Transition Matrices Between Young's Natural and Seminormal Representations

We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.

Inverse K-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

Generic Newton points and the Newton poset in Iwahori-double cosets

We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.

Flip Posets of Bruhat Intervals

In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph $B(u,v)$ of a Bruhat interval $[u,v]$ into rank posets $P_{i}$ in a way that each $P_{i}$ has a unique maximal chain that is rising under a reflection order. In the case where each $P_{i}$ has rank three, the construction yields a combinatorial description of some terms of the complete $\textbf{cd}$-index as a sum of ordinary $\textbf{cd}$-indices of Eulerian posets obtained from each of the $P_{i}$.

Hultman Elements for the Hyperoctahedral Groups

Hultman, Linusson, Shareshian, and Sjöstrand gave a pattern avoidance characterization of the permutations for which the number of chambers of its associated inversion arrangement is the same as the size of its lower interval in Bruhat order. Hultman later gave a characterization, valid for an arbitrary finite reflection group, in terms of distances in the Bruhat graph. On the other hand, the pattern avoidance criterion for permutations had earlier appeared in independent work of Sjöstrand and of Gasharov and Reiner. We give characterizations of the elements of the hyperoctahedral groups satisfying Hultman's criterion that is in the spirit of those of Sjöstrand and of Gasharov and Reiner. We also give a pattern avoidance criterion using the notion of pattern avoidance defined by Billey and Postnikov.