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The quantum groups of finite and affine type A admit geometric realizations in terms of partial flag varieties of finite and affine type A. Recently, the quantum group associated to partial flag varieties of finite type B/C is shown to be a coideal subalgebra of the quantum group of finite type A. In this paper the authors study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type C. The authors show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine mathfrak{sl} and mathfrak{gl} types, respectively. In this way, the authors provide geometric realizations of eight quantum symmetric pairs of affine types. The authors construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine mathfrak{sl} type, the authors establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, the authors obtain a new and geometric construction of the idempotented quantum affine mathfrak{gl} and its canonical basis.

Flag manifolds. --- Affine algebraic groups. --- Quantum groups. --- Schur complement. --- Kazhdan-Lusztig polynomials. --- Algebra, Homological.

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Coxeter groups are of central importance in several areas of algebra, geometry, and combinatorics. This clear and rigorous exposition focuses on the combinatorial aspects of Coxeter groups, such as reduced expressions, partial order of group elements, enumeration, associated graphs and combinatorial cell complexes, and connections with combinatorial representation theory. While Coxeter groups have already been exposited from algebraic and geometric perspectives, this text is the first one to focus mainly on the combinatorial aspects of Coxeter groups. The first part of the book provides a self-contained introduction to combinatorial Coxeter group theory. The emphasis here is on the combinatorics of reduced decompositions, Bruhat order, weak order, and some aspects of root systems. The second part deals with more advanced topics, such as Kazhdan-Lusztig polynomials and representations, enumeration, and combinatorial descriptions of the classical finite and affine Weyl groups. A wide variety of exercises, ranging from easy to quite difficult are also included. The book will serve graduate students as well as researchers. Anders Björner is Professor of Mathematics at the Royal Institute of Technology in Stockholm, Sweden. Francesco Brenti is Professor of Mathematics at the University of Rome.

Agrotechnology and Food Sciences. Toxicology --- Toxicity of Pesticides. --- Coxeter groups --- Groupes de Coxeter --- Théorie combinatoire des groupes --- Coxeter groups. --- Kazhdan-Lusztig polynomials. --- Combinatorial groups --- Groups, Combinatorial --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Mathematics. --- Group theory. --- Topological groups. --- Lie groups. --- Combinatorics. --- Topological Groups, Lie Groups. --- Group Theory and Generalizations. --- Combinatorial group theory. --- Combinatorial analysis --- Group theory --- Combinatorial group theory --- Théorie combinatoire des groupes --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Topological Groups. --- Groups, Topological --- Continuous groups --- Combinatorics --- Algebra --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics) --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups