Weak Bruhat Order Research Articles

Combinatorial Results on Barcode Lattices

AbstractA barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.

Multipermutations and Stirling Multipermutations

We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are given. We also study special multipermutations called Stirling multipermutations and their properties.

Permutation graphs and the weak Bruhat order

Permutation graphs are graphs associated with permutations where edges represent inversions. We study different classes of permutation graphs and  isomorphic permutation graphs. A complete answer of a basic isomorphism question is given for trees. A connection to the majorization order for integer partitions is established, in connection with the notion of inversion vectors. Finally, we study special transitivity properties related to inversions of permutations.

Three Fuss-Catalan posets in interaction and their associative algebras

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $\delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak Bruhat order of permutations, we construct algebras whose products form intervals of the lattices of $\delta$-cliff. We provide necessary and sufficient conditions on $\delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $\delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra.

Atoms for signed permutations

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general formula for the cohomology class of a K-orbit closure in an arbitrary flag variety, where K is a symmetric subgroup of a complex algebraic group. In type A, the terms in this formula are indexed by atoms for permutations. We study the combinatorics of atoms for involutions in the group of signed permutations. In particular, we give a compact description of the atom set for any signed involution and endow it with the structure of a graded poset. Our main result, as an application, is to identify explicitly the terms in Brion’s cohomology formula in types B and C. These descriptions apply to all K-orbits in these types and are the first of their kind outside of type A.

Local finiteness of the twisted Bruhat orders on affine Weyl groups

Abstract In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra 163 1994, 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set B in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed B-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.

Lattice bijections for string modules, snake graphs and the weak Bruhat order

In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module we define a Coxeter element in a symmetric group. We then establish a bijection between the submodule lattice of the string module and the lattice given by the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondence between string modules and snake graphs, we give a new concise formulation of snake graph calculus.

The continuous weak order

The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet Σ={x,y,z,…}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card(Σ)=2, as lattices of lattice paths. By interpreting the letters x,y,z,… as axes, these words can be interpreted as discrete increasing paths on a grid of a d-dimensional cube, with d=card(Σ).We show how to extend this ordering to images of continuous monotone functions from the unit interval to a d-dimensional cube and prove that this ordering is a lattice, denoted by L(Id). This construction relies on a few algebraic properties of the quantale of join-continuous functions from the unit interval of the reals to itself: it is cyclic ⋆-autonomous and it satisfies the mix rule.We investigate structural properties of these lattices, which are self-dual and not distributive. We characterize join-irreducible elements and show that these lattices are generated under infinite joins from their join-irreducible elements, they have no completely join-irreducible elements nor compact elements. We study then embeddings of the d-dimensional multinomial lattices into L(Id). We show that these embeddings arise functorially from subdivisions of the unit interval and observe that L(Id) is the Dedekind-MacNeille completion of the colimit of these embeddings. Yet, if we restrict to embeddings that take rational values and if d>2, then every element of L(Id) is only a join of meets of elements from the colimit of these embeddings.

Stirling Pairs of Permutations

Stirling permutations are permutations $$\pi $$ of the multiset $$\{1,1,2,2,\ldots ,n,n\}$$ in which those integers between the two occurrences of an integer are greater than it. We identify a permutation $$\pi $$ of $$\{1,1,2,2,\ldots ,n,n\}$$ as a pair of permutations $$(\pi _1,\pi _2)$$ which we call a Stirling pair. We characterize Stirling pairs using the weak Bruhat order and the notion of a 312-avoiding permutation. We give two algorithms to determine if a pair of permutations is a Stirling pair.

Correlation for permutations

In this note we investigate correlation inequalities for ‘up-sets’ of permutations, in the spirit of the Harris–Kleitman inequality. We focus on two well-studied partial orders on Sn, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on Sn, up-sets are positively correlated (in the Harris–Kleitman sense). Thus, for example, for a (uniformly) random permutation π, the event that no point is displaced by more than a fixed distance d and the event that π is the product of at most k adjacent transpositions are positively correlated. In contrast, under the weak Bruhat order we show that this completely fails: surprisingly, there are two up-sets each of measure 1/2 whose intersection has arbitrarily small measure.We also prove analogous correlation results for a class of non-uniform measures, which includes the Mallows measures. Some applications and open problems are discussed.

A combinatorial duality between the weak and strong Bruhat orders

In recent work, the authors used an order lowering operator ∇, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted ∇ as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator Δ for the strong Bruhat order on the symmetric group, which is in many ways dual to ∇. We prove a Schubert identity dual to that of Hamaker et al. and derive formulas for counting weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas for the weak order, providing a strong order analog of Macdonald's reduced word identity. We also show that powers of ∇ and Δ have the same Smith normal forms, which we describe explicitly, answering a question of Stanley.

Separable elements in Weyl groups

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, and that the product of their rank generating functions gives that of the whole group, answering an open problem of Fan Wei. We also prove that separable elements are characterized by pattern avoidance in the sense of Billey and Postnikov.

K-automorphisms of a weak-crossed product F-algebra over a Galois extension K/F

We compute the K-automorphism group of AK=K⊗FAf of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent part and the reductive part Hˆ. Automorphisms in Hˆ are computed via their restriction to semi-simple part of AK. There is a strong relationship between Hˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of Hˆ, namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.

The equational theory of the weak Bruhat order on finite symmetric groups

It is well known that the weak Bruhat order on the symmetric group on a finite number n of letters is a lattice, denoted by \mathsf{P}(n) and often called the permutohedron on n letters , of which the Tamari lattice \mathsf{A}(n) is a lattice retract. The equational theory (or word problem ) of a class of lattices is the set of all lattice identities satisfied by all members of that class. Our main results imply, as particular cases, the following. Theorem I. The equational theories of all \mathsf{P}(n) and of all \mathsf{A}(n) are both decidable. Theorem I. There exists a lattice identity that holds in all \mathsf{P}(n) , but fails in a certain 3,338-element lattice. Theorem III. The equational theory of all extended permutohedra, on arbitrary (possibly infinite) posets, is trivial. The proofs of Theorems I and II involve reductions of algebraic statements to certain tiling properties of finite chains.

Poset edge densities, nearly reduced words, and barely set-valued tableaux

In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young's lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-Hecke algebras; that is, factorizations that are one letter longer than the Coxeter group length.

From the weak Bruhat order to crystal posets

We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak order via the key map. First, a crystal theoretic analogue of the statement that any two reduced expressions for the same Coxeter group element are related by Coxeter moves is proven for all lower intervals $$[\hat{0},v]$$ in a simply or doubly laced crystal. On the other hand, it is shown that no finite set of moves exists, even in type A, for arbitrary crystal graph intervals. In fact, it is shown that there are relations of arbitrarily high degree amongst crystal operators that are not implied by lower degree relations. Second, for crystals associated to Kac–Moody algebras it is shown for lower intervals that the Mobius function is always 0 or ±1, and in finite type this is also proven for upper intervals, with a precise formula given in each case. Moreover, the order complex for each of these intervals is proven to be homotopy equivalent to a ball or to a sphere of some dimension, despite often not being shellable. For general intervals, examples are constructed with arbitrarily large Mobius function, again even in type A. Any interval having Mobius function other than 0 or ±1 is shown to contain within it a relation amongst crystal operators that is not implied by the relations giving rise to the local structure of the crystal, making precise a tight relationship between the Mobius function and these somewhat unexpected relations appearing in crystals. New properties of the key map are also derived. The key is shown to be determined entirely by the edge-colored poset-theoretic structure of the crystal, and a recursive algorithm is given for calculating it. In finite types, the fiber of the longest element of any parabolic subgroup of the Weyl group is also proven to have a unique minimal and a unique maximal element; this property fails for more general elements of the Weyl group.

Higher Bruhat Orders in Type B

Motivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer $n \geq 1$ a hierarchy of finite partially ordered sets $B(n, k),$ indexed by positive integers $k$, called the higher Bruhat orders. The poset $B(n, 1)$ is naturally identified with the weak left Bruhat order on the symmetric group $S_n$, each $B(n, k)$ has a unique maximal and a unique minimal element, and the poset $B(n, k + 1)$ can be constructed from the set of maximal chains in $B(n, k)$. Ben Elias has demonstrated a striking connection between the posets $B(n, k)$ for $k = 2$ and the diagrammatics of Bott-Samelson bimodules in type A, providing significant motivation for the development of an analogous theory of higher Bruhat orders in other Cartan-Killing types, particularly for $k = 2$. In this paper we present a partial generalization to type B, complete up to $k = 2$, prove a direct analogue of the main theorem of Manin and Schechtman, and relate our construction to the weak Bruhat order and reduced expression graph for Weyl group $B_n$.

A positive Grassmannian analogue of the permutohedron

The classical permutohedron Perm is the convex hull of the points (w(1),...,w(n)) in R^n where w ranges over all permutations in the symmetric group. This polytope has many beautiful properties -- for example it provides a way to visualize the weak Bruhat order: if we orient the permutohedron so that the longest permutation w_0 is at the top and the identity e is at the bottom, then the one-skeleton of Perm is the Hasse diagram of the weak Bruhat order. Equivalently, the paths from e to w_0 along the edges of Perm are in bijection with the reduced decompositions of w_0. Moreover, the two-dimensional faces of the permutohedron correspond to braid and commuting moves, which by the Tits Lemma, connect any two reduced expressions of w_0. In this note we introduce some polytopes Br(k,n) (which we call bridge polytopes) which provide a positive Grassmannian analogue of the permutohedron. In this setting, BCFW bridge decompositions of reduced plabic graphs play the role of reduced decompositions. We define Br(k,n) and explain how paths along its edges encode BCFW bridge decompositions of the longest element pi(k,n) in the circular Bruhat order. We also show that two-dimensional faces of Br(k,n) correspond to certain local moves for plabic graphs, which by a result of Postnikov [Pos06], connect any two reduced plabic graphs associated to pi(k,n). All of these results can be generalized to the positive parts of Schubert cells. A useful tool in our proofs is the fact that our polytopes are isomorphic to certain Bruhat interval polytopes. Conversely, our results on bridge polytopes allow us to deduce some corollaries about the structure of Bruhat interval polytopes.

Modules of the 0-Hecke algebra and quasisymmetric Schur functions

We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable.From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containing a canonical tableau we discover a new basis for the Hopf algebra of quasisymmetric functions, and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by Panova and Adin–King–Roichman.Generalizing our 0-Hecke action to one on skew standard reverse composition tableaux, we derive 0-Hecke modules whose quasisymmetric characteristics are the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric Schur functions, which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for Schur functions.

Permutation statistics and weak Bruhat order in permutation tableaux of type [formula omitted

Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type B: Alignments, crossings, and inversions of signed permutations are realized in the corresponding permutation tableaux of type B, and the cycles of signed permutations are understood in the corresponding bare tableaux of type B. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type B in terms of permutation tableaux of type B.