154 research outputs found
Zelevinsky's Involution At Roots of Unity
No full textWe give a combinatorial algorithm for computing Zelevinsky's involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra b Hm (t) when t is a primitive nth root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of Z=nZ-graded vector spaces. 1 Introduction In [32], Zelevinsky has introduced an involution of the Grothendieck group of the category of complex smooth representations of finite length of G = GL(m;F ), where F is a p-adic field, and conjectured that this involution permutes the classes of irreducible representations ([32], 9.17). In [33], Zelevinsky further conjectured a geometric description of this involution in terms of the graded nilpotent orbits that parametrize the simple G-modules. Both conjectures have been proved by Moeglin and Waldspurger [22] for the category of admissible representations of G generated by their space of I-fixed vectors, where I is an Iwahori subgroup of G. ..
High-precision spectroscopy of ultracold molecules in an optical lattice
No full textThe study of ultracold molecules tightly trapped in an optical lattice can expand the frontier of precision measurement and spectroscopy, and provide a deeper insight into molecular and fundamental physics. Here we create, probe, and image microkelvin 88Sr2 molecules in a lattice, and demonstrate precise measurements of molecular parameters as well as coherent control of molecular quantum states using optical fields. We discuss the sensitivity of the system to dimensional effects, a new bound-to-continuum spectroscopy technique for highly accurate binding energy measurements, and prospects for new physics with this rich experimental system
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The saturation conjecture (after A. Knutson and T. Tao)
No full textIn this exposition we give a simple and complete treatment of A. Knutson and T.
Tao's recent proof (http://front.math.ucdavis.edu/math.RT/9807160) of the saturation
conjecture, which asserts that the Littlewood-Richardson semigroup is saturated. The main
tool is Knutson and Tao's hive model for Berenstein-Zelevinsky polytopes. In an appendix of
W. Fulton it is shown that the hive model is equivalent to the original
Littlewood-Richardson rule
The number of (0,1) - Matrices with fixed row and column sums
No full textLet R and S be non-negative and non-increasing vectors of order m and n respectively. We consider the set A(R, S) of all m x n matrices with entries restricted to {0, 1}. We give an alternative proof of the Gale-Ryser theorem, which determines when A(R, S) is non-empty. We show conditions for R and S so that ∣A(R, S) ∣ ∈ {1, n!}. We also examine the case where ∣A(R, S) ∣ = 2 and describe the structure of those matrices. We show that for each positive integer k, there is a possible choice of R and S so that ∣A(R, S) ∣ = k. Furthermore, we explore gm,n(x; y), the generating function for the cardinality ∣A(R, S) ∣ of all possible combinations of R and S
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