Double Bruhat Cells and Total Positivity Sergey Fomin and Andrei Zelevinsky
نویسنده
چکیده
0. Introduction 2 1. Main results 3 1.1. Semisimple groups 3 1.2. Factorization problem 4 1.3. Total positivity 5 1.4. Generalized minors 5 1.5. The twist maps 6 1.6. Formulas for factorization parameters 7 1.7. Total positivity criteria 8 1.8. Fundamental determinantal identities 9 2. Preliminaries 10 2.1. Involutions 10 2.2. Commutation relations 10 2.3. Generalized determinantal identities 12 2.4. Affine coordinates in Schubert cells 15 2.5. y-coordinates in double Bruhat cells 17 2.6. Factorization problem in Schubert cells 18 2.7. Totally positive bases for N−(w) 20 2.8. Total positivity in y-coordinates 22 3. Proofs of main results 23 3.1. Proofs of Theorems 1.1, 1.2, and 1.3 23 3.2. Proofs of Theorems 1.6 and 1.7 25 3.3. Proof of Theorem 1.9 26 3.4. Proofs of Theorems 1.11 and 1.12 28 4. GLn theory 29 4.1. Bruhat cells and double Bruhat cells for GLn 30 4.2. Factorization problem for GLn 30 4.3. The twist maps for GLn 33 4.4. Double pseudoline arrangements 34 4.5. Solution to the factorization problem 36 4.6. Applications to total positivity 40 References 44
منابع مشابه
Double Bruhat Cells and Total Positivity
0. Introduction 336 1. Main results 337 1.1. Semisimple groups 337 1.2. Factorization problem 338 1.3. Total positivity 339 1.4. Generalized minors 339 1.5. The twist maps 340 1.6. Formulas for factorization parameters 341 1.7. Total positivity criteria 343 1.8. Fundamental determinantal identities 343 2. Preliminaries 344 2.1. Involutions 344 2.2. Commutation relations 345 2.3. Generalized det...
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In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
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