Total positivity in loop groups, I: Whirls and curls

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1 Avalable onlne at wwwscencedrectcom Advances n Mathematcs 230 (2012) wwwelsevercom/locate/am Total postvty n loop groups, I: Whrls and curls Thomas Lam a, Pavlo Pylyavskyy b, a Department of Mathematcs, Unversty of Mchgan, 530 Church St, Ann Arbor, MI 48109, USA b Department of Mathematcs, Unversty of Mnnesota, 206 Church St SE, Mnneapols, MN 55455, USA Receved 1 June 2011; accepted 9 March 2012 Avalable onlne 23 Aprl 2012 Communcated by Andre Zelevnsky Abstract Ths s the frst of a seres of papers where we develop a theory of total postvty for loop groups In ths paper, we completely descrbe the totally nonnegatve part of the polynomal loop group GL n (R[t, t 1 ]), and for the formal loop group GL n (R((t))) we descrbe the totally nonnegatve ponts whch are not totally postve Furthermore, we make the connecton wth networks on the cylnder Our approach nvolves the ntroducton of dstngushed generators, called whrls and curls, and we descrbe the commutaton relatons amongst them These matrces play the same role as the poles and zeros of the Edre Thoma theorem classfyng totally postve functons (correspondng to our case n = 1) We gve a soluton to the factorzaton problem usng lmts of ratos of mnors Ths s n a smlar sprt to the Berensten Fomn Zelevnsky Chamber Ansatz where ratos of mnors are used A bratonal symmetrc group acton arsng n the commutaton relaton of curls appeared prevously n Noum Yamada s study of dscrete Panlevé dynamcal systems and Berensten Kazhdan s study of geometrc crystals c 2012 Elsever Inc All rghts reserved Keywords: Total postvty; Loop groups Contents 1 Introducton Total postvty n loop groups Total postvty n GL n (R)1224 Correspondng author E-mal addresses: tfylam@umchedu (T Lam), pylyavskyy@gmalcom,ppylyavs@umnedu (P Pylyavskyy) /$ - see front matter c 2012 Elsever Inc All rghts reserved do:101016/jam

2 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Totally postve functons Whrls and curls The totally postve part GL n (R((t))) > Canoncal form From planar networks to cylndrc networks The factorzaton problem Loop symmetrc functons Curl commutaton relatons, bratonal R-matrx, and dscrete Panlevé systems Future drectons The totally nonnegatve part of the loop group Formal and polynomal loop groups Totally nonnegatve matrces Semgroup generators for GL n (R[t, t 1 ]) Cylndrc networks and total postvty Cylndrc networks Cylndrc Lndström Lemma GL n (R[t, t 1 ]) 0 and cylndrc networks Determnant of the foldng Upper trangular matrces and a reducton result Upper trangular matrces Reducton to U Convergence n U The operaton c Whrls, curls, and ASW factorzaton Whrls and curls ϵ-sequence Fntely supported TNN matrces Totally postve matrces Extenson to the whole formal loop group Whrl and curl relatons Infnte products of whrls and curls Infnte whrls and curls Loop symmetrc functons Basc propertes of nfnte whrls and curls Canoncal form Whrl and curl components Doubly entre matrces as exponentals Infnte products of Chevalley generators Factorzaton of doubly entre TNN matrces Commutng through nfnte whrls and curls (Lmt) semgroups of nfnte whrls and curls Chevalley generators out of whrls Not all Chevalley generators at once Pure whrls and curls Mnor rato lmts Rato lmt nterpretaton and the factorzaton problem Invarance Some open problems 1268 Acknowledgments1269 References 1270

3 1224 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Introducton A matrx wth real entres s totally nonnegatve f all of ts mnors are nonnegatve 11 Total postvty n loop groups Suppose A(t) s a matrx wth entres whch are real polynomals, or real power seres When do we say that A(t) s totally nonnegatve? Frst assocate to A(t) an nfnte perodc matrx X, as n the followng example: 1 + 9t t 1 2t 3t t 4t 2 = t + t A(t) We declare that A(t) s totally nonnegatve f and only f X s totally nonnegatve We use ths to defne and study the totally nonnegatve part of the loop groups GL n (R[t, t 1 ]) and GL n (R((t))) Here R((t)) denotes the feld of formal Laurent seres We let GL n (R((t))) 0 denote the totally nonnegatve part of GL n (R((t))) Our man am s to unfy and generalze two classcal subjects: total postvty n GL n (R) and totally postve functons 12 Total postvty n GL n (R) The theory of totally postve matrces began n the 1930 s n the works of Schoenberg [36] and Gantmacher Kren [14] who dscovered that totally postve matrces had remarkable spectral propertes and a varaton-dmnshng property, cf [18] Let e (a) GL n (R) (resp f (a) GL n (R)) be the Chevalley generators, whch dffer from the dentty matrx by a sngle entry n the -th row (resp column) equal to a R mmedately above (resp below) the dagonal From our pont of vew, the most mportant classcal result s: Theorem 11 (Loewner Whtney Theorem [29,41]) The space of non-sngular totally nonnegatve matrces GL n (R) 0 s the multplcatve semgroup generated by Chevalley generators e (a), f (a) wth postve parameters, and postve dagonal matrces Theorem 11 led Lusztg [31] to hs ground-breakng generalzaton of total postvty to reductve groups Lusztg dscovered deep connectons between the theory of total postvty and hs own theory of canoncal bases n quantum groups [30] In another drecton, Fomn and Zelevnsky [12,11] studed the problem of parametrzng and testng for totally nonnegatve matrces Ther attempt to classfy the ways to test whether a matrx s totally nonnegatve eventually led to the theory of cluster algebras [13] Our frst theorem (Theorem 26) establshes the analogue of Theorem 11 for the totally nonnegatve part GL n (R[t, t 1 ]) 0 of the polynomal loop group, usng the affne Chevalley generators Note that the polynomal loop group tself s not generated by the torus and affne Chevalley generators wth arbtrary parameters X

4 13 Totally postve functons T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) A formal power seres a(t) = 1 + a 1 t + a 2 t 2 + R[[t]] can be consdered a 1 1 matrx We may then apply the defnton of total nonnegatvty n GL 1 (R((t))) of Secton 11 to defne when a formal power seres s totally nonnegatve Tradtonally, formal power seres a(t) whch are totally nonnegatve are called totally postve functons The coeffcents {a 1, a 2, } are sad to form a Polya frequency sequence, see [5] Totally postve functons were classfed ndependently by Edre and Thoma [9,39] Theorem 12 (Edre Thoma Theorem) Every totally postve functon a(t) has a unque expresson as (1 + α t) a(t) = e γ t (1 β t), where α, β and γ are nonnegatve parameters satsfyng α 1 α 2, β 1 β 2 and α + β < In partcular, totally postve functons are meromorphc functons, holomorphc n a neghborhood of 0 Thoma [39] showed that the classfcaton of totally postve functons was equvalent to the classfcaton of characters of the nfnte symmetrc group S Ths connecton was made more robust when Vershk and Kerov [40] nterpreted the zeros and poles n Theorem 12 as asymptotc frequences occurrng n the representaton theory of S No completely elementary proof of Theorem 12 seems to be known For example, the orgnal proofs of Edre and Thoma use Nevanlnna theory from complex analyss, whle Okounkov s proofs [33] rely on the connecton wth asymptotc representaton theory One of the man themes of our work s the parallel between Theorems 11 and 12: (1 + αt), 1/(1 βt), and e γ t can be thought of as semgroup generators for totally postve functons, when we also allow takng lmts of products We begn by consderng the analogues of these generators for n > 1 14 Whrls and curls We ntroduce matrces M(a 1, a 2,, a n ) GL n (R((t))) called whrls, and N(b 1, b 2,, b n ) GL n (R((t))), called curls, dependng on n real (usually nonnegatve) parameters For n = 2, ther nfnte perodc presentatons look lke 1 a a M(a 1, a 2 ) = a a

5 1226 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) b 1 b 1 b 2 b 1 2 b 2 b 2 1 b2 0 1 b 2 b 1 b 2 b 1 b 2 2 N(b 1, b 2 ) = b 1 b 1 b b Unlke Theorem 12, our theory s not commutatve when n > 1 We study whrls and curls n detal In Secton 6, we descrbe the commutaton relatons for whrls and curls In Secton 9, we defne the noton of nfnte products of whrls or curls, and show (see Theorems 91, 95 and 96) the followng Theorem (Structure of Infnte Whrls and Curls) Infnte products of whrls (or curls) form semgroups whch are closed under multplcaton by Chevalley generators on one sde 15 The totally postve part GL n (R((t))) >0 If X s an nfnte perodc matrx correspondng to A(t) GL n (R((t))), then every suffcently southwest entry of X s necessarly equal to 0 Thus X s never totally postve n the usual sense, whch requres all mnors to be strctly postve We defne A GL n (R((t))) 0 to be totally postve f t s totally nonnegatve, and n addton, all suffcently northeast mnors (see Secton 22 for the precse defnton) of the correspondng nfnte perodc matrx are strctly postve We show (Theorem 514): Theorem (Matrces of Fnte Type) The set GL n (R((t))) 0 GL n (R((t))) >0 of totally nonnegatve matrces n the formal loop group whch are not totally postve s a semgroup generated by postve Chevalley generators, whrls, curls, shft matrces (defned n Secton 4), and dagonal matrces 16 Canoncal form For smplcty, we restrct (usng Theorem 42) to the subsemgroup U 0 GL n (R((t))) 0 consstng of matrces A(t) wth upper trangular nfnte perodc representatons In Theorems 83 and 88, we establsh a partal generalzaton of Theorem 12 to n > 1 (t s n fact a rather precse generalzaton of the result of Assen et al [1]) We call a matrx Y U 0 entre f all n 2 matrx entres are entre functons The followng results are our man theorems Theorem (Canoncal Form I) Every X U 0 has a unque factorzaton as X = Z exp(y )W, where Z s a (possbly nfnte) product of curls, W s a (possbly nfnte) product of whrls, and Y s entre such that exp(y ) U 0 The lmts of products A and B n the followng theorem are not necessarly sngle nfnte products Theorem (Canoncal Form II) Every matrx exp(y ) U 0 wth Y entre, has a factorzaton as exp(y ) = AV B, where A and B are both lmts of products of Chevalley generators, and V U 0 s regular

6 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) In [21], we strengthen ths result by showng that the matrces A, V, B n the above theorem are unque The noton of regular totally nonnegatve matrces s ntroduced and dscussed n Secton 8 These results establsh that every X U 0 has three components : (a) a whrl and curl component, (b) a component consstng of products of Chevalley generators, and (c) a regular totally nonnegatve matrx We study (a) n detal here, but leave (b) and (c) for subsequent papers [21,22] 17 From planar networks to cylndrc networks A fundamental property of totally postve matrces s ther realzablty by planar weghted networks, connectng total postvty wth combnatorcs By the Lndström theorem [28] and Theorem 11 (see also [6]) a matrx X GL n (R) s totally nonnegatve f and only f t s realzable by a planar weghted drected acyclc network In Secton 3, we prove (Theorem 34) an analogous statement for loop groups: a matrx X GL n (R[t, t 1 ]) s totally nonnegatve f and only f t s realzable by a weghted drected acyclc network on a cylnder (see for example Fg 4) In the classcal (planar) case, the mnors of the matrx X GL n (R) are nterpreted n terms of non-ntersectng famles of paths Usng the wndng number of paths on a cylnder, we defne a noton of pars of paths beng uncrossed (not the same as non-crossng) The analogous nterpretaton (Theorem 32) of mnors of X GL n (R[t, t 1 ]) 0 nvolves uncrossed famles of paths on the cylnder, and ncludes some paths whch do ntersect The dea of usng a chord on a cylnder to keep track of the wndng number, as t s done n ths paper, appeared frst n the work of Gekhtman et al [15], whch remaned unpublshed for some tme 18 The factorzaton problem In [3], Berensten et al study the problem of fndng an expresson for the parameters t 1, t 2,, t l R >0 n terms of the matrx entres of X = e 1 (t 1 )e 2 (t 2 ) e l (t l ) They solve the problem by wrtng the parameters t as ratos of mnors of the twsted matrx of X Ths nverse problem led to the study of double wrng dagrams and double Bruhat cells [11], and later contrbuted to the dscovery of cluster algebras [13] In Secton 10, we pose and solve a smlar queston n our settng For a matrx X whch s an nfnte product of curls, we dentfy a partcular factorzaton nto curls, called the ASW factorzaton Roughly speakng, the ASW factorzaton has curls ordered by radus of convergence We express (Theorem 101 and Corollary 102) the parameters of the curls n the ASW factorzaton as lmts of ratos of mnors of X Other factorzatons of X nto curls are obtaned from the ASW factorzaton by the acton of the nfnte symmetrc group S 19 Loop symmetrc functons One of the techncal tools we use throughout the paper s a theory of tableaux for a Hopf algebra we call loop symmetrc functons, denoted LSym For n = 1, we obtan the usual symmetrc functons Roughly speakng, LSym generalzes usual symmetrc functons n the same way matrx multplcaton generalzes scalar multplcaton The ponts of GL n (R((t))) 0 are n bjecton wth algebra homomorphsms φ : LSym R whch take nonnegatve values on a partcular spannng set of LSym We leave the detaled nvestgaton of LSym for future work,

7 1228 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) see [20] for a short survey In the present artcle we defne LSym analogues of homogeneous and elementary symmetrc functons, tableaux, and Schur functons, and gve a Jacob Trud formula (Theorem 74) 110 Curl commutaton relatons, bratonal R-matrx, and dscrete Panlevé systems The commutaton relatons for curls gve rse to a bratonal acton of the symmetrc group on a polynomal rng, for whch LSym s the rng of nvarants Ths bratonal acton was studed extensvely by Noum and Yamada [32,42] n the context of dscrete Panlevé dynamcal systems (see also [19]) It also occurs as a bratonal R-matrx n the Berensten Kazhdan [4] theory of geometrc crystals (see also [10]) The tropcalzaton of ths bratonal acton s the combnatoral R-matrx of affne crystals, studed n [17] We hope to clarfy these unexpected connectons n the future The current progress n these drectons s as follows In [25,23] we construct certan affne geometrc crystals n the unpotent loop group and use them to gve a subtracton-free formula for energy functon of some classcal affne crystals The curl commutaton relaton plays a crucal role In [24] we study total postvty and geometrc crystals on arbtrary orentable surfaces, wth the cylnder correspondng to the current case of loop groups We use our network technques to generalze a result of Kajwara et al [16] on commutng R-matrx actons The network model of [24] also leads to a generalzaton of dscrete dynamcal systems called ball-box systems Ths generalzaton s studed n a jont work wth Sakamoto [27] Fnally, a remarkable smlarty between totally postve networks and electrcal networks s explored n [26] There we use an analog of curl commutaton relaton to solve the nverse Drchlet-to-Neumann problem for certan cylndrcal electrcal networks 111 Future drectons Our work suggests many future drectons For example: What asymptotc representaton theory corresponds to total nonnegatvty of the formal loop group? (see [39,40,34,33]) How does our work generalze to loop groups of other types? (see [31]) Is there an asymptotc noton of a cluster algebra? (see [13]) We also gve a lst of precse problems, conjectures and questons n Secton 11 2 The totally nonnegatve part of the loop group 21 Formal and polynomal loop groups An nteger n 1 s fxed throughout the paper If Z, we wrte ī for the mage of n Z/nZ Occasonally, ī s treated as an element of Z, n whch case we pck the representatves of Z/nZ n {1, 2,, n} Let GL n (R((t))) denote the formal loop group, consstng of n n matrces A(t) = (a j (t)) n, j=1 whose entres are formal Laurent seres of the form a j(t) = k N b k t k, for some real numbers b k R and an nteger N, and such that det(a(t)) R((t)) s a non-zero formal Laurent seres We let GL n (R[t, t 1 ]) GL n (R((t))) denote the polynomal loop group, consstng of n n matrces wth Laurent polynomal coeffcents, such that the determnant s

8 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) a non-zero monomal We wll allow ourselves to thnk of the rows and columns of A(t) to be labeled by Z/nZ, and f no confuson arses we may wrte a j (t) for aī j (t), where, j Z To a matrx A(t) = (a j (t)) GL n (R((t))), we assocate a doubly-nfnte, perodc, real matrx X = (x, j ), j= satsfyng x +n, j+n = x, j for any, j, called the unfoldng of A(t), defned va the relaton: a j (t) = x, j+kn t k k= We call A(t) the foldng of X, and wrte A(t) = X(t) for ths relaton Clearly, X(t) and X determne each other and furthermore we have XY = Z f and only f X(t)Y (t) = Z(t) We abuse notaton by wrtng X GL n (R((t))) or X GL n (R[t, t 1 ]) f the same s true for X(t) If X GL n (R((t))), we also wrte det(x) for det(x(t)) We defne the support of X to be the set supp(x) = {(, j) Z 2 x j 0} Example 21 For n = 2, an element of GL n (R((t))) and ts unfoldng are cosh( abt) a/bt snh( abt) bt/a snh( abt) cosh( abt) ab a 2 b a 2 b 2 1 a ab ab b 2 6 ab a b Example 22 For n = 3, an element of GL n (R[t, t 1 ]) and ts unfoldng are t t t For a real parameter a R and an nteger k, we defne e k (a) = (x, j ), j= GL n (R[t, t 1 ]) to be the matrx gven by

9 1230 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) f = j, x, j = a f j = + 1 and ī = k, 0 otherwse Smlarly, defne f k (a) GL n (R[t, t 1 ]) to be the transpose of e k (a) We call the e k -s and f k -s Chevalley generators 22 Totally nonnegatve matrces If X GL n (R((t))), and I Z and J Z are fnte sets of equal cardnalty, we wrte I,J (X) for the mnor of X obtaned from the rows ndexed by I and columns ndexed by J We wrte X I,J to denote a submatrx, so that det(x I,J ) = I,J (X) Let us say that X GL n (R((t))) s totally nonnegatve, or TNN for short, f every fnte mnor of X s nonnegatve We wrte GL n (R((t))) 0 for the set of totally nonnegatve elements of GL n (R((t))) Smlarly, we defne GL n (R[t, t 1 ]) 0 We say that X GL n (R((t))) 0 s totally postve f there exsts an nteger k such that for every par of subsets I = { 1 < 2 < < r } Z and J = { j 1 < j 2 < < j r } Z satsfyng t j t + k for each t [1, r], we have I,J (X) > 0 In other words, X s totally postve f every suffcently northeast mnor s strctly postve We denote the totally postve part of GL n (R((t))) by GL n (R((t))) >0 Note that GL n (R((t))) >0 GL n (R[t, t 1 ]) = Example 23 The matrces n both Examples 21 and 22 are totally nonnegatve The matrx n Example 21 can be shown to be totally postve Lemma 21 The sets GL n (R((t))) 0, GL n (R((t))) >0 and GL n (R[t, t 1 ]) 0 are semgroups Proof Follows mmedately from the Cauchy Bnet formula whch states that I,J (XY ) = K I,K (X) K,J (Y ) (1) where the sum s over subsets K Z wth the same cardnalty as I and J Lemma 22 Suppose X GL n (R((t))) Then the rows of X, consdered as vectors n R, are lnearly ndependent Proof Assume the statement s false and I p r = 0, where I s a fnte set of rows, p R are real coeffcents, and r denotes the -th row of X Then the rows r j of the foldng X satsfy I p t r ī = 0, where s defned by n {1, 2,, n} But ths mples that the rows of X are lnearly dependent over R((t)), contradctng the assumpton that det(x) s non-vanshng A sold mnor of a matrx s a mnor consstng of consecutve rows and columns A row-sold mnor (resp column-sold mnor) s a mnor consstng of consecutve rows (resp consecutve columns) Lemma 23 Suppose X GL n (R((t))) Then X s TNN f ether all row-sold mnors of X, or all column-sold mnors of X, are nonnegatve Proof Let M be a rectangular matrx wth at least as many columns as rows By a theorem of Cryer [7], such a matrx M of maxmal rank s totally nonnegatve f all ts row-sold mnors are totally nonnegatve, cf [2, Theorem 21] By Lemma 22 we know that every mnor of X s

10 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) contaned n a fnte matrx of maxmal rank formed by several consecutve rows of X, and we may assume that ths fnte matrx has more columns than rows Thus to conclude nonnegatvty of ths mnor t suffces to know nonnegatvty of the row-sold mnors of X The same argument proves the statement for column-sold mnors Throughout ths paper, we wll use the followng nave topology on GL n (R((t))) Let X (1), X (2), be a sequence of nfnte perodc matrces n GL n (R((t))) Then lm k X (k) = X f and only f lm k x (k) j = x j for every, j We wll show later n Proposton 44 that ths seemngly weak noton of convergence mples much stronger notons for convergence n the case of TNN matrces Lemma 24 Suppose X s the lmt of a sequence X (1), X (2), of TNN matrces Then X s TNN Proof We must prove that every fnte mnor I,J (X) of X s nonnegatve But each such mnor nvolves only fntely many entres Thus I,J (X) = lm I,J (X () ) 0 For X, Y GL n (R((t))), we wrte X Y, f the same nequalty holds for every entry We note the followng statement, whch s used repeatedly Lemma 25 Suppose X, Y and Z are nonnegatve, upper-trangular matrces wth 1 s on the dagonal Then XY Z X Z 23 Semgroup generators for GL n (R[t, t 1 ]) 0 Let T GL n (R) GL n (R((t))) denote the subgroup of dagonal matrces wth real entres Let T >0 denote those dagonal matrces wth postve real entres Let S = (s j ), j= GL n (R((t))) denote the shft matrx, defned by 1 f j = + 1, s j = 0 otherwse The followng s the loop group analogue of the Loewner Whtney theorem (Theorem 11) Theorem 26 The semgroup GL n (R[t, t 1 ]) 0 s generated by shft matrces, the postve torus T >0 and Chevalley generators wth postve parameters {e 1 (a), e 2 (a),, e n (a) a > 0} { f 1 (a), f 2 (a),, f n (a) a > 0} Proof Frst, usng a (possbly negatve) power of the shft matrx we can reduce to the case when the determnant of an element of GL n (R[t, t 1 ]) 0 s a non-zero real number Next, we recall (see [2]) that f A B M = C D s a block decomposton of a fnte square matrx M such that D s nvertble, then the Schur complement S(M, D) of the block D s the matrx A B D 1 C whch has dmensons equal to that of A It s clear that all the generators stated n the Theorem do le n GL n (R[t, t 1 ]) 0 Now let X GL n (R[t, t 1 ]) 0 Call a non-zero entry x, j of X a NE corner (northeast corner) f

11 1232 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) x, j+k = x k, j = 0 for k 1 If x, j s a NE corner then t follows from the TNN condton for sze two mnors that all entres strctly NE of x, j all vansh A NE corner x, j s specal f x +1, j+1 s not a NE corner We clam that ether x, j = 0 for all j >, or there exsts a specal NE corner Indeed, f t was not so, that s f all NE corners le along a dagonal j = c > 0 for some fxed c, then entres on ths dagonal would contrbute to det(x(t)) a monomal wth a postve power of t not acheved by any other term n det(x(t)), leadng to a contradcton Let x, j be a specal NE corner, whch we may pck to be on a dagonal as NE as possble We clam that x +1, j > 0 Indeed, f x +1, j = 0 then by nonnegatvty of all 2 2 mnors n rows, + 1 we conclude that all entres n row + 1 of X are zero, contradctng the assumpton that X GL n (R[t, t 1 ]) Now, let X = e x, j x +1, j X We clam that X s agan TNN (and t s clear that X GL n (R[t, t 1 ])) By Lemma 23 t suffces to check nonnegatvty of row-sold mnors, and n fact one only needs to check the row-sold mnors contanng row of X but not the row + 1 Assume we have a row-sold mnor wth rows I = [, ] and column set J We may assume that max(j ) j, for otherwse ths mnor wll be 0 n both X and X Now pck a set of columns J = [ j, j] contanng J Let Y be the rectangular submatrx of X wth row set [, + 1] and columns set [ j, j] Complete t to a square matrx Z by addng zero rows or columns on the top or on the left By constructon Z s TNN and contans the row-sold mnor we are nterested n Suppose that Z s a m m matrx Let Z be obtaned from Z by subtractng x, j x +1, j tmes the last row (ndexed by +1) from the second last row (ndexed by ) Then the top left (m 1) (m 1) submatrx of Z s by defnton equal to the Schur complement of x +1, j n Z It follows from [2, Theorem 33] that Z s also TNN, and thus the mnor of X we are nterested n has nonnegatve determnant Note that the part of the support of X above the man dagonal s strctly contaned n that of X On the other hand, the support below the man dagonal has not ncreased, as can be seen by lookng agan of postvty of 2 2 mnors n rows, + 1 Snce after quotentng out by the perodcty the set supp(x) s fnte, ths process, when repeated, must termnate That s, at some pont we have x, j = 0 for all j > A smlar argument wth SW corners, and multplcaton by f j -s reduces X to a TNN matrx wth entres only along the man dagonal What remans s an element of T >0, provng the theorem Example 24 The matrx n Example 22 factors as f 3 (2) f 1 (1)e 2 (1)e 1 (1)e 3 (1) 3 Cylndrc networks and total postvty 31 Cylndrc networks Let C be a cylnder (that s, S 1 [0, 1]) and consder an orented weghted network N = (G, w, h) on t defned as follows G s a fnte acyclc orented graph embedded nto C, havng n sources {v } n =1 on one of the two boundary components of C, and havng n snks {w } n =1 on the other boundary component Sources and snks are numbered n counterclockwse order (we vsualze the cylnder drawn standng wth sources on the bottom and snks on the top; counterclockwse s when vewed from above) We may, as usual, thnk of the sources and snks as labeled by {v, w Z/nZ} and wrte v when we mean vī The chord h s a sngle edge connectng the two boundary components, startng on the arc v n v 1 and endng on the arc w n w 1 We assume h s chosen so that no vertex of G les on t

12 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Fg 1 A network on a cylnder The weght functon w : E(G) R + assgns to every edge e of G a real nonnegatve weght w(e) The weght w(p) of a path p s the product e p w(e) of weghts of all edges along the path For a collecton P = {p} of paths we let w(p) = p P w(p) For a path p let the rotor of p, denoted rot(p), be the number of tmes p crosses h n the counterclockwse drecton mnus the number of tmes p crosses h n the clockwse drecton If x, y are two vertces on a path p, we let p [x,y] denote the part of the path p between the ponts x and y, and let denote ether the begnnng or the end of a path For example, p [x, ] denotes the part of p from x to the end of p For an nteger, let us defne α() = ( ī)/n, where ī s to be taken n {1, 2,, n} For two ntegers and j, an (, j)-path s a path n G whch (1) starts at the source vī; (2) ends at the snk w j ; (3) has rotor equal to α( j) α() Defne an nfnte matrx X (N) = (x, j ), j= by settng x, j to be the sum of weghts over all (, j)-paths n G Note that by defnton X (N) s perodc: x, j = x +n, j+n for any, j Let p be an (, j)-path and let q be an (, j )-path Assume c s a pont of crossng of p and q Let p and q be the two paths obtaned by swappng p and q at c: that s followng one of them untl pont c and the other afterward Although p starts at vī and ends at w j, t s not necessarly an (, j )-path, snce rot( p) may not be equal to α( j ) α() Example 31 Let m = n = 4 and consder two paths shown n Fg 1, one an (1, 6)-path and one an (2, 1)-path Then f we swap the two paths at the marked pont of crossng, we do not get a (1, 1)-path and a (2, 6)-path Instead we get a (1, 5)-path and a (2, 2)-path Lemma 31 Let c be a pont of ntersecton of p and q Then the path p s a (, j )-path f and only f q s a (, j)-path Ths happens when rot(p [c, ] ) rot(q [c, ] ) = α( j) α( j ) In the case of Lemma 31, we say that c s a proper crossng of p and q Two paths that do not have a proper crossng we call an uncrossed par of paths Thus, the crossng marked n Fg 1 s not proper Ths par of paths s however not uncrossed snce the other crossng, not marked on the fgure, happens to be proper 32 Cylndrc Lndström Lemma Let I = 1 < < K and J = j 1 < < j K be two sets of ndexes of equal (fnte) cardnalty K Let Φ(I, J) denote the set of all famles P = {p k } k=1 K of paths such that

13 1234 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) (1) each p k s an ( k, j k )-path; (2) every par of paths n P are uncrossed The followng theorem s a cylndrc analogue of Lndström s Lemma [28] Theorem 32 We have I,J (X (N)) = P Φ(I,J) w(p) Frst we prove the followng lemma Lemma 33 If < and j < j then every (, j)-path p properly crosses every (, j )-path q Proof We make use of the followng observaton: assume p and q are two paths that do not cross each other but mght have one or two common endponts Then rot(p) rot(q) can only take values 1, 0, or 1 Indeed, cut C along p, vewng the result as a rectangle wth a par of opposte vertcal sdes dentfed Snce q never crosses p, t follows that q remans strctly nsde the rectangle Chord h s represented nsde the rectangle by at least rot(p) + 1 dsjont segments We can gnore the segments whch have a crossng wth the same vertcal sde of a rectangle, snce ther ntersectons wth q contrbute 0 to rot(q) What remans are exactly rot(p) + 1 segments, all but the frst and the last of whch connect the two vertcal sdes of the rectangle Those rot(p) 1 segments must be crossed by any path nsde the rectangle, n partcular by q The frst and the last segments of p however may or may not be crossed, dependng on relatve poston of endponts of p and q Ths mples the needed statement concernng rot(p) rot(q) We frst clam that p and q have at least one pont of ntersecton Ths follows easly from unfoldng the cylnder repeatedly Let c 1,, c k be all the crossngs of p and q arranged n order Now, by the argument above each of the quanttes a 0 = rot(p [,c1 ]) rot(q [,c1 ]), a 1 = rot(p [c1,c 2 ]) rot(q [c1,c 2 ]),, a k = rot(p [ck, ]) rot(q [ck, ]) s equal to 1, 0 or 1 Snce k rot(p) rot(q) = a m = α( j) α() α( j ) + α( ) α( j) α( j ) 0 m=0 there must be an ndex l {0, 1,, k + 1} such that k m=l a m = α( j) α( j ) If l {0, k + 1} then c l s a proper crossng by Lemma 31 If l = k + 1 then α( j) = α( j ) and j > j, so as a result a k 0 Smlarly, f l = 0 then α( ) α() = 0 and ī > ī, so as a result a 0 0 In both cases there exsts at least one other ndex l {1,, k} such that k m=l a m = α( j) α( j ) It s easy to see that the resultng c l s a proper crossng Now we are ready to prove Theorem 32 Proof Let P be a collecton of K paths each of whch s an ( k, j l )-path for some k, l so that each element of I and J s used once Pck the frst proper crossng c of two paths p, q P (f t exsts), where we choose an order on vertces of G accordng to some heght functon We assume that the heght functon s chosen so that along any path the vertces are encountered n order of ncreasng heght We can of course assume wthout loss of generalty that no two vertces of G have the same heght Now swap p and q after c, obtanng two new paths p and q Let P be the collecton obtaned from P by replacng p, q wth p, q We clam that n P, c s agan the frst proper crossng of any par of paths

14 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Fg 2 Networks for Chevalley generators Assume p s an ( k, j l )-path and q s an ( k, j l )-path Frst, c s clearly a proper crossng of p and q We need to argue that t s stll the frst proper crossng Suppose t s not Snce p and q are the only two paths n P that changed, any possble new proper crossng c precedng c must belong ether to p or to q or to both If c s a proper crossng of p and q then from rot(p [c, ] ) rot(q [c, ] ) = α( j l ) α( j l ) and rot( q [ c, ] ) rot( p [ c, ] ) = α( j l ) α( j l ) we obtan rot(p [ c,c] ) = rot(q [ c,c] ), from whch t follows that c should have been a proper crossng of p and q ths contradcts the orgnal choce of c Smlarly, suppose c s a proper crossng of say q and some r, whch s an ( k, j l )-path Then rot(q [c, ] ) rot(p [c, ] ) = α( j l ) α( j l ) and rot( q [ c, ] ) rot(r [ c, ] ) = α( j l ) α( j l ) mply rot(q [ c, ] ) rot(r [ c, ] ) = α( j l ) α( j l ) and c should have been a proper crossng of q and r Thus we have obtaned a weght preservng nvoluton on collectons P of paths whch have proper crossngs We observe lookng at the correspondng terms of I,J (X (N)) that ths nvoluton s sgn-reversng Thus, the correspondng contrbutons to the determnant cancel To get the summaton over Φ(I, J) t remans to check that a collecton of paths s parwse uncrossed only f each path n t s an ( k, j k )-path for some k Ths follows from Lemma 33 Remark 31 Theorem 32 and the other results n ths secton can be generalzed to the case of n sources {v } =1 n and m snks {w j} m j=1 n the obvous manner 33 GL n (R[t, t 1 ]) 0 and cylndrc networks Theorem 34 Let X GL n (R[t, t 1 ]) Then X s equal to X (N) for some cylndrc network N wth nonnegatve weght functon, f and only f X GL n (R[t, t 1 ]) 0 Proof From Theorem 32 t follows that every X GL n (R[t, t 1 ]) that arses from a cylndrc network s TNN Further, concatenaton of a cylndrc network N and one of the specal buldng block networks as shown n Fgs 2 and 3 corresponds to multplcaton of X (N) by a Chevalley generator and by a shft matrx respectvely We conclude by Theorem 26 that every element of g 0 can be represented by a cylndrc network 34 Determnant of the foldng Let N be a cylndrc network We now gve a combnatoral nterpretaton for the coeffcents of the determnant det(x (N)(t)) Let {v } =1 n and {w } =1 n be the sources and snks of N as before Then x j (t) enumerates the weghts of paths from v to w j wth an extra factor t rot(p) keepng track of how many tmes the path p crossed the chord h n the counterclockwse drecton Let Γ k be the set of famles P = (p 1,, p n ) of paths, satsfyng: (a) the path p connects v and w +k, (b) no par of paths ntersect n the nave sense (rather than n the sense of uncrossed of Secton 31), and (c) and there are k (net) counterclockwse crossngs of paths n P wth h

15 1236 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Fg 3 A network for the shft matrx Theorem 35 Let N be a cylndrc network Then det(x (N)(t)) = ( 1) k(n 1) w(p) t k k Z P Γ k Proof We proceed usng the usual argument n Lndström s lemma Suppose P = (p 1, p 2,, p n ) s a famly of paths such that p goes from v to w σ () for some permutaton σ S n, and so that there are k (net) counterclockwse crossngs of paths n P wth h If p and p j ntersect at a vertex c, swappng the two paths after c wll gve another famly P wth the same weght, and stll k (net) counterclockwse crossngs wth h Applyng the usual sgn-reversng nvoluton argument (see the proof of Theorem 32), we see that the coeffcent of t k n det(x (N)(t)) s equal to the weght generatng functons of such famles P wth the addtonal requrement that no par of paths ntersect We now observe such famles P exst only f σ s a power of the long cycle, that s, belong to Γ k The sgn of the correspondng permutaton σ s ( 1) k(n 1) Example 32 Consder the network gven n Fg 4, where all edges are orented upwards and have weght 1 One can check that the assocated element of GL n (R((t))) and ts foldng are gven by t t t 1 + 4t t t The determnant of the folded matrx equals 6 t The non-crossng subnetwork correspondng to the t term s shown on the rght of Fg 4 Corollary 36 If X = X (N) arses from a cylndrc network N, then the odd mnors of X(t) have nonnegatve coeffcents, the even mnors have sgn-alternatng coeffcents 4 Upper trangular matrces and a reducton result 41 Upper trangular matrces Let U GL n (R((t))) be the subgroup of the formal loop group consstng of nfnte perodc matrces whch are upper trangular, and such that all dagonal entres are equal to 1 We denote the totally nonnegatve matrces n U by U 0, and the totally postve matrces n U by U >0

16 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Fg 4 A cylndrc network and a non-crossng subnetwork We say that X U 0 s fntely supported f fntely many of dagonals of X, gven by j = constant, are non-zero Otherwse we say that X s not fntely supported Lemma 41 If X U 0 s not fntely supported then all of ts entres above the man dagonal are non-zero Proof Suppose some entry x, j = 0 By usng the nonnegatvty of the 2 2 mnors nvolvng x, j and ether x, or x j, j we deduce that x,k = 0 for k > j and x k, j = 0 for k < Thus all the entres northeast of x, j are 0 Snce the entres of X are perodc, we deduce that X s fntely supported Thus f X s totally nonnegatve but not fntely supported, then the entres of the foldng of X are all nfnte power seres 42 Reducton to U 0 Theorem 42 Every X GL n (R((t))) 0 has a unque factorzaton of the form X = F S k Y where F s the product of an element n T >0 and some f (a)-s, k s an nteger, and Y U 0 Proof We frst prove exstence By the defnton of GL n (R((t))), the matrx X has at least one SW-corner, where SW-corner s defned n obvous analogy wth the NE-corners used n the proof of Theorem 26 Argung as n that proof, ether (a) one can wrte X = f j (a)x where X GL n (R((t))) and a > 0, or (b) the southwestmost non-zero of dagonal of X s completely flled wth non-zero entres If we are n Case (b), then we can use the shft matrx S to shft the southwest-most dagonal to the central dagonal, and then multply by a matrx n T >0 to obtan the desred matrx Y U 0 In Case (a), we repeatedly factor out Chevalley generators f j (a), whch n partcular does not change the determnant det(x) We must eventually encounter Case (b), for otherwse we wll have reduced the support of X to so far n the northeast that the lowest degree monomal n det(x) cannot be obtaned Ths establshes exstence We now prove unqueness We frst note that ST >0 S 1 T >0 and that S f (a)s 1 = f 1 (a) Suppose we have F S k Y = F S k Y Then one has Y = S k F where Y U, F s a product of f (a)-s wth possbly negatve parameters, and k Z But det(y ) 1 + tr[[t]] and det(f ) R, so we conclude that k = 0 But F s lower trangular, and Y s upper trangular, so F = Y s the dentty matrx Ths mples that k = k, F = F, and Y = Y For the rest of ths secton, and most of the rest of the paper, we focus on the semgroup U 0

17 1238 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Convergence n U 0 Let a(t) = 1 + a 1 t + a 2 t 2 + be a formal power seres wth real coeffcents Then a(t) s a totally postve functon f a(t) = X(t) for some X U 0 wth n = 1 Note that wth ths termnology, we do not make the usual dstncton between totally nonnegatve and totally postve As we have mentoned, the Edre Thoma theorem (Theorem 12) classfes totally postve functons Proposton 43 Suppose X U 0 Then the entres of X(t) are meromorphc functons holomorphc n a neghborhood of 0 Proof Apply Theorem 12 to each entry of X(t) (See also the proof of Proposton 44) The radus of convergence of X, denoted r(x), s the mnmum of the rad of convergence of the entres of X(t) The followng proposton shows that our weak noton of convergence automatcally mples stronger convergence Proposton 44 Suppose X (1), X (2), s a sequence of matrces n U 0 wth lmt X Then there s a neghborhood V C of 0so that (1) every matrx amongst X () (t) and X(t) s holomorphc n V (2) every matrx entry of X () (t) approaches the correspondng entry of X(t) unformly, consdered as holomorphc functons on V Proof It s enough to prove the statement for the case n = 1, that s, for totally postve functons If a(t) = 1 + a 1 t + s a totally postve functon, then lookng at 2 2 mnors we have a 1 a 2 /a 1 a 3 /a 2, whenever the ratos are defned Thus f a(t) s not a polynomal, the radus of convergence r(a) of a(t) s at least a /a +1 and we have r = lm a /a +1 Now suppose that a (1) (t), a (2) (t), converge to a(t) Then there s a suffcently large N so that for k > N, a (k) 1 a 1 1 It follows that r(a (k) (t)) > 1/(a 1 + 1) for all k > N and so there exsts a neghborhood V of 0 wth property (1) To see that a () (t) approaches a(t) unformly n a possbly smaller neghborhood V, we note that for t < R we have a t a k R k k k a k 1 R k (a 1 R) k 1 a 1 R Fx some R 1/a 1 It follows that for any l 0, the value of a(t) a (l) (t) for t < R can be approxmated by throwng away all but the frst k terms But for l suffcently large, the frst k terms of a(t) and a (l) (t) are arbtrarly close Ths shows that a () (t) approaches a(t) unformly n the doman t < R Note that nether concluson of Proposton 44 holds for general meromorphc functons 44 The operaton c We defne X c U to be the matrx obtaned by applyng to X U the transformaton x, j ( 1) j x, j A specal role n what follows s played by the operaton c -nverse gven by X (X c ) 1 Abusng notaton slghtly, we shall also wrte X c := (X c ) 1 Note that (X c ) 1 = (X 1 ) c Also note that the operaton X X c s an nvoluton, and that (XY ) c = Y c X c

18 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Lemma 45 Suppose X U 0 Then X c U 0 Proof It suffces to show that X c I,I s TNN for every nterval I = [a, b], snce every mnor of X c s contaned n such a submatrx Let Y = X I,I and m = I Then Y GL m (R) 0 GL m (R[t, t 1 ]) 0 By Theorem 26 (or Theorem 11), Y s a product of Chevalley generators {e (a) = 1, 2,, m 1} wth postve parameters We now observe that e (a) c = e (a) Usng (W V ) c = W c V c, we deduce that Y c s also a product of Chevalley generators wth postve parameters But then X c I,I = Y c s TNN Suppose, j, k are ntegers such that j k 1 and k 0 Let X, j,k denote the sold submatrx of X obtaned from the rows, +1,, j k and the columns +k, +k +1,, j In the followng proposton we use the conventon that the determnant of a 0 0 matrx s 1 Proposton 46 Let X U Then det(x, j,k ) = det(x c, j, j+1 k ) for j + k 1 Proof If k = 0, then det(x, j,k ) = 1 = det(x c, j, j+1 k ) Consder now k = 1 That s, we need to show (X c ), j = det(x, j,1 ) Expandng det(x, j,k ) nto smaller mnors usng the frst row, we obtan det(x, j,1 ) = ( 1) r x,+r+1 det(x +r+1, j,1 ) j 1 r=0 The clam then follows from the defnton of X c and nducton on j We now allow k to be arbtrary We wll prove the equalty as a polynomal dentty Recall that for an n n matrx M, Dodgson s condensaton lemma [8] says (M) {2,3,,n 1},{2,3,,n 1} (M) = {1,,n 1},{1,,n 1} (M) {2,,n},{2,,n} (M) {2,,n},{1,,n 1} (M) {1,,n 1},{2,,n} (M) (2) Applyng ths and proceedng by nducton on k, we calculate det(x, j,k+1 ) = det(x, j 1,k) det(x +1, j,k ) det(x, j,k ) det(x +1, j 1,k ) det(x +1, j 1,k 1 ) c c c c det(x, j 1, j k ) det(x+1, j, j k ) det(x, j, j+1 k ) det(x+1, j 1, j 1 k = ) det(x c +1, j 1, j k ) = det(x, c j, j k ) Note that the equaltes hold as polynomals when appled to a matrx X consstng of varables x, j Thus the dvsons n the calculaton are always legtmate Lemma 47 We have det(x)(t) det(x c )(t) = det(x)( t) f n s even, f n s odd Proof Suppose X(t) = (x j (t)) Then X c (t) has entres ( 1) j x j (( 1) n t)

19 1240 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Whrls, curls, and ASW factorzaton 51 Whrls and curls Let a 1,, a n be n real parameters We defne a whrl to be a matrx M = (m, j ), j= = M(a 1,, a n ) wth m, = 1, m,+1 = a and the rest of the entres equal to zero Here, the ndexng of the parameters are taken modulo n Note that the Chevalley generator e (a) s gven by M(0,, 0, a, 0,, 0) where the a s n the -th poston If at least one of the parameters a n a whrl s zero, then we call the whrl degenerate A degenerate whrl always factors nto Chevalley generators Furthermore, f the orgnal parameters are nonnegatve then the parameters n factorzaton are also nonnegatve We defne a curl to be a matrx N of the form N(a 1,, a n ) := M(a 1,, a n ) c Examples of whrls and curls were gven n Secton 1 Lemma 51 The folded determnants of whrls and curls are gven by n det(m(a 1,, a n )) = 1 + ( 1) n+1 a t =1 1 det(n(a 1,, a n )) = n 1 a t =1 52 ϵ-sequence Let X U 0 Defne ϵ = ϵ (X) = lm j x, j x +1, j Clearly ϵ depends only on ī Smlarly, defne x j,+1 µ = µ (X) = lm j x j, Example 51 Let n = 2 Consder the followng matrx 1 + 2t (1 t)(1 2t) (1 t)(1 2t) t 1 + t (1 t)(1 2t) (1 t)(1 2t) Ths matrx s n fact the product N(1, 1)N(1, 2) of two curls, and thus s totally nonnegatve Then ϵ 1 = lm (2 1) = 4 3 Smlarly one computes ϵ 2 = 3 2

20 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Lemma 52 Suppose X U 0 and not fntely supported Then the lmts ϵ and µ exst Furthermore, 1/( n =1 ϵ ) = 1/( n =1 µ ) s the radus of convergence of every entry of the foldng X(t) x, j x +1, j x, j+1 x +1, j+1 Proof The nequalty follows from the nonnegatvty of the 2 2 mnor x, j x +1, j+1 x +1, j x, j+1 of X A non-ncreasng sequence of nonnegatve real numbers has a lmt, gvng the frst statement of the lemma The second statement follows from the observaton that x, j+n x, j = x n 1 n, j x +k n, j = x, j x k=0 +k+1 n, j Although we often omt t from the notaton, the ϵ -s depend on X We call (ϵ 1,, ϵ n ) the ϵ-sequence of X Assen et al [1] used a factorzaton procedure as a frst step toward the Edre Thoma theorem We now descrbe a generalzaton of t to n > 1 We call ths generalzaton ASW factorzaton Lemma 53 Suppose X U 0 s not fntely supported Let X = M( ϵ 1,, ϵ n )X Then X U 0 Proof Let J = j 1 < j 2 < < j k be a set of column ndces We have x, j1 x, j2 x, jk x,l x +1, j1 x +1, j2 x +1, jk x +1,l det x +k, j1 x +k, j2 x +k, jk x +k,l x +k+1, j1 x +k+1, j2 x +k+1, jk x +k+1,l lm l x +k+1,l x, j1 x, j2 x, jk x,l /x +k+1,l x +1, j1 x +1, j2 x +1, jk x +1,l /x +k+1,l = lm det l x +k, j1 x +k, j2 x +k, jk x +k,l /x +k+1,l x +k+1, j1 x +k+1, j2 x +k+1, jk 1 x, j1 x, j2 x, j2 ϵ ϵ +k x +1, j1 x +1, j2 x +1, j2 ϵ +1 ϵ +k = det x +k, j1 x +k, j2 x +k, j2 ϵ +k x +k+1, j1 x +k+1, j2 x +k+1, j2 1 x, j1 ϵ x +1, j1 x, j2 ϵ x +1, j2 x, jk ϵ x +1, jk 0 x +1, j1 ϵ +1 x +2, j1 x +1, j2 ϵ +1 x +2, j2 x +1, jk ϵ +1 x +2, jk 0 = det x +k, j1 ϵ +k x +k+1, j1 x +k, j2 ϵ +k x +k+1, j2 x +k, jk ϵ +k x +k+1, jk 0 x +k+1, j1 x +k+1, j2 x +k+1, jk 1 x, j1 ϵ x +1, j1 x, j2 ϵ x +1, j2 x, jk ϵ x +1, jk x +1, j1 ϵ +1 x +2, j1 x +1, j2 ϵ +1 x +2, j2 x +1, jk ϵ +1 x +2, jk = det x +k, j1 ϵ +k x +k+1, j1 x +k, j2 ϵ +k x +k+1, j2 x +k, jk ϵ +k x +k+1, jk

21 1242 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Ths s a mnor of X, and every row-sold mnor of X can be presented as a lmt n ths way Snce a lmt of a nonnegatve quantty s nonnegatve, we conclude that all row-sold mnors of X are nonnegatve By Lemma 23, we conclude that X s totally nonnegatve We can rewrte the defnton of X as X = N(ϵ 1,, ϵ n )X Ths gves a factorzaton of X nto a product of two TNN matrces Note that the radus of convergence of X s at least as large as that of X Thus, f we repeat the ASW factorzaton to obtan X = N(ϵ 1,, ϵ n )N(ϵ 1,, ϵ n )X then we must have n =1 ϵ n =1 ϵ We also note that the factorzaton n Lemma 53 nvolves the bggest whrl Lemma 54 Suppose X U 0 s not fntely supported Suppose that X = M( a 1,, a n )X s TNN Then a ϵ for each Furthermore, f a < ϵ for some then r(x ) = r(x) Proof We obtan X = (x, j ) from X = (x, j) by subtractng a tmes the ( + 1)-th row from the -th row But the rato x, j /x +1, j approaches ϵ, so x, j 0 mples that a ϵ For the last statement, suppose that a < ϵ Snce r(m( a 1,, a n )) =, we have r(x ) r(x) But usng Lemma 52, we have r(n(a 1,, a n )) = 1 a > 1 ϵ = r(x) so that from X = N(a 1,, a n )X, we have r(x) r(x ) Thus r(x ) = r(x) Example 52 In Example 51, t was computed that the curl N( 4 3, 2 3 ) can be factored out on the left One can check that the remanng totally nonnegatve matrx s the curl N( 3 2, 3 2 ) Thus the ASW factorzaton of the matrx n ths example s N( 4 3, 2 3 )N(2 3, 2 3 ) 53 Fntely supported TNN matrces Theorem 55 The semgroup U fn 0 of fntely supported matrces n U 0 s generated by whrls and Chevalley generators wth nonnegatve parameters Proof It s clear that the semgroup generated by whrls and Chevalley generators wth nonnegatve parameters les nsde U 0 fn Now let X U fn 0 Frst suppose that X c s fntely supported In ths case, the entres of X c (t) are polynomals, and n partcular, entre But then both 1/ det(x c (t)) = det(x c (t)) and det(x c (t)) are polynomals, so we conclude that det(x c (t)) and by Lemma 47 det(x) s a constant By Theorem 26, we deduce that X fan be factored nto a fnte number of nonnegatve Chevalley generators Now suppose that X c s not fntely supported Apply Lemmas 45 and 53 to obtan X c = N(a 1,, a n )Y, where the parameters a = ϵ (X c ) are nonnegatve and Y s totally nonnegatve If at least one of parameters a s zero, by Lemma 52 the entres of X c are entre, and the determnant s entre We may then proceed as n the case that X c s fntely supported Thus we may assume that all a are strctly postve Then X = Y c M(a 1,, a n ), where both X and Y c are fntely supported TNN matrces One observes that the number of nonzero dagonals of Y c must be strctly smaller than that of X Now repeat the applcaton of Lemma 53 to Y c Snce the number of non-zero dagonals of X s fnte, n a fnte number of steps we must obtan the stuaton n one of the two prevous paragraphs Thus we obtan a factorzaton of X nto a fnte number of whrls and Chevalley generators wth nonnegatve parameters

22 T Lam, P Pylyavskyy / Advances n Mathematcs 230 (2012) Fg 5 Networks for a whrl and a curl Snce whrls are representable by cylndrc networks, as shown on the left n Fg 5, we mmedately get the followng corollary Corollary 56 Every X U 0 fn s representable by a cylndrc network 54 Totally postve matrces For I = { 1 < 2 < < k } and J = { j 1 < j 2 < < j k } we defne I J f t j t for each t [1, k] Theorem 57 Let X U 0 Then X U >0 f and only f X s a fnte product of whrls and curls (ncludng Chevalley generators) In other words, the semgroup generated by whrls and curls s exactly the set U 0 \ U >0 We start by provng the followng lemma Lemma 58 Suppose X has a vanshng mnor I,J (X) = 0 for I J Assume that (I, J) s chosen so that I = J = k s mnmal Then X has a sold vanshng mnor I,J (X) = 0 of sze k wth I J Furthermore, all mnors I,J (X) for I I and J J vansh Proof For k = 1 the statement s already proved n Lemma 41, so assume k > 1 If I = 1 < < k and J = j 1 < < j k then k < j k snce otherwse there s a smaller sngular mnor Look at the submatrx X I { jk },J { j k +1} Wrtng down Dodgson s condensaton (2) for ths matrx we get I { jk } { 1 },J (X) I,J { jk +1} { j 1 }(X) = I { jk },J { j k +1}(X) I {1 },J { j 1 }(X) Ths mples that the left-hand sde must be zero, snce t s non-postve and the rght-hand sde s nonnegatve If I jk { 1 },J (X) = 0 then the (k 1) (k 1) mnor I {1 },J { j k }(X) vanshes If I = I { 1 } and J = J { j k } satsfes I J then ths contradcts the mnmalty of k Otherwse we would have t+1 > j t for some t [1, k 1], mplyng that the submatrx X I,J s block upper trangular Agan ths would mply a smaller vanshng mnor, contradctng the mnmalty of k Thus I,J { jk +1} { j 1 }(X) = 0 Repeatng ths k tmes, the column ndexng set becomes sold, and smlarly, we may move the rows up to obtan a sold row ndexng set The second clam s proved n a smlar manner Corollary 59 Suppose X U 0 Then X U >0 f and only f all mnors I,J (X) > 0 for I J Lemma 510 If X U 0 s not fntely supported and has a vanshng sold mnor then all ϵ -s are postve