On uniqueness of tensor products of irreducible categorifications

In this paper, we propose an axiomatic definition for a tensor product categorification. A tensor product categorification is an abelian category with a categorical action of a Kac-Moody algebra g in the sense of Rouquier or Khovanov-Lauda whose Grothendieck group is isomorphic to a tensor product of simple modules. However, we require a much stronger structure than a mere isomorphism of representations; most importantly, each such categorical representation must have standardly stratified structure compatible with the categorification functors, and with combinatorics matching those of the tensor product. With these stronger conditions, we recover a uniqueness theorem similar in flavor to that of Rouquier for categorifications of simple modules. Furthermore, we already know of an example of such a categorification: the representations of algebras T^\lambda previously defined by the second author using generators and relations. Next, we show that tensor product categorifications give a categorical realization of tensor product crystals analogous to that for simple crystals given by cyclotomic quotients of KLR algebras. Examples of such categories are also readily found in more classical representation theory; for finite and affine type A, tensor product categorifications can be realized as quotients of the representation categories of cyclotomic q-Schur algebras.Comment: 27 pages; v2 28 pages, minor correction

A stress-based macroscopic approach for microcracks unilateral effect

The question of the nonlinear response of brittle materials undergoing elastic damage is investigated here. Owing to the specific nature of microcracking, the macroscopic behaviour of these materials is complex, generally anisotropic owing to the possible preferential orientation of defects and multilinear because of the unilateral effect due to the transition between open and closed state of microcracks. A new three-dimensional macroscopic model outlined by Welemane and Cormery [1] has been proposed to account simultaneously for these both aspects. This paper intends to present in details the principles of such approach and to demonstrate its applicability to a stress-based framework. Based on a fabric tensor representation of the damage density distribution, the model provides a continuum and rigorous description of the contribution of defaults which avoids classical spectral decompositions and related inconsistencies. The model is also strongly micromechanically motivated, especially to handle the elastic moduli recovery that occurs at the closure of microcracks. The macroscopic theoretical framework proposed constitutes a general approach that leads in particular to predictions of a class of micromechanical models. The capacities of the approach are illustrated and discussed on various cases of damage configurations and opening 13closure states, with a special attention to the differences with the strain-based framework and to the influence of the damage variables order

Second-order Gauge Invariant Cosmological Perturbation Theory: -- Einstein equations in terms of gauge invariant variables --

Along the general framework of the gauge invariant perturbation theory developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723; {\it ibid}, {\bf 113} (2005), 481.], we formulate the second order gauge invariant cosmological perturbation theory in a four dimensional homogeneous isotropic universe. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We derive the all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. All equations are derived in terms of gauge invariant variables without any gauge fixing. These equations imply that the second order vector and tensor modes may be generated due to the mode-mode coupling of the linear order scalar perturbations. We also briefly discuss the main progress of this work by the comparison with some literatures.Comment: 58 pages, no figure. Complete version of gr-qc/0605107; some typos are corrected (v2); References and some typos are corrected. To be appeard Progress of Theoretical Physic