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We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, imply the existence of the (non-homogeneous) c d -index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete c d -index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the c d -index for acyclic digraphs having a balanced linear edge labeling. [ABSTRACT FROM AUTHOR]

The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a characterization of the elements for which the reflection length coincides with both depth and length. These are the boolean elements. [ABSTRACT FROM AUTHOR]

The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length.

We define the shortest path poset SP( u, v) of a Bruhat interval [ u, v], by considering the shortest u- v paths in the Bruhat graph of a Coxeter group W, where u, v∈ W. We consider the case of SP( u, v) having a unique rising chain under a reflection order and show that in this case SP( u, v) is a Gorenstein poset. This allows us to derive the nonnegativity of certain coefficients of the complete cd-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected reductive linear algebraic group over with an involution θ. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety are indexed by the twisted identities in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a “Bruhat graph” whose vertices form a subset of ι. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on ι is rank symmetric. In the special case , , we strengthen our criterion by showing that only the degree of a single vertex, the “bottom one”, needs to be examined. This generalises a result of Deodhar for type A Schubert varieties. [Copyright &y& Elsevier]

Abstract: Let W be a finite Coxeter group. For a given , the following assertion may or may not be satisfied: [(⁎)] The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w. We present a type independent combinatorial criterion which characterises the elements that satisfy (⁎). A couple of immediate consequences are derived: [(1)] The criterion only involves the order ideal of w as an abstract poset. In this sense, (⁎) is a poset-theoretic property. [(2)] For W of type A, another characterisation of (⁎), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjöstrand. We obtain a short and simple proof of that result. [(3)] If W is a Weyl group and the Schubert variety indexed by is rationally smooth, then w satisfies (⁎). [Copyright &y& Elsevier]

We connect different results about irreducible components of the Springer fibers of type A. Firstly, we show a relation between the Spaltenstein partition of the fibers and a total order $${\prec}$$ on the set of standard Young tableaux. Next, using a result of Steinberg, we connect a work of the first author to the Robinson–Schensted map. We also perform the Spaltenstein study of the relative position of the Springer fibers and $${\mathbb{P}^1}$$ -fibrations of the flag manifold. This leads us to consider the adjacency relation on the set of standard Young tableaux and to define oriented and labeled graphs with the standard Young tableaux as vertices. Using this adjacency relation, we describe some smooth irreducible components of the Springer fibers. Finally, we show that these graphs can be identified with some full subgraphs of the Bruhat graph. [ABSTRACT FROM AUTHOR]

We provide an explicit product formula for the number of spanning trees of the Bruhat graph of the symmetric group, that is, the graph where two permutations π and σ are connected with an edge if π σ − 1 is a transposition. We also give the number of spanning trees for the graph where the two permutations are connected if π σ − 1 is an r -cycle for r even. For r odd we obtain the similar result for the alternating group. [ABSTRACT FROM AUTHOR]