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Kac-Moody Groups, their Flag Varieties and Representation Theory
Shrawan Kumar
- 10 Sep 2002
994
TL;DR: In this article, Kac-Moody Lie Algebra Homology and Cohomology has been studied in the context of representation theory of kac-moody groups.
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Abstract: Introduction * Kac--Moody Algebras -- Basic Theory * Representation Theory of Kac--Moody Algebras * Lie Algebra Homology and Cohomology * An Introduction to ind-Varieties and pro-Groups * Tits Systems -- Basic Theory * Kac--Moody Groups -- Basic Theory * Generalized Flag Varieties of Kac--Moody Groups * Demazure and Weyl--Kac Character Formulas * BGG and Kempf Resolutions * Defining Equations of G/P and Conjugacy Theorems * Topology of Kac-Moody Groups and Their Flag Varieties * Smoothness and Rational Smoothness of Schubert Varieties * An Introduction to Affine Kac-Moody Lie Algebras and Groups * Appendix A. Results from Algebraic Geometry * Appendix B. Local Cohomology * Appendix C. Results from Topology * Appendix D. Relative Homological Algebra * Appendix E. An Introduction to Spectral Sequences * Bibliography * Index of Notation * Index
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Citations
On Generalized Minors and Quiver Representations
TL;DR: In this article, it was shown that cluster variables of pre-projective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight group representations, generalizing results of Yang-Zelevinsky in finite type.
Cluster algebras in algebraic Lie theory
TL;DR: In this paper, the authors survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and Kac-Moody groups and also review a quantized version of these results.
Richardson Varieties Have Kawamata Log Terminal Singularities
Shrawan Kumar,Karl Schwede +1 more
TL;DR: In particular, this paper showed that the Richardson varieties are Frobenius split with Kawamata log terminal singularities in the case when the Richardson variety is a Richardson variety associated to a symmetrizable Kac-Moody group.
K-theoretic analogues of factorial Schur P- and Q-functions
Takeshi Ikeda,Hiroshi Naruse +1 more
TL;DR: In this paper, the authors introduce two families of symmetric functions generalizing the factorial Schur P -and Q -functions due to Ivanov, and show that these functions represent the Schubert classes in the K-theory of torus equivariant coherent sheaves on the maximal isotropic Grassmannians of symplectic and orthogonal types.
On the topologies on ind-varieties and related irreducibility questions
TL;DR: In this paper, a large class of affine ind-varieties where these two topologies differ is given, and a counter-example of a supposed irreducibility criterion given in Shafarevich (1981) [Sha81] is given.