Generalized Projective Representations for sl(n+1)

It is well known that $n$-dimensional projective group gives rise to a non-homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of $sl(n+1)$ to a non-homogenous representation on the tensor space of any finite-dimensional irreducible $gl(n)$-module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible $sl(n+1)$-modules, which are in general not highest-weight type, for any given finite-dimensional irreducible $sl(n)$-module. The results could also be used to study the quantum field theory with the projective group as the symmetry.Comment: 24page

Stress-induced permeability evolution in coal: Laboratory testing and numerical simulations

Mining operations produce a multiscale network of fractures in the coal seams. Permeability evolution in rocks is important for coal bed methane (CBM) and shale gas exploitation as well as for greenhouse gas storage. Therefore, this work presents laboratory tests and a coupled model using PFC3D and FLAC3D to simulate the stress induced permeability evolution in coal samples. Basic mechanical properties are determined via lab testing. The spatial distributions of different components inside the reconstructed samples produce a significant heterogeneity based on CT technique. A newly developed experimental system is employed to perform 3-dimensional loading and to measure the flow rate simultaneously. The evolution process is described by 5 distinct phases in terms of permeability and deformation. Triaxial tests are simulated with PFC3D using a novel flexible wall boundary method. Gas seepage simulations are performed with FLAC3D. Relations between hydraulic properties and fracture data are established. Permeability and volumetric strain show good nonlinear exponential relation after a newly introduced expansion point. Piecewise relations fit the whole process, the expansion point can be treated as critical point. The structural characteristics of the samples influence this relation before and after the expansion point significantly

Descriptions of strongly multiplicity free representations for simple Lie algebras

Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers $\mathbb{C}$. Let $Z(\mathfrak{g})$ be the center of the universal enveloping algebra $U(\mathfrak{g})$. Denote by $V_\lambda$ the finite-dimensional simple $\mathfrak{g}$-module with highest weight $\lambda$. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivited by studying the structure of the endomorphism algebras $End _{ U(\mathfrak{g})}(V_\lambda^{\otimes r})$in terms of the quotients of Kohno's infinitesimal braid algebra. Kostant introduced the $\mathfrak{g}$ -invariant endomorphism algebras $R_\lambda= (End V_\lambda\otimes U(\mathfrak{g}))^\mathfrak{g}$ and $R_{\lambda,\pi}=(End V_\lambda\otimes \pi[U(\mathfrak{g})])^\mathfrak{g}.$ In this paper, we give some other criterion for a multiplicity free representation to be a strongly multiplicity free representation for simple Lie algebras by classifing the pairs $(\mathfrak{g}, V_\lambda)$, which are multiplicity free irreducible modules and for such pairs, $R_\lambda$ and $R_{\lambda,\pi}$ are generated by generalizations of the quadratic Casimimir elements of $Z(\mathfrak{g})$

The effect of substrate temperature on cadmium telluride films in high temperature vapor deposition process

Physical vapor high-temperature deposition of CdTe thin films is one of the main methods for preparing high-efficiency CdTe solar cells, but high-temperature deposition also has an impact on the internal structure of the film. The difference in thermal expansion coefficients between the substrate and CdTe leads to the generation of internal stress in the CdTe thin film during the cooling process. In this work, we prepared thin films with different substrate temperatures using a homemade GVD device, and observed by SEM that the crystallization quality of the film gradually improved with the increase of substrate temperature, but accompanied by the shift of XRD peak position. We calculated the internal stress situation of the film by the shift amount, and the possible causes of stress generation were speculated by the results of TEM and SAED to be the combined effects of the different thermal expansion coefficients between the substrate and the film and the stacking fault defects inside the film

Strongly Commuting Ring and The Prounet-Tarry-Escott Problem

In 1935, Wright conjectured that ideal solutions to the PTE problem in Diophantine number theory should exist. In this paper, we prove Wright's conjecture holds true based on the the representation theory of the minuscule strongly commuting ring introduced by Kostant in 1975 and the complex coefficient cohomology ring structures of the Grassmannian variey