Cohen–Macaulay binomial edge ideals and accessible graphs

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

One-relator groups and algebras related to polyhedral products

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

Stability of characters and filters for weighted semilattices

We continue the study of the AMNM property for weighted semilattices that was initiated in Choi (J Aust Math Soc 95(1):36–67, 2013. 10.1017/S1446788713000189). We reformulate this in terms of stability of filters with respect to a given weight function, and then provide a combinatorial condition which is necessary and sufficient for this “filter stability” property to hold. Examples are given to show that this new condition allows for easier and unified proofs of some results in loc. cit., and furthermore allows us to verify the AMNM property in situations not covered by the results of that paper. As a final application, we show that for a large class of semilattices, arising naturally as union-closed set systems, one can always construct weights for which the AMNM property fails.

How to have more things by forgetting how to count them

Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.

An asymptotic property of large matrices with identically distributed Boolean independent entries

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically [Formula: see text]-diagonal and Boolean independent from each other. This paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that the random matrices considered are asymptotically Boolean independent from some of their partial transposes. The main results of the paper are based on combinatorial techniques.

INTERTWINING SEMISIMPLE CHARACTERS FOR -ADIC CLASSICAL GROUPS

Let $G$ be an orthogonal, symplectic or unitary group over a non-archimedean local field of odd residual characteristic. This paper concerns the study of the “wild part” of an irreducible smooth representation of $G$, encoded in its “semisimple character”. We prove two fundamental results concerning them, which are crucial steps toward a complete classification of the cuspidal representations of $G$. First we introduce a geometric combinatorial condition under which we prove an “intertwining implies conjugacy” theorem for semisimple characters, both in $G$ and in the ambient general linear group. Second, we prove a Skolem–Noether theorem for the action of $G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of $G$ which have the same characteristic polynomial must be conjugate under an element of $G$ if there are corresponding semisimple strata which are intertwined by an element of $G$.

The topological strong spatial mixing property and new conditions for pressure approximation

In the context of stationary $\mathbb{Z}^{d}$ nearest-neighbour Gibbs measures $\unicode[STIX]{x1D707}$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property) on the support of $\unicode[STIX]{x1D707}$ that is sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of topological strong spatial mixing for studying strong spatial mixing on systems with hard constraints. We also show that topological strong spatial mixing is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by Gamarnik and Katz [Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys.137(2) (2009), 205–232], and Marcus and Pavlov [An integral representation for topological pressure in terms of conditional probabilities. Israel J. Math.207(1) (2015), 395–433], who gave a special representation of topological pressure in terms of conditional probabilities.

Rigged Configurations for all Symmetrizable Types

In an earlier work, the authors developed a rigged configuration model for the crystal $B(\infty)$ (which also descends to a model for irreducible highest weight crystals via a cutting procedure). However, the result obtained was only valid in finite types, affine types, and simply-laced indefinite types. In this paper, we show that the rigged configuration model proposed does indeed hold for all symmetrizable types. As an application, we give an easy combinatorial condition that gives a Littlewood-Richardson rule using rigged configurations which is valid in all symmetrizable Kac-Moody types.

A Criterion for a Monomial Ideal to have a Linear Resolution in Characteristic 2

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2.