In this paper, we give an explicit construction of a quasi-idempotent in the $q$ -rook monoid algebra $R_{n}(q)$ and show that it generates the whole annihilator of the tensor space $U^{\otimes n}$ in $R_{n}(q)$ .

We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.

In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$ .

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

This paper contributes to the regular covers of a complete bipartite graph minus a matching, denoted $K_{n,n}-nK_{2}$ , whose fiber-preserving automorphism group acts 2-arc-transitively. All such covers, when the covering transformation group $K$ is either cyclic or $\mathbb{Z}_{p}^{2}$ with $p$ a prime, have been determined in Xu and Du [‘2-arc-transitive cyclic covers of $K_{n,n}-nK_{2}$ ’, J. Algebraic Combin.39 (2014), 883–902] and Xu et al. [‘2-arc-transitive regular covers of $K_{n,n}-nK_{2}$ with the covering transformation group $\mathbb{Z}_{p}^{2}$ ’, Ars. Math. Contemp.10 (2016), 269–280]. Finally, this paper gives a classification of all such covers for $K\cong \mathbb{Z}_{p}^{3}$ with $p$ a prime.

A $(d-1)$ -dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$ -coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated lower bound theorem (LBT) for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; and we propose the balanced analog of the generalized lower bound conjecture (GLBC) and establish some related results. We close with constructions of balanced manifolds with few vertices.

We prove that among all flag triangulations of manifolds of odd dimension $2r-1$ , with a sufficient number of vertices, the unique maximizer of the entries of the $f$ -, $h$ -, $g$ - and $\unicode[STIX]{x1D6FE}$ -vector is the balanced join of $r$ cycles. Our proof uses methods from extremal graph theory.

Motivated by a problem of characterising a family of Cayley graphs, we study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\mathsf{Aut}(G)$. It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.

Let $G$ be a finite group and ${\rm\Gamma}$ a $G$-symmetric graph. Suppose that $G$ is imprimitive on $V({\rm\Gamma})$ with $B$ a block of imprimitivity and ${\mathcal{B}}:=\{B^{g};g\in G\}$ a system of imprimitivity of $G$ on $V({\rm\Gamma})$. Define ${\rm\Gamma}_{{\mathcal{B}}}$ to be the graph with vertex set ${\mathcal{B}}$ such that two blocks $B,C\in {\mathcal{B}}$ are adjacent if and only if there exists at least one edge of ${\rm\Gamma}$ joining a vertex in $B$ and a vertex in $C$. Xu and Zhou [‘Symmetric graphs with 2-arc-transitive quotients’, J. Aust. Math. Soc. 96 (2014), 275–288] obtained necessary conditions under which the graph ${\rm\Gamma}_{{\mathcal{B}}}$ is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to ${\rm\Gamma}$ and ${\mathcal{B}}$ and show that there is a unique graph corresponding to this case.

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.

A companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a $\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in type A (and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation type A or D.

We consider the Jack–Laurent symmetric functions for special values of parameters p 0=n+k −1 m, where k is not rational and m and n are natural numbers. In general, the coefficients of such functions may have poles at these values of p 0. The action of the corresponding algebra of quantum Calogero–Moser integrals $\mathcal{D}$ (k, p 0) on the space of Laurent symmetric functions defines the decomposition into generalised eigenspaces. We construct a basis in each generalised eigenspace as certain linear combinations of the Jack–Laurent symmetric functions, which are regular at p 0=n+k −1 m, and describe the action of $\mathcal{D}$ (k, p 0) in these eigenspaces.

Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities.

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

Let $\mathbb{A}=(A,+)$ be a (possibly non-commutative) semigroup. For $Z\subseteq A$, we define $Z^{\times }:=Z\cap \mathbb{A}^{\times }$, where $\mathbb{A}^{\times }$ is the set of the units of $\mathbb{A}$ and

$$\begin{eqnarray}{\it\gamma}(Z):=\sup _{z_{0}\in Z^{\times }}\inf _{z_{0}\neq z\in Z}\text{\text{ord}}(z-z_{0}).\end{eqnarray}$$

${\it\gamma}(\cdot )$

$\mathbb{A}$

$X,Y\subseteq A$

$$\begin{eqnarray}|X+Y|\geqslant \min ({\it\gamma}(X+Y),|X|+|Y|-1).\end{eqnarray}$$

$\mathbb{A}$

${\it\gamma}(X+Y)$

$|S|$

$S$

$\mathbb{A}$

Let $X$ be a simple, connected, $p$-valent, $G$-arc-transitive graph, where the subgroup $G\leq \text{Aut}(X)$ is solvable and $p\geq 3$ is a prime. We prove that $X$ is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that $G$ is at most 3-arc-transitive on $X$ and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all $G$-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.

In this paper, we combine group-theoretic and combinatorial techniques to study $\wedge$-transitive digraphs admitting a cartesian decomposition of their vertex set. In particular, our approach uncovers a new family of digraphs that may be of considerable interest.