This book gives a comprehensive treatment of the Grassmannian varieties and their Schubert subvarieties, focusing on the geometric and representation-theoretic aspects of Grassmannian varieties. Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this text uniquely presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. The standard monomial theory (SMT) for the Grassmannian varieties and their Schubert subvarieties are introduced and the text presents some important applications of SMT including the Cohen-Macaulay property, normality, unique factoriality, Gorenstein property, singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. This text would serve well as a reference book for a graduate work on Grassmannian varieties and would be an excellent supplementary text for several courses including those in geometry of spherical varieties, Schubert varieties, advanced topics in geometric and differential topology, representation theory of compact and reductive groups, Lie theory, toric varieties, geometric representation theory, and singularity theory. The reader should have some familiarity with commutative algebra and algebraic geometry.

"Singular Loci of Schubert Varieties" is a unique work at the crossroads of representation theory, algebraic geometry, and combinatorics. Over the past 20 years, many research articles have been written on the subject in notable journals. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this important subdiscipline of Schubert varieties - namely singular loci. The main focus, therefore, is on the computations for the singular loci of Schubert varieties and corresponding tangent spaces. The methods used include standard monomial theory, the nil Hecke ring, and Kazhdan-Lusztig theory. New results are presented with sufficient examples to emphasize key points. A comprehensive bibliography, index, and tables - the latter not to be found elsewhere in the mathematics literature - round out this concise work. After a good introduction giving background material, the topics are presented in a systematic fashion to engage a wide readership of researchers and graduate students.

Schubert varieties provide an inductive tool for studying flag varieties. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties on the other.

Schubert varieties provide an inductive tool for studying flag varieties. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties on the other.

"Singular Loci of Schubert Varieties" is a unique work at the crossroads of representation theory, algebraic geometry, and combinatorics. Over the past 20 years, many research articles have been written on the subject in notable journals. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this important subdiscipline of Schubert varieties a" namely singular loci. The main focus, therefore, is on the computations for the singular loci of Schubert varieties and corresponding tangent spaces. The methods used include standard monomial theory, the nil Hecke ring, and Kazhdan-Lusztig theory. New results are presented with sufficient examples to emphasize key points. A comprehensive bibliography, index, and tables a" the latter not to be found elsewhere in the mathematics literature a" round out this concise work. After a good introduction giving background material, the topics are presented in a systematic fashion to engage a wide readership of researchers and graduate students.

This book gives a comprehensive treatment of the Grassmannian varieties and their Schubert subvarieties, focusing on the geometric and representation-theoretic aspects of Grassmannian varieties. Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this text uniquely presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. The standard monomial theory (SMT) for the Grassmannian varieties and their Schubert subvarieties are introduced and the text presents some important applications of SMT including the Cohen-Macaulay property, normality, unique factoriality, Gorenstein property, singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. This text would serve well as a reference book for a graduate work on Grassmannian varieties and would be an excellent supplementary text for several courses including those in geometry of spherical varieties, Schubert varieties, advanced topics in geometric and differential topology, representation theory of compact and reductive groups, Lie theory, toric varieties, geometric representation theory, and singularity theory. The reader should have some familiarity with commutative algebra and algebraic geometry.

Flag varieties are important geometric objects and their study involves an interplay of geometry, combinatorics, and representation theory. This book is a detailed account of this interplay. In the area of representation theory, the book presents a discussion of complex semisimple Lie algebras and of semisimple algebraic groups; in addition, the representation theory of symmetric groups is also discussed. In the area of algebraic geometry, the book gives a detailed account of Grassmann varieties, flag varieties, and their Schubert subvarieties. Because of their connections with root systems, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This is shown to be a consequence of standard monomial theory (abbreviated SMT). Thus the book includes SMT and some important applications - singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. In this second edition, two recent results on Schubert varieties in the Grassmannian have been added, and some errors in the first edition corrected.

Over the past fifty years, C.S. Seshadri has been a towering figure in the mathematical horizon, and his contributions have been central to the development of moduli problems and geometric invariant theory as well as representation theory of algebraic groups. The two volumes of the collected papers have been organised in accordance with the subject matter, reflecting faithfully the diversity of his mathematical contributions. These volumes will achieve the objective of inspiring future generations of mathematicians and provide insights into the unique mathematical personality of C.S. Seshadri. Table of Contents Volume 1: Vector Bundles and Invariant Theory Preface Curriculum Vitae of C S Seshadri List of Publications Acknowledgements 1. V. Balaji and V. Lakshmibai, C.S. Seshadri's work on vector bundles and invariant theory 2. David Gieseker and Jun Li, Non-abelian Jacobians and gauge theory 3. Generalised multiplicative meromorphic functions on a complex analyticmanifold Jour. Ind. Math. Soc., 21(1957), 149-175. 4. Triviality of vector bundles over the affine space K2, Proc. Nat. Aca. Sci., U.S.A. 44 (1958), 456-458. 5. L' operation de Cartier: Applications, Seminaire C. Chevalley, 3e annee, 1958-1959. 6. Diviseurs en geometrie algebrique, Seminaire C. Chevalley, 3e annee, 1958-1959. 7. Diviseurs en geometrie algebrique (Suite), Seminaire C. Chevalley, 3e annee, 1958-1959. 8. Algebraic vector bundles over the product of an a ne curve and the afffine line, Proc. Amer. Math. Soc. 10 (1959), 670-673. 9. Variete de Picard d'une variete complete, Annali di Mat.Italy IV, Vol. LVII (1962), 117-142. 10. On a theorem of Weitzenbock in invariant theory, J. Math, Kyoto Univ. 1, No.3, (1962), 403-409. 11. Some results on the quotient space by an algebraic group of automorphisms, Math. Annalen, 149 (1963), 286-301. 12. Quotient space by an Abelian variety, Math. Annalen, 152 (1963), 185-194. 13. (with M.S. Narasimhan) Holomorphic vector bundles on a compact Riemann surface, Math. Ann., 155 (1964), 69-80. 14. (with M.S. Narasimhan) Stable bundles and unitary bundles on a compact Riemann surface, Proc. Nat. Acad. Soc., 52 (1964), 207-211. 15. (with M.S. Narasimhan) Stable bundles and unitary vector bundles on a compact Riemann surface, Annals of Math., 82 (1965), 540-567. 16. Universal property of the Picard variety of a complete variety, Math. Ann., 156 (1965), 293-296. 17. Space of unitary vector bundles on a compact Riemann surface, Annals of Math., 85 (1967), 303-335. 18. Mumford's conjecture for GL(2) and applications, Proc. Intern. Colloquium on Algebraic Geometry, Bombay (1968), 347-371. 19. Moduli of vector bundles over an algebraic curve, Questions On algebraic Varieties, C.I.M.E, Varenna, (1969), 139-261. 20. Quotient spaces modulo reductive algebraic groups, Annals. of Math., 95, No.3 (1972) 511-556. 21. Errata to `Quotient spaces modulo reductive algebraic groups', Annals. of Math. Vol.96, (1972), p.599. 22. Theory of moduli, Proc. Symp. in Pure Mathematics, Algebraic Geometry, Arcata, 1974, Amer. Math. Soc. (1975), 265-304. 23. Geometric reductivity over arbitrary base, Advances in Maths., 26 (1977) 225-274. 24. Moduli of vector bundles on curves with parabolic structures, Bulletin of the Amer. Math. Soc., 83 (1977) 124-126. 25. Desingularisation of the moduli varieties of vector bundles on curves, Proc. Tokyo Symposium on Algebraic Geometry, (1977), 155-184. 26. (with T. Oda), Compactications of the generalized Jacobian variety, Trans. Amer. Math. Soc. Vol.253, (1979), 1-90. 27. (with V.B. Mehta), Moduli of vector bundles on curves with parabolic structures, Math. Ann. 228 (1980) 205-239. 28. (with V. Balaji), Cohomology of a moduli space of vector bundles, The Grothendieck Festschrift, Volume 1, Birkhauser, 87-120. 29. (with V. Balaji) Poincare polynomials of some moduli varieties, Algebraic Geometry and Analytic Geometry, Springer Verlag (1991) 1-25. 30. Vector bundles on curves, Contemporary Mathematics, Volume 153 (1993), 163-200. 31. (with D.S. Nagaraj) Degenerations of the moduli spaces of vector bundles on curves I, Proc. Indian Acad. Sci. (Math. Sci.) 107 (1997), 101-137. 32. (with D.S. Nagaraj), Degenerations of the moduli spaces of vector bundles on curves II, Proc. Indian Acad. Sci. (Math. Sci.) 109, No.2, (1999), 165-201. 33. Degenerations of the moduli spaces of vector bundles on curves, ICTP Lecture Notes 1, (2000), 205-265. 34. (with V. Balaji), Semistable principal bundles{I, Journal of Algebra, 258, (2002), 321-347. 35. Geometric reductivity (Mumford's Conjecture) revisited, Contemporary Mathematics, Volume 390 (2005), 137-145. 36. (with P. Sastry) Geometric Reductivity: A quotient space approach, Journal of the Ramanujan Math. Soc., (to appear in 2011). Volume 2: Schubert Geometry and Representation Theory Preface Curriculum Vitae of C S Seshadri List of Publications Acknowledgements 1. Standard Monomial Theory: A Historical Account 2. (with V. Lakshmibai & C. Musili) Cohomology of line bundles on G=B, Annales Scientifiques de l'E.N.S, 4 Series, (1974), 89-138. 3. Correction to `Cohomology of line bundles on G=B', Annales Scientifiques de l'E.N.S, 4 Series, (1974). 4. Cohomology of line bundles on SL3=B, (unpublished), Talk given at the Institute for Advanced Study, (1976). 5. Geometry of G=P{I (Theory of standard monomials for minuscule representation), C.P. Ramanujam { A Tribute, TIFR Publication, (1978), 207-239. 6. (with V. Lakshmibai), Geometry of G=P{II (The work of De Concini and Procesi and the basic conjectures), Proc. Indian Acad. Sci., 87 A, No.2, (1978), 1-54. 7. (with V. Lakshmibai & C. Musili), Geometry of G=P{III (Standard Monomial Theory for a quasi-minuscule P), Proc. Indian Acad. Sci. Vol.87 A, (1979), 93-177. 8. (with V. Lakshmibai & C. Musili), Geometry of G=P{IV (Standard Monomial Theory for classical types), Proc. Indian Acad. Sci., Vol. 88 A, (1979), 279-362. 9. (with V. Lakshmibai & C. Musili), Geometry of G=P, Bulletin of the Amer. Math. Soc. Vol 1, (1979), 432-435. 10. (with C. Musili) Standard Monomial Theory, Lecture Notes in Mathematics, Springer Verlag No. 867, 441-476. 11. (with C. Musili), Schubert varieties and the variety of complexes, Volume dedicated to Prof. Shafarevich on his 60th birthday, Birkhauser, 329-359. 12. (with V. Lakshmibai), Singular Locus of a Schubert Variety, Bulletin of the Amer. Math. Soc., (1984), 363-366. 13. Line bundles on Schubert varieties, International Colloquium on `Vector bundles on algebraic varieties', TIFR, (1984). 14. (with V. Lakshmibai), Geometry of G=P{V, Journal of Algebra, Vol. 100, (1986), 462-557. 15. The work of P. Littelmann and Standard Monomial Theory, Recent Trends in Mathematics and Physics: A Tribute to Harish Chandra, Narosa, (1995), 178-197. 16. (with P. Littelmann), A Pieri{Chevalley type formula for K(G=B) and Standard Monomial Theory, Studies in Memory of Issai Schur, Birkhauser, (2003), 155-176. 17. Chevalley: Some Reminiscences, Transformation Groups, No.2-3, (1999), 119-125. 18. George Kempf (unpublished). 19. M.S.Narasimhan, Collected Papers, Hindustan Book Agency, (2009).

Over the past two decades, and more intensely in recent years, the algebro-geometric study of Schubert Varieties has had considerable impact on the theory of algebraic groups. One of the most interesting developments in the theory has been the construction of natural bases of representations of the full linear group $GL(n)$, the orthogonal group, and the symplectic group. This construction gives character formulas of these representations which are quite different in spirit from the famous character formulas of H. Weyl. In fact, they connect to monomial theory and the work of Hodge which was done more than fifty years ago, and to the very recent developments in path models, Frobenius splittings, and quantum groups. Written by three of the world's leading mathematicians in algebraic geometry, group theory, and combinatorics, this excellent self- contained exposition on Schubert Varieties unfolds systematically, from relevant introductory material on commutative algebra and algebraic geometry. First-rate text for a graduate course or for self-study.