Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2

In this paper various representations of the exceptional Lie algebra G_2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G_2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the elementary representations of G_2 are investigated in detail, and the corresponding six-boson realization is given. After obtaining explicit forms of all twelve extremal vectors of the elementary representation with the highest weight {\Lambda}, all representations with their respective highest weights related to {\Lambda} are systematically discussed. For one of these representations the corresponding five-boson realization is constructed. Moreover, a new three-fermion realization from the fundamental representation (0,1) of G_2 is constructed also.Comment: 29 pages, 4 figure

Spectrally-Corrected and Regularized Global Minimum Variance Portfolio for Spiked Model

Considering the shortcomings of the traditional sample covariance matrix estimation, this paper proposes an improved global minimum variance portfolio model and named spectral corrected and regularized global minimum variance portfolio (SCRGMVP), which is better than the traditional risk model. The key of this method is that under the assumption that the population covariance matrix follows the spiked model and the method combines the design idea of the sample spectrally-corrected covariance matrix and regularized. The simulation of real and synthetic data shows that our method is not only better than the performance of traditional sample covariance matrix estimation (SCME), shrinkage estimation (SHRE), weighted shrinkage estimation (WSHRE) and simple spectral correction estimation (SCE), but also has lower computational complexity