NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN

Transcription

1 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN KELLI TALASKA AND LAUREN WILLIAMS Abstract Deodhar ntroduced hs decomoston of artal flag varetes as a tool for understandng Kazhdan-Lusztg olynomals The Deodhar decomoston of the Grassmannan s also useful n the context of solton solutons to the KP equaton, as shown by Kodama and the second author Deodhar comonents R D of the Grassmannan are n bjecton wth certan tableaux D called Go-dagrams, and each comonent s somorhc to (K ) a (K) b for some non-negatve ntegers a and b Our man result s an exlct arameterzaton of each Deodhar comonent n the Grassmannan n terms of networks More secfcally, from a Go-dagram D we construct a weghted network N D and ts weght matrx W D, whose entres enumerate drected aths n N D By lettng the weghts n the network vary over K or K as arorate, one gets a arameterzaton of the Deodhar comonent R D One alcaton of such a arameterzaton s that one may mmedately determne whch Plücker coordnates are vanshng and nonvanshng, by usng the Lndstrom-Gessel-Vennot Lemma We also gve a (mnmal) characterzaton of each Deodhar comonent n terms of Plücker coordnates A man tool for us s the work of Marsh and Retsch [8] on Deodhar comonents n the flag varety Contents Introducton Background on the Grassmannan 3 3 The man result: network arameterzatons from Go-dagrams 6 4 The Deodhar decomoston of the Grassmannan 0 5 Formulas for entres of the MR-matrces 3 6 Proof of the man result 0 7 A characterzaton of Deodhar comonents n terms of Plücker coordnates 4 References 7 Introducton There s a remarkable subset of the real Grassmannan Gr k,n (R) called ts totally non-negatve art (Gr k,n ) 0 [7, 9], whch may be defned as the subset of the real Grassmannan where all Plücker coordnates have the same sgn Postnkov showed that (Gr k,n ) 0 has a decomoston nto ostrod cells, whch are ndexed by certan tableaux called -dagrams He also gave exlct arameterzatons of each cell In artcular, he showed that from each -dagram one can roduce a lanar network, and that one can wrte down a arameterzaton of the corresondng cell usng the weght matrx of that network Ths arameterzaton shows that the cell s somorhc to R d >0 for some d Such a arameterzaton s convenent, because for examle, one may read off formulas for Plücker coordnates from non-ntersectng aths n the network, usng the Lndstrom-Gessel-Vennot Lemma Γ Γ Date: February 7, 03 The frst author was artally suorted by NSF Grant DMS The second author was artally suorted by an NSF CAREER award and an Alfred Sloan Fellowsh

2 KELLI TALASKA AND LAUREN WILLIAMS A natural queston s whether these network arameterzatons for ostrod cells can be extended from (Gr k,n ) 0 to the entre real Grassmannan Gr k,n (R) In ths aer we gve an affrmatve answer to ths queston, by relacng the ostrod cell decomoston wth the Deodhar decomoston of the Grassmannan Gr k,n (K) (here K s an arbtrary feld) The comonents of the Deodhar decomoston are not n general cells, but nevertheless have a smle toology: by [, 3], each one s somorhc to (K ) a (K) b The relaton of the Deodhar decomoston of Gr k,n (R) to Postnkov s cell decomoston of (Gr k,n ) 0 s as follows: the ntersecton of a Deodhar comonent R D = (R ) a (R) b wth (Gr k,n ) 0 s recsely one ostrod cell somorhc to (R >0 ) a f b = 0, and s emty otherwse In artcular, when one ntersects the Deodhar decomoston wth (Gr k,n ) 0, one obtans the ostrod cell decomoston of (Gr k,n ) 0 There s a related ostrod stratfcaton of the real Grassmannan, and each ostrod stratum s a unon of Deodhar comonents As for the combnatorcs, comonents of the Deodhar decomoston are ndexed by dstngushed subexressons [, 3], or equvalently, by certan tableaux called Go-dagrams [6], whch generalze - dagrams In ths aer we assocate a network to each Go-dagram, and wrte down a arameterzaton of the corresondng Deodhar comonent usng the weght matrx of that network Our constructon generalzes Postnkov s, but our networks are no longer lanar n general Our man results can be summed u as follows See Theorems 36 and 78 and the constructons recedng them for comlete detals Theorem Let K be an arbtrary feld Every ont n Gr k,n (K) can be realzed as the weght matrx of a unque network assocated to a Go-dagram, and we can exlctly construct the corresondng network The networks corresondng to onts n the same Deodhar comonent have the same underlyng grah, but dfferent weghts Every Deodhar comonent may be characterzed by the vanshng and nonvanshng of certan Plücker coordnates Usng ths characterzaton, we can also exlctly construct the network assocated to a ont gven ether by a matrx resresentatve or by a lst of Plücker coordnates To llustrate the man results, we rovde a small examle here More comlcated examles may be seen throughout the rest of the aer Γ D a 4 a 4 a 3 a 3 D + + c 4 a 4 a 3 3 Fgure The dagrams and networks assocated to R D and R D n Examle Examle Consder the Grassmannan Gr,4 The large Schubert cell n ths Grassmannan can be characterzed as Ω λ = {A Gr,4, (A) 0}, where J denotes the Plücker coordnate corresondng to the column set J n a matrx reresentatve of a ont n Gr,4 Ths Schubert cell contans multle ostrod strata, ncludng S I, where I s the Grassmann necklace I = (, 3, 34, 4) Ths ostrod stratum can also be characterzed by the vanshng of certan Plücker coordnates: S I = {A Gr,4, (A) 0,,3 (A) 0, 3,4 (A) 0,,4 (A) 0}

3 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 3 Fgure shows two Go-dagrams D and D and ther assocated networks Note that the network on the rght s not lanar The weght matrces assocated to these dagrams are ( ) ( ) 0 a3 (a 3 a 4 + a 3 a ) 0 a3 a and 3 c 4 0 a a a 0 0 a The ostrod stratum S I s the dsjont unon of the two corresondng Deodhar comonents R D and R D, whch can be characterzed n terms of vanshng and nonvanshng of mnors as: R D = {A S I,3 0} and R D = {A S I,3 = 0} Note that f one lets the a s range over K and lets c 4 range over K, then we see that R D = (K ) 4 and R D = (K ) K There are several alcatons of our constructon Frst, as a secal case of our theorem, one may arameterze all k n matrces usng networks Second, by alyng the Lndstrom-Gessel-Vennot Lemma to a gven network, one may wrte down exlct formulas for Plücker coordnates n terms of collectons of non-ntersectng aths n the network Thrd, buldng uon work of [6], we obtan (mnmal) descrtons of Deodhar comonents n the Grassmannan, n terms of vanshng and nonvanshng of Plücker coordnates It follows that each Deodhar comonent s a unon of matrod strata Although less well known than the Schubert decomoston and matrod stratfcaton, the Deodhar decomoston s very nterestng n ts own rght Deodhar s orgnal motvaton for ntroducng hs decomoston was the desre to understand Kazhdan-Lusztg olynomals In the flag varety, one may ntersect two ooste Schubert cells, obtanng a Rchardson varety, whch Deodhar showed s a unon of Deodhar comonents Each Rchardson varety R v,w (q) may be defned over a fnte feld K = F q, and n ths case, the number of onts determnes the R-olynomals R v,w (q) = #(R v,w (F q )), ntroduced by Kazhdan and Lusztg [4] to gve a recursve formula for the Kazhdan-Lusztg olynomals Snce each Deodhar comonent s somorhc to (F q) a (F q ) b for some a and b, f one understands the decomoston of a Rchardson varety nto Deodhar comonents, then n rncle one may comute the R-olynonomals and hence Kazhdan-Lusztg olynomals Another reason for our nterest n the Deodhar decomoston s ts relaton to solton solutons of the KP equaton It s well-known that from each ont A n the real Grassmannan, one may construct a solton soluton u A (x, y, t) of the KP equaton It was shown n recent work of Kodama and the second author [6] that when the tme varable t tends to, the combnatorcs of the soluton u A (x, y, t) deends recsely on whch Deodhar comonent A les n The outlne of ths aer s as follows In Secton, we gve some background on the Grassmannan and ts decomostons, ncludng the Schubert decomoston, the ostrod stratfcaton, and the matrod stratfcaton In Secton 3, we resent our man constructon: we exlan how to construct a network from each dagram, then use that network to wrte down a arameterzaton of a subset of the Grassmannan that we call a network comonent In Secton 4 we defne Deodhar s decomoston of the flag varety, and ts rojecton to the Grassmannan We also descrbe arameterzatons of Deodhar comonents n the flag varety whch are due to Marsh and Retsch [8] In Sectons 5 and 6 we rove that after a ratonal transformaton of varables, our network arameterzatons concde wth the rojectons of the Marsh-Retsch arameterzatons Fnally n Secton 7 we gve a characterzaton of Deodhar comonents n terms of the vanshng and nonvanshng of certan Plücker coordnates Acknowledgements: We are grateful to Sara Blley for numerous helful comments on the frst verson of ths aer LW s also grateful to Yuj Kodama for ther jont work on solton solutons of the KP equaton, whch rovded motvaton for ths roject Background on the Grassmannan The Grassmannan Gr k,n s the sace of all k-dmensonal subsaces of an n-dmensonal vector sace K n In ths aer we wll usually let K be an arbtrary feld, though we wll often thnk of t as

4 4 KELLI TALASKA AND LAUREN WILLIAMS R or C An element of Gr k,n can be vewed as a full-rank k n matrx modulo left multlcaton by nonsngular k k matrces In other words, two k n matrces reresent the same ont n Gr k,n f and only f they can be obtaned from each other by row oeratons Let ( ) [n] k be the set of all k-element subsets of [n] := {,, n} For I ( ) [n] k, let I (A) be the Plücker coordnate, that s, the maxmal mnor of the k n matrx A located n the column set I The ma A ( I (A)), where I ranges over ( [n] ) k, nduces the Plücker embeddng Grk,n KP (nk) nto rojectve sace We now descrbe several useful decomostons of the Grassmannan: the Schubert decomoston, the ostrod stratfcaton, and the matrod stratfcaton Note that the matrod stratfcaton refnes the ostrod stratfcaton, whch refnes the Schubert decomoston The man subject of ths aer s the Deodhar decomoston of the Grassmannan, whch refnes the ostrod stratfcaton, and s refned by the matrod stratfcaton (as we rove n Corollary 79) The Schubert decomoston of Gr k,n Throughout ths aer, we dentfy arttons wth ther Young dagrams Recall that the arttons λ contaned n a k (n k) rectangle are n bjecton wth k-element subset I [n] The boundary of the Young dagram of such a artton λ forms a lattce ath from the uer-rght corner to the lower-left corner of the rectangle Let us label the n stes n ths ath by the numbers,,n, and defne I = I(λ) as the set of labels on the k vertcal stes n the ath Conversely, we let λ(i) denote the artton corresondng to the subset I Defnton For each artton λ contaned n a k (n k) rectangle, we defne the Schubert cell Ω λ = {A Gr k,n I(λ) s the lexcograhcally mnmal subset such that I(λ) (A) 0} As λ ranges over the arttons contaned n a k (n k) rectangle, ths gves the Schubert decomoston of the Grassmannan Gr k,n, e Gr k,n = Ω λ λ (n k) k We now defne the shfted lnear order < (for [n]) to be the total order on [n] defned by < + < + < < n < < < One can then defne cyclcally shfted Schubert cells as follows Defnton For each artton λ contaned n a k (n k) rectangle, and each [n], we defne the cyclcally shfted Schubert cell Ω λ = {A Gr k,n I(λ) s the lexcograhcally mnmal subset wth resect to < such that I(λ) 0} The ostrod stratfcaton of Gr k,n The ostrod stratfcaton of the Grassmannan Gr k,n s obtaned by takng the smultaneous refnement of the n Schubert decomostons wth resect to the n shfted lnear orders < Ths stratfcaton was frst consdered by Postnkov [9], who showed that the strata are convenently descrbed n terms of Grassmann necklaces, as well as decorated ermutatons and -dagrams Postnkov coned the termnology ostrod because the ntersecton of the ostrod stratfcaton of the real Grassmannan wth the totally non-negatve art of the Grassmannan (Gr k,n ) 0 gves a cell decomoston of (Gr k,n ) 0 (whose cells are called ostrod cells) Γ Defnton 3 [9, Defnton 6] A Grassmann necklace s a sequence I = (I,, I n ) of subsets I r [n] such that, for [n], f I then I + = (I \ {}) {j}, for some j [n] (j may concde wth ); and f / I then I + = I (Here ndces are taken modulo n) In artcular, we have I = = I n, whch s equal to some k [n] We then say that I s a Grassmann necklace of tye (k, n) Examle 4 I = (345, 3456, 3456, 4567, 4567, 467, 478, 348) s an examle of a Grassmann necklace of tye (4, 8)

5 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 5 Lemma 5 [9, Lemma 63] Gven A Gr k,n, let I(A) = (I,, I n ) be the sequence of subsets n [n] such that, for [n], I s the lexcograhcally mnmal subset of ( ) [n] k wth resect to the shfted lnear order < such that I (A) 0 Then I(A) s a Grassmann necklace of tye (k, n) The ostrod stratfcaton of Gr k,n s defned as follows Defnton 6 Let I = (I,, I n ) be a Grassmann necklace of tye (k, n) The ostrod stratum S I s defned to be S I = {A Gr k,n I(A) = I} Equvalently, each ostrod stratum s an ntersecton of n cyclcally shfted Schubert cells, that s, n S I = Ω λ(i ) = Grassmann necklaces are n bjecton wth tableaux called Γ -dagrams Defnton 7 [9, Defnton 6] Fx k, n A -dagram (λ, D) k,n of tye (k, n) s a artton λ contaned n a k (n k) rectangle together wth a fllng D : λ {0, +} of ts boxes whch has the -roerty: there s no 0 whch has a + above t and a + to ts left (Here, above means above and n the same column, and to ts left means to the left and n the same row) In Fgure we gve an examle of a Γ Γ -dagram Γ Fgure A Le-dagram L = (λ, D) k,n 3 The matrod stratfcaton of Gr k,n Defnton 8 A matrod of rank k on the set [n] s a nonemty collecton M ( ) [n] k of k-element subsets n [n], called bases of M, that satsfes the exchange axom: For any I, J M and I there exsts j J such that (I \ {}) {j} M Gven an element A Gr k,n, there s an assocated matrod M A whose bases are the k-subsets I [n] such that I (A) 0 Defnton 9 Let M ( ) [n] k be a matrod The matrod stratum SM s defned to be S M = {A Gr k,n I (A) 0 f and only f I M} Ths gves a stratfcaton of Gr k,n called the matrod stratfcaton, or Gelfand-Serganova stratfcaton Remark 0 Clearly the matrod stratfcaton refnes the ostrod stratfcaton, whch n turn refnes the Schubert decomoston Ths forbdden attern s n the shae of a backwards L, and hence s denoted Γ and ronounced Le

6 6 KELLI TALASKA AND LAUREN WILLIAMS 3 The man result: network arameterzatons from Go-dagrams In ths secton we defne certan tableaux called Go-dagrams, then exlan how to arameterze the Grassmannan usng networks assocated to Go-dagrams Frst we wll defne more general tableaux called dagrams 3 Dagrams and networks Defnton 3 Let λ be a artton contaned n a k (n k) rectangle A dagram n λ s an arbtrary fllng of the boxes of λ wth luses +, black stones, and whte stones To each dagram D we assocate a network N D as follows Defnton 3 Let λ be a artton wth l boxes contaned n a k (n k) rectangle, and let D be a dagram n λ Label the boxes of λ from to l, startng from the rghtmost box n the bottom row, then readng rght to left across the bottom row, then rght to left across the row above that, etc The (weghted) network N D assocated to D s a drected grah obtaned as follows: Assocate an nternal vertex to each + and each ; After labelng the southeast border of the Young dagram wth the numbers,,, n (from northeast to southwest), assocate a boundary vertex to each number; From each nternal vertex, draw an edge rght to the nearest +-vertex or boundary vertex; From each nternal vertex, draw an edge down to the nearest +-vertex or boundary vertex; Drect all edges left and down After dong so, k of the boundary vertces become sources and the remanng n k boundary vertces become snks If e s a horzontal edge whose left vertex s a +-vertex (resectvely a -vertex) n box b, assgn e the weght a b (resectvely c b ) We thnk of a b and c b as ndetermnates, but later they wll be elements of K and K resectvely If e s a vertcal edge, assgn e the weght Note that n general such a drected grah s not lanar, as two edges may cross over each other wthout meetng at a vertex See Fgure 3 for an examle of a dagram and ts assocated network a a 8 c 5 + a 8 a a 0 a 9 c 7 a 6 3 a Fgure 3 An examle of a dagram and ts assocated network We now exlan how to assocate a weght matrx to such a network Defnton 33 Let N D be a network as n Defnton 3 Let I = { < < < k } [n] denote the sources If P s a drected ath n the network, let w(p) denote the roduct of all weghts along P

7 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 7 If P s the emty ath whch starts and ends at the same boundary vertex, we let w(p) = If r s a source and s s any boundary vertex, defne W rs = ± P w(p), where the sum s over all aths P from r to s The sgn s chosen (unquely) so that I\{r} {s} (W D ) = P W D = (W rs ) w(p), where s the k (n k) weght matrx We make the conventon that the rows of W D are ndexed by the sources,, k from to to bottom, and ts columns are ndexed by,,,n from left to rght An equvalent way to defne the sgn of W rs s to let q = {s +, s +,, t } I, e the number of sources whch are strctly between s and t Then the sgn of W rs s () q Examle 34 The weght matrx assocated to the network n Fgure 3 s a a 9 a 0 0 a 9 a 0 (a + c 7 ) a 9 a 0 (a a + a c 5 + a 8 + c 7 c 5 ) a 6 0 a 6 c 7 a 6 a 8 + a 6 c 7 c a 4 a 4 c a 3 Dstngushed exressons We now revew the noton of dstngushed subexressons, as n [] and [8] Ths defnton wll be essental for defnng Go-dagrams We assume the reader s famlar wth the (strong) Bruhat order < on W = S n, and the bascs of reduced exressons, as n [] Let w := s s m be a reduced exresson for w W A subexresson v of w s a word obtaned from the reduced exresson w by relacng some of the factors wth For examle, consder a reduced exresson n S 4, say s 3 s s s 3 s s 3 Then s 3 s s 3 s s a subexresson of s 3 s s s 3 s s 3 Gven a subexresson v, we set v (k) to be the roduct of the leftmost k factors of v, f k, and v (0) = Defnton 35 [8, ] Gven a subexresson v of a reduced exresson w = s s s m, we defne J v := {k {,, m} v (k) < v (k) }, J + v := {k {,, m} v (k) = v (k) }, J v := {k {,, m} v (k) > v (k) } The exresson v s called non-decreasng f v (j) v (j) for all j =,,m, eg J v = Defnton 36 (Dstngushed subexressons) [, Defnton 3] A subexresson v of w s called dstngushed f we have (3) v (j) v (j) s j for all j {,,m} In other words, f rght multlcaton by s j decreases the length of v (j), then n a dstngushed subexresson we must have v (j) = v (j) s j We wrte v w f v s a dstngushed subexresson of w Defnton 37 (Postve dstngushed subexressons) We call a subexresson v of w a ostve dstngushed subexresson (or a PDS for short) f (3) v (j) < v (j) s j for all j {,,m} In other words, t s dstngushed and non-decreasng Lemma 38 [8] Gven v w and a reduced exresson w for w, there s a unque PDS v + for v n w

8 8 KELLI TALASKA AND LAUREN WILLIAMS s 5 s 4 s 3 s s s 6 s 5 s 4 s 3 s s 7 s 6 s 5 s 4 s Fgure 4 The labelng of a the boxes of a artton by smle generators s, and two readng orders 33 Go-dagrams In ths secton we exlan how to ndex dstngushed subexressons by certan tableaux called Go-dagrams, whch were ntroduced n [6] Go-dagrams are fllngs of Young dagrams by luses +, black stones, and whte stones Fx k and n Let W k = s, s,, ŝ n k,, s n be a arabolc subgrou of W = S n Let W k denote the set of mnmal-length coset reresentatves of W/W k Recall that a descent of a ermutaton π s a oston j such that π(j) > π(j + ) Then W k s the subset of ermutatons of S n whch have at most one descent; and that descent must be n oston n k It s well-known that elements w of W k can be dentfed wth arttons λ w contaned n a k (n k) rectangle: f w = (w,,w n ) W k then {w n k+, w n k+,,w n } s a subset of sze k, whch gves rse to a artton, as descrbed at the begnnng of Secton We refer to ths artton as λ w Moreover, t follows from [] and [0] that the reduced exressons of w W k corresond to certan readng orders of the boxes of the artton λ w Secfcally, let Q k be the oset whose elements are the boxes of a k (n k) rectangle; f b and b are two adjacent boxes such that b s mmedately to the left or mmedately above b, we have a cover relaton b b n Q k The artal order on Q k s the transtve closure of Now label the boxes of the rectangle wth smle generators s as n the fgure below If b s a box of the rectangle, then let s b denote ts label by a smle generator Let w0 k W k denote the longest element n W k Then the set of reduced exressons of w0 k can be obtaned by choosng a lnear extenson of Q k and wrtng down the corresondng word n the s s We call such a lnear extenson a readng order; two lnear extensons are shown n the fgure below Addtonally, gven a artton λ w contaned n the k (n k) rectangle (chosen so that the uer-left corner of ts Young dagram s algned wth the uer-left corner of the rectangle), and a lnear extenson of the sub-oset of Q k comrsed of the boxes of λ, the corresondng word n s s s a reduced exresson of the ermutaton w W k Moreover, all reduced exressons of w can be obtaned by varyng the lnear extenson Defnton 39 [6, Secton 4] Fx k and n Let w W k, let w be a reduced exresson for w, and let v be a dstngushed subexresson of w Then w and w determne a artton λ w contaned n a k (n k) rectangle together wth a readng order of ts boxes The Go-dagram assocated to v and w s a fllng of λ w wth luses and black and whte stones, such that: for each k Jv we lace a whte stone n the corresondng box; for each k J v we lace a black stone n the corresondng box of λ w ; and for each k J v + we lace a lus n the corresondng box of λ w Remark 30 By [6, Secton 4], whether or not a fllng of a artton λ w s a Go-dagram does not deend on the choce of readng order of the boxes of λ w Defnton 3 We defne the standard readng order of the boxes of a artton to be the readng order whch starts at the rghtmost box n the bottom row, then reads rght to left across the bottom In KW, we used a slghtly dfferent conventon and used blank boxes n lace of + s

9 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 9 row, then rght to left across the row above that, then rght to left across the row above that, etc Ths readng order s llustrated at the rght of the fgure below By default, we wll use the standard readng order n ths aer Examle 3 Let k = 3 and n = 7, and let λ = (4, 3, ) The standard readng order s shown at the rght of the fgure below s 4 s 3 s s s 5 s 4 s 3 s Then the followng dagrams are Go-dagrams of shae λ They corresond to the exressons s 6 s 3 s 4 s 5 s s s 3 s 4, s 6 s 4 s s 3, and s 3 s 4 s s 4 The frst and second are ostve dstngushed subexressons (PDS s), and the thrd one s a dstngushed subexresson (but not a PDS) Note that the followng dagram of shae λ s not a Go-dagram It corresonds to the word s 4 s s, whch s not dstngushed Remark 33 The Go-dagrams assocated to PDS s are n bjecton wth -dagrams, see [6, Secton 4] Note that the Go-dagram assocated to a PDS contans only luses and whte stones Ths s recsely a -dagram Γ If we choose a readng order of λ w, then we wll also assocate to a Go-dagram of shae λ w a labeled Go-dagram, as defned below Equvalently, a labeled Go-dagram s assocated to a ar (v,w) Defnton 34 [6, Defnton 45] Gven a readng order of λ w and a Go-dagram of shae λ w, we obtan a labeled Go-dagram by relacng each wth a, relacng each box b contanng a wth a and an m, and relacng each box b contanng a + by a, where the subscrt corresonds to the label of b nherted from the readng order The labeled Go-dagrams corresondng to the examles above usng the standard readng order are: Γ

10 0 KELLI TALASKA AND LAUREN WILLIAMS m The man result To state the man result, we now consder Go-dagrams (not arbtrary dagrams), the corresondng networks (Go-networks), and the corresondng weght matrces Defnton 35 Let D be a Go-dagram contaned n a k (n k) rectangle We defne a subset R D of the Grassmannan Gr k,n by lettng each varable a of the weght matrx (Defnton 33) range over all nonzero elements K, and lettng each varable c of the weght matrx range over all elements K We call R D the network comonent assocated to D Theorem 36 Let D be a Go-dagram contaned n a k (n k) rectangle Suose that D has t luses and u black stones Then R D s somorhc to (K ) t K u Furthermore, Gr k,n s the dsjont unon of the network comonents R D, as D ranges over all Go-dagrams contaned n a k (n k) rectangle In other words, each ont n the Grassmannan Gr k,n can be reresented unquely by a weghted network assocated to a Go-dagram A more refned verson of Theorem 36 s gven n Corollary 69 Corollary 37 Every matrx can be reresented by a unque weghted network assocated to a Godagram Proof Suose M = (m,j ) s a k (n k) matrx Let A(M) = (a,j ) be the full rank k n matrx wth an dentty submatrx n the frst k columns and the remanng columns gven by a,j+k = () + m n+,j Then A(M) reresents an element n the Grassmannan Gr k,n, so Theorem 36 ales Moreoever, the mnors of M are n bjecton wth the k k mnors of A(M), so f A(M) s reresented by the network N, we see that m,j enumerates aths from the boundary source to the boundary vertex j n N We wll rove Theorem 36 by showng that each network comonent R D from a Go-dagram concdes wth a (rojected) Deodhar comonent P v,w n the Grassmannan (Therefore we may refer to each R D as a Deodhar comonent) More secfcally, such Deodhar comonents have arameterzatons due to Marsh and Retsch [8], and we wll show that after an nvertble transformaton of varables, our network arameterzatons concde wth thers 4 The Deodhar decomoston of the Grassmannan In ths secton we revew Deodhar s decomoston of the flag varety G/B [], and the arameterzatons of the comonents due to Marsh and Retsch [8] The Deodhar decomoston of the Grassmannan s obtaned by rojectng the Deodhar decomoston of G/B to the Grassmannan [3] 4 The flag varety Let K be a feld, and let G denote the secal lnear grou SL n = SL n (K) Fx a maxmal torus T, and ooste Borel subgrous B + and B, whch consst of the dagonal, uertrangular, and lower-trangular matrces, resectvely Let U + and U be the unotent radcals of B + and B ; these are the subgrous of uer-trangular and lower-trangular matrces wth s on the

11 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN dagonals For each n we have a homomorhsm φ : SL SL n such that ( ) a b φ = a b c d c d SL n, ( ) a b that s, φ relaces a block of the dentty matrx wth Here a s at the (+)st dagonal c d entry countng from the southeast corner (Corresondngly, we wll label the rows of such a matrx from bottom to to, and the columns of such a matrx from rght to left) We have -arameter subgrous of G defned by ( ) ( ) m 0 x (m) = φ and y 0 (m) = φ, where m K m Let W denote the Weyl grou N G (T)/T, where ( N G (T) ) s the normalzer of T The smle reflectons 0 s W are gven by s := s T where s := φ and any w W can be exressed as a roduct 0 w = s s s l wth l = l(w) factors We set ẇ = ṡ ṡ ṡ l In our settng W s somorhc to S n, the symmetrc grou on n letters, and s s the transoston exchangng and + We can dentfy the flag varety G/B wth the varety B of Borel subgrous, va gb g B + := gb + g We have two ooste Bruhat decomostons of B: B = B + ẇ B + = B v B + w W We defne the ntersecton of ooste Bruhat cells v W R v,w := B + ẇ B + B v B +, whch s nonemty recsely when v w The strata R v,w are often called Rchardson varetes 4 Deodhar comonents n the flag varety We now descrbe the Deodhar decomoston of the flag varety Marsh and Retsch [8] gave exlct arameterzatons for each Deodhar comonent, dentfyng each one wth a subset n the grou Defnton 4 [8, Defnton 5] Let w = s s m be a reduced exresson for w, and let v be a dstngushed subexresson Defne a subset G v,w n G by (4) G v,w := g = g g l = x l (m l )ṡ l f l J v g g m, g l = y l ( l ) f l J v +, for l K, m l K g l = ṡ l f l Jv, There s an obvous ma (K ) J+ v K J v G v,w defned by the arameters l and m l n (4) For v = w = we defne G v,w = {} Examle 4 Let W = S 8, w = s 6 s 7 s 4 s 5 s 6 s 3 s 4 s 5 s s s 3 s 4 and v = s 6 s 4 s 6 s 4 Ths s the dstngushed exresson v encoded by the dagram from Fgure 3 (whch s a Go-dagram) Then the corresondng element g G v,w (the MR-matrx) s gven by (4) g = ṡ 6 y 7 ( )ṡ 4 y 5 ( 4 )x 6 (m 5 )ṡ 6 y 3( 6 )x 4 (m 7 )ṡ 4 y 5( 8 )y ( 9 )y ( 0 )y 3 ( )y 4 ( ),

12 KELLI TALASKA AND LAUREN WILLIAMS whch s m m 7 8 m The followng result from [8] gves an exlct arametrzaton for the Deodhar comonent R v,w We use Prooston 43 as the defnton of R v,w Prooston 43 [8, Prooston 5] The ma (K ) J+ v K J v G v,w from Defnton 4 s an somorhsm The ma g g B + defnes an somorhsm (43) G v,w Rv,w between the subset G v,w of the grou, and the Deodhar comonent R v,w n G/B Suose that for each w W we choose a reduced exresson w for w Then t follows from [] and [8, Secton 44] that (44) R v,w = R v,w and G/B = ( ) R v,w, v w where n the frst sum v ranges over all dstngushed subexressons for v n w, and n the second sum v ranges over all dstngushed subexressons of w These two decomostons are called the Deodhar decomostons of R v,w and G/B Remark 44 Although the Deodhar decomoston of R v,w deends on the choce w of reduced exresson for w, ts rojecton to the Grassmannan does not deend on w [6, Prooston 46] 43 Projectons of Deodhar comonents to the Grassmannan Followng [6], we now consder the rojecton of the Deodhar decomoston to the Grassmannan Gr k,n for k < n Gven w = (w(), w(),, w(n)) W k, we let I(w) = {w(n k + ), w(n k + ),, w(n)} The ma I gves a bjecton between W k and k-element subsets of [n] Let π k : G/B Gr k,n be the rojecton from the flag varety to the Grassmannan; ths s an somorhsm on each R v,w For each w W k and v w, defne P v,w = π k (R v,w ) Then by [7] we have a decomoston (45) Gr k,n = w W k For each reduced decomoston w for w W k, and each v w, we defne P v,w = π k (R v,w ) Now f for each w W k we choose a reduced decomoston w, then we have (46) P v,w = P v,w and Gr k,n = ( ) P v,w, v w v w P v,w w W w W k where n the frst sum v ranges over all dstngushed subexressons for v n w, and n the second sum v ranges over all dstngushed subexressons of w v w v w

13 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 3 Prooston 43 gves a concrete way to thnk about the rojected Deodhar comonents P v,w The rojecton π k : G/B Gr k,n mas g B + R v,w (where g G v,w ) to the san of the leftmost k columns of g More secfcally, t mas g = g n,n g n,n k+ g n, g,n g,n k+ g, M = g,n k+ g n,n k+ g,n g n,n We call the resultng k n matrx M = (M st ) the MR-matrx To smlfy the notaton later, we wll label ts rows from to to bottom by,,, k, where { < < k } = I(w) Remark 45 Recall from Secton 33 that n the Grassmannan settng (e W k = s, s,,ŝ n k,,s n s a arabolc subgrou of W = S n ), the dstngushed subexressons of W k are n bjecton wth Godagrams Therefore each Go-dagram gves rse to an MR-matrx Examle 46 We contnue Examle 4 Note that w W k, where k = Then the ma π : G v,w Gr,5 s gven by m g = m 7 8 m m M = m m We label the rows of M from to to bottom by the ndex set {, 3, 4, 6}, and the columns from left to rght by the ndex set {,,, 8}, so eg M 34 = m 7 8 The followng lemma s a consequence of [6, Secton 5] and n artcular [6, Corollary 58] Lemma 47 Let M = M D be the MR-matrx assocated to the dagram D The leftmost nonzero entry n row l of M s n column l Furthermore, that entry s equal to () b, where b s the number of black stones n the row l of D, and the roduct s over all boxes n the row l of the labeled Go-dagram of D contanng a 5 Formulas for entres of the MR-matrces In ths secton we consder arbtrary dagrams (not necessarly Go-dagrams) contaned n a k (n k) rectangle and the corresondng MR-matrces, obtaned by multlyng factors ṡ, y ( j ), x (m j )ṡ as secfed by the fllng of the dagram, and then rojectng the resultng n n matrx to a k n matrx We wll gve formulas for the entres of the MR-matrces n terms of seudoaths n the corresondng network For the urose of gvng ths formula, we wll relace weghts a and c j on the edges of the network by weghts and m j Recall that f D s a dagram, ts network N D has three tyes of vertces: +-vertces, -vertces, and boundary vertces A ste on a network s an edge between two vertces Let W denote a sngle ste west, S denote a sngle ste south, and E denote ether a sngle ste east, or an east-west combnaton ste consstng of a ste east, followed by a ste west endng at a Let A ndcate 0 or more nstances of a ste of tye A

14 4 KELLI TALASKA AND LAUREN WILLIAMS Defnton 5 A seudoath P on a network s a ath on the (undrected verson of the) network such that: t starts and ends at two dfferent boundary vertces, or else s the emty ath from a boundary vertex to tself; t does not cross the same edge twce; ts sequence of stes (for a nonemty ath) has the form WW S(EE S) E In artcular, a seudoath may not take two consecutve stes south Defnton 5 The weght w( P) of a seudoath P n a network s a Laurent monomal n s and m j s, whch s obtaned by multlyng the followng terms: for every ste west along an edge weghted ; for every ste east along an edge weghted whch s receded by a ste east; m j for every ste west along an edge weghted m j ; () b+w, where b (resectvely w) s the number of black (res whte) stones that the seudoath sks over n the horzontal (res vertcal) drecton, when we suermose the Godagram onto the network Examle 53 In Fgure 5, there are two seudoaths from to 4, wth weghts 9 0 and and there s one seudoath from to 5, wth weght 9 0 m 7 9 0, m m m m m m Fgure 5 The two seudoaths from to 4 and the unque seudoath from to 5, ndcated n bold Note that the seudoath n the mddle fgure contans an east-west combnaton ste Defnton 54 If M s an MR-matrx, we wll let M denote the matrx obtaned from M by rescalng rows so that the leftmost nonzero entry n each row s Defnton 55 If D s a dagram contaned n a k (n k) rectangle, then we let < < < k denote the labels of the sources n the corresondng network If M and M are the corresondng k n MR and rescaled MR-matrces assocated to D, then we wll ndex ther rows by,, k from to to bottom, and ther columns by,,, n from left to rght

15 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 5 Theorem 56 Let D be a dagram contaned n a k (n k) rectangle, and let M = ( M st ) be the corresondng k n rescaled MR-matrx Then M st = e P w( P), where the sum s over all seudoaths from the source s to the boundary vertex t n the network The roof of Theorem 56 wll follow from Theorem 50 and Lemma 5 Examle 57 The MR-matrx M from Examle 46 corresonds to the network from Fgure 5 The rows of M are ndexed by, 3, 4, 6 from to to bottom Note that after we rescale the rows of M, obtanng M, we have M m 4 = , and M 5 = 9 0 Ths agrees wth our seudoath comutaton from Examle 53 Next we wll gve a formula for entres of MR-matrces, n terms of seudoaths n modfed networks Defnton 58 Gven a network N D wth k sources labeled,, k and n boundary vertces, we obtan from t a corresondng modfed network N D, by: addng k new boundary vertces to the left of N D, labeled,, k from to to bottom; addng a new horzontal edge whch connects j to the leftmost nternal vertex of the network See Fgure 6 for the modfed network assocated to the network from Fgure 5 Defnton 59 A seudoath P on a modfed network s a ath on the modfed network whch: starts at one of the boundary vertces labeled,, k, and ends at one of the boundary vertces labeled,,,n; takes a sequence of stes whch has the form (EE S) E The arrows n Fgure 6 ndcate the allowed drectons n whch a ath may travel The weght of a seudoath n a modfed network s defned the same way as the weght of a seudoath n a network (see Defnton 5) Note that snce a seudoath n a modfed network does not contan stes west along edges weghted, ts weght s a monomal (not a Laurent monomal) m m Fgure 6 Examle of a modfed network

16 6 KELLI TALASKA AND LAUREN WILLIAMS Theorem 50 Let D be a dagram contaned n a k (n k) rectangle, and let M = (M st ) be the corresondng k n MR-matrx Then M st = P w(p), where the sum s over all seudoaths n the modfed network from the boundary vertex s to the boundary vertex t Lemma 5 Theorems 56 and 50 are equvalent Proof There s an obvous bjecton between seudoaths n a network startng at boundary vertex s, and seudoaths n the corresondng modfed network startng at boundary vertex s The weghts of the corresondng seudoaths are the same excet for a factor of () b, where b s the number of -vertces n row s of the network, and the s range over all edge weghts n row s On the other hand, Lemma 47 mles that the leftmost nonzero entry of row s of the MR-matrx M s recsely the quantty () b above Therefore Theorems 56 and 50 are equvalent By Lemma 5, t suffces to rove just one of the two theorems We wll rove Theorem 50 Our strategy wll be to nterret entres of the MR-matrx n terms of aths n a ch network, and then construct a weght-reservng bjecton between these aths and between seudoaths n the modfed network Defnton 5 A ch s one of the three confguratons shown n Fgure 7 We call the three confguratons y ()- or y -chs, s -chs, and x (m)- or x -chs, resectvely m n n n y () y -ch ṡ s -ch x (m)ṡ x -ch Fgure 7 The three tyes of chs Defnton 53 A ch network s a concatenaton of chs Note that t has n boundary vertces at the left and n boundary vertces at the rght Let g be a roduct of factors of the form y (), ṡ, and x (m)ṡ We assocate a ch network C g to every such roduct g by concatenatng the chs corresondng to the factors of g n the order gven by the factorzaton Examle 54 The ch network C g assocated to the roduct g from (4) s shown n Fgure 8

17 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN m m Fgure 8 The ch network corresondng to the roduct g = ṡ 6 y 7 ( )ṡ 4 y 5 ( 4 )x 6 (m 5 )ṡ 6 y 3( 6 )x 4 (m 7 )ṡ 4 y 5( 8 )y ( 9 )y ( 0 )y 3 ( )y 4 ( ) Defnton 55 A route Q n a ch network s a ath n the network whose stes all travel east (or southeast or northeast for slanted edges) The weght w(q) of such a route s the roduct of all weghts on ts edges To each ch network C we assocate a weght matrx x(c) = x j where x j = w(q) Q and the sum s over all routes from the boundary vertex at the west to the boundary vertex j at the east It s smle to verfy the followng result Recall our conventon from Secton 4 that the rows of g are labeled from bottom to to, and the columns of g are labeled from rght to left Lemma 56 Let g be a roduct of factors of the form y (), ṡ, and x (m)ṡ Then the weght matrx x(c g ) of the ch network C g assocated to g concdes wth the matrx g We now rove Theorem 50 Proof Let D be a dagram contaned n a k (n k) rectangle, and N D the corresondng modfed network Let < < k be the labels of the sources of N D Let g be the roduct of factors of the form y (), ṡ, and x (m)ṡ whch s encoded by D, and let M be the corresondng MR-matrx, whose rows are ndexed from to to bottom by,, k Recall that the rojecton from g to M swtches rows and columns, and the columns labeled n k +, n k +,, n n g become rows labeled,,, k n M Therefore to rove Theorem 50, t suffces to rove that for all t n and s k we have (5) g t,s+(n k) = w(p), P where the sum s over all seudoaths P from s to t n the modfed network By Lemma 56, the matrx g concdes wth the weght matrx x(c g ) = (x st ) of the ch network assocated to g Therefore by (5) t suffces to rove that for all t n and s k we have (5) x t,s+(n k) = w(p), P where the sum s over all seudoaths P from s to t n the modfed network Recall from Defnton 55 that x j = w(q), where the sum s over all routes Q n the ch network Q from the boundary vertex at the west to the boundary vertex j at the east To rove (5), we wll

18 8 KELLI TALASKA AND LAUREN WILLIAMS gve a weght-reservng bjecton between seudoaths P n the modfed network from s to t, and routes Q n the ch network from the boundary vertex t at the west to the boundary vertex s+(n k) at the east More secfcally, gven a seudoath P, we wll examne ts sequence of stes from source to snk, and exlan how to buld the corresondng route Q n the ch network As llustrated n Fgure 9, each ste n a seudoath corresonds to a orton of a route n a ch network (Note that our bjecton wll buld the route n the ch network from east to west, rather than west to east) Recall from Fgure 4 that each modfed network comes from a dagram, and that every box of a dagram s naturally assocated wth a smle generator s Therefore every nternal vertex n a modfed network s naturally assocated wth a smle generator s for some We wll call ths the oston of the vertex Let us consder the varous knds of stes n a seudoath Such stes naturally fall nto one of the followng categores (llustrated n Fgure 9): 0 A sngle ste east, whch starts at a source, and ends at oston s I A sngle ste east, whch s receded by a south ste, and followed by an east or south ste Such a ste starts and ends at ostons s and s j (for > j), and s labeled by some weght It may sk over some (ostons corresondng to) whte and black stones n the Go-dagram II An east-west combnaton ste, whch s receded by a south ste, and travels from oston s to s k to s j (where > j > k) The two comonents of such a ste are labeled by some weghts and m, and may sk over some whte and black stones III A sngle ste east, whch s receded by an east ste, and followed by an east or south ste Such a ste starts and ends at ostons s and s j (for > j), and s labeled by some weght It may sk over some whte and black stones IV An east-west combnaton ste, whch s receded by an east ste, and travels from oston s to s k to s j (where > j > k) The two comonents of such a ste are labeled by some weghts and m, and may sk over some whte and black stones V A south ste, whch starts and ends at ostons s and s j (for < j) Such a ste may sk over some whte and black stones The above stes n a seudoath corresond to the followng orton of a route n a ch network: 0 Start at the boundary vertex + at the east of the ch network I Start at level, then travel west straght across the y () ch For each whte or black stone (say n oston s l ) whch les n between ostons s and s j, travel northwest through the corresondng s l or x l -ch, endng at level j + II Start at level, then travel west straght across the y () ch For each whte or black stone (say n oston s l ) whch les n between ostons s and s j, travel northwest through the corresondng s l or x l -ch Fnally, travel along the -edge and then the m-edge of the x j (m) ch, endng at level j + III Start at level +, then travel northwest along the -edge n the y () ch For each whte or black stone (say n oston s l ) whch les n between ostons s and s j, travel northwest through the corresondng s l or x l -ch, endng at level j + IV Start at level +, then travel northwest along the -edge n the y () ch For each whte or black stone (say n oston s l ) whch les n between ostons s and s j, travel northwest through the corresondng s l or x l -ch Fnally, travel along the -edge and the m-edge of the x j (m) ch, endng at level j + V Start at level + For each whte or black stone (say n oston s l ) whch les n between ostons s and s j, travel southwest through the corresondng s l or x l -ch, endng at level j It s a straghtforward exercse to verfy that ths ma s a bjecton between seudoaths P from s to t n the modfed network, and routes Q between the t vertex at the west and the s + (n k) vertex

19 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN 9 (I) s s j j j + m j + + j j + j + + (II) s s j m s k j j + j + + m m j j + j + + (III) s s j j j + m j + + j j + j + + (IV) s s j m s k j j + j + + m m j j + j + + (V) s + + m + + s j j j j + j j j + Fgure 9 Stes n seudoaths and ther corresondng fragments of the ch network at the east n the ch network Moreover, the weghts of P and Q are equal See Fgure 0 for examle of entre seudoaths and routes

20 0 KELLI TALASKA AND LAUREN WILLIAMS m m m m m m m m Fgure 0 Pseudoaths n the modfed network and ther corresondng routes n the ch network 6 Proof of the man result Let D be a dagram whch contans t luses and u black stones In Secton 6 we wll defne an somorhsm Ψ = Ψ D : (K ) t K u (K ) t K u whch mas each arameter from the weght matrx of the network N D to a Laurent monomal n the arameters used n the MR-matrx M = M D Then n Secton 6, we wll show that after alyng Ψ, our network arameterzaton of the network comonent R D concdes wth the corresondng MRarameterzaton of the rojected Deodhar comonent P v,w Combnng ths fact wth Prooston 43 yelds Theorem 36 6 A ratonal transformaton of arameters Defnton 6 Let D be a dagram, and let b 0 be a box of D contanng a + or Let b be the nearest box to the rght of b 0 whch contans a + (f t exsts) Let R t be the set of boxes n the same row as b 0 whch are to the rght of b 0 and left of b Let R l be the set of boxes n the same column as b 0 and below b 0 If b exsts, let R r be the set of boxes n the same column as b and below b (otherwse R r = ) (See Fgure ) Let R + r (res R + l ) be the set of boxes n R r (res R l ) contanng a + Let R = R r R l R t and let R be the set of boxes n R contanng a If b 0 contans a +, then defne Ψ(a b0 ) = () R b R + b r b0 b b R + l

21 NETWORK PARAMETERIZATIONS FOR THE GRASSMANNIAN And f b 0 contans a, then defne Ψ(c b0 ) = m b 0 () R b R + b r b R + l b We also extend the defnton of Ψ to all olynomals n the a b s and c b s by requrng t to be a rng homomorhsm b 0 R t b R l R r Fgure The defnton of b 0, b, R t, R l, and R r Remark 6 Clearly Ψ = Ψ D s an somorhsm from (K ) t K u to tself Examle 63 Consder the network from Fgure 3 (shown agan n Fgure ) Then we have Ψ(a ) =, Ψ(a 4 ) = 4, Ψ(c 5 ) = m5, Ψ(a 6 ) = 6, Ψ(c 7 ) = m7 4, Ψ(a 8 ) = 8, Ψ(a 9 ) = 9, Ψ(a 0 ) = 6 0, Ψ(a ) = 6 4, and Ψ(a ) = 4 8 From Defnton 6, one may easly deduce a formula for Ψ(w(P)), where P s a ath n N D We wll state ths formula n terms of the Go-dagram Note that one may dentfy a ath P n N D wth a connected sequence P of boxes n the Go-dagram, where any two adjacent boxes must share a sde We call a box b n P a corner box f the ath P turns from west to south, or from south to west, at the vertex assocated to b (Such a box b n D must contan a + or ) Prooston 64 Let P be a ath n the network N D, whch we dentfy wth a sequence P of boxes n the Go-dagram D Among the boxes n P, let B denote the subset contanng a +; let B denote the subset contanng a whch are corner boxes of P; and let B 3 denote the subset contanng a whch are not corner boxes of P Then (6) Ψ(w(P)) = () B3 b B m b b B b Proof We leave ths as an exercse for the reader It s a smle consequence of Defnton 6 Examle 65 Let P and P be the aths shown n Fgure Then Ψ(w(P )) = Ψ(a 9 a 0 a c 5 ) = m and Ψ(w(P )) = Ψ(a 9 a 0 c 5 c 7 ) = m5m Alyng row oeratons to the rescaled MR-matrx Theorem 66 Let D be a dagram, let M = M D be the corresondng MR-matrx, and let L be the matrx we obtan by uttng M nto reduced row-echelon form Let W D = (W j ) be the weght matrx assocated to D, and let Ψ(W D ) be the matrx obtaned from W D by alyng the ratonal ma Ψ to each entry Then Ψ(W D ) = L

22 KELLI TALASKA AND LAUREN WILLIAMS Fgure Two aths n N D and ther corresondng sequences of boxes n D Proof To rove Theorem 66, we start by consderng the rescaled MR-matrx M Its rows are ndexed by the set,, k, the set of sources of the network N D, and the leftmost nonzero entry n every row s a Moreover, by Lemma 47, the n row l s located n column l The entres of the reduced row echelon matrx L obtaned from M are gven by the formula (6) L st = M st + s<j < <j r<t () r Msj Mjj M jrt, where the sum ranges over all nonemty subsets {j,,j r } {,, k } of sources of the network between s and t By Theorem 56, the entry M st equals e P w( P), where the sum s over all seudoaths n the network N D from the source s to the boundary vertex t Therefore the rght-hand sde of (6) can be nterreted as a generatng functon for all concatenatons of seudoaths, where the frst seudoath starts at s, and the last seudoath ends at t Let us dentfy a seudoath wth ts collecton of drected edges Gven a set of seudoaths on N D, we defne ts sgned unon to be the unon of drected edges that one obtans by takng the multset of all drected edges n the seudoaths, and then cancellng ars whch traverse the same edge but n ooste drectons We defne the weght of a set of seudoaths to be the roduct of the weghts of each seudoath n the set Our goal s to show that after cancellaton, the only terms whch survve on the rght-hand sde of (6) corresond to concatenatons of seudoaths whose sgned unon forms a drected ath (and not merely a seudoath) from s to t n N D Ths wll allow us to relate (6) to W st, whch s defned as a sum over all aths from s to t n N D Defnton 67 A U-turn seudoath n a network s a seudoath whose sequence of stes has the form (WW S)E Frst note that any ath P n a network N D has a unque decomoston as a sgned unon of U-turn seudoaths Moreover, the roducts of the weghts of the seudoaths s recsely the quantty on the rght-hand sde of (6) See Fgure 3 for an examle of the decomoston nto U-turn seudoaths Ths observaton on the decomoston of aths may be generalzed to seudoaths Consder a seudoath P whch s not a ath, and turns from south to east recsely q tmes (for q ) Then for each 0 r q, there are ( q r) decomostons of P as a sgned unon of r seudoaths Moreover, each set of seudoaths formng a decomoston of P has the same weght See Fgure 4 for an examle of all decomostons of a seudoath as a sgned unon of seudoaths It s easy to check that each decomoston has the same weght