This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig's seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group.Part I Preliminaries.- 1 Preorders on Bases of Algebras.- 2 Lusztig's Lemma.- Part II Coxeter Systems, Hecke Algebras.- 3 Coxeter Systems.- 4 Hecke Algebras.- Part III Kazhdan–Lusztig Cells.- 5 The Kazhdan–Lusztig Basis.- 6 Kazhdan–Lusztig Cells.- 7 Semicontinuity.- Part IV General Properties of Cells.- 8 Cells and Parabolic Subgroups.- 9 Descent Sets, Knuth Relations and Vogan Classes.- 10 The Longest Element and Duality in Finite Coxeter Groups.- 11 The Guilhot Induction Process.- 12 Submaximal Cells (Split Case).- 13 Submaximal Cells (General Case).- Part V Lusztig's a-Function.- 14 Lusztig's Conjectures.- 15 Split and quasi-split cases.- Part VI Applications of Lusztig's Conjectures.- 16 Miscellanea.- 17 Multiplication by Tw0.- 18 Action of the Cactus Group.- 19 Asymptotic Algebra.- 20 Automorphisms.- Part VII Examples.- 21 Finite Dihedral Groups.- 22 The Symmetric Group.- 23 Affine Weyl Groups of Type A2.- 24 Free Coxeter Groups.- 25 Rank 3.- 26 Some Bibliographical Comments.- Appendices.- A Symmetric Algebras.- B Reflection Subgroups of Coxeter Groups.- References.- Index.
Size : 9,33 MB
Format : Adobe PDF
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This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig's seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group.Part I Preliminaries.- 1 Preorders on Bases of Algebras.- 2 Lusztig's Lemma.- Part II Coxeter Systems, Hecke Algebras.- 3 Coxeter Systems.- 4 Hecke Algebras.- Part III Kazhdan–Lusztig Cells.- 5 The Kazhdan–Lusztig Basis.- 6 Kazhdan–Lusztig Cells.- 7 Semicontinuity.- Part IV General Properties of Cells.- 8 Cells and Parabolic Subgroups.- 9 Descent Sets, Knuth Relations and Vogan Classes.- 10 The Longest Element and Duality in Finite Coxeter Groups.- 11 The Guilhot Induction Process.- 12 Submaximal Cells (Split Case).- 13 Submaximal Cells (General Case).- Part V Lusztig's a-Function.- 14 Lusztig's Conjectures.- 15 Split and quasi-split cases.- Part VI Applications of Lusztig's Conjectures.- 16 Miscellanea.- 17 Multiplication by Tw0.- 18 Action of the Cactus Group.- 19 Asymptotic Algebra.- 20 Automorphisms.- Part VII Examples.- 21 Finite Dihedral Groups.- 22 The Symmetric Group.- 23 Affine Weyl Groups of Type A2.- 24 Free Coxeter Groups.- 25 Rank 3.- 26 Some Bibliographical Comments.- Appendices.- A Symmetric Algebras.- B Reflection Subgroups of Coxeter Groups.- References.- Index.
Size : 9,33 MB
Format : Adobe PDF
