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29 March 2024
 
  » arxiv » math-ph/9806014

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Fundamental Weights, Permutation Weights and Weyl Character Formula
Hasan R. Karadayi ; Meltem Gungormez ;
Date 19 Jun 1998
Journal J.Phys. A32 (1999) 1701-1707
Subject Mathematical Physics; Representation Theory; Group Theory | math-ph hep-th math.GR math.MP math.RT
AbstractFor a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(Lambda^{++})$ of strictly dominant weights $Lambda^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(Lambda^{++})$, there is a very peculiar subset $wp(Lambda^{++})$ for which we always have $$ dimwp(Lambda^{++})=dimW(G_N)/dimW(A_{N-1}) . $$ For any dominant weight $ Lambda^+ $, the elements of $wp(Lambda^+)$ are called {f Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $wp(Lambda^{++})$ and $wp( ho)$ where $ ho$ is the Weyl vector of $G_N$. The concept of signature factor which enters in Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant $Lambda^+$, calculation of the character $ChR(Lambda^+)$ for irreducible representation $R(Lambda^+)$ will then be provided by $A_N$ multiplicity rules governing generalized Schur functions. The main idea is again to express everything in terms of the so-called {f Fundamental Weights} with which we obtain a quite relevant specialization in applications of Weyl character formula.
Source arXiv, math-ph/9806014
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