SERGEY FOMIN AND ANDREI ZELEVINSKY Tota Positivity: Tests and Param etrizat io n s
Introduction positivity and canonical bases for quantum groups, dis- A matrix is totally positive (resp. totally non-negative) if covered by G. Lusztig [33] (cf. also the surveys in [31, all its minors are positive (resp. non-negative) real num- 34]). Among other things, he extended the subject by bers. The first systematic study of these classes of matri- defining totally positive and totally non-negative elements ces was undertaken in the 1930s by F. R. Gantmacher and for any reductive group. Further development of these M. G. Krein [20-22], who established their remarkable spec- ideas in [3, 4, 15, 17] aims at generalizing the whole body tral properties (in particular, an n X n totally positive ma- of classical determinantal calculus to any semisimple trix x has n distinct positive eigenvalues). Earlier, I. J. group. Schoenberg [41] had discovered the connection between As often happens, putting things in a more general per- total non-negativity and the following variation-dimin- spective shed new light on this classical subject. In the next ishing property: the number of sign changes in a vector two sections, we provide serf-contained proofs (many of does not increase upon multiplying by x. them new) of the fundamental results on problems 1 and Total positivity found numerous applications and was 2, due to A. Whitney [46], C. Loewner [32], C. Cryer [9, 10], studied from many different angles. An incomplete list in- and M. Gasca and J. M. Pefia [23]. The rest of the article cludes oscillations in mechanical systems (the original mo- presents more recent results obtained in [15]: a family of tivation in [22]), stochastic processes and approximation efficient total positivity criteria and explicit formulas for theory [25, 28], P61ya frequency sequences [28, 40], repre- expanding a generic matrix into a product of elementary sentation theory of the infmite symmetric group and the Jacobi matrices. These results and their proofs can be gen- Edrei-Thoma theorem [13, 44], planar resistor networks eralized to arbitrary semisimple groups [4, 15], but we do [11], unimodality and log-concavity [42], and theory of im- not discuss this here. manants [43]. Further references can be found in S. Karlin's Our approach to the subject relies on two combinator- book [28] and in the surveys [2, 5, 38]. ial constructions. The first one is well known: it associates In this article, we focus on the following two problems: a totally non-negative matrix to a planar directed graph with positively weighted edges (in fact, every totally non- 1. parametrizing all totally non-negative matrices negative matrix can be obtained in this way [6]). Our sec- 2. testing a matrix for total positivity ond combinatorial tool was introduced in [15]; it is a par- Our interest in these problems stemmed from a surpris- ticular class of colored pseudoline arrangements that we ing representation-theoretic connection between total call the double wiring diagrams.
9 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 1, 2000 23 1 f 1 det(x) = Z Z sgn(w) eK~r), (2) W oT the sum being over all permutations w in the symmetric group Sn and over all collections of paths Ir = (1rl,..., ~n) 2 e 2 1 1 such that vi joins the source i with the sink w(i). Any col- lection ~r of vertex-disjoint paths is associated with the identity permutation; hence, eK~r) appears in (2) with the 1 9 d 91 positive sign. We need to show that all other terms in (2) Figure 1. A planar network. cancel out. Deforming F a bit ff necessary, we may assume that no two vertices lie on the same vertical line. This makes the following involution on the non-vertex-disjoint Planar Networks collections of paths well defmed: take the leftmost com- To the uninitiated, it might be unclear that totally positive mon vertex of two or more paths in ~r, take two smallest matrices of arbitrary order exist at all. As a warm-up, we indices i and j such that 7ri and ~. contain v, and switch invite the reader to check that every matrix given by the parts of ~ri and ~. lying to the left of v. This involution d dh dhi 1 preserves the weight of ~r while changing the sign of the associated permutation w; the corresponding pairing of bd bdh + e bdhi + eg + ei J , (1) terms in (2) provides the desired cancellation. [] abd abdh + ae + ce abdhi + (a + c)e(g + i) + f COROLLARY 2. If a planar network has non-negative where the numbers a, b, c, d, e,f, g, h, are i are positive, is real weights, then its weight matrix is totally non-nega- totally positive. It will follow from the results later that tive. every 3 x 3 totally positive matrix has this form. We will now describe a general procedure that produces As an aside, note that the weight matrix of the network totally non-negative matrices. In what follows, a planar network (F, to) is an acyclic directed planar graph F whose edges e are assigned scalar weights to(e). In all of our ex- amples (cf. Figures 1, 2, 5), we assume the edges of F di- rected left to right. Also, each of our networks will have n sources and n sinks, located at the left (resp. right) edge of the picture, and numbered bottom to top. (with unit edge weights) is the "Pascal triangle" The weight of a directed path in F is defmed as the prod- uct of the weights of its edges. The weight matrix x(F, to) 10000 ... is an n • n matrix whose (i, 2)-entry is the sum of weights 11000 ... of all paths from the source i to the sink j; for example, the weight matrix of the network in Figure 1 is given by (1). 12100 The minors of the weight matrix of a planar network 13310--. have an important combinatorial interpretation, which can 14641..- be traced to B. Lindstr0m [30] and further to S. Karlin and G. McGregor [29] (implicit), and whose many applications were given by I. Gessel and G. X. Viennot [26, 27]. which is totally non-negative by Corollary 2. Similar argu- In what follows, hi, j(x) denotes the minor of a matrix ments can be used to show total non-negativity of various x with the row set I and the column set J. other combinatorial matrices, such as the matrices of q-bino- The weight of a collection of directed paths in F is de- mial coefficients, Stiding numbers of both kinds, and so forth. fmed to be the product of their weights. We call a planar network F totally connected if for any LEMMA 1 (Lindstr0m's Lemma). A minor AI, J of the two subsets/, J C [1, n] of the same cardinality, there ex- weight matmx of a planar network is equal to the sum of ists a collection of vertex-disjoint paths in F connecting the weights of aU collections of vertex-disjoint paths that con- sources labeled by I with the sinks labeled by J. nect the sources labeled by I with the sinks labeled by J. COROLLARY 3. If a totally connected planar network has To illustrate, consider the matrix x in (1). We have, for positive weights, then its weight matrix is totally positive. example, A23,23(X) = bcdegh + bdfh + fe, which also For any n, let F0 denote the network shown in Figure 2. equals the sum of the weights of the three vertex-disjoint Direct inspection shows that F0 is totally connected. path collections in Figure 1 that connect sources 2 and 3 to sinks 2 and 3. COROLLARY 4. For any choice of positive weights eke), the weight matrix x(F0, to) is totally positive. Proof. It suffices to prove the lemma for the determinant of the whole weight matrix x = x(F, to) (i.e., for the case It turns out that this construction produces al/totally I = J = [1, n]). Expanding the determinant, we obtain positive matrices; this result is essentially equivalent to
24 THE MATHEMATICAL INTELLIGENCER To illustrate Lemma 6, consider the special case n = 3. The network F0 is shown in Figure 1; its essential edges have the weights a, b,..., i. The weight matrix x(F0, w) is given in (1). Its initial minors are given by the monomials
hl, 1 = d, A1,2 = dh, Z~l,3 = dhi, A2,1 = bd, A12,12 ---- d_e, A12,23 = degh, 5a,1 = abd, A23,12 = bc_de, A123,123= del, where for each minor A, the "leading entry" w(e(A)) is un- Figure 2. Planar network s derlined. To complete the proof of Theorem 5, it remains to show that every totally positive matrix x has the form x(F0, ~o) for A. Whitney's Reduction Theorem [46] and can be sharp- some essential positive weighting w. By Lemma 6, such an ened as follows. Call an edge of F0 essential if it either is w can be chosen so that x and x(s w) will have the same slanted or is one of the n horizontal edges in the middle of initial minors. Thus, our claim will follow from Lemma 7. the network. Note that F0 has exactly n 2 essential edges. A weighting w of F0 is essential if w(e) r 0 for any essen- LEMMA 7. A square matrix x is uniquely determined by tial edge e and w(e) = 1 for all other edges. its initial minors, provided all these minors are nonzero. TItEOREM 5. The map ~o ~ X(Fo, w) restricts to a bijec- Proof. Let us show that each matrix entry x~j ofx is uniquely tion between the set of all essential positive weightings determined by the initial minors. If i = 1 orj = 1, there is of F0 and the set of all totally positive n x n matrices. nothing to prove, since xij is itself an initial minor. Assume that min(i,j) > 1. Let A be the initial minor whose last row The proof of this theorem will use the following notions. is i and last column is j, and let A' be the initial minor ob- A minor A[,j is called solid if both I and J consist of sev- tained from A by deleting this row and this column. Then, eral consecutive indices; if, furthermore, I U J contains 1, h = h'xij + P, where P is a polynomial in the matrix en- then hz,z is called initial (see Fig. 3). Each matrix entry is tries xi,j, with (i',j') r (i,3~ and i' <- i and j' <- j. Using the lower-right corner of exactly one initial minor; thus, induction on i + j, we can assume that each xi,j, that oc- the total number of such minors is n 2. curs in P is uniquely determined by the initial minors, so LEMMA 6. The n 2 weights of essential edges in an es- the same is true for xij = (h -- p)/h'. This completes the sential weighting w of F0 are related to the n 2 initial mi- proofs of Lemma 7 and Theorem 5. [] nors of the weight matrix x = x(F0, w) by an invertible Theorem 5 describes a parametrization of totally pos- monomial transformation. Thus, an essential weighting itive matrices by n2-tuples of positive reals, providing a par- w of F0 is uniquely recovered from x. tial answer (one of the many possible, as we will see) to the first problem stated in the Introduction. The second Proof. The network F0 has the following easily verified problem--that of testing total positivity of a matrix--can property: For any set I of k consecutive indices in [1, n], also be solved using this theorem, as we will now explain. there is a unique collection of k vertex-disjoint paths con- An n x n matrix has altogether (2nn) - 1 minors. This necting the sources labeled by [1, k] (resp. by/) with the makes it impractical to test positivity of every single mi- sinks labeled by I (resp. by [1, k]). These paths are shown nor. It is desirable to find efficient criteria for total posi- by dotted lines in Figure 2, for k = 2 and I = [3, 4]. By tivity that would only check a small fraction of all minors. Lindstr6m's lemma, every initial minor h of x(F0, w) is equal to the product of the weights of essential edges cov- EXAMPLE 8, A 2 x 2 matrix ered by this family of paths. Note that among these edges, there is always a unique uppermost essential edge e(A) (in- x:[: :] dicated by the arrow in Figure 2). Furthermore, the map h ~ e(h) is a bijection between initial minors and essen- has (4) - 1 = 5 minors: four matrix entries and the deter- tial edges. It follows that the weight of each essential edge minant A = oxt - bc. To test that x is totally positive, it is e = e(h) is equal to h times a Laurent monomial in some enough to check the positivity of a, b, c, and A; then, d = initial minors h', whose associated edges e(A') are located (A + bc)/a > O. below e. [] The following theorem generalizes this example to ma- trices of arbitrary size; it is a direct corollary of Theorem 5 and Lemmas 6 and 7.
TIIEOREM 9. A square matrix is totally positive if and only if all its initial minors (see Fig. 3) are positive.
This criterion involves n 2 minors, and it can be shown that this number cannot be lessened. Theorem 9 was Figure 3. Initial minors. proved by M. Gasca and Pefia [23, Theorem 4.1] (for rec-
VOLUME 22, NUMBER 1, 2000 25 tangular matrices); it also follows from Cryer's results in and [9]. Theorem 9 is an enhancement of the 1912 criterion by xi(t) = I + tEi+l, i = (xi(t)) T M. Fekete [14], who proved that the positivity of all solid minors of a matrix implies its total positivity. (the transpose of xi(t)). Also, for i = 1,..., n and t r 0, let
Theorems of Whitney and Loewner x| = I + (t - 1)El,i, In this article, we shall only consider invertible totally non- the diagonal matrix with the ith diagonal entry equal to t negative n x n matrices. Although these matrices have real and all other diagonal entries equal to 1. Thus, elementary entries, it is convenient to view them as elements of the Jacobi matrices are precisely the matrices of the form x i(t), general linear group G = GLn(C). We denote by G_>0 (resp. x~(t), and xQ(t). An easy check shows that they are totally G>0) the set of all totally non-negative (resp. totally posi- non-negative for any t > 0. tive) matrices in G. The structural theory of these matri- For any word i = (il, 9 9 9 il) in the alphabet ces begins with the following basic observation, which is ~g={1, ...,n-l,(D,...,(~,l,...,n-1}, (3) an immediate corollary of the Binet-Cauchy formula. PROPOSITION 10. Both G>0 and G>0 are closed under we define the product map xi : (C\{0})L --~ G by matrix multiplication. Furthermore, if x ~ G>_o and y xi(tl, ..., tl) = xil(tl)'"xi~(tl). (4) G>0, then both xy and yx belong to G>0. (Actually, xi(tl, 9 9 9 tl) is well defined as long as the right- Combining this proposition with the foregoing results, hand side of (4) does not involve any factors of the we will prove the following theorem of Whitney [46]. form x| To illustrate, the word i = (~) 1 (~) 1 gives rise THEOREM 11. (Whitney's theorem). Every invertible to- to tally non-negative matrix is the limit of a sequence of to- tally positive matrices. x tt2,3t, = [: :] [,: :] Thus, G_>0 is the closure of G>0 in G. (The condition of ::1 [: 1 invertibility in Theorem ll can, in fact, be lifted.) =[t tit4] Proof. First, let us show that the identity matrix I lies in t2 t2t4 + t3 the closure of G>0. By Corollary 4, it suffices to show that We will interpret each matrix xi(tl, 9 9tl) as the weight I = llnlg___.~ x(l~0, (ON) for some sequence of positive weight- matrix of a planar network. First, note that any elemen- ings Wg of the network F0. Note that the map w ~ x(F0, w) tary Jacobi matrix is the weight matrix of a "chip" of one is continuous and choose any sequence of positive weight- of the three kinds shown in Figure 4. In each "chip," all ings that converges to the weighting w0 defined by w0(e) = edges but one have weight 1; the distinguished edge has 1 (resp. 0) for all horizontal (resp. slanted) edges e. Clearly, weight t. Slanted edges connect horizontal levels i and x(F0, wo) =/, as desired. i § 1, counting from the bottom; in all examples in Figure To complete the proof, write any matrix x E G_>0 as x = 4, i=2. limg_~ X" X(F0, raN), and note that all matrices x- x(F0, WN) The weighted planar network (F(i), w(tl, 9 9 9 , tl)) is then are totally positive by Proposition 10. [] constructed by concatenating the "chips" corresponding to The following description of the multipllcative monoid consecutive factors xik (tk), as shown in Figure 5. It is easy G_>0 was first given by Loewner [32] under the name to see that concatenation of planar networks corresponds "Whitney's Theorem"; it can indeed be deduced from [46]. to multiplying their weight matrices. We conclude that the product xi(tl, 9 9 9 tl) of elementary Jacobi matrices equals THEOREM 12 (Loewner-Whitney theorem). Any invert- the weight matrix x(F(i), w(tl,..., tt)). ible totally non-negative matrix is a product of elemen- In particular, the network (F0, co) appearing in Figure tary Jacobi matrices with non-negative matmx entries. 2 and Theorem 5 (more precisely, its equivalent defor- Here, an "elementary Jacobi matrix" is a matrix x @ G mation) corresponds to some special word imax of length that differs from I in a single entry located either on the n2; instead of defining imax formally, we just write it for main diagonal or immediately above or below it. n=4:
Proof. We start with an inventory of elementary Jacobi ma- ima~ = (3,2,3, 1, 2, 3, (~), (~), (~),(~), 3, 2, 3, 1,2,3). trices. Let Eij denote the n • n matrix whose (i,2)-entry is 1 and all other entries are 0. For t E C and i = 1,..., n - 1, let In view of this, Theorem 5 can be reformulated as follows. 1 "" 0 O "'" O THEOREM 13. The product map Xim~, restricts to a bi-
: : : : : : jection between n2-tuples of positive real numbers and to- tally positive n • n matrices. 0 " 1 t -.. 0 xi(t) = I + tEi,i+l = We will prove the following refinement of Theorem 12, 0 ..- 0 1 --- 0 which is a reformulation of its original version [32]. : : : : : : TIIEOREM 14. Every matrix x ~ G>_o can be written as 0 ... 0 0 ... 1 x = xim~(tl, . .. , tn2),for some tl, 9 9 9 , tn2 >- O.
26 THE MATHEMATICAL INTELLIGENCER (Since x is invertible, we must in fact have tk ~ 0 for diagram consists of two families of n piecewise-straight n(n - 1)/2 < k <-- n(n + 1)/2 (i.e., for those indices k for lines (each family colored with one of the two colors), the which the corresponding entry of ima~ is of the form O).) crucial requirement being that each pair of lines of like color intersect exactly once. Proof. The following key lemma is due to Cryer [9]. The lines in a double wiring diagram are numbered sep- LEMMA 15. The leading principal minors h[1,k],[l,k] of a arately within each color. We then assign to every chamber matrix x ~ G >_ o are positive for k = 1,..., n. of a diagram a pair of subsets of the set [1, n] = {1,..., n}: each subset indicates which lines of the corresponding Proof. Using induction on n, it suffices to show that color pass below that chamber; see Figure 7. A[1,n-1],[1,n-1](X ) > O. Let hiJ(x) [resp. hii',JJ'(x)] denote the Thus, every chamber is naturally associated with a mi- minor of x obtained by deleting the row i and the column nor AI, g of an n • n matrix x = (xij) (we call it a chamber j (resp. rows i and i', and columnsj andj'). Then, for any minor) that occupies the rows and columns specified by 1 -< i < i' -< n and 1 -
VOLUME 22, NUMBER 1, 2000 27 Figure 6. Double wiring diagram. Figure 4. Elementary "chips." Laurent polynomials in question. We note a case in which the conjecture is true and the desired expressions can be e.g., [18, (15.53)]), and were obtained by Desnanot alongside given: the "lexicographically minimal" double wiring dia- (5) in the same 1819 publication we mentioned earlier. gram whose chamber minors are the initial minors. Indeed, Theorem 17 is now proved as follows. We first note that a generic matrix x can be uniquely written as the product any minor appears as a chamber minor in some double xim~x (tl,. 9 tn2) of elementary Jacobi matrices (cf. Theorem wiring diagram. Therefore, it suffices to show that the 13); then, each minor ofx can be written as a polynomial in chamber minors of one diagram can be written as sub- the tk with non-negative integer coefficients (with the help traction-free rational expressions in the chamber minors of LindstrSm's lemma), whereas each & is a Laurent mono- of any other diagram. This is a direct corollary of Lemma mial in the initial minors of x, by Lemma 6. 18 combined with the fact that any two diagrams are re- It is proved in [15, Theorem 1.13] that every minor can lated by a sequence of local moves: indeed, each local move be written as a Laurent polynomial with integer (possibly replaces Y by (AC + BD)/Z, or Z by (AC + BD)/Y. [] negative) coefficients in the chamber minors of a given di- agram. Note, however, that this result combined with Implicit in the above proof is an important combinato- Theorem 17, does not imply Conjecture 19, because there rial structure lying behind Theorems 16 and 17: the graph do exist subtraction-free rational expressions that are (I)~, whose vertices are the (isotopy classes of) double Laurent polynomials, although not with non-negative coef- wiring diagrams and whose edges correspond to local ficients (e.g., think of (p3 + q3)/(p + q) = p2 _ pq + q2). moves. The study of (Pn is an interesting problem in itself. The following special case of Conjecture 19 can be de- The first nontrivial example is the graph r shown in rived from [3, Theorem 3.7.4]. Figure 10. It has 34 vertices, corresponding to 34 different THEOREM 20. Conjecture 19 holds for all wiring dia- total positivity criteria. Each of these criteria tests nine mi- grams in which all intersections of one color precede the nors of a 3 • 3 matrix. Five of these minors [viz., x3t, x13, intersections of another color. A23,12 , A12,23, and det(x)] correspond to the "unbounded" chambers that lie on the periphery of every double wiring We do not know an elementary proof of this result; the diagram; they are common to all 34 criteria. The other four proof in [3] depends on the theory of canonical bases for minors correspond to the bounded chambers and depend quantum general linear groups. on the choice of a diagram. For example, the criterion de- rived from Figure 7 involves "bounded" chamber minors Digression: Somos sequences A3,2, A1,2, At3,12, and At3,23. In Figure 10, each vertex of qb3 The three-term relation AC + BD = YZ is surrounded by is labeled by the quadruple of "bounded" minors that ap- some magic that eludes our comprehension. We cannot re- pear in the corresponding total positivity criterion. sist mentioning the related problem involving the Somos- We suggest the following refinement of Theorem 17. 5 sequences [19]. (We thank Richard Stanley for telling us about them.) These are the sequences al, a2, 9 9 9in which CONJECTURE 19. Every minor of a generic square ma- any six consecutive terms satisfy this relation: trix can be written as a Laurent polynomial with non- negative integer coefficients in the chamber minors of an anan+5 = an+lan+4 + an+2an+3. (6) arbitrary double wiring diagram. Each term of a Somos-5 sequence is obviously a subtrac- Perhaps more important than proving this conjecture tion-free rational expression in the fn~st five terms al, 9 9 9 would be to give explicit combinatorial expressions for the a5. It can be shown by extending the arguments in [19, 35]
Figure 5. Planar network s Figure 7. Chamber minors.
28 THE MATHEMATICALINTELUGENCER li - Jl ~ 2, and sisysi = sjsisj for li - Jl = 1. A reduced word for a permutation w @ Sn is a word j = (Jb - - 9Jl) of the shortest possible length I = ((w) that satisfies w = sj~...sj~. The munber ((w) is called the length of w (it is the num- ber of inversions in w). The group Sn has a unique element w0 of maximal length: the order-reversing permutation of 1, ..., n; it gives ((w0) = (~). Figure 8. Lexicographically minimal diagram. It is straightforward to verify that the encodings of dou- ble wiring diagrams are precisely the shuffles of two re- that each an is actually a Laurent polynomial in ai, duced words for w0, in the barred and (mbarred entries, re- .. 9a5. This property is truly remarkable, given the nature spectively; equivalently, these are the reduced words for of the recurrence, and the fact that, as n grows, these the element (w0, w0) of the Coxeter group Sn X Sn. Laurent polynomials become huge sums of monomials in- volving large coefficients; still, each of these sums cancels DEFINITION 22. A word i in the alphabet ~ (see (3)) is out from the denonfinator of the recurrence relation an+5 = called afactorization scheme if it contains each circled en- try C) exactly once, and the remaining entries encode the (an+lan+4 + an+2an+3)/an. We suggest the following analog of Conjecture 19. heights of intersections in a double wiring diagram. Equivalently, a factorization scheme i is a shuffle of two CONJECTURE 21. Every term of a Somos-5 sequence is reduced words for w0 (one barred and one unbarred) and a Laurent polynomial with non-negative integer coeffi- an arbitrary permutation of the entries (~),..., (~). In par- cients in the first five terms of the sequence. ticular, i consists of n 2 entries.
FactorizaUon Schemes To illustrate, the word i = 2 1 (~) 2 T (~) 2 1 (~), appear- According to Theorem 16, every double wiring diagram ing in Figure 5 is a factorization scheme. gives rise to an "optimal" tot~ positivity criterion. We will _An important example of a factorization scheme is the now show that double wiring diagrams can be used to ob- word imax introduced in Theorem 13. Thus, the following tain a family of bijective parametrizations of the set G>0 of result generalizes Theorem 13. all totally positive matrices; this family will include the pa- rametrization in Theorem 13 as a special case. We encode a double wiring diagram by the word of length n(n- 1) in the alphabet {1, . . . , n- 1, t, . . . , n - 1} obtained by recording the heights of intersections of pseudolines of like color (traced left to right; barred dig- its for red crossings, unbarred for blue). For exam_pie, the diagram in Figure 6 is encoded by the word 2 1 2 1 2 1. The words that encode double wiring diagrams have an alternative description in terms of reduced expressions in the symmetric group S~. Recall that by a famous theorem of E. H. Moore [36], S.~ is a Coxeter group of type An-l; that is, it is generated by the involutions sl,. 9 9 Sn-1 (ad- jacent transpositions) subject to the relations sisj = sjsi for
Figure 9. Local "moves." Figure 10. Total positivity criteria for GL3.
VOLUME22, NUMBER1, 2000 ~19 THEOREM 23 [15]. For an arbitrary factorization Theorem 23 suggests an alternative approach to total scheme i = (il, 9 9 9 , in2), the product map xi given by (4) positivity criteria via the followingfactorization problem: restricts to a bijection between n2-tuples of positive real for a given factorization scheme i, fend the genericity con- numbers and totally positive n • n matrices. ditions on a matrix x assuring that x can be factored as
Proof. We have already stated that any two double wiring x = xi(tl, 9 9 9 , tn2) = xil(tl)'"x~2(tn2), (9) diagrams are connected by a succession of the local and compute explicitly the factorization parameters tk as "moves" shown in Figure 9. In the language of factoriza- functions of x. Then, the total positivity of x will be equiv- tion schemes, this translates into any two factorization alent to the positivity of all these functions. Note that the schemes being connected by a sequence of local transfor- criterion in Theorem 9 was essentially obtained in this way: mations of the form for the factorization scheme imax, the factorization para- 9..ij i... ~-~ ...j ij..., ii = 1, meters tk are Laurent monomials in the initial minors of x -Jl (7) 9..ij i... ~.~ ...j ij..., ii -Jl = 1, (cf. Lemma 6). A complete solution of the factorization problem for an or of the form arbitrary factorization scheme was given in [15, Theorems 9..ab... ~ ...ba..., (8) 1.9 and 4.9]. An interesting (and unexpected) feature of this solution is that, in general, the tk are not Laurent mono- where (a, b) is any pair of symbols in si different from (i, mials in the minors of x; the word imax is quite exceptional i _+ 1) or (i, i -+ 1). (This statement is a special case of Tits's in this respect. It turns out, however, that the tk are Laurent theorem [45], for the Coxeter group Sn X Sn • ($2)~.) monomials in the minors of another matrix x' obtained In view of Theorem 13, it suffices to show that if from x by the following birational transformation: Theorem 23 holds for some factorization scheme i, then it also holds for any factorization scheme i' obtained from i X' = [xTwo] +Wo(XT)- lwo[ woxT]-. (10) by one of the transformations (7) and (8). To see this, it is Here, x T denotes the transpose of x, and w0 is the permu- enough to demonstrate that the collections of parameters tation matrix with l's on the antidiagonal; finally, y = {tk} and {t'k} in the equality [Y]-[Y]0[Y] + denotes the Gaussian (LDU) decomposition of Xil(tl)'"Xin2 (tn 2) = xi,(t'l)'"xi;2 (t'~2) a square matrix y provided such a decomposition exists. In the special cases n = 2 and n = 3, the transformation are related to each other by (invertible) subtraction-free x ~ x' is given by rational transformations. The latter is a direct consequence of the commutation relations between elementary Jacobi [XllXllx211 X211 ] matrices, which can be found in [15, Section 2.2 and (4.17)]. X' = L X121 X22 det(x) -1 The most important of these relations are the following. First, fori=l,...,n-landj=i+ 1, we have and xi(tl) x| x~)(t3) x;(t4) = x-i(t~) x| x~)(t~) xi(t~), xll A1243 1 where X31 Xt3 X31 A12,23 X31 A13,12 x33A12,12-- det(x) x32 t3t4 t2t3 tit3 X r = t~ - T , t'2 = T, t3' - T , t~ - T , T = t2 + tlt3t4. Xl3 A23,12 A23,12 A12,23 A23,12 1 x23 A23,23 The proof of this relation (which is the only nontrivial re- x13 A12,23 det(x) lation associated with (8)) amounts to verifying that The following theorem provides an alternative explana- [: t:][t: t:][t: :]= [t~ :][~2 t0][: t14]. tion for the family of total positivity criteria in Theorem 16.
THEOREM 24 [15]. The right-hand side of (10) is well Also, for any i and j such that li -Jl = 1, we have the defined for any x E G>o; moreover, the "twist map" x ~ x' following relation associated with (7): restricts to a bijection of G>0 with itself. xi(h)xj(t2)xi(t3) = xj(tl)xi(t~)xj(t~), Let x be a totally positive n x n matrix, and i a fac- x~(h)x](t2)x~(t3) = x](t'l)x~(t~)x](t~), torization scheme. Then, the parameters tl, 9 9 9 tn2 ap- pearing in (9) are related by an invertible monomial where transformation to the n 2 chamber minors (for the double t2t3 tit2 wiring diagram associated with i) of the twisted matrix t'1 - T ' t' 2 = T, t3' - T , T = tl + t3. x' given by (10). One sees that in the commutation relations above, the for- In [15], we explicitly describe the monomial transfor- mulas expressing the t], in terms of the tl are indeed sub- mation in Theorem 24, as well as its inverse, in terms of traction-free. [] the combinatorics of the double wiring diagram.
30 THE MATHEMATICAL INTELLIGENCER Double Bruhat Cells nonvanishing of all "antiprincipal" minors All ,i l, In - i +1 ,n] (X) Our presentation in this section will be a bit sketchy; de- and A[n-i+l,nl,[1,i](x) for i = 1, . .., n - 1. tails can be found in [15]. Theorem 23 provides a family of bijective (and biregu- THEOREM 27 [15]. A totally non-negative matrix is of lar) parametrizations of the totally positive variety G>0 by type (u, v) if and only if it belongs to the double Bruhat n2-tuples of positive real numbers. The totally non-nega- cell G u,v- tive variety G_>0 is much more complicated (note that the In view of (11), Theorem 27 provides the following sim- map in Theorem 14 is surjective but not injective). In this ple test for total positivity of a totally non-negative matrix. section, we show that G_>0 splits naturally into "simple pieces" corresponding to pairs of permutations from Sn. COROLLARY 28 [23]. A totally non-negative matrix x is totally positive if and only if A[1,i],[n-i+l,n](X) r 0 and THEOREM 25 [15]. Let x E G>_o be a totally non-negative A[n-i+l,n],[1,i](x) r Ofor i = 1, . . . , n. matrix. Suppose that a word i in the alphabet ~ is such that x can be factored as x = xi(tl, . 9 tin) with positive The results obtained above for GW~'wo = G>0 (as well as tl, 9 9 tin, and i has the smallest number of uncircled en- their proofs) extend to the variety G>'~ for an arbitrary pair tries among all words with this property. Then, the sub- of permutations u, v E Sn. In particular, the factorization word of i formed by entries from {1, . . . , n- 1} (resp. schemes for (u, v) (or rather their uncircled parts) can be from { 1,..., n - 1 }) is a reduced word for some permu- visualized by double wiring diagrams of type (u, v) in the tation u (resp. v) in Sn. Furthermore, the pair (u, v) is same way as before, except now any two pseudolines in- uniquely determined by x (i.e., does not depend on the tersect at most once, and the lines are permuted "accord- choice of i). ing to u and v." Every such diagram has f(u) + s + n chamber minors, and their positivity provides a criterion In the situation of Theorem 25, we say that x is of type for a matrix x E G u,v to belong to G>0.U~V The factorization (u, v). Let G>'~ C G_>0 denote the subset of all totally non- problem and its solution provided by Theorem 24 extend negative matrices of type (u, v); thus, G_>0 is the disjoint to any double Bruhat cell, with an appropriate modifica- union of these subsets. tion of the twist map x ~ x'. The details can be found in Every subvariety G>'~ has a family of parametrizations [15]. similar to those in Theorem 23. Generalizing Defmition 22, If the double Bmhat cell containing a matrix x ~ G is let us call a word i in the alphabet ~ afactorization scheme not specified, then testing x for total non-negativity be- of type (u, v) if it contains each circled entry (~) exactly comes a much harder problem; in fact, every known crite- once, and the barred (resp. unbarred) entries of i form a rion involves exponentially many (in n) minors. (See [8] reduced word for u (resp. v); in particular, i is of length for related complexity results.) The following corollary of e(u) + e(v) + n. a result by Cryer [10] was given by Gasca and Pefia [24].
THEOREM 26 [15]. For an arbitrary factorization THEOREM 29. An invertible square matrix is totally scheme i of type (u, v), the product map xi restricts to a non-negative if and only if all its minors occupying sev- bijection between (g(u) + e(v) + n)-tuples of positive real eral initial rows or several initial columns are non-neg- numbers and totally non-negative matrices of type (u, v). ative, and all its leading principal minors are positive.
Comparing Theorems 26 and 23, we see that This criterion involves 2 n+l - n - 2 minors, which is G>~'w~ = V>0; (11) roughly the square root of the total number of minors. We do not know whether this criterion is optimal. that is, the totally positive matrices are exactly the totally non-negative matrices of type (w0, w0). Oscillatory Matrices We now show that the splitting of G_>0 into the union of We conclude the article by discussing the intermediate varieties G>~ is closely related to the well-known Bruhat class of oscillatory matrices that was introduced and in- decompositions of the general linear group G = GLn. Let tensively studied by Gantmacher and Krein [20, 22]. A ma- B (resp. B ) denote the subgroup of upper-triangular (resp. trix is oscillatory if it is totally non-negative while some lower-triangular) matrices in G. Recall (see, e.g., [1, w power of it is totally positive; thus, the set of oscillatory that each of the double coset spaces B~G/B and B_\G/B_ matrices contains G>0 and is contained in G_>0. The fol- has cardinality n!, and one can choose the permutation ma- lowing theorem provides several equivalent characteriza- trices w E Sn as their common representatives. To every tions of oscillatory matrices; the equivalence of (a)-(c) was two permutations u and v we associate the double Bruhat proved in [22], and the rest of the conditions were given in cell G u,v = BuB A B vB_; thus, G is the disjoint union of [17]. the double Bruhat cells. Each set G u,v can be described by equations and in- THEOREM 30 [17,22]. For an invertible totally non-neg- equalities of the form A(x) = 0 and/or A(x) r 0, for some ative n • n matrix x, the following are equivalent: collection of minors A. (See [15, Proposition 4.1] or [16].) In (a) x is oscillatory; particular, the open double Bnthat cell G wo'wo is given by Co) xi,i+ 1 > 0 and xi+l,i > O for i = 1,..., n - 1;
VOLUME 22, NUMBER 1, 2000 31 (C) X n-1 is totally positive; any two consecutive levels i and i + 1. These three state- (d) x is not block-triangular (cf. Figure 11); ments are easily seen to be equivalent. By Theorem 27, (e) ~ (f). It remains to show that (e) $$$$$ **00 0- (c). In view of Theorem 26 and (i1), this can be restated
$$$~$ as follows: given any permutation j of the entries 1,... , **00 0 n - 1, prove that the concatenation j n-1 ofn - 1 copies of 00"** j contains a reduced word for w0. Let j' denote the subse- * 0 0555 quence ofj n-t constructed as follows. First, j' contains all n - 1 entries ofj ~-1 which are equal to n - 1. Second, j' 8885 * _0 contains the n - 2 entries equal to n - 2 which interlace Figure 11. Block-triangular matrices. the n - 1 entries chosen at the previous step. We then in- clude n - 3 interlacing entries equal to n - 3, and so forth. (e) x can be factored as x = xi(tl, 9 9 9 tt), for positive tl, The resulting word j' of length (~) will be a reduced word 9 9 9tt , and a word i that contains every symbol of the for w0, for it will be equivalent, under the transformations form i or i at least once; (8), to the lexicographically maximal reduced word (n - 1, (f) x lies in a double Bruhat cell G u,v, where both u and n-2, n-l,n-3, n-2, n-1,...). [] v do not fix any set {1, . . . , i},fori = l, . . . , n- 1.
Proof. Obviously, (c) ~ (a) ~ (d). Let us prove the equiv- ACKNOWLEDGMENTS alence of (b), (d), and (e). By Theorem 12, x can be rep- We thank Sara Billey for suggesting a number of editorial resented as the weight matrix of some planar network F(i) improvements. This work was supported in part by NSF with positive edge weights 9Then, Co) means that sink i 4- grants DMS-9625511 and DMS-9700927. 1 (resp. i) can be reached from source i (resp. i + 1), for all i; (d) means that for any i, at least one sink j > i is reachable from a source h -< i, and at least one sink h -< i REFERENCES is reachable from a source j > i; and (e) means that F(i) 1. J.L. Alperin and R.B. Bell, Groups and Representations, Springer- contains positively and negatively sloped edges connecting Verlag, New York, 1995.
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