Weyl Chamber Research Articles

Asymptotics for the number of walks in a Weyl chamber of type B

ABSTRACTWe consider lattice walks in confined to the region with fixed (but arbitrary) starting and end points. These walks are assumed to be such that their number can be counted using a reflection principle argument. The main results are asymptotic formulas for the total number of walks of length n with either a fixed or a free end point for a general class of walks as n tends to infinity. As applications, we find the asymptotics for the number of k‐non‐crossing tangled diagrams as well as asymptotics for two k‐vicious walkers models subject to a wall restriction. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 261–305, 2014

BKM Lie superalgebra for the Z 5-orbifolded CHL string

We study the Z_5-orbifolding of the CHL string theory by explicitly constructing the modular form tilde{Phi}_2 generating the degeneracies of the 1/4-BPS states in the theory. Since the additive seed for the sum form is a weak Jacobi form in this case, a mismatch is found between the modular forms generated from the additive lift and the product form derived from threshold corrections. We also construct the BKM Lie superalgebra, tilde{G}_5, corresponding to the modular form tilde{Delta}_1 (Z) = tilde{Phi}_2 (Z)^{1/2} which happens to be a hyperbolic algebra. This is the first occurrence of a hyperbolic BKM Lie superalgebra. We also study the walls of marginal stability of this theory in detail, and extend the arithmetic structure found by Cheng and Dabholkar for the N=1,2,3 orbifoldings to the N=4,5 and 6 models, all of which have an infinite number of walls in the fundamental domain. We find that analogous to the Stern-Brocot tree, which generated the intercepts of the walls on the real line, the intercepts for the N >3 cases are generated by linear recurrence relations. Using the correspondence between the walls of marginal stability and the walls of the Weyl chamber of the corresponding BKM Lie superalgebra, we propose the Cartan matrices for the BKM Lie superalgebras corresponding to the N=5 and 6 models.

Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $\alpha<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using this information, construct a conditioned process which lives on a partial compactification of the Weyl chamber.

Fluctuations of central measures on partitions

We study the fluctuations of models of random partitions $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ stemming from the representation theory of the infinite symmetric group. Using the theory of polynomial functions on Young diagrams, we establish a central limit theorem for the values of the irreducible characters $χ ^λ$ of the symmetric groups, with $λ$ taken randomly according to the laws $\mathbb{P}_n,ω$ . This implies a central limit theorem for the rows and columns of the random partitions, and these ``geometric'' fluctuations of our models can be recovered by relating central measures on partitions, generalized riffle shuffles, and Brownian motions conditioned to stay in a Weyl chamber. Nous étudions les fluctuations de modèles de partitions aléatoires $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ issus de la théorie des représentations du groupe symétrique infini. En utilisant la théorie des fonctions polynomiales sur les diagrammes de Young, nous établissons un théorème central limite pour les valeurs des caractères irréductibles $χ ^λ$ des groupes symétriques, avec $λ$ pris aléatoirement suivant les lois $\mathbb{P}_n,ω$ . Ceci implique un théorème central limite pour les lignes et les colonnes des partitions aléatoires, et ces fluctuations ``géométriques'' de nos modèles peuvent être retrouvées en reliant les mesures centrales sur les partitions, les battages généralisés de cartes, et les mouvements browniens conditionnés à rester dans une chambre de Weyl.

Affine Deligne–Lusztig varieties associated to additive affine Weyl group elements

Affine Deligne–Lusztig varieties can be thought of as affine analogs of classical Deligne–Lusztig varieties, or Frobenius-twisted analogs of Schubert varieties. We provide a method for proving a non-emptiness statement for affine Deligne–Lusztig varieties inside the affine flag variety associated to affine Weyl group elements satisfying a certain length additivity hypothesis. In particular, we prove that non-emptiness holds whenever it is conjectured to do so for alcoves in the shrunken dominant Weyl chamber, providing a partial converse to the emptiness results of Görtz, Haines, Kottwitz, and Reuman. Our technique involves the work of Geck and Pfeiffer on cuspidal conjugacy classes, in addition to an analysis of the combinatorics of certain fully commutative elements in the finite Weyl group.

Green functions for killed random walks in the Weyl chamber of Sp(4)

Dans cet article, nous considérons une famille de marches aléatoires tuées au bord de la chambre de Weyl du dual de Sp(4), qui vérifie en outre la propriété suivante : pour tout n ≥ 3, il y a, dans cette famille, une marche ayant un groupe de réflexions d’ordre 2n. De plus, le cas n = 4 correspond à un processus bien connu apparaissant lors de l’étude des marches aléatoires quantiques sur le dual de Sp(4). Pour tous les processus de cette famille, nous trouvons l’asymptotique exacte des fonctions de Green selon toutes les trajectoires, ainsi que l’asymptotique des probabilités d’absorption sur le bord.

Compactification d’espaces de représentations de groupes de type fini

Let $\Gamma$ be a finitely generated group and $G$ be a noncompact semisimple connected real Lie group with finite center. We consider the space $\mathcal X$ of conjugacy classes of reductive representations of $\Gamma$ into $G$. We define the {\it translation vector} of an element $g$ in $G$, with values in a Weyl chamber, as a refinement of the translation length in the associated symmetric space. We construct a compactification of $\mathcal X$, induced by the marked translation vector spectrum, generalizing Thurston's compactification of the Teichm\"uller space. We show that the boundary points are projectivized marked translation vector spectra of actions of $\Gamma$ on affine buildings with no global fixed point. An analoguous result holds for any reductive group $G$ over a local field.

Random walks in Weyl chambers and crystals

We use Kashiwara crystal basis theory to associate a random walk 𝒲 to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non-uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitman transform defined for similar random walks with uniform distributions yields yet another Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as 𝒲 conditioned to never exit the cone of dominant weights. For the defining representation V of 𝔤𝔩n, we notably recover the main result of O’Connell [Trans. Amer. Math. Soc. 355 (2003) 3669–3697]. At the heart of our proof is a quotient version of a renewal theorem which holds for a general random walk on a lattice. This theorem also has applications in representation theory since it permits the behaviour of some outer multiplicities for large dominant weights to be described.

BKM Lie superalgebras from counting twisted CHL dyons

Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko[arXiv:0812.3962]. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to a periodic table of BKM Lie superalgebras with properties that are consistent with physical expectations.

Brownian Motion, Reflection Groups and Tanaka Formula

In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Tanaka's formula for linear Brownian motion. The paper is closed with a description of the boundary process through the local times of the distances from the initial process to the facets.

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Let $p,q$ be positive integers. The groups $U_p(\mathbb{C})$ and $U_p(\mathbb{C})\times U_q(\mathbb{C}) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\mathbb{C})\times \mathbb{R}$ canonically as groups of automorphisms, where $M_{p,q}(\mathbb{C})$ is

Local Rigidity of Partially Hyperbolic Actions. II: The Geometric Method and Restrictions of Weyl Chamber Flows on SL(n, R)/

We consider the restriction α0,G of the Weyl chamber flow on SL(n,R)/Γ (where Γ is a cocompact lattice) to a closed subgroup G isomorphic to Z × R, k + l ≥ 2 of the group D+ of positive diagonal matrices which contains a lattice in a twodimensional plane in general position. We prove that any C small smooth perturbation of the action α0,G is differentiably conjugate to a standard perturbation which arises from a perturbation of the embedding Z × R → D+. We introduce a new method in rigidity of actions of higher rank abelian groups based on the study of combinatorial structure of the the web of Lyapunov (unipotent) foliations. Insights from the classical algebraic K-theory play a crucial role in establishing stability properties of that web. The method has applications to other classes of partially hyperbolic algebraic actions and is complementary to the other new analytic method which we introduced in the first paper of this series.

Log-convexity properties of Schur functions and generalized hypergeometric functions of matrix argument

We establish a positivity property for the difference of products of certain Schur functions, sλ(x), where λ varies over a fundamental Weyl chamber in ℝn and x belongs to the positive orthant in ℝn. Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions of two Hermitian matrix arguments, and we show how that result may be extended to derive higher-order log-convexity properties.

Folded fans and string functions

Folded fans Fψ describe recursion properties of weights of integrable highest weight modules L u . Being considered simultaneously for the set of string functions $$ \sigma_s^u $$ belonging to a fundamental Weyl chamber and corresponding to the same congruence class, the system of recursion relations gives rise to an equation which connects the string functions and the power series depending on multiplicities of weights of a folded fan Fψ. We apply these equations to study properties of string functions $$ \sigma_s^u $$ associated with integrable modules for affine Lie algebras. New important relations for string functions are thus obtained. The set of folded fans provides a compact and effective tool to study them. Bibliography: 7 titles.

Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of a complex reductive group H. We assume that the action of a maximal compact subgroup U of H is Hamiltonian and that G is compatible with a Cartan decomposition of H. We have an associated gradient map μ p : Z → p where g = k ⊕ p is the Cartan decomposition of g . For a G-stable subset Y of Z we consider convexity properties of the intersection of μ p ( Y ) with a closed Weyl chamber in a maximal abelian subspace a of p . Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z = P ( V ) where V is a unitary representation of U.

BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N

We show that the generating function of electrically charged 1/2-BPS states in N=4 supersymmetric Z_N-CHL orbifolds of the heterotic string on T^6 are given by multiplicative eta-products. The eta-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/0609109] for Z_N orbifolds when N is non-prime. We study the Z_4 CHL orbifold in detail and show that the associated Siegel modular forms, \Phi_3(Z) and \widetilde{\Phi}_3(Z), are given by the square of the product of three even genus-two theta constants. Extending work by us[arXiv:0807.4451] as well as Cheng and Dabholkar[arXiv:0809.4258], we show that their `square roots' appear as the denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie superalgebras. The BKM Lie superalgebra associated with the generating function of 1/4-BPS states, i.e., \widetilde{\Phi}_3(Z) has a parabolic root system with a light-like Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the 1/4-BPS states.

Determinant formulas relating to tableaux of bounded height

Chen et al. recently established bijections for ( d + 1 ) -noncrossing/nonnesting matchings, oscillating tableaux of bounded height d, and oscillating lattice walks in the d-dimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length n and of any shape. We find a determinant formula for the exponential generating function. The same idea applies to prove Gessel's remarkable determinant formula for permutations with bounded length of increasing subsequences. We also give short algebraic derivations for some results of the reflection principle.

Generic two-qubit photonic gates implemented by number-resolving photodetection

We combine numerical optimization techniques [Uskov et al., Phys. Rev. A 79, 042326 (2009)] with symmetries of the Weyl chamber to obtain optimal implementations of generic linear-optical KLM-type two-qubit entangling gates. We find that while any two-qubit controlled-U gate, including CNOT and CS, can be implemented using only two ancilla resources with success probability S > 0.05, a generic SU(4) operation requires three unentangled ancilla photons, with success S > 0.0063. Specifically, we obtain a maximal success probability close to 0.0072 for the B gate. We show that single-shot implementation of a generic SU(4) gate offers more than an order of magnitude increase in the success probability and two-fold reduction in overhead ancilla resources compared to standard triple-CNOT and double-B gate decompositions.

Asymptotics for Walks in a Weyl chamber of Type $B$ (extended abstract)

We consider lattice walks in $\mathbb{R}^k$ confined to the region $0 &lt; x_1 &lt; x_2 \ldots &lt; x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asymptotic formula for the total number of walks of length $n$ with fixed but arbitrary starting and end point for a general class of walks as the number $n$ of steps tends to infinity. As applications, we find the asymptotics for the number of $k$-non-crossing tangled diagrams on the set $\{1,2, \ldots,n\}$ as $n$ tends to infinity, and asymptotics for the number of $k$-vicious walkers subject to a wall restriction in the random turns model as well as in the lock step model. Asymptotics for all of these objects were either known only for certain special cases, or have only been partially determined.

Random walks conditioned to stay in Weyl chambers of type C and D

We construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob $h$-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an $h$-transform in the Weyl chamber of type C.