d\geq 4$ with ${\tau }_{3} - {\tau }_{2} = 1$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is not very ample.

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.

Local models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure of integral models of certain $p$ -adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at ${ \mathbb{Q} }_{p} $ is ramified, quasi-split $G{U}_{n} $ , Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper, we proved that their new local models are topologically flat when $n$ is odd. In the present paper, we prove topological flatness when $n$ is even. Along the way, we characterize the $\mu $ -admissible set for certain cocharacters $\mu $ in types $B$ and $D$ , and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.

Let Ω be a finite set and let G be a permutation group acting on it. A subset H of G is called t-intersecting if any two elements in H agree on at least t points. Let SDn and SBn be the classical Coxeter group of type Dn and type Bn, respectively. We show that the maximum-sized (2t)-intersecting families in SDn and SBn are precisely cosets of stabilizers of t points in [n] ≔ {1, 2, …, n}, provided n is sufficiently large depending on t.

In this short paper, we characterise graphs of order $pq$ with $p, q$ prime which are self-complementary and vertex-transitive.

We study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In these examples, the alphabet size always divides the length of the code. We show that there is no elusive pair for the smallest set of parameters that does not satisfy this condition. We also pose several questions regarding elusive pairs.

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such as principal coefficients.