Singular Loci of Schubert Varieties

by Billey, Sara; Lakshmibai, V.

• ISBN13: 9780817640927
• ISBN10: 0817640924
• Format: Hardcover
• Copyright: 2000-09-01
• Publisher: Birkhauser

Summary

"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.

Table of Contents

Preface	xi
Introduction	1	(6)
Generalities on G/B and G/Q	7	(16)
Abstract root systems	7	(2)
Root systems of algebraic groups	9	(1)
Root subgroups	10	(1)
Parabolic subgroups	11	(1)
The Weyl group of a parabolic subgroup	12	(1)
Schubert varieties	12	(1)
The Bruhat-Chevalley order	13	(1)
Line bundles on G/Q	13	(2)
Geometric properties of Schubert varieties	15	(1)
Equations defining a Schubert variety	16	(1)
Representations of semisimple algebraic groups	17	(6)
Specifics for the Classical Groups	23	(14)
The Grassmannian variety Gd,n	23	(3)
The special linear group SL (n)	26	(3)
The symplectic group Sp(2n)	29	(2)
The odd orthogonal group SO(2n+1)	31	(2)
The even orthogonal group SO(2n)	33	(4)
The Tangent Space and Smoothness	37	(10)
The Zariski tangent space	37	(1)
Smooth and singular points	37	(1)
The space T(w,T)	38	(1)
A canonical affine neighborhood of a T-fixed point	39	(1)
Tangent cone and Jacobian criteria for smoothness	40	(1)
Discussion of smoothness at a T-fixed point	41	(1)
Multiplicity at a point P on a variety X	42	(2)
Degree of X(w)	44	(2)
Summary of smoothness criteria	46	(1)
Root System Description of T(w,T)	47	(24)
Polo's results	47	(2)
Bases Bλ Bλ* for Vk(λ) and H0(G/B, Lλ)	49	(3)
Description of T(w,id)	52	(4)
Description of T(w,T)	56	(7)
Tangent space and certain weight multiplicities	63	(4)
The B-module T(w,id)	67	(1)
Two smoothness criteria of Carrell-Kuttler	68	(3)
Rational Smoothness and Kazhdan-Lusztig Theory	71	(20)
Kazhdan-Lusztig polynomials	72	(5)
Carrell-Peterson's criteria	77	(4)
Combinatorial formulas for Kazhdan-Lusztig polynomials	81	(10)
Nil-Hecke Ring and the Singular Locus of X(w)	91	(12)
The nil-Hecke ring	91	(3)
Criteria for smoothness and rational smoothness	94	(3)
Representation-theoretic results on the tangent cone	97	(1)
Proof of smoothness criterion	98	(5)
Patterns, Smoothness and Rational Smoothness	103	(16)
Type A: criterion in terms of patterns	103	(1)
Conjecture in type A	104	(2)
Types B, C, D: criterion in terms of patterns	106	(9)
Type C results of Lakshmibai-Song using permutations	115	(4)
Minuscule and cominuscule G/P	119	(40)
Results on small resolutions	122	(9)
Brion-Polo results	131	(7)
Irreducible components of SingX(w) in special cases	138	(6)
Multiplicity at a singular point	144	(11)
The symplectic Grassmannian Sp(2n)/Pn	155	(4)
Rank Two Results	159	(10)
Kumar's method	159	(2)
Tangent space computations	161	(8)
Related Combinatorial Results	169	(6)
Factoring the Poincare polynomial of a Schubert variety	169	(1)
Structure of Bruhat intervals	170	(2)
Generating function for smooth permutations	172	(1)
Bona's results	172	(3)
Related Varieties	175	(32)
Opposite cells in Schubert varieties in SL(n)/B	175	(5)
Determinantal varieties	180	(5)
Ladder determinantal varieties	185	(16)
Quiver varieties	201	(2)
Variety of complexes	203	(4)
Addendum	207	(32)
Dynkin Diagrams	207	(1)
Summary of Smoothness Criteria	208	(2)
Table of Minimal Bad Patterns	210	(2)
Singular loci of A5, B4, C4, D4	212	(27)
Bibliography	239	(8)
Index	247