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Grassmannian Variety: Geometric and Representation-Theoretic Aspects 1st ed. 2015 [Kietas viršelis]

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  • Formatas: Hardback, 172 pages, aukštis x plotis: 235x155 mm, weight: 4026 g, 39 Illustrations, color; 84 Illustrations, black and white; X, 172 p. 123 illus., 39 illus. in color., 1 Hardback
  • Serija: Developments in Mathematics 42
  • Išleidimo metai: 26-Sep-2015
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493930818
  • ISBN-13: 9781493930814
Kitos knygos pagal šią temą:
Grassmannian Variety: Geometric and Representation-Theoretic Aspects 1st ed. 2015Didesnis paveikslėlis
  • Formatas: Hardback, 172 pages, aukštis x plotis: 235x155 mm, weight: 4026 g, 39 Illustrations, color; 84 Illustrations, black and white; X, 172 p. 123 illus., 39 illus. in color., 1 Hardback
  • Serija: Developments in Mathematics 42
  • Išleidimo metai: 26-Sep-2015
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493930818
  • ISBN-13: 9781493930814
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781493930814.html
Raktažodžiai:

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.

Recenzijos

The present book gives a detailed treatment of the standard monomial theory (SMT) for the Grassmannians and their Schubert subvarieties along with several applications of SMT. It can be used as a reference book by experts and graduate students who study varieties with a reductive group action such as flag and toric varieties. (Valentina Kiritchenko, zbMATH 1343.14001, 2016)

The book under review is more elementary; it is exclusively devoted to Grassmannians and their Schubert subvarieties. The book is divided into three parts. This is a nicely written book, one that may appeal to students and researchers in related areas. (Felipe Zaldivar, MAA Reviews, maa.org, December, 2015)

1 Introduction
1(6)
Part I Algebraic Geometry: A Brief Recollection
2 Preliminary Material
2.1 Commutative Algebra
7(4)
2.2 Affine Varieties
11(4)
2.2.1 Zariski topology on A"
11(2)
2.2.2 The affine algebra K[ X]
13(1)
2.2.3 Products of affine varieties
14(1)
2.3 Projective Varieties
15(1)
2.3.1 Zariski topology on Pn
15(1)
2.4 Schemes --- Affine and Projective
16(2)
2.4.1 Presheaves
16(1)
2.4.2 Sheaves
16(1)
2.4.3 Sheafification
17(1)
2.4.4 Ringed and geometric spaces
17(1)
2.5 The Scheme Spec(A)
18(1)
2.6 The Scheme Pro((S)
19(1)
2.6.1 The cone over X
20(1)
2.7 Sheaves of Ox-Modules
20(3)
2.7.1 The twisting sheaf Ox(1)
21(1)
2.7.2 Locally free sheaves
22(1)
2.7.3 The scheme V(Ω) associated to a rank n locally free sheaf Ω
22(1)
2.7.4 Vector bundles
22(1)
2.8 Attributes of Varieties
23(4)
2.8.1 Dimension of a topological space
23(1)
2.8.2 Geometric properties of varieties
24(1)
2.8.3 The Zariski tangent space
24(1)
2.8.4 The differential (dφ)x
25(2)
3 Cohomology Theory
27(12)
3.1 Introduction to Category Theory
27(1)
3.2 Abelian Categories
28(4)
3.2.1 Derived Functors
31(1)
3.3 Enough Injective Lemmas
32(5)
3.4 Sheaf and Local Cohomology
37(2)
4 Grobner Bases
39(12)
4.1 Monomial Orders
39(3)
4.2 Grobner Basis
42(1)
4.3 Compatible Weight Orders
43(3)
4.4 Flat Families
46(5)
Part II Grassmann and Schubert Varieties
5 The Grassmannian and Its Schubert Varieties
51(22)
5.1 Grassmannian and Flag Varieties
51(2)
5.2 Projective Variety Structure on Gd.n
53(4)
5.2.1 Plucker coordinates
53(1)
5.2.2 Plucker Relations
54(2)
5.2.3 Plucker coordinates as T-weight vectors
56(1)
5.3 Schubert Varieties
57(4)
5.3.1 Dimension of Xw
59(1)
5.3.2 Integral Schemes
60(1)
5.4 Standard Monomials
61(4)
5.4.1 Generation by Standard Monomials
62(1)
5.4.2 Linear Independence of Standard Monomials
63(2)
5.5 Unions of Schubert Varieties
65(2)
5.5.1 The Picard Group
67(1)
5.6 Vanishing Theorems
67(6)
6 Further Geometric Properties of Schubert Varieties
73(22)
6.1 Cohen--Macaulay
73(4)
6.2 Lemmas on Normality and Factoriality
77(8)
6.2.1 Factoriality
82(3)
6.3 Normality
85(3)
6.3.1 Stability for multiplication by certain parabolic subgroups
86(2)
6.4 Factoriality
88(2)
6.5 Singular Locus
90(5)
7 Flat Degenerations
95(22)
7.1 Grobner basis
95(2)
7.2 Toric Degenerations
97(6)
7.3 Monomial Scheme Degenerations
103(1)
7.4 Application to the Degree of Xw
104(4)
7.5 Gorenstein Schubert Varieties
108(9)
Part III Flag Varieties and Related Varieties
8 The Flag Variety: Geometric and Representation Theoretic Aspects
117(12)
8.1 Definitions
117(1)
8.2 Standard Monomials on the Flag Variety
118(3)
8.3 Toric Degeneration for the Flag Variety
121(1)
8.4 Representation Theoretic Aspects
122(2)
8.4.1 Application to Gd.n
124(1)
8.5 Geometric Aspects
124(5)
8.5.1 Description of the tangent space
125(1)
8.5.2 Pattern avoidance
125(4)
9 Relationship to Classical Invariant Theory
129(14)
9.1 Basic Definitions in Geometric Invariant Theory
129(2)
9.1.1 Reductive Groups
130(1)
9.2 Categorical Quotient
131(6)
9.3 Connection to the Grassmannian
137(6)
10 Determinantal Varieties
143(12)
10.1 Determinantal Varieties
143(2)
10.1.1 The determinantal variety Dt
143(1)
10.1.2 Relationship between determinantal varieties and Schubert varieties
144(1)
10.2 Standard Monomial Basis for K[ Dt]
145(2)
10.2.1 The partial order ≥
146(1)
10.2.2 Cogeneration of an Ideal
147(1)
10.3 Grobner Bases for Determinantal Varieties
147(2)
10.4 Connections with Classical Invariant Theory
149(6)
10.4.1 The First and Second Fundamental Theorems of Classical Invariant Theory (cf. [ 88]) for the action of GLn(K)
150(5)
11 Related Topics
155(8)
11.1 Standard Monomial Theory for a General G/Q
155(1)
11.2 The Cohomology and Homology of the Flag Variety
156(2)
11.2.1 A Z-basis for H*(Fl(n))
156(1)
11.2.2 A presentation for the Z-algebra H*(Fl(n))
156(1)
11.2.3 The homology H*(Fl(n))
157(1)
11.2.4 Schubert classes and Littlewood-Richardson coefficients
157(1)
11.3 Free Resolutions
158(1)
11.4 Bott--Samelson Scheme of G
158(1)
11.5 Frobenius-Splitting
159(1)
11.6 Affine Schubert Varieties
160(1)
11.7 Affine Flag and Affine Grassmannian Varieties
161(2)
References 163(4)
List of Symbols 167(2)
Index 169