Bruhat order
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On the Weak Order of Coxeter Groups Open
This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…
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The equational theory of the weak Bruhat order on finite symmetric groups Open
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 latt…
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Poset edge densities, nearly reduced words, and barely set-valued tableaux Open
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. …
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Weak generalized lifting property, Bruhat intervals, and Coxeter matroids Open
We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also de…
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Maximal Newton Points and the Quantum Bruhat Graph Open
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…
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Noncrossing partitions, Bruhat order and the cluster complex Open
We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular, we study the restriction of these orders to noncrossing partitions and sh…
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Combinatorial invariance of Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions Open
When acts on the flag variety of , the orbits are in bijection with fixed point free involutions in the symmetric group . In this case, the associated Kazhdan-Lusztig-Vogan polynomials can be indexed by pairs of fixed point free involution…
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The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders Open
We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff-Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserv…
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Affine Bruhat order and Demazure products Open
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 Iwah…
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Symplectic keys and Demazure atoms in type C Open
We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More p…
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Intersecting principal Bruhat ideals and grades of simple modules Open
prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's a-function. Our arguments are combinatorial, and include a …
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Lattice bijections for string modules, snake graphs and the weak Bruhat order Open
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…
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Doubly stochastic matrices and the Bruhat order Open
The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order n which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, …
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Extremal matrices for the Bruhat-graph order Open
UID/MAT/00297/2019 UID/MAT/04721/2019
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The classification of blocks in BGG category O Open
We classify all equivalences between the indecomposable abelian categories which appear as blocks in BGG category O for reductive Lie algebras. Our classification implies that a block in category O only depends on the Bruhat order of the r…
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Symplectic Keys and Demazure Atoms in Type C Open
We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More p…
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Nilpotent orbits of height 2 and involutions in the affine Weyl group Open
Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B⊂G be a Borel subgroup. Then B acts with finitely many orbits on the variety N2⊂g of the nilpotent elements whos…
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Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Classical Groups and Monomial Realizations of Demazure Crystals Open
Let |$G$| be a simply connected simple algebraic group over |$\mathbb{C}$|, |$B$| and |$B_-$| its two opposite Borel subgroups, and |$W$| the associated Weyl group. It is shown that the coordinate ring |${\mathbb C}[G^{u,v}]$| (|$u$|, |…
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Phase portraits of the generalized full symmetric Toda systems on rank 2 groups Open
In this paper we continue investigations that we began in our previous works, where we proved, that the phase diagram of Toda system on special linear groups can be identified with the Bruhat order on symmetric group, when all the eigenval…
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SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements Open
In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset…
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Poset associahedra. Open
For each poset $P$, we construct a polytope $A(P)$ called the $P$-associahedron. Similarly to the case of graph associahedra, the faces of $A(P)$ correspond to certain tubings of $P$. The Stasheff associahedron is a compactification of the…
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The Bruhat order on classes of isotopic Latin squares Open
In a previous paper, the authors introduced and studied the Bruhat order in the class of Latin squares of order n. In this paper, we investigate the restriction of the Bruhat order in a class of isotopic Latin squares. We present equivalen…
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On the double-affine Bruhat order: the conjecture and classification of covers in ADE type Open
For any Kac–Moody group , we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for is strictly compatible with a -valued length function. We conjecture in general and prove for of affine ADE type t…
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The weak Bruhat order for random walks on Coxeter groups Open
We show that for the simple random walk on a Coxeter group generated by the Coxeter generators and identity, the likelihoods of being at any pair of states respect the weak Bruhat order. That is, after any number of steps, the most likely …
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The $cd$-Index: A Survey Open
This is a survey of the $cd$-index of Eulerian partially ordered sets. The $cd$-index is an encoding of the numbers of chains, specified by ranks, in the poset. It is the most efficient such encoding, incorporating all the affine relations…
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On the little secondary Bruhat order Open
Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More …
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A Kazhdan-Lusztig Atlas on G/P Open
A stratified variety has a Kazhdan-Lusztig atlas if it can be locally modelled with Kazhdan-Lusztig varieties stratified by Schubert varieties in some Kac-Moody flag manifold via stratified isomorphisms. In this paper, we show that the par…
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The facial weak order and its lattice quotients Open
We investigate a poset structure that extends the weak order on a finite Coxeter group $W$ to the set of all faces of the permutahedron of $W$. We call this order the facial weak order. We first provide two alternative characterizations of…
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Classification of Cocovers in the Double Affine Bruhat Order Open
We classify cocovers of a given element of the double affine Weyl semigroup $W_{\mathcal{T}}$ with respect to the Bruhat order, specifically when $W_{\mathcal{T}}$ is associated to a finite root system that is irreducible and simply laced.…
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Double Affine Bruhat Order Open
We classify cocovers and covers of a given element of the double affine Weyl semigroup W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approach…