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We study the back stable Schubert calculus of the infinite flag variety. Our main results are:
– a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;
– a novel definition of double and triple Stanley symmetric functions;
– a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;
– the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;
– the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;
– equivariant Pieri rules for the homology of the infinite Grassmannian;
– homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.
Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least
$$|S| - m\ln |G|$$
elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least
$$|G{|^{1 - c{ \in ^l}}}$$
for certain c=c(m) and
$$ \in = \in (m) < 1$$
; we use the probabilistic method to give sharper values of c(m) and
$$ \in (m)$$
in the case when G is a vector space; and we give new proofs of related known results.
A family of vectors in [k]n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k]n invariant under a transitive group of symmetries is o(kn), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.
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.
For an indifference graph G, we define a symmetric function of increasing spanning forests of G. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular $\textrm {LLT}$ polynomials. As a consequence, we give a combinatorial interpretation of the coefficients of the $\textrm {LLT}$ polynomial in the elementary basis (up to a factor of a power of $(q-1)$), strengthening the description given in [4].
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$
.
If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.
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.