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We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra.
      
This study was continued in the paper [FKRW] in the framework of vertex algebra theory.
      
As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus.
      
We also show how to distinguish examples of open subsets with a good quotient coming from Mumford's theory and give examples of open subsets with non-quasi-projective quotients.
      
The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K?hler manifolds.
      
Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K].
      
Quantum integrable systems and differential Galois theory
      
We relate the invariant theory of cones of highest weight vectors to weight multiplicities and theirq-analogs.
      
Classical invariant theory for finite reflection groups
      
We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations.
      
The main idea for our approach relies on a study of the boundary theory we established for the general CAT(-1) spaces.
      
We see that the theory ofn-valued groups is distinct from that of groups with a given automorphism group.
      
These results are applied to the theory ofcompactly causal symmetric spaces: we describe explicitly the complex domain Ξ associated to such a space.
      
This paper shows how the Kazhdan-Lusztig theory of cells can be directly applied to establish the quasi-heredity ofq-Schur algebras.
      
A new approach to standard monomial theory for classical groups
      
As an immediate application we obtain a new proof of the main theorem of standard monomial theory for classical groups.
      
There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns.
      
Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semi-simple algebraic groups.
      
We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties.
      
Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem.
      
 

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