Ebook: Singular Loci of Schubert Varieties

"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.

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"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.

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"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.

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-5
Generalities on G / B and G / Q ....Pages 7-21
Specifics for the Classical Groups....Pages 23-36
The Tangent Space and Smoothness....Pages 37-46
Root System Description of T(w, ?) ....Pages 47-69
Rational Smoothness and Kazhdan—Lusztig Theory....Pages 71-89
Nil-Hecke Ring and the Singular Locus of X(w)....Pages 91-102
Patterns, Smoothness and Rational Smoothness....Pages 103-117
Minuscule and cominuscule G/P ....Pages 119-158
Rank Two Results....Pages 159-168
Related Combinatorial Results....Pages 169-173
Related Varieties....Pages 175-206
Addendum....Pages 207-237
Back Matter....Pages 239-251

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"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.

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-5
Generalities on G / B and G / Q ....Pages 7-21
Specifics for the Classical Groups....Pages 23-36
The Tangent Space and Smoothness....Pages 37-46
Root System Description of T(w, ?) ....Pages 47-69
Rational Smoothness and Kazhdan—Lusztig Theory....Pages 71-89
Nil-Hecke Ring and the Singular Locus of X(w)....Pages 91-102
Patterns, Smoothness and Rational Smoothness....Pages 103-117
Minuscule and cominuscule G/P ....Pages 119-158
Rank Two Results....Pages 159-168
Related Combinatorial Results....Pages 169-173
Related Varieties....Pages 175-206
Addendum....Pages 207-237
Back Matter....Pages 239-251
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