Abstract Schubert polynomials are distinguished representatives of Schubert cycles in the cohomology of the flag variety. In the spirit of Bergeron and Sottile, we use the Bruhat order to give $(n-1)!$ different combinatorial formulas for the Schubert polynomial of a permutation in $S_{n}$. By the work of Lenart and Sottile, one extreme of the formulas recover the classical pipedream (PD) formula. We prove the other extreme corresponds to bumpless pipedreams (BPDs). We give two applications of this perspective to view BPDs: using the Fomin–Kirillov algebra, we solve the problem of finding a BPD analogue of Fomin and Stanley’s algebraic construction on PDs; we also establish a bijection between PDs and BPDs using Lenart’s growth diagram, which conjecturally agrees with the existing bijection of Gao and Huang.

Orbits on a product of two flags and a line and the Bruhat order, I

Abstract Let $$G=G_{n}=GL(n)$$ be the $$n\times n$$ complex general linear group and embed $$G_{n-1}=GL(n-1)$$ in the top left hand corner of $$G$$ . The standard Borel subgroup of upper triangular matrices $$B_{n-1}$$ of $$G_{n-1}$$ acts on the flag variety $$\mathcal {B}_{n}$$ of $$G$$ with finitely many orbits. In this paper, we show that each $$B_{n-1}$$ -orbit is the intersection of orbits of two Borel subgroups of $$G$$ acting on $$\mathcal {B}_{n}$$ . This allows us to give a new combinatorial description of the $$B_{n-1}$$ -orbits on $$\mathcal {B}_{n}$$ by associating to each orbit a pair of Weyl group elements. The closure relations for the $$B_{n-1}$$ -orbits can then be understood in terms of the Bruhat order on the symmetric group, and the Richardson-Springer monoid action on the orbits can be understood in terms of a well-understood monoid action on the symmetric group. This approach makes the closure relation more transparent than in Magyar (J. Algebraic Combin 21:71–101, 2005) and the monoid action significantly more computable than in our papers (Colarusso and Evens, J. Algebra 596:128–154, 2022) and (Colarusso and Evens, J. Algebra 619:249–297, 2023), and also allows us to obtain new information about the orbits including a simple formula for the dimension of an orbit.

In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung–Li and of Postnikov to reduce quantum products by hook Schur polynomials to the (known) classical product.

In this paper, we show that the Bruhat order on any sect of a symmetric variety of type $AIII$ is lexicographically shellable. Our proof proceeds from a description of these posets as rook placements in a partition shape which fits in a $p \times q$ rectangle. This allows us to extend an EL-labeling of the rook monoid given by Can to an arbitrary sect. As a special case, our result implies that the Bruhat order on matrix Schubert varieties is lexicographically shellable.

Counting Boolean intervals in the weak Bruhat order of a finite Coxeter group

Rectangulotopes

Abstract The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup‐ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders, we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. Defining this correspondence involves interpreting the coproducts geometrically in terms of zonotopal tilings, which allows us to give conceptual proofs of their properties and show that all reasonable coproducts arise from our construction.

We study the restriction of the strong Bruhat order on an arbitrary Coxeter group W to cosets xW L θ , where x is an element of W and W L θ the subgroup of fixed points of an automorphism θ of order at most two of a standard parabolic subgroup W L of W. When θ≠id, there is in general more than one element of minimal length in a given coset, and we explain how to relate elements of minimal length. We also show that elements of minimal length in cosets are exactly those elements which are minimal for the restriction of the Bruhat order.

Abstract Let be the ring of polynomials in variables and consider the ideal generated by quasisymmetric polynomials without constant term. It was shown by J. C. Aval, F. Bergeron, and N. Bergeron that the th Catalan number. In the present work, we explain this phenomenon by defining a set of permutations with the following properties: first, is a basis of the Temperley–Lieb algebra , and second, when considering as a collection of points in , the top‐degree homogeneous component of the vanishing ideal is . Our construction has a few byproducts that are independently noteworthy. We define an equivalence relation on the symmetric group using weak excedances and show that its equivalence classes are naturally indexed by noncrossing partitions. Each equivalence class is an interval in the Bruhat order between an element of and a 321‐avoiding permutation. Furthermore, the Bruhat order induces a well‐defined order on . Finally, we show that any section of the quotient gives an (often novel) basis for .

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext space between a simple module and a Verma module. We give a conjectural complete description of such socles which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan–Lusztig polynomials.

A 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.

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the $(N-1)$-simplex equation can be regarded as a realization of the higher Bruhat order $B(N,N-2)$, the $N$-gon equation is a realization of the higher Tamari order $T(N,N-2)$. The latter and its dual $\tilde T(N,N-2)$, associated with which is the dual $N$-gon equation, have been shown to arise as suborders of $B(N,N-2)$ via a ''three-color decomposition''. There are two different reductions of $T(N,N-2)$ and $\tilde T(N,N-2)$, to ${T(N-1,N-3)}$, respectively $\tilde T(N-1,N-3)$. In this work, we explore the corresponding reductions of (dual) polygon equations, which lead to relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.

Higher Bruhat orders of types B and C

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G . By fixing w from the Weyl group, we can define MV polytopes whose highest vertex is labelled by w . We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w^{-1} . To do this, we define a collection of generalized minor functions \Delta_{\gamma}^{\mathrm{new}} which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex w . We also describe the combinatorial structure of MV polytopes of highest vertex w . We explicitly describe the map from the Weyl group to the subset of elements bounded by w in the Bruhat order which sends u \mapsto v if the vertex labelled by u coincides with the vertex labelled by v for every MV polytope of highest vertex w . As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than w in the Bruhat order.

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.

Abstract We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

Abstract Chow rings of flag varieties have bases of Schubert cycles $\sigma _u $ , indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood–Richardson rules solve this problem for special products $\sigma _u \cdot \sigma _v$ , where u and v are p-Grassmannian permutations. Building on work of Wyser, we introduce backstable clans to prove such a rule for the problem of computing the product $\sigma _u \cdot \sigma _v$ when u is p-inverse Grassmannian and v is q-inverse Grassmannian. By establishing several new families of linear relations among structure constants, we further extend this result to obtain a positive combinatorial rule for $\sigma _u \cdot \sigma _v$ in the case that u is covered in weak Bruhat order by a p-inverse Grassmannian permutation and v is a q-inverse Grassmannian permutation.

Let A(n,k) represent the collection of all n×n zero-and-one matrices, with the sum of all rows and columns equalling k. This set can be ordered by an extension of the classical Bruhat order for permutations, seen as permutation matrices. The Bruhat order on A(n,k) differs from the Bruhat order on permutations matrices not being, in general, graded, which results in some intriguing issues. In this paper, we focus on the maximum length of antichains in A(n,k) with the Bruhat order. The crucial fact that allows us to obtain our main results is that two distinct matrices in A(n,k) with an identical number of inversions cannot be compared using the Bruhat order. We construct sets of matrices in A(n,k) so that each set consists of matrices with the same number of inversions. These sets are hence antichains in A(n,k). We use these sets to deduce lower bounds for the maximum length of antichains in these partially ordered sets.