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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 $\mathcal {Z}_K$.
For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.
A connected graph G is
$\mathcal {CF}$
-connected if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete bipartite graph
$K_{m,n}$
is
$\mathcal {CF}$
-connected if and only if it does not contain a subgraph of
$K_{3,6}$
or
$K_{4,4}$
. We establish the validity of this conjecture for all complete bipartite graphs
$K_{m,n}$
for any
$m,n$
with
$\min \{m,n\}\leq 6$
, and conditionally for
$m,n\geq 7$
on the assumption of Zarankiewicz’s conjecture that
$\mathrm {cr}(K_{m,n})=\big \lfloor \frac {m}{2} \big \rfloor \big \lfloor \frac {m-1}{2} \big \rfloor \big \lfloor \frac {n}{2} \big \rfloor \big \lfloor \frac {n-1}{2} \big \rfloor $
.
Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.
Let I be a zero-dimensional ideal in the polynomial ring
$K[x_1,\ldots ,x_n]$
over a field K. We give a bound for the number of roots of I in
$K^n$
counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.
Assume that G is a graph with edge ideal
$I(G)$
and star packing number
$\alpha _2(G)$
. We denote the sth symbolic power of
$I(G)$
by
$I(G)^{(s)}$
. It is shown that the inequality
$ \operatorname {\mathrm {depth}} S/(I(G)^{(s)})\geq \alpha _2(G)-s+1$
is true for every chordal graph G and every integer
$s\geq 1$
. Moreover, it is proved that for any graph G, we have
$ \operatorname {\mathrm {depth}} S/(I(G)^{(2)})\geq \alpha _2(G)-1$
.
We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.
It is well known that the pair
$(\mathcal {S}_n,\mathcal {S}_{n-1})$
is a Gelfand pair where
$\mathcal {S}_n$
is the symmetric group on n elements. In this paper, we prove that if G is a finite group then
$(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$
where
$G\wr \mathcal {S}_n$
is the wreath product of G by
$\mathcal {S}_n,$
is a Gelfand pair if and only if G is abelian.
We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.
Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers.
This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra.
We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.
The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space
${\mathbb{R}}^d$
. We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than
$k+1$
to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of
$n^{-1/d}$
, the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is
$o(n^{-1/d})$
, i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.
Let
$X$
be a nonempty set and
${\mathcal{P}}(X)$
the power set of
$X$
. The aim of this paper is to identify the unital subrings of
${\mathcal{P}}(X)$
and to compute its cardinality when it is finite. It is proved that any topology
$\unicode[STIX]{x1D70F}$
on
$X$
such that
$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$
, where
$\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$
, is a unital subring of
${\mathcal{P}}(X)$
. It is also shown that
$X$
is finite if and only if any unital subring of
${\mathcal{P}}(X)$
is a topology
$\unicode[STIX]{x1D70F}$
on
$X$
such that
$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$
if and only if the set of unital subrings of
${\mathcal{P}}(X)$
is finite. As a consequence, if
$X$
is finite with cardinality
$n\geq 2$
, then the number of unital subrings of
${\mathcal{P}}(X)$
is equal to the
$n$
th Bell number and the supremum of the lengths of chains of unital subalgebras of
${\mathcal{P}}(X)$
is equal to
$n-1$
.
We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.
One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995): consider a plane partition P in an
$a \times b \times c$
box
${\sf B}$
. Let
$\Psi (P)$
denote the smallest plane partition containing the minimal elements of
${\sf B} - P$
. Then if
$p= a+b+c-1$
is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the
$\Psi $
-orbit of P is always a multiple of p.
This conjecture was established for
$p \gg 0$
by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of p in work of K. Dilks, J. Striker and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.
Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts.
We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.
Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.
The
$W$
-operator,
$W([n])$
, generalises the cut-and-join operator. We prove that
$W([n])$
can be written as the sum of
$n!$
terms, each term corresponding uniquely to a permutation in
$S_{\!n}$
. We also prove that there is a correspondence between the terms of
$W([n])$
with maximal degree and noncrossing partitions.
This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.
We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$-theoretic deformation of the quasi-key basis and also a lift of the $K$-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy.
The second new basis is the kaon basis, a $K$-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
Throughout, we explore how the relationships among these $K$-analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.
We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.
In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.
For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.