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Resources Arun Ram

Department of Mathematics and Statistics University of Melbourne Parkville VIC 3010 Australia aram@unimelb.edu.au

Resources: like the books in the library, you have the choice whether to open them and read them, or let them sit on the shelf. Professors and teachers fit in this category too.

Some of the material available from the links below is based upon work supported by the Australian Research Council ARC grants DP0986774 and DP087995 and the US National Science Foundation under Grant No. 0353038 and earlier awards. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of these agencies.

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E. Brieskorn and K. Saito, Artin Groups and Coxeter Groups, (with C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett and D. Jackson): An English translation, with notes, of the paper Artin-gruppen und Coxeter-gruppen Inv. Math. 17 (1972) 245-271, by E. Brieskorn and K. Saito.

• Brieskorn-Saito translated: Introduction
• Brieskorn-Saito translated Section 1: Definition of Artin groups
• Brieskorn-Saito translated Section 2: Reduction rule
• Brieskorn-Saito translated Section 3: The division algorithm
• Brieskorn-Saito translated Section 4: Divisibility theory
• Brieskorn-Saito translated Section 5: The fundamental element
• Brieskorn-Saito translated Section 6: The word problem
• Brieskorn-Saito translated Section 7: The centre
• Brieskorn-Saito translated Section 8: The conjugation problem
• Brieskorn-Saito translated: Bibliography

I.V. Cherednik, An analogue of the character formula for Hecke algebras, Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 21, No. 2, pp. 94-95, April-June, 1987.

• An analogue of the character formula for Hecke algebras

I.V. Cherednik, Lectures on affine Khnizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras and Macdonald theory, in collaborations wiht E. Date, K. Iohara, M. Jimbo, M. Kashiwara, T. Miwa, M. Noumi and Y. Saito, from lectures at IIAS Kyoto Japan 1997.

• Acknowledgement
• Introduction: Hecke algebras in representation theory
• The affine Khnizhnik-Zamolodchikov equation
• Isomorphism theorems for the AKZ equation
• Isomorphism theorems for the QAKZ equation
• Double affine Hecke algebras and Macdonald polynomials
• Bibliography

I.V. Cherednik, Computation of monodromy of certain W-invariant local systems of types B, C, and D, Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 1, pp. 88-89, January-March, 1990.

• Computation of monodromy of certain W-invariant local systems of types B, C, and D

Ivan Cherednik, Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics), Lecture notes from lectures at Bonn University May-June 1990.

• Abstract
• Acknowledgments
• Section 1: Introduction
• Section 2: Yang-Baxter identities
• Section 3: Factorization
• Section 4: An algebraic interpretation
• Section 5: Yang's S-matrix
• Section 6: Quantization of angles
• Section 7: Yangians
• Section 8: Mirrors and Polarization
• Section 9: Towards the CFT
• Bibliography
• Figures

The thesis of P.N. Hoefsmit: Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type, University of British Columbia, 1974.

• Contents,
• Acknowledgments
• Introduction
• Preliminaries
• Representations of the generic ring corresponding to a Coxeter system of classical type
• Degrees of the irreducible constituents of 1_B^G
• Bibliography

Steven A. Johnson, The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices, Ph.D. Thesis, University of California, Santa Barbara, 1979, written under the supervision of Robert C. Thompson.

• Title
• Dedication
• Abstract
• Table of Contents
• Section 0: Introduction
• Section 1: Notation, Conventions
• Section 2: Horn's Conjecture and the Hersch-Zwahlen methods
• Section 3: Algebraic Geometry, Intersection Theory, and the Schubert Calculus
• Section 4: The Symmetric Algebra and Schur Functions
• Section 5: Proof of the main result
• Bibliography
• Vita

V.F.R Jones, The Potts model and the symmetric group Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), River Edge, NJ, World Sci. Publishing, 1994, pp. 259–267.

• Abstract
• Introduction
• Section 1: Two bases
• Section 2: Turning into a planar form
• Section 3: Proof of the theorem
• References

David W. Koster's thesis: Complex Reflection groups, University of Wisconsin, Madison 1975, supervised by Louis Solomon

• Acknowledgment,
• Introduction,
• Section I,
• Section II,
• Section III,
• Section IV

Kumar, Shrawan, Fusion product of positive level representations and Lie algebra homology, from "Geometry and Physics" (ed. by J.E. Andersen et. al.), Lecture Notes in Pure and Applied Mathematics, vol. 184, Marcel Dekker, inc. New York-Basel-Hong Kong (1997), 253-259.

• Introduction
• Section 1. Preliminaries and notation
• Section 2. A certain complex and Lie algebra homology
• Section 3. A new geometric definition of fusion product
• Section 4. Comparison of the two fusion products
• References

van der Lek's PhD Thesis: University of Nijmegen, September 1983, Supervised by E.J.N. Looijenga

• The thesis of van der Lek: Contents and Summary
• The thesis of van der Lek - Chapter 1: Hyperplane systems and galleries
• The thesis of van der Lek - Chapter 2: Artin groups
• The thesis of van der Lek - Chapter 3: Extended Artin groups
• The thesis of van der Lek - Chapter 4: The K(π,1) problem

I.G. Macdonald's book: Algebraic Geometry: Introduction to schemes, originally published by W.A. Benjamin, Inc 1968

• Contents, Foreword, Chapter 1: Introduction,
• Chapter 2: Irreducible and Noetherian topological spaces,
• Chapter 3: The spectrum of a commutative ring,
• Chapter 4: Presheaves and Sheaves,
• Chapter 5: Affine Schemes,
• Chapter 6: Preschemes,
• Chapter 7: Operations on Sheaves, quasicoherent and coherent sheaves,
• Chapter 8: Sheaf cohomology,
• Chapter 9: Cohomology of Affine Schemes,
• Chapter 10: The Riemann-Roch Theorem,
• Bibliography.

I.G. Macdonald's Notes on Kac-Moody Lie Algebras, a typed version of handwritten lecture notes from 1983

• Introduction,
• Chapter 1,
• Chapter 2,
• Chapter 3,
• Chapter 4

I.G. Macdonald's Notes on Reflection Groups, a typed version of handwritten lecture notes from a course at University of California, San Diego, January-March 1991

• Chapter 1: Root systems,
• Chapter 2: Classification,
• Chapter 3: Invariants,
• Chapter 4: Exponentials,
• Chapter 5: Orthogonal polynomials

I.G. Macdonald's Notes on Schubert polynomials, originally published by LACIM, UQAM ????

• Contents,
• Foreword,
• Notes and References,
• Chapter 1: Permutations,
• Chapter 2: Divided differences,
• Chapter 3: MultiSchur functions,
• Chapter 4: Schubert polynomials 1,
• Chapter 5: Orthogonality,
• Chapter 6: Double Schubert polynomials,
• Chapter 7: Schubert polynomials 2,
• Appendix, Schubert varieties,
• Appendix: Combinatorial construction of Schubert polynomials,

I.G. Macdonald, Spherical functions on a Group of p-adic type, originally published by the University of Madras in November 1971.

• Preface,
• Contents,
• Chapter I: Basic properties of spherical functions,
• Chapter II: Group of p-adic type,
• Chapter III: Spherical functions on a group of p-adic type,
• Chapter IV: Calculation of the spherical functions,
• Chapter V: Plancherel measure,
• Notes,
• References,

Dale Peterson's MIT lecture notes on Quantum Cohomology of G/P, from Spring 1997

• Quantum cohomology of G/P: Lectures 1-5
• Quantum cohomology of G/P: Lectures 6-10
• Quantum cohomology of G/P: Lectures 11-15
• Quantum cohomology of G/P: Lectures 16-18

V.L. Popov, Discrete complex reflection groups, Lectures delivered at the Mathematical Institute, Rijksuniversteit Utrecht, October 1980.

• Introduction
• Table of contents
• Notation and formulation of the problem
• Formulation of the results
• Several auxiliary results and the classification of irreducible infinite noncrystallographic complex r-groups
• Invariant lattices
• The structure of r-groups in the case s=n+1
• References

L.D. Fadeev, N.Yu. Reshetikhin and L.A. Takhtajan, Quantization of Lie groups and Lie algebras, LOMI preprint 1987.

• Quantization of Lie groups and Lie bialgebras

N.Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint 1987.

• Introduction
• Contents
• Section 1: Solutions of the Yang-Baxter equation connected with the q-deformation of the universal enveloping algebras of simple Lie algebras
• Section 2: Graphical representation of the R-matrices
• Section 3: The q-analog of Brauer-Weyl duality
• Section 4: sl(n)
• Section 5: so(2n+1) and so(2n)
• Section 6: sp(2n)
• Section 7: G₂
• Appendix
• References

Schwalm, Elliptic functions sn, cn, dn, as trigonometry

• Elliptic functions sn, cn, dn, as trigonometry

J.P. Serre, Groupe finis d'automorphismes d'anneaux locaux réguliers, Ecole Normale Supérieure de Jeunes Filles, Colloque d'Algèbre [1967. Paris], no. 8 11 p. (translated to English by Arun Ram)

• Finite groups of automorphisms of local regular rings

Steinberg's Yale Lecture Notes: Lectures on Chevalley groups, by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson.

• Chapter 1: A basis for L,
• Chapter 2: A basis for U,
• Chapter 3: The Chevalley groups,
• Chapter 4: Simplicity of G,
• Chapter 5: Chevalley groups and algebraic groups,
• Chapter 6: Generators and relations,
• Chapter 7: Central extensions,
• Chapter 8: Variants of the Bruhat lemma,
• Chapter 9: The orders of the finite Chevalley groups,
• Chapter 10: Isomorphisms and Automorphisms,
• Chapter 11: Some twisted groups,
• Chapter 12: Representations,
• Chapter 13: Representations continued,
• Chapter 14: Representations concluded,

Thiem, Nat, Unipotent Hecke algebras: the structure, representation theory, and combinatorics, PhD Thesis, University of Wisconsin--Madison, 2004.

• Abstract
• Acknowledgements
• Contents
• Chapter 1: Introduction
• Chapter 2: Preliminaries
• Chapter 3: Unipotent Hecke algebras
• Chapter 4: A basis with multiplication in the G = GL_n(F_q) case
• Chapter 5: Representation theory in the G = GL_n(F_q) case
• Chapter 6: The representation theory of the Yokonuma algebra
• Appendix A: Commutation relations
• Bibliography

J. Tits, Les groupes simple de Suzuki et Ree, Séminaire N. Bourbaki, 1960-1961, exp. no. 210, p. 65-82, http://www.numdam.org/item?id=SB_1960-1961__6__65_0 (translated to English by Arun Ram)

• The simple groups of Suzuki and Ree

Verma, Daya-Nand (1975), Role of Affine Weyl groups in the representation theory of Chevalley groups and their Lie algebras in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653-705.

• Contents
• Summary
• Introduction
• List of notations
• Part I. A conjecture on Weyl's dimension polynomial
• Affine Weyl groups
• Harmonic polynomials and conjecture I
• Evidence in low ranks
• Part II. Representations and decompositions in characteristic $p\ne 0$
• Fundamental constants $c_{\lambda \mu }$ and the character formula
• The "Harish-Chandra principle" and two conjectures on $c_{\lambda \mu }$
• PIM's of the $u$-algebra and the Humphrey's numbers $d_{\lambda }$
• Conjecture on the decomposition of induced modules $Z_{\lambda }$
• Epilogue; A metamathematical (?) conjecture
• References

E. Wiesner, Translation functors and the Shapovalov determinant, PhD thesis, University of Wisconsin - Madison, 2005

• Abstract
• Contents
• Introduction
• Chapter 1. Basics and Background
• Chapter 2. Lie algebras with Triangular Decomposition
• Chapter 3. The Virasoro Algebra
• Chapter 4. Quantum Groups
• Bibliography

A.V. Zelevinskii, Two remarks on graded nilpotent classes, Uspekhi Mat. Nauk 40 (1985), no. 1(241), 199–200. English translation: Russian Math. Surveys 40 (1985), no. 1, 249–250.

• Two remarks on graded nilpotent classes

A.V. Zelevinskii, Resolvents, dual pairs, and character formulas, Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 21, No. 2, pp. 74-75, April-June, 1987.

• Resolvents, dual pairs, and character formulas

A.V. Zelevinskii, p-adic analog of the Kazhdan-Lusztig Hypothesis, Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 15, No. 2, pp. 9-21, April-June, 1981.

• Introduction
• Section 1: Basic definitions and formulation of the hypothesis
• Section 2: Contiguity Theorem
• Section 3. Examples
• Section 4. Duality
• Literature Cited