S. Fomin and A. Zelevinsky. Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 (1999), no. 2, 335--380.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the reader to the references [9] 10] 11] 12] Instead, I will include in the last section some remarks on an application of birational Weyl group actions to totally positive matrices; this part, motivated by a series of works by G. Lusztig [4] A. Berenstein, S. Fomin and A. Zelevinsky [2] [3], is based on a discussion with Y. Yamada. Notes: By a birational W action on an ane space X , W being any group, we mean a realization of W as a group of birational transformations of X . It is equivalent to giving a group homomorphism : W Aut(K(X) from W to the group of automorphisms of ....

.... . For comparison, we will show how Young diagrams are generated by the action of simple re ections in the cases of A1 and A 2 (n = 3) See Figure 2. In this section we give an application of our framework of birational Weyl group actions to some problems which have been discussed by [4] 2] [3] in relation to the parametrization of totally positive matrices. 39 A1 0 0 R 0 0 1 0 2 0 1 2 1 0 1 2 2 0 1 2 0 1 0 0 1 2 2 1 0 1 2 0 Figure 2. Generating Young diagrams We keep the notation of the semisimple Lie algebra g associated with Cartan matrix ....

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S. Fomin and Z. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12(1999), 335-380.

....and any reduced decomposition i = i 1 ,i 2 , i N) of the permutation w # S n , the map P i : x 1 ,x 2 , x N)## P i 1 (x 1 ) P i 2 (x 2 ) P i N (x N) is a bijection from R N to the Schubert cell Cw # GL n (R) Theorem 1.1 is known when R is a field (cf. Sp] Lemma 10.2. 6; see also [FZ], Proposition 2.11) The proof we give for an arbitrary ring in Section 2.5 is based on Relations (1.4a b) and on a matrix identity (given in Proposition 2.1) which is of independent interest. Let us illustrate this theorem with the permutation w = # 12345 45321 # #S 5 . 1.5) 3 It is ....

S. Fomin, A. Zelevinsky, Double Bruhat cells and total positivity, preprint IRMA Strasbourg

....we will demonstrate that the complexity of the two problems is the same: informally speaking, it takes as much time to recognize a cell as it takes to describe it. Our interest in these problems was originally motivated by their relevance to the theory of total positivity criteria. As shown in [5], these criteria take different form Date: July 16, 1998. 1991 Mathematics Subject Classification. Primary 14M15, Secondary 05E15, 06A07, 20F55. Key words and phrases. Schubert cell, Schubert variety, flag manifold, Plucker coordinates, Bruhat cell, vanishing pattern. The authors were supported ....

S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, University of Strasbourg, preprint IRMA-98008.

....between total positivity and canonical bases for quantum groups, discovered by G. Lusztig [33] cf. also the surveys [31, 34] Among other things, he extended the subject by defining totally positive and totally nonnegative elements for any reductive group. Further development of these ideas in [3, 4, 15, 17] aims at generalizing the whole body of classical determinantal calculus to any semisimple group. As it often happens, putting things in a more general perspective shed new light on this classical subject. In the next two sections of this paper, we provide self contained proofs (many of them new) ....

....In the next two sections of this paper, we provide self contained proofs (many of them new) of the fundamental results on problems (i) ii) due to A. Whitney [46] C. Loewner [32] C. Cryer [9, 10] and M. Gasca and J. M. Pe na [23] The rest of the paper presents more recent results obtained in [15]: a family of efficient total positivity criteria, and explicit formulas for expanding a generic matrix into a product of elementary Jacobi matrices. These results and their proofs can be generalized to arbitrary semisimple groups [4, 15] but we do not discuss this here. Our approach to the ....

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S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12 (1999), 335--380.

....Plenary and Invited Lectures 13. Combinatorics classical theory of totally positive matrices and its recent generalization to an arbitrary reductive group due to Lusztig [9] give rise to other algebraic and combinatorial problems. If time allows, some results in this direction obtained in [1, 3, 4] will be presented. ....

S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, Pr'e--publication 1998/08, IRMA, Universit'e Louis Pasteur, Strasbourg, 1998.

....properties of the canonical basis from geometric properties of totally positive varieties. To do this, we rely on (and further develop) the calculus of generalized minors and its applications to the study of totally positive varieties in Schubert cells and double Bruhat cells developed in [3, 9, 11, 12]. An intriguing feature of our results is that combinatorial parametrizations of the canonical basis and related expressions for tensor product multiplicities for a semisimple Lie algebra g are expressed in terms of geometry of totally positive varieties in the Langlands dual semisimple group L ....

....geometric counterpart of Kashiwara s crystal operators which play an important part in our arguments. The remaining sections contain the proofs of all the results in this paper. The proofs are not very di#cult because most of the needed geometric technique was developed in the preceding papers [3, 9, 11]. However we need a number of important modifications to the geometric setup developed in the previous papers: for example, we replace double Bruhat cells from [11] by reduced double Bruhat cells introduced in Section 4 below. 2. Tensor product multiplicities 2.1. Background on semisimple Lie ....

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S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), no. 2, 335--380.

....from geometry. In the case when W is the ( nite) Weyl group of a simply laced root system, we expect that the i (F 2 ) orbits in F m 2 enumerate connected components of the real part of the reduced double Bruhat cell corresponding to (u; v) Double Bruhat cells were introduced and studied in [4] as a natural framework for the study of total positivity in semisimple groups; as explained to us by N. Reshetikhin, they also appear naturally in the study of symplectic leaves in semisimple groups (see [6] Let us brie y recall their de nition. Let G be a split simply connected semisimple ....

....u;v by left (or right) translations, and L u;v is a section of this action. Thus G u;v is biregularly isomorphic to H L u;v , and all properties of G u;v can be translated in a straightforward way into the corresponding properties of L u;v (and vice versa) In particular, Theorem 1. 1 in [4] implies that L u;v is biregularly isomorphic to a Zariski open subset of an ane space of dimension (u) v) Conjecture 7.1. For every two elements u and v in W , and every reduced word i 2 R(u; v) the connected components of L u;v (R) are in a natural bijection with the i (F 2 ) orbits ....

[Article contains additional citation context not shown here]

S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335-380.

....we will demonstrate that the complexity of the two problems is the same: informally speaking, it takes as much time to recognize a cell as it takes to describe it. Our interest in these problems was originally motivated by their relevance to the theory of total positivity criteria. As shown in [5], these criteria take different form in different Bruhat cells BwB, so one has to first find out which cell an element g 2 G is in. Date: July 17, 1998. 1991 Mathematics Subject Classification. Primary 14M15, Secondary 05E15, 06A07, 20F55. Key words and phrases. Schubert cell, Schubert variety, ....

S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, University of Strasbourg, preprint IRMA-98008.

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S. Fomin and A. Zelevinsky. Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 (1999), no. 2, 335--380.

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