Separable elements in Weyl groups

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

Minuscule Elements of Weyl Groups, the Numbers Game, andd-Complete Posets

We develop a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups.

Inversion tables and minimal left coset representatives for Weyl groups of classical type

We characterize, in the case of sl∧(2),the crystal base of the Demazure module Ew(Λ) in terms of extended Young diagrams or paths for any dominant integral weight Λ and Weyl group element w. Its character is evaluated via two expressions, “bosonic” and “fermionic.”

The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ is known to be gamma positive for all $n$. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all $n$.
 We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively. We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$. 
 We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.

We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of noncommutative crepant resolutions.

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.

Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

Looping of the numbers game and the alcoved hypercube

A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D. We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.

We define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements of a particular Schubert cell, indexed by an element of the Weyl group. They are generally not fully automorphic. We will develop some results and methods for ${\rm GL}_3$ that may be suggestive about the general case. The six Schubert Eisenstein series are shown to have meromorphic continuation and some functional equations. The Schubert Eisenstein series $E_{s_1s_2}$ and $E_{s_2s_1}$ corresponding to the Weyl group elements of order three are particularly interesting: at the point where the full Eisenstein series is maximally polar, they unexpectedly become (with minor correction terms added) fully automorphic and related to each other.

We give a representation-theoretic (or rather, crystal-theoretic) proof of Cherednik-Orr's recursion formula of Demazure type for the specialization at t = ∞ of the nonsymmetric Macdonald polynomials Ewλ(q, t), w ∈ W, where λ is a dominant integral weight and W is a finite Weyl group.

Let A be a subspace arrangement and let χ(A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from \({\mathcal{L}}({\mathcal{B}}_n)\), where \({\mathcal{B}}_n\) is the type B Weyl arrangement, then χ(A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of χ(A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in \({\mathcal{B}}_n\) while the second deals with all finite Weyl groups but only their hyperplane arrangements.