scholarly journals Young Diagrams and Intersection Numbers for Toric Manifolds associated with Weyl Chambers

10.37236/4307 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Hiraku Abe
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Ring Structure ◽  
Classical Type ◽  
Toric Manifold ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Intersection Numbers ◽  
Toric Manifolds

We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold.

scholarly journals Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Cristian Lenart
Keyword(s):  
Weyl Group ◽  
Type A ◽  
Macdonald Polynomials ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Affine Weyl Group ◽  
Littlewood Polynomials ◽  
Combinatorial Formulas ◽  
International Audience ◽  
Type C

International audience A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

scholarly journals Combinatorics of Generalized Exponents

2018 ◽  
Vol 2020 (16) ◽  
pp. 4942-4992 ◽  
Author(s):  
Cédric Lecouvey ◽  
Cristian Lenart
Keyword(s):  
Root System ◽  
Lie Algebras ◽  
Finite Type ◽  
Irreducible Representations ◽  
Combinatorial Proof ◽  
Combinatorial Formula ◽  
Branching Rules ◽  
Crystal Graphs ◽  
Type C ◽  
Generalized Exponents

Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $A_{n-1}$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $C_{n}$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $A_{2n-1}$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $t$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $C$ branching rules. Our methods are expected to extend to the orthogonal types.

Admissible Sequences for Twisted Involutions in Weyl Groups

2011 ◽  
Vol 54 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Ruth Haas ◽  
Aloysius G. Helminck
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Weyl Groups ◽  
One To One ◽  
Admissible Sequences

AbstractLetW be a Weyl group, Σ a set of simple reflections inW related to a basis Δ for the root system Φ associated with W and θ an involution such that θ(Δ) = Δ. We show that the set of θ- twisted involutions in W, = {w ∈ W | θ(w) = w–1} is in one to one correspondence with the set of regular involutions . The elements of are characterized by sequences in Σ which induce an ordering called the Richardson–Springer Poset. In particular, for Φ irreducible, the ascending Richardson–Springer Poset of , for nontrivial θ is identical to the descending Richardson–Springer Poset of .

scholarly journals The canonical spectrum of projective toric manifolds

2018 ◽  
Vol 29 (07) ◽  
pp. 1850048
Author(s):  
Mounir hajli
Keyword(s):  
Spectral Theory ◽  
Spectral Properties ◽  
Kähler Manifolds ◽  
Combinatorial Structure ◽  
Kahler Manifolds ◽  
Toric Manifold ◽  
Closed Formula ◽  
Toric Manifolds ◽  
Complex Projective

Let [Formula: see text] be a complex projective toric manifold. We associated to [Formula: see text], a positive and closed [Formula: see text]-current called the canonical toric Kähler current of [Formula: see text] denoted by [Formula: see text], and a new invariant called the canonical spectrum of [Formula: see text]. This spectrum is obtained as the set of the eigenvalues of a singular Laplacian defined by [Formula: see text] and which is described uniquely by the combinatorial structure of [Formula: see text]. The construction of this Laplacian and the study of its spectral properties are the consequence of a generalized spectral theory of Laplacians on compact Kähler manifolds that we develop in this paper.

Sign in / Sign up

Export Citation Format

Share Document