Topic
Hyperoctahedral group
About: Hyperoctahedral group is a research topic. Over the lifetime, 320 publications have been published within this topic receiving 5571 citations. The topic is also known as: O(n;ℤ).
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TL;DR: Using the theory of symmetric functions, a formula is found for r(w) when W is the symmetric group Sn and for the element w0 ∈ Sn of longest length and certain other w ∉ Sn the formula is particularly simple.
415 citations
TL;DR: In this article, it was shown that if x E X,,,, then x-lW,x n W, = W, for some subset L of S, then X, = X, for the distinguished cross section of W/W,.
354 citations
Book•
01 Jan 2002
TL;DR: A Glance at Group Representations as discussed by the authors presents a general overview of group representation and its relation to algebraic elements, including rings, modules and algebraic algebras.
Abstract: Preface. 1. Groups. 2. Rings, Modules and Algebras. 3. Group Rings. 4. A Glance at Group Representations. 5. Group Characters. 6. Ideals in Group Rings. 7. Algebraic Elements. 8. Units of Group Rings. 9. The Isomorphism Problem. 10. Free Group of Units. 11. Properties of the Unit Group. Bibliography. Index.
309 citations
TL;DR: This paper gives what seems to be the simplest proof to date that the q-binomial coefficient k+l k has unimodal coefficients and investigates how these representations are related to one another and to the structure of P.
277 citations
TL;DR: In this article, it is shown 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.
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.
217 citations