Classical Weyl Research Articles

The prediction of façade effects from a point source above an impedance ground

In many environmental noise prediction schemes, an empirical correction factor of 2.5 or 3.0 dB is normally added for the calculation of noise levels 1 meter in front of a reflecting façade. In this paper, theoretical and experimental studies have been conducted to examine the validity and accuracy of such approximations for a point source. The theoretical analysis involves the extension of the classical Weyl–van der Pol formula to include the effect of a reflecting façade. Experimental measurements have been conducted in an anechoic chamber to validate the theoretical predictions. It has been demonstrated that the simple empirical correction factor is adequate in cases for the assessment of the façade noise levels near the ground. However, this correction factor is not sufficient to predict the noise levels high above the ground. Some adjustments of the correction factor are required for a better estimation of sound fields in front of a reflecting façade above an impedance ground.

The Flag Major Index and Group Actions on Polynomial Rings

A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.

Classical Weyl correspondences of some two-mode quantum operators and their application

Abstract Using the Fan Hong-yi representations ξ = exp − 1 2 ξ +ξa 1 † +ξ ∗ a 2 † −a 1 † a 2 † 00 , and η = exp − 1 2 η +ηa 1 † −η ∗ a 2 † +a 1 † a 2 † 00 we give two kinds of two-mode Wigner operators and generally give the formulae of the classical Weyl correspondences of some two-mode quantum operators. Making use of the new expressions we calculate the classical Weyl correspondences of the squeezed Shapiro–Wagner (SW) phase operators.

Predicted ground effects for the sound fields near a reflecting wall

In many environmental noise prediction schemes, an empirical correction factor of 2.5 or 3.0 dB(A) is to be added for the calculation of noise 1 m in front of a reflecting facade. In this paper, theoretical and experimental studies have been conducted to examine the validity and accuracy of such approximations. The theoretical analysis is based on the representation of the sound waves in their integral forms. By imposing the impedance boundary conditions on the absorbing ground and the vertical wall, it is possible to extend the classical Weyl–Van der Pol formula for inclusion of the effect of a reflecting facade. As expected, the analytical solution is composed of four terms (a source and three image sources) as a result of the absorbing ground and the reflecting facade. Experimental measurements have been conducted indoors to validate the theoretical predictions. It has been demonstrated that the simple empirical correction factor is not adequate in most cases for the assessment of the facade noise levels above an absorbing ground. [Work supported by the Hong Kong Polytechnic University.]

Brownian motion in a Weyl chamber, non-colliding particles, and random matrices

Abstract Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An−1, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t −n(n−1) 4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a semisimple Lie algebra, viewing it as a vector space with the Killing form. Since the Killing form is invariant under the adjoint, the motion induces a process in the Weyl chamber of the Lie algebra, giving a Brownian motion conditioned never to exit the chamber. If there are m roots in n dimensions, this shows that the radial part of the conditioned process is the same as the n + 2m-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for An−1 corresponding to un of n particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them.

Exterior power operations in the representation theory of the classical weyl groups

First, we introduce a class of operations, called ø-operations, on the repre-sentation rings of the classical Weyl groups W(Bk ) and W(Dk ) These operations are shown to generate the exterior power operations in the representation rings R(W(Bk )) and R(W(Dk )). Given integers l, h satisfying l + h = k, let β be a partition of l and α a partition of h. The main theorem shows that induced representations of the form Where W βα is a product of Wely groups, can be expressed as polynomials in the ø poerations acting on the two canonical induced representations Next, we show that the set which consists of elements of the form is a basis of Q ⊗ R(W(Bk )). Since the ø-operations generate the λ-operations, one can deduce that Q ⊗ R(W( k )) is generated as a λ-ring over Q by the elements 1 ⊗ xk and 1 ⊗ Yk . By applying a result of Lusztig which characterizes the irreducible representations of the Wely groups W(Bk ) and W(Dk ) ir follows, as a corollary, that Q ⊗ R(W(Bk )) is generated by two elements as a λ-ring over Q.

Exterior power operations in the representation theory of the classical Weyl groups

First, we introduce a class of operations, called ø-operations, on the representation rings of the classical Weyl groups and W(B k) and W(D k). These operations are shown to generate the exterior power operations in the representation rings R(W(B k)) and R(W(D k)). Given integers l h satisfying l+h =k, let β be a partition of l and α a partition of h. The main theorem shows that induced representations of the form where W βα is a product of Weyl groups, can be expressed as polynomials in the cooperations acting on the two canonical induced representations. , Next, we show that the set which consists of elements of the form. .is a basis of Q⊗R(W(B k)). Since the ø-operations generate the λ-operations, one can deduce that Q⊗R(W(B k)) is generated as a λ-ring over Q by the elements 1 ⊗ X k and 1 ⊗ Y k By applying a result of Lusztig which characterizes the irreducible representations of the Weyl groups W:(B k) and W(D k) it follows, as a corollary, that Q ⊗ R(W(D k)) is generated by two elements as λ-ring over Q.

METHODS OF PERTURBATION THEORY IN THE QUALITATIVE ESSENTIAL SPECTRAL ANALYSIS OF DIFFERENTIAL OPERATORS

The qualitative properties of perturbed differential equations are investigated. The analogies of the classical Weyl theorem are proved.

Methods of perturbation theory in the qualitative essential spectral analysis of differential operators

Abstract The qualitative properties of perturbed differential equations are investigated. The analogies of the classical Weyl theorem are proved.

Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

On Uniqueness of Commutative Rings of Weyl Group Invariant Differential Operators

The uniqueness of commutative rings of classical Weyl group invariant differential operators is discussed. We show this uniqueness for trigonometric or elliptic potential cases under some order conditions. For rational potential cases, counter examples are constructed.

Separation of variables and the computation of Fourier transforms on finite groups, I

This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroup-adapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easy-to-use upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the best known fast transforms for some abelian groups, the symmetric groups and their wreath products, and the classical Weyl groups. Beyond this, we obtain greatly improved upper bounds for the general linear and unitary groups over a finite field, and for the classical Chevalley groups over a finite field. We also apply these techniques to obtain analogous results for homogeneous spaces. This is part I of a two part paper. Part II will present a refinement of these techniques which yields further savings.

Yangians: Their representations and characters

The finite-dimensional irreducible representations of the Yangian of sl2 are parametrized by their highest weights, which are monic polynomials in one variable. In this paper, we give a formula for the character of such a representation which depends only on its highest weight, and is an analogue of the classical Weyl character formula.

Representation Theory of a Hecke Algebra of G( r, p, n)

In our previous paper [1], we introduced a notion of a Hecke algebra for a series of complex reflection groups (Z/rZ) ≀Kn, which is the group of n by n permutation matrices with entries running through rth roots of unity. The ordinary representation of the algebra is also studied in it. The results are direct generalizations of Hoefsmit′s [4] for Hecke algebras of classical Weyl groups. In this paper, we establish a similar theory for the remaining series of complex reflection groups G(r, p, n) (p\r). p = 1 is the case we previously considered in [1]. The construction of the Hecke algebra was soon generalized to the case p = 2 by Broué and Malle [2], motivated by a much deeper background. In [2], they conjectured that these Hecke algebras over a ring of l-adic integers are candidates for endomorphism rings of certain Deligne-Lusztig modules over the ring, which arise in modular representation theory related to algebraic groups. They also defined Hecke algebras of complex reflection groups for (not all but various kinds of) exceptional and two-dimensional cases in the same context. The first section starts with the definition of the Hecke algebra and then we prove that they actually have desirable properties. The second section is for explicit construction of irreducible representations. We call them the representations of semi-normal form. The author deeply thanks Professor Broué and Professor Malle, since he was able to learn much from them, and they hinted him to the problem.

The cohomology of the classical and quantum Weyl CCR in curved spaces

An analysis is presented of the cohomological underpinnings for the Weyl group of the canonical commutation relations on manifolds of constant negative curvature. Several uniqueness results are obtained leading from purely classical considerations to the group associated with the systems of imprimitivity of the orthodox approach to quantum mechanics.

Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form σ πϵW χ(π)t d(π) where W is a classical Weyl group, X is a one-dimensional character of W , and d ( π ) is the descent statistic . This completes a picture which is known when W is the symmetric group S n (the Weyl group A n −1 ). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B n (≊C n ) and D n . The B n , case uses sign-reversing involutions, while the D n case follows from a result of independent interest relating statistics for all three groups.

Differential geometry and quantization on a locally compact group

For an arbitrary locally compact group G, we describe the structure of the Lie algebra χ(G) of vector fields, the exterior algebra Λ(G) of differential forms, and the Poisson algebra of symbols on G polynomial with respect to the momenta. A continuous left-invariant qp-quantizaton is constructed, giving rise to a one-to-one correspondence between symbols and differential operators on G. It is demonstrated that neither of the other two classical quantizations, namely, the pq and Weyl quantizations, can be constructed on an infinite group G if the same properties are to be retained.

Reflection decompositions in the classical Weyl groups

In this paper, we present formulas for the number of decompositions of elements of the Weyl groups of type A n , D n and B n as products of a number of reflections that is not necessarily minimal. For this purpose, we consider the poset of conjugacy classes of W introduced in Bédard and Goupil (1992) for the symmetric group. This poset describes the action of the set of reflections of a reflection group on its conjugacy classes. In particular, we show how the reflection decompositions in the symmetric group %plane1D;50A; n are related to the reflection decompositions in D n .

Macdonald's evaluation conjectures and difference Fourier transform

In the previous author's paper the Macdonald norm conjecture (including the famous constant term conjecture) was proved. This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation theorem is in fact a $q,t$-generalization of the classic Weyl dimension formula. As to the duality theorem, it states that the generalized trigonometric-difference zonal Fourier transform is self-dual (at least formally). We define this transform in terms of double affine Hecke algebras related to elliptic braid groups. The duality appeared to be directly connected with the transposition of the periods of an elliptic curve.

Weyl-minkowski duality for integarl monoids*

The abstract linear duality model of Carvalho and Trotter is extended to accornodate a duality relation shared by point sets and families of (possibly) nonlinear functions. Elementary properties of the general (linear) are established along with results linking finiteness conditions which correspond to well-known (linear) relations of Weyl, Farkas, Minkowski, Lehman and Fulkerson. The particular instance of integer programming duality serves as the main motivation for the general model; duality between finitely generated integral monoids and finite families of Chvatal functions is established in direct analogy to the classical weyl- Minkowski auaiity of iiniteiy generated and polyhedral cones. Affine versions of the iniegrai monoid duality results are developed, i.e., duality relations for the integral elements of rational polyhedra.