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Parametrizations of Canonical Bases and Totally Positive Matrices Arkady Berenstein

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Parametrizations of Canonical Bases and Totally Positive Matrices Arkady Berenstein Advances in Mathematics AI1567 advances in mathematics 122, 49149 (1996) article no. 0057 Parametrizations of Canonical Bases and Totally Positive Matrices Arkady Berenstein Department of Mathematics, Northeastern University, Boston, Massachusetts 02115 Sergey Fomin* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; and Theory of Algorithms Laboratory, St. Petersburg Institute of Informatics, Russian Academy of Sciences, St. Petersburg, Russia and Andrei Zelevinsky- Department of Mathematics, Northeastern University, Boston, Massachusetts 02115 Received October 24, 1995; accepted November 27, 1995 Contents. 1. Introduction and main results. 2. Lusztig variety and Chamber Ansatz. 2.1. Semifields and subtraction-free rational expressions. 2.2. Lusztig variety. 2.3. Pseudo-line arrangements. 2.4. Minors and the nilTemperleyLieb algebra. 2.5. Chamber Ansatz. 2.6. Solutions of the 3-term relations. 2.7. An alternative description of the Lusztig variety. 2.8. Trans- 0 n n ition from n . 2.9. Polynomials T J and Za . 2.10. Formulas for T J . 2.11. Sym- metries of the Lusztig variety. 3. Total positivity criteria. 3.1. Factorization of unitriangular matrices. 3.2. General- izations of Fekete's criterion. 3.3. Polynomials TJ (x). 3.4. The action of the four- group on N >0. 3.5. The multi-filtration in the coordinate ring of N. 3.6. The S3 _ZÂ2Z symmetry. 3.7. Dual canonical basis and positivity theorem. 4. Piecewise-linear minimization formulas. 4.1. Polynomials over the tropical semi- field. 4.2. Multisegment duality. 4.3. Nested normal orderings. 4.4. Quivers and boundary pseudo-line arrangements. 5. The case of an arbitrary permutation. 5.1. Generalization of Theorem 2.2.3. 5.2. Extending the results of Sections 2.3, 2.4. 5.3. Chamber sets and Chamber Ansatz for arbitrary w. 5.4. Generalizations of Theorems 3.1.1 and 3.2.5. Appendix. Connections with the YangBaxter equation. * Partially supported by the NSF Grant, DMS-9400914. - Partially supported by the NSF Grant, DMS-9304247. 49 0001-8708Â96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 607J I56701 . By:BV . Date:26:08:96 . Time:13:44 LOP8M. V8.0. Page 01:01 Codes: 4487 Signs: 2113 . Length: 50 pic 3 pts, 212 mm 50 BERENSTEIN, FOMIN, AND ZELEVINSKY 1. INTRODUCTION AND MAIN RESULTS Let N be the maximal unipotent subgroup of a semisimple group of simply-laced type; let m=dim N. Recently, Lusztig [29, 30] discovered a remarkable parallelism between (1) labellings (by m-tuples of nonnegative integers) of the canonical basis B of the quantum group corresponding to N, and (2) parametrizations of the variety N>0 of totally positive elements in N by m-tuples of positive real numbers. In each case, there is a natural family of parametrizations, one for each reduced word of the element of maximal length in the Weyl group. In this paper, the following two problems are solved for the type A. Problem I. For any pair of reduced words, find an explicit formula for the transition map that relates corresponding parametrizations of B or N>0 . Problem II. Each parametrization in (2) is a restriction of a birational isomorphism Cm Ä N. For any reduced word, find an explicit formula for the inverse map N Ä Cm. Our results have the following applications: (i) a new proof and generalization of piecewise-linear minimization formulas for quivers of type A given in [23, 24], and (ii) a new family of criteria for total positivity generalizing and refining the classical Fekete criterion [13]. In order to treat Problems I and II simultaneously, we develop a general framework for studying Lusztig's transition maps, in which the role of scalars is played by an arbitrary zerosumfree semifield. Our methods rely on an interpretation of reduced words via pseudo-line arrangements. The solutions of the above problems are then obtained by means of a special substitution that we call the Chamber Ansatz. We now begin a systematic account of our main results. Since we shall only treat the Ar type, N will be the group of unipotent upper triangular r+1 matrices of order r+1; its dimension is equal to m=( 2 ). Then Problem II can be reformulated as the following problem in linear algebra. 1.1. Problem. For a generic unipotent upper-triangular matrix x of order r+1, find explicit formulas for the factorizations of x into the mini- mal number of elementary Jacobi matrices. File: 607J I56702 . By:BV . Date:26:08:96 . Time:13:44 LOP8M. V8.0. Page 01:01 Codes: 2737 Signs: 2118 . Length: 45 pic 0 pts, 190 mm CANONICAL BASES AND TOTAL POSITIVITY 51 To be more precise, by an elementary Jacobi matrix we mean a matrix of the form 1 . 1 t 1+te = i 1 . \ 1+ where ei is an (i, i+1) matrix unit. The minimal possible number of such matrices needed for factoring a generic x # N is easily seen to be dim N= r+1 m=( 2 ). Then, for a sequence h=(h1, ..., hm) of indices, we consider fac- torizations x=(1+t1eh1)}}}(1+tmehm). (1.1) It is not hard to show that such a factorization of a generic matrix x # N exists and is unique if and only if h is a reduced word of the element w0 of maximal length in the symmetric group Sr+1, i.e., w0=sh1 }}}shm where the si=(i, i+1) are adjacent transpositions. Moreover, for a given reduced word h, each coefficient tk in (1.1) is a h rational function in the matrix entries xij that we denote by tk to emphasize the role of h. Thus the map t [ x=x h(t) defined by (1.1) is a birational isomorphism Cm Ä N. In this notation, Problem 1.1 asks for explicit com- putation of the inverse rational map t h: N Ä Cm. Example. 1 2 Let r=2. In this case, there are two reduced words for w0 : h=121, h$=212. h h$ Denote tk(x)=tk, tk (x)=t$k for k=1, 2, 3. Thus we are interested in the factorizations x=(1+t1e1)(1+t2e2)(1+t3e1)=(1+t$1e2)(1+t$2e1)(1+t$3 e2) (1.2) of a generic unitriangular 3_3-matrix x. Multiplying the matrices in (1.2), we obtain 1 x12 x13 1 t1+t3 t1 t2 1 t$2 t$2 t$3 x= 01x23 = 01 t2= 01t$1+t$3 . (1.3) \00 1+ \ 0 0 1+ \ 00 1+ File: 607J I56703 . By:BV . Date:26:08:96 . Time:13:44 LOP8M. V8.0. Page 01:01 Codes: 2910 Signs: 1377 . Length: 45 pic 0 pts, 190 mm 52 BERENSTEIN, FOMIN, AND ZELEVINSKY Solving these equations for the tk and t$k , we find x13 x12 x23&x13 t1= , t2=x23 , t3= (1.4) x23 x23 and x12 x23&x13 x13 t$1= , t$2=x12 , t$3= . (1.5) x12 x12 Our general solution of Problem 1.1 will require two ingredients. The first is the following combinatorial construction. Let h=(h1, ..., hm)bea reduced word for w0 . For each entry hk of h, define the set L/[1, ..., r+1] and two integers i and j by L=shm }}}shk+1([1, ..., hk&1]) i=shm }}}shk+1(hk) (1.6) j=shm }}}shk+1(hk+1) For example, for h=213231 and k=3, we obtain (i, j)=(1, 3) and L=[2, 4]. As k ranges from 1 to m, the pair (i, j) given by formulas (1.6) ranges over the m pairs of integers (i, j) satisfying 1i< jr+1, each of these pairs appearing exactly once. This is a special case of a result in [5, VI.1.6], if one identifies such pairs (i, j) with the positive roots of type Ar . Another ingredient of our solution of Problem 1.1 is a certain birational transformation of the group N which is defined as follows. For a matrix g, let [ g]+ denote the last factor u in the Gaussian LDU-decomposition g=vT } d } u where u # N, v # N, and d is diagonal. (Such decomposition exists and is unique for a sufficiently generic g # GLr+1.) The matrix entries of [ g]+ are rational functions of g; explicit formulas can be written in terms of the minors of g. The following linear-algebraic fact is not hard to prove. Lemma. T 1 3 The map y [ x=[w0y ]+ is a birational automorphism of N.(Here w0 is identified with the corresponding permutation matrix.) The inverse birational automorphism x [ y is given by &1 &1 T y=w0 [xw0 ] + w0 . File: 607J I56704 . By:BV . Date:26:08:96 . Time:13:44 LOP8M. V8.0. Page 01:01 Codes: 2529 Signs: 1564 . Length: 45 pic 0 pts, 190 mm CANONICAL BASES AND TOTAL POSITIVITY 53 For example, in the case r=2, the birational automorphism x [ y is given by x 1 1 23 x12 x23&x13 x13 1 x12 x13 x= 0 1 x23 [ x12 = y. \0 0 1 + 01 x13 \00 1+ In what follows, [i, j] will stand for [a # Z: ia j]. For a subset J J/[1, r+1] and a matrix g # GLr+1, let 2 (g) denote the minor of g that occupies several first rows and whose column set is J. We are now in a position to present a solution to Problem 1.1 (cf. Theorem 3.1.1). 1.4. Theorem. For x # N, define the matrix y as in Lemma 1.3.
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