Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the Riemann hypothesis if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system A _{n – 1}. The proof is based on an explicit formula [1, 2, 11] for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.     

Poincaré Series of the Weyl Groups of the Elliptic Root Systems A 1 (1,1), A 1 (1,1)* and A 2 (1,1)

We calculate the Poincaré series of the elliptic Weyl group W(A ₂ ^(1,1)), which is the Weyl group of the elliptic root system of type A ₂ ^(1,1). The generators and relations of W(A ₂ ^(1,1)) have been already given by K. Saito and the author.     

Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

We give several necessary and sufficient conditions for the existence of the presentation by conjugation for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.     

Poincaré series and an application to Weyl algebras

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

On a Jacobian Identity Associated with Real Hyperplane Arrangements

For the mapping is onto R. It was shown by G. Boole in the 1850's that We give an n-dimensional analogue of this result. The proof makes use of the Griffiths–Harris residue theorem from algebraic geometry.     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

A Root System Criterion for Fully Commutative and Short Braid-Avoiding Elements in Affine Weyl Groups

We provide simple characterizations of short-braid avoiding and fully commutative elements in an affine Weyl group W, generalizing results of Fan and Stembridge for finite Weyl groups. Our results rely on the combinatorics of the compatible subsets of the root system of W.     

Upper Bounds in Affine Weyl Groups under the Weak Order

Björner and Wachs proved that under the weak order every quotient of a Coxeter group is a meet semi-lattice, and in the finite case is a lattice. In this paper, we examine the case of an affine Weyl group W with corresponding finite Weyl group W ₀. In particular, we show that the quotient of W by W ₀ is a lattice and that up to isomorphism this is the only quotient of W which is a lattice. We also determine that the question of which pairs of elements of W have upper bounds can be reduced to the analogous question within a particular finite subposet.     

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from , where is the type B Weyl arrangement, then (A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of (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 while the second deals with all finite Weyl groups but only their hyperplane arrangements.     

Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id,π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id,π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind, and for the group An putting q=1 yields the poly-Bernoulli numbers defined by Kaneko.     

On the Automorphism Group of the First Weyl Algebra

Let A ₁: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut_𝕜(A ₁) be the automorphism group of the ring A ₁. One can associate to each right ideal I of A ₁ a subgroup of Aut_𝕜(A ₁) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A ₁ contains a given isomorphism subgroup.     

Ixr(a)的有限型Ixor(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型IC or (a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数IXr(a)的Weyl群W同构于有限型IXr(a)的未定Weyl群.     

(A)3型仿射Weyl群最低双边胞腔上的Kazhdan-Lusztig系数

对于Kazhdan-Lusztig多项式Py,w(q),μ(y,w)为它的首项系数(简称KL系数).首项系数在李代数及其表示理论中起着重要的作用.在文章中,W为?3型仿射Weyl群,通过它对应的Hecke代数的性质及其KL基{Cw}的乘积计算,以及不可约模在张量积中的重数公式,给出了?3型仿射Weyl群最低双边胞腔上的...     

Isomorphism problems and groups of automorphisms for generalized Weyl algebras

We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of by finding sets of generators for the group of automorphisms.

     

Signed graphs and the freeness of the Weyl subarrangements of type Bℓ

A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell }$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of their graph. In addition, the Weyl subarrangements of type $B_{\ell }$ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type $A_{\ell -1}$ and type $B_{\ell }$. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type $B_{\ell }$ under certain assumption.     

On the Affine Completeness of Lattice Ordered Groups

In the paper it is proved that a nontrivial direct product of lattice ordered groups is never affine complete.     

On holonomy of Weyl connections in Lorentzian signature

Holonomy algebras of Weyl connections in Lorentzian signature are classified. In particular, examples of Weyl connections with all possible holonomy algebras are constructed.     

D_l和E_6型Weyl群的扭子群的扩群

本文决定了 Dl 和 E6 型 Weyl群扭子群的所有扩群 ,这为确定相应 Chevalley群扭子群的所有扩群奠定了基础 .