Double Bruhat Cells and Total Positivity
نویسندگان
چکیده
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 determinantal identities 346 2.4. Affine coordinates in Schubert cells 349 2.5. y-coordinates in double Bruhat cells 351 2.6. Factorization problem in Schubert cells 353 2.7. Totally positive bases for N−(w) 354 2.8. Total positivity in y-coordinates 356 3. Proofs of the main results 357 3.1. Proofs of Theorems 1.1, 1.2, and 1.3 357 3.2. Proofs of Theorems 1.6 and 1.7 359 3.3. Proof of Theorem 1.9 360 3.4. Proofs of Theorems 1.11 and 1.12 363 4. GLn theory 364 4.1. Bruhat cells and double Bruhat cells for GLn 364 4.2. Factorization problem for GLn 365 4.3. The twist maps for GLn 367 4.4. Double pseudoline arrangements 369 4.5. Solution to the factorization problem 371 4.6. Applications to total positivity 374 References 379
منابع مشابه
Double Bruhat Cells and Total Positivity Sergey Fomin and Andrei Zelevinsky
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 1...
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