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.

We prove a conjecture of Kontsevich, which asserts that the iterations of the non-commutative rational map Fr:(x,y)→(xyx−1,(1+yr)x−1) are given by non-commutative Laurent polynomials with non-negative integer coefficients.

We compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of ℛ-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

We construct and classify all groups given by triangular presentations associated to the smallest thick generalized quadrangle that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification yields 23 non-isomorphic torsion-free groups (which were obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the case, we find both torsion and torsion-free groups acting on the same building.

This paper aims to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category for the rational Cherednik algebra of type G(r, p, n). As a first application, a self-contained and elementary proof of the analogue for the groups G(r, p, n), with r > 1, of Gordon's Theorem (previously Haiman's Conjecture) on the diagonal co-invariant ring is given. No restriction is imposed on p; the result for p ≠ r has been proved by Vale using a technique analogous to Gordon's. Because of the combinatorial application to Haiman's Conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon's proof of his theorem.

For each finite real reflection group W, we identify a copy of the type-W simplicial generalized associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalized associahedron and the elements of the type-W non-crossing partition lattice that is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for W.

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G, including the orbits of K(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

A new type of basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key ingredient in the basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new Selberg integral for Jack polynomials.

We give counterexamples to Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients.

A close connection is uncovered between the lower central series of the free associative algebra of countable rank and the descending Loewy series of the direct sum of all Solomon descent algebras Δn, n ∈ ℕ0. Each irreducible Δn-module is shown to occur in at most one Loewy section of any principal indecomposable Δn-module.A precise condition for his occurence and formulae for the Cartan numbers are obtained.