Set Of Positive Roots Research Articles

Hardy and BMO spaces on Weyl chambers

Abstract Let W be a finite reflection group associated with a root system R in ℝ d {\mathbb{R}^{d}} . Let C + {C_{+}} denote a positive Weyl chamber distinguished by a choice of R + {R_{+}} , a set of positive roots. We define and investigate Hardy and BMO spaces on C + {C_{+}} in the framework of boundary conditions given by a homomorphism η ∈ Hom ⁡ ( W , ℤ ^ 2 ) {\eta\in\operatorname{Hom}(W,\widehat{\mathbb{Z}}_{2})} which attaches the ± {\pm} signs to the facets of C + {C_{+}} . Specialized to orthogonal root systems, atomic decompositions in H η 1 {H^{1}_{\eta}} and h η 1 {h^{1}_{\eta}} are obtained and the duality problem is also treated.

Explicit expressions for Šapovalov elements in Type A

We give explicit expressions for Šapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in [11] Section 9, using non-commutative determinants, and in fact our first main results, Theorems 2.3 and 2.6 can be viewed as complete expansions of these determinants. But we give new proofs, which seem easier because they avoid induction and cofactor expansion. We also describe Šapovalov elements for gl(m,n) with respect to an arbitrary Borel subalgebra in Theorem 3.7, and interpret Šapovalov elements in Type A as determinants of Hessenberg matrices in Theorems 4.5 and 4.9. The exact form of the explicit expressions depends on an ordering on the set of positive roots, and Hessenberg matrices are useful in changing the ordering. Having explicit expressions for Šapovalov elements allows us to give easy proofs of several results in representation theory, see Section 6 and Subsection 4.6.

The Laplacian with mixed Dirichlet-Neumann boundary conditions on Weyl chambers

Let W be a finite reflection group associated with a root system R in Rd. Let C+ denote a positive Weyl chamber distinguished by a choice of R+, a set of positive roots. We investigate realizations in L2(C+) of the Laplacian on C+, subject to mixed Dirichlet-Neumann boundary conditions imposed on the facets of C+. These conditions are determined by a homomorphism η∈Hom(W,Zˆ2), where Zˆ2={1,−1} with multiplication. The essential part of the paper contains thorough analysis of the corresponding η-heat kernels together with proof of their positivity on C+.

Picture groups and maximal green sequences

<p style='text-indent:20px;'>We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).</p><p style='text-indent:20px;'>Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.</p>

Glorious pairs of roots and Abelian ideals of a Borel subalgebra

Let $${{\mathfrak {g}}}$$ be a simple Lie algebra with a Borel subalgebra $${{\mathfrak {b}}}$$ . Let $$\Delta ^+$$ be the corresponding (po)set of positive roots and $$\theta $$ the highest root. A pair $$\{\eta ,\eta '\}\subset \Delta ^+$$ is said to be glorious, if $$\eta ,\eta '$$ are incomparable and $$\eta +\eta '=\theta $$ . Using the theory of abelian ideals of $${{\mathfrak {b}}}$$ , we (1) establish a relationship of $$\eta ,\eta '$$ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types $${{\mathbf {\mathsf{{{DE}}}}}}_{}$$ , we prove that if $$\{\eta ,\eta '\}$$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $$\eta \wedge \eta '$$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type $${{\mathbf {\mathsf{{{A}}}}}}_{}$$ . As an application, we describe the minimal non-abelian ideals of $${{\mathfrak {b}}}$$ .

Abelian Ideals of a Borel Subalgebra and Root Systems, II

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$ and $\mathfrak{Ab}$ the set of abelian ideals of $\mathfrak b$. Let $\Delta^+$ be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of $\mathfrak{Ab}$ parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if $\mathfrak g\ne\mathfrak{sl}_n$. We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of $\Delta^+$ that are modular lattices.

Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

Let (W,S) be a finite Coxeter system with root system R and with set of positive roots R+. For α∈R, v,w∈W, we denote by ∂α, ∂w and ∂w/v the divided difference operators and skew divided difference operators acting on the coinvariant algebra of W. Generalizing the work of Liu [15], we prove that ∂w/v can be written as a polynomial with nonnegative coefficients in ∂α where α∈R+. In fact, we prove the stronger and analogous statement in the Nichols–Woronowicz algebra model for Schubert calculus on W after Bazlov [4]. We draw consequences of this theorem on saturated chains in the Bruhat order, and partially treat the question when ∂w/v can be written as a monomial in ∂α where α∈R+. In an appendix, we study related combinatorics on shuffle elements and Bruhat intervals of length two.

Arrangements of ideal type

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type AI stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that AI is free if the root system is of classical type or G2 and conjectured that this is also the case for all types. This was established only very recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement AI is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro–Steinberg–Kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated Weyl group.Our first aim in this paper is to investigate a stronger freeness property of the AI. We show that all AI are inductively free, with the possible exception of some cases in type E8.In the same paper from 2006, Sommers and Tymoczko define a Poincaré polynomial I(t) associated with each ideal I which generalizes the Poincaré polynomial W(t) for the underlying Weyl group W. Solomon showed that W(t) satisfies a product decomposition depending on the exponents of W for any Coxeter group W. Sommers and Tymoczko showed in a case by case analysis in types An, Bn and Cn, and some small rank exceptional types that a similar factorization property holds for the Poincaré polynomials I(t) generalizing the formula of Solomon for W(t). They conjectured that their multiplicative formula for I(t) holds in all types. In our second aim to investigate this conjecture further, the same inductive tools we develop to obtain inductive freeness of the AI are also employed. Here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions.

The freeness of ideal subarrangements of Weyl arrangements

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers–Tymoczko. In particular, when an ideal subarrangement is equal to the entireWeyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula

The freeness of ideal subarrangements of Weyl arrangements

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula. Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule.

Generalised Kostka–Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles

Let G be a simple algebraic group and B a Borel subgroup. We consider generalisations of Lusztig’s q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a B-submodule N of an arbitrary finite-dimensional G-module V. The corresponding polynomials in q are called generalised Kostka–Foulkes polynomials (gKF). We prove vanishing theorems for the cohomology of line bundles on G × B N and derive from this a sufficient condition for the non-negativity of the coefficients of gKF. We also consider in detail the case in which V is the simple G-module whose highest weight is the short dominant root and N is the B-submodule whose weights are all short positive roots.

On orbits of antichains of positive roots

For any finite poset P , there is a natural operator, X = X P , acting on the set of antichains of P . We discuss conjectural properties of X for some graded posets associated with irreducible root systems. In particular, if Δ + is the set of positive roots and Π is the set of simple roots in Δ + , then we consider the cases P = Δ + and P = Δ + ∖ Π . For the root system of type A n , we consider an X -invariant integer-valued function on the set of antichains of Δ + and establish some properties of it.

Auslander algebras and initial seeds for cluster algebras

Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q, a rigid Λ-module IQ is produced with r = |Π| pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of k Q to Λ. If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type |Q|, then the coordinate ring ℂ[N] is an upper cluster algebra. It is shown that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of IQ coincide with certain generalized minors which form an initial cluster for ℂ[N] and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of EndΛ(IQ). Finally, the fact that the categories of injective modules over Λ and over its covering Λ̃ are triangulated is exploited in order to show several interesting identities in the respective stable module categories.

REFLEXIVE NUMBERS AND BERGER GRAPHS FROM CALABI–YAU SPACES

We review the Batyrev approach to Calabi–Yau spaces based on reflexive weight vectors. The Universal CY algebra gives a possibility to construct the corresponding reflexive numbers in a recursive way. A physical interpretation of the Batyrev expression for the Calabi–Yau manifolds is presented. Important classes of these manifolds are related to the simple-laced and quasi-simple-laced numbers. We discuss the classification and recurrence relations for them in the framework of quantum field theory methods. A relation between the reflexive numbers and the so-called Berger graphs is studied. In this correspondence the role played by the generalized Coxeter labels is highlighted. Sets of positive roots are investigated in order to connect them to possible new algebraic structures stemming from the Berger matrices.

Weighted Cohomology of Arithmetic Groups

Let G be a semisimple algebraic group defined over the rational numbers, K a maximal compact subgroup ofG = G(R), and Γ ⊂ G(Q) a neat arithmetic subgroup. LetX = Γ\G/K be the locally symmetric space associated to Γ, and E the local system on X constructed out of a finite-dimensional, irreducible, algebraic representation E of G. Fix a maximally Q-split torus S in G; S is assumed to be nontrivial, so that X is noncompact. Let A = S(R)0 and, by a slight abuse of notation, let X∗(A) denote the group of rational characters of S. Choose a minimal rational parabolic subgroup P0 ⊃ S; the choice determines a notion of positivity in X∗(A) and a set of positive roots among the roots of S in G. Let ρ0 be the half-sum of the positive roots. Weighted cohomology is an invariant of Γ introduced by M. Goresky, G. Harder, and R. MacPherson [12] in the study ([15], [16], [12], [14]) of the trace of Hecke operators in the cohomology of Γ. For each p ∈ X∗(A)⊗Q there are weight profiles p and p (upper and lower p; see 1.5) and groups W pH i(Γ, E) and W H i(Γ, E). (The notion of profile here differs from (but is equivalent to) that of [12]; see Remark 1.2 below.) If p+q = −2ρ0 then the profiles p and q are dual and the corresponding groups are in Poincare duality. For very positive p one gets the compactly supported cohomology H i c(X,E) and dually, for very negative p, the full cohomology H i(X,E). The definition of the weighted cohomology groups is somewhat involved, so I will attempt to give some idea of the motivation behind it; detailed definitions are recalled in Section 1. Associated to p and p are complexes of sheaves WpC•(E) and WpC•(E) on the reductive Borel-Serre compactification X of X. Let P be a rational parabolic subgroup, NP its unipotent radical, SP the split centre of P/NP and XP the P-boundary component in X. The idea behind the construction in [12] is as follows: The stalk cohomology of the direct image Ri∗E (by i : X ↪→ X) at x ∈ XP is

A New Proof of a Theorem of Solomon

The main purpose of this note is to give another proof and a different view of a theorem of Solomon [3] on Coxeter groups (finite groups of symmetries of R generated by reflections). A secondary purpose is to give more publicity to the fascinating special case of symmetric groups. This case was handled by Etienne whose recent paper [2] kindled my interest in this subject; I owe to Michelle Wachs my awareness of Solomon's earlier and more general theorem. The new proof is shorter and apparently more elementary than the proof of [3] (and the proof in the appendix to [3] due to Tits); nor does it reduce to Etienne's proof in the special case of the symmetric group. We begin by recalling some of the terminology of Coxeter groups. By definition they are finite groups of symmetries of n-dimensional real Euclidean space generated by those elements which are reflections in hyperplanes. Each reflecting hyperplane has two unit vectors r and — r orthogonal to it called roots and the set of roots has a reasonably canonical partition into positive roots and negative roots. The set of positive roots has a distinguished subset called the set of fundamental roots and the associated set of fundamental reflections (the Coxeter generators) also generates the group; with respect to this generating set the group has a very simple set of defining relations. Suppose that {T15 T2, ..., rm} is a set of Coxeter generators for a Coxeter group G associated with the set of fundamental roots {rx, r2,.. . , rm}. For any geG let X(g) denote the length of a minimal word in the generators which represents g. It is known [1, 6.1.1] that X(Zig) = X(g)± 1, with the positive sign if rtg is a positive root and the negative sign if rtg is a negative root. If S is any subset of {1, 2, ..., m) the signature class associated with S is the set