The Saga of Reduced Factorizations of Elements of the Symmetric Group

Transcription

1 Topics in Algebric Combintorics LECTURE NOTES April, 00 The Sg of Reduced Fctoriztions of Elements of the Symmetric Group by A. M. Grsi Abstrct These notes cover the contents of series of lectures in Topics in Algebric Combintorics course given t UCSD in Winter 00. The initil effort ws prompted by desire to understnd the connections between the theory of reduced decompositions strted by the pioneering pper [] of R. Stnley nd the theory of blnced tbloids studied by C. Green et l. [] [] However soon it ppered quite cler tht deeper understnding of the subject requires prllel understnding of the Lscoux-Schützenberger theory of Schubert polynomils. These notes should offer glimpse of the fscinting combintoril connections between these theories. The presenttion is generlly self contined. The notes culminte with wht should be firly lucid nd illuminting proof of the Schur positivity of the Stnley symmetric functions. TABLE OF CONTENTS. Reduced Fctoriztions. Nottion. The Mtrix Approch. Blnced Lbelled Circle Digrms. From Mtrices to Tbloids. Descents nd Kevin Kdel s ZIGZAGs. Specil Circle Digrms. The Lscoux Schützenberger Tree of Generl Permuttion. Symmetric Functions nd Schubert Polynomils. Stnley s Theory of P-Prtitions. The Stnley Symmetric function of Permuttion. Divided Differences nd Schubert Polynomils REFERENCES ()

2 Topics in Algebric Combintorics LECTURE NOTES April, 00 Introduction In 98 R. Stnley initited the study of reduced decompositions of elements of S n. Centrl to his wor ws the introduction of fmily of symmetric functions indexed by permuttions. He conjectured these functions to be Schur positive nd proved number of their interesting properties including the enumertion of certin clsses of reduced decompositions. Over the yers tht followed severl wors hve ppered with different proofs of the Stnley conjecture by vrious methods which rnge from the purely combintoril to the purely lgebric. Circ 98 in completely independent development Lscoux nd Schützenberger founded the Theory of Schubert polynomils. Centrl to their study were some combintoril consequences of Pieri-lie result for Schubert polynomils which they clled Mon s rule. This led to the definition of tree ssocited to every permuttion σ S n. Unbenown to them t the time nd to mny even t the present time, the LS tree of permuttion is, in sense tht cn be mde precise, purely combintoril version of the Stnley symmetric function. Using this tree nd severl combintoril properties of reduced decompositions, the Schur positivity of the Stnley symmetric function follows in remrbly illuminting mnner. In these notes we present the contents of series of lectures in Topics in Algebric Combintorics Course given t UCSD in Winter 00. The mteril by no mens covers ll the spects of the fscinting subject of reduced decompositions tht hve been developed over the lst two decdes. The choice of topics, limited by the time vilble, follows the tste of the uthor nd wht ppered to be nturl pth through luscious forest of remrble combintoril discoveries. We strived throughout to me our presenttion s self-contined s possible. Some of the lter proofs tht ppered in the literture fter the originl ppers re so elegnt nd simple tht we were forced to reproduced them here lmost verbtim. We clim no credit here for ny of the results presented. This in only n expository wor. Our min effort hs been concentrted into providing novel nd illuminting wy to develop the mteril. Our originl stimulus for choosing this topic cme from severl exciting exchnges with Kevin Kdel visitor t UCSD for the cdemic yer We lso benefitted immensely from some of the insights he provided us in the study nd developments connecting reduced decompositions to blnced tbloids. ()

3 Topics in Algebric Combintorics LECTURE NOTES April, 00 ()

4 Topics in Algebric Combintorics LECTURE NOTES my, 00 The Sg of Reduced Fctoriztions of Elements of the Symmetric Group by A. M. Grsi. Reduced Fctoriztions. Nottion It is customry to interpret permuttion σ S n s bijection of {,,...,n} onto itself nd we often write it in the form ( )... n σ =, σ σ σ... σ n mening tht σ i is the imge of i under σ. In this vein to compute the product θ σ we proceed from right to left nd obtin (... n θ σ θ σ θ σ... θ σn ) = (... n θ θ θ... θ n ) ( )... n σ σ σ... σ n Keeping this in mind, it will be convenient nd economicl with spce to omit the Þrst line nd simply write σ = σ σ σ σ n viewing σ s word in the letters,,,...,n. Here nd fter we let s i (for i n ) represent the simple trnsposition ( ) i i+ n s i =(i, i +)=.. i+ i n Note tht multipliction of σ on the right by s i results in the interchnge of the elements σ i,σ i+. Thus in our shorthnd we my write σ σ σ i+ σ i σ n = σ σ σ i σ i+ σ n s i. Let us recll tht the number of inversions of σ is given by the sum inv(sig) = χ(σ i >σ j ). i<j n It is cler tht right multipliction of σ by ny simple trnsposition increses the number of inversions by one if σ i <σ i+ nd decreses it by one if σ i >σ i+. Let us recll tht n index i such tht σ i >σ i+ is clled ÒdescentÓ ofσnd correspondingly D(σ) = { i n :σ i >σ i+ } is usully referred to s the Òdescent setóofσ. This given, if we wnt to express n element σ s product of simple reßections the number of fctors required should be t the very lest inv(σ). For this reson, inv(σ) is often referred to s the ÒlengthÓ nd brießy lso denoted by l(σ). Note tht it is lwys possible (in fct in

5 Topics in Algebric Combintorics LECTURE NOTES my, 00 mny wys) to express σ s product of l(σ) simple trnspositions. To do this we simply strt with σ = σ (o) nd construct sequence of permuttions 78 7 σ (r+) σ (r) σ () σ () σ (o) with σ (r+) = σ (r) s i nd where i is only chosen by the requirement tht i be 78 in the descent set of σ (r), tht is σ (r) 78 i >σ (r) i+. Since this requirement ssures tht l(σ (r+) )=l(σ (r) ) the sequence will stop fter exctly l(σ) steps with σ (l(σ)) = n, (the identity permuttion). In the disply on the right we illustrte such 78 sequence for the permuttion σ = 78. Here the lbels on the right of the 78 dividing line give the indices i for which the correspondind s i ws chosen. It should 78 be pprent from this exmple tht ech time we hve vriety of choices, (one for 78 ech element of the descent set of the current permuttion). 78 Fctoriztions of permuttion σ s product of l(σ) reßections re clled ÒreducedÓ nd the word in the letters,,..., n giving the successive indices of the fctors is clled the Òreduced word Ó corresponding to the fctoriztion. Thus for the fctoriztion bove 78 = s s s s s s s s s s s 7.. the corresponding reduced word is 7 Fctoriztions into simple reßections whether reduced or not re best studied by mens of line digrm which exhibits the trjectories of ech of the lbels,,...,ns we proceed in our construction of the trget permuttion. In the disply below we illustrte the digrm corresponding to the fctoriztion illustrted bove A close exmintion of this disply revels one fundmentl property of digrms corresponding to reduced fctoriztions: for ny pir of indices i<j n: the i-line nd j-line cross t most once. The reson for this is quite simple: once we interchnge i nd j, doing it gin would decrese the number of inversions, nd we never do tht to get reduced fctoriztion.

6 Topics in Algebric Combintorics LECTURE NOTES my, 00 We should mention tht there is systemtic wy of getting reduced fctoriztion for ny permuttion σ = σ σ σ n. Strting from the identity permuttion, we me Þrst the interchnges tht bring σ to Þrst position, then those tht bring σ to second position, then those tht bring σ to third position nd so on until we rech σ. This is best understood by n exmple. In the next disply we hve illustrted this process pplied to σ =... We thus obtin the fctoriztion = s s s s s s s s. It is esily seen tht, in generl, the resulting fctoriztion will be of the form σ = n i= ( si s i s i s i+ s i ).. with i i (note tht i = i must be included for the cses when the the corresponding fctor should be ten equl to (i.e. missing). Here nd fter these fctoriztions will be clled ÒcnoniclÓ. A momentõs reßection should revel tht these observtions yield the following bsic identity Theorem.. σ = σ S n n i= ( +si +s i+ s i + s i+ s i+ s i + +s n s i+ s i+ s i ).. It should be understood tht the fctors in the right hnd side of.. re to be ten from left to right s i goes from to n. This given, interpreting the left hnd side s n element of the group lgebr of S n, then the identity simply sserts tht ech σ S n hs fctoriztion of the form given in... The following bsic identities will ply fundmentl role in the sequel, they re usully referred to s the ÒCoxeter ReltionsÓ. Proposition.. ) s i = id i n, ) s i s i+ s i = s i+ s i s i+ i n, ) s i s j = s j s i if i j...7

7 Topics in Algebric Combintorics LECTURE NOTES my, 00 The Þrst nd lst follow immeditely from the deþnitions of the s i. The middle one just expresses the fct tht the permuttion θ i = ( i i+ i+ ) n i+ i+ i n hs two reduced decompositions. We should lso point out tht the right hnd side of..7 ) is in fct the cnonicl decomposition of θ i. A visul understnding of this reltion my lso be provided by the following disply b b c c c b c b b c b c c c b b This is but n instnce of the more generl result which my be stted s follows Theorem.. We my pss from ny reduced fctoriztion to ny other of given permuttion σ by sequence of pplictions of identities..7 ) & ). The inclusion of..7 ) is only necessry to pss from non-reduced fctoriztion of σ to reduced one. It is sufþcient to show tht we cn pss from ny fctoriztion of σ to cnonicl one. To this end our Þrst step is to show tht we my pss from ny fctoriztion which does not contin s,s,...,s i to one which contins t most one occurrence of s i. We cn prove this by descent induction on i. Clerly the ssertion is trivil for i = n. So let us ssume tht it is true for i +,i+,...,n nd let W be fctoriztion which contins no occurrences of s,s,...,s i. Suppose W contins two occurrences of s i nd let us write it in the form W = W s i W s i W..8 with no occurrences of s,s,...,s i in W. So by induction we chnge W to expression W which contins no occurrences of s i+ or one of the form W = W s i+ W with W nd W not contining ny occurrences of s,s,...,s i+. In the Þrst cse, by successive uses of the Coxeter reltions we cn crry out the three trnsitions W = W s i W s i W W s i W s i W W s i s i W W W W W. In fct, the second trnsition only needs successive uses of..7 ). Clerly, this cse only occurs when W is not reduced. In the other cse, using the Coxeter reltions we Þrst crry out the trnsition W = W s i W s i W W s i W s i+ W s i W.

8 Topics in Algebric Combintorics LECTURE NOTES my, 00 Since W nd W hve only occurrences of s j with j>i+, by successive uses of..7 ) we cn then crry out the trnsition W s i W s i+ W s i W W W s i s i+ s i W W nd Þnlly use of..7 ) completes the sequence W = W s i W s i W W s i W s i+ W s i W W W s i s i+ s i W W W W s i+ s i s i+ W W reducing by one the number of occurrence of s i in W. Proceeding in this mnner we cn rrive t point where either there is only one s i left or none t ll. This completes our induction. This given, strting from ny fctoriztion W, by mens of the Coxeter reltions we cn eliminte ltogether ll the occurrences of s or crry out the trnsition W W s W with W nd W contining no occurrences of s. By further sequence of steps we cn crry out one of the two trnsitions W s W W s W s W or W s W W s W with no occurrences of s or s in W or W. In ech cse successive uses of..7 ) will complete the succession of trnsitions W W s W W s W s W W s s W W or W W s W W s W s W W. Since there re no other occurrences of s in either cse nd no ocurrences of s or s in W in the Þrst cse, we see tht the pttern typicl of cnonicl fctoriztion is beginning to emerge. Indeed the next step is to wor on W nd obtin one of the trnsitions W W s W or W s W with no occurrennces of s,s,s in W. This gives the trnsitions W W s s W W W s W s s W W W W s s s W W or W W s s W W s W s s W W s s s W W W. We need not sy ny more here. The reder should hve no difþculty understnding how this process cn be continued to yield in the end cnonicl decomposition of the permuttion σ corresponding to the fctoriztion W. To cler up ny remining uncertinties it my be pproprite to crry out the ll the steps necessry in prticulr instnce. A good cse in point is the fctoriztion in... In the disply below the lbels on the right of the verticl line indicte which of the Coxeter reltions re used in tht prticulr trnsition the boxes pper s soon s one of the descent strings typicl of cnonicl fctoriztions is formed

9 Topics in Algebric Combintorics LECTURE NOTES my, 00 The min gol of these notes is to present some of the min results obtined in the description nd enumertion of ll reduced decompositions of ny given permuttion. Nevertheless, we should note t this point tht, t lest for smll n, these reduced words cn be constructed by computer in reltively simple mnner. This construction is bsed on the following identity. Theorem.. If for given σ S n, we denote by RED(σ) the collection of ll words corresponding to reduced fctoriztions of σ then w = w i..9 w RED(σ) i D(σ) w RED(σs i) It might be good to strt by explining the nottion used in..9. To begin with the left hnd side should be interpreted s the forml sum of ll the elements of R(σ). Thus to prove..9 we only hve to show tht ech summnd occurring in the left hnd side occurs once nd only once on the right hnd side. Finlly, we should note tht the symbol Òw i Ó simply mens the word obtined by ppending the letter i to the word w. Now note tht if W = W s i is reduced fctoriztion of σ then we must necessrily hve σ i >σ i+ nd W will necessrily be reduced fctoriztion of σ = σs i. This is becuse W is fctoriztion of σ nd the number of its fctors is l(σ) =l(σ ). Now if w is the word corresponding to W nd w is the word corresponding to W we hve w = w i. This given we see tht ll w RED(σ) do occur in the right hnd side nd they occur only once for the simple reson tht ech sum w RED(σs i) w i consists of distinct words nd different vlues of Òi Ó yield different sums of words. It will be instructive t this point to show how this identity cn be trnslted into MAPLE progrm. However, before implementing. we need three uxiliry procedures ÒsigctÓ, ÒpredsÓ, ÒcoctÓ. The Þrst hs input vribles, n index i nd permuttion σ. Then sigct returns the permuttion σ = σs i. The procedure preds tes permuttion σ s input nd returns ll the ÒpredecessorsÓ of σ, tht is the collection PRED(σ) = {σ : σ = σs i & σ i >σ i+ }..0 Finlly, coct tes two input vribles, n index s nd list of words L. Its output is the list of ll words obtined by ppending the index s to ech word of L. These three procedures re given below sigct:=proc(i,sig) locl j,out; out:=[seq(sig[j],j=..i-), sig[i+],sig[i],seq(sig[j], j=i+..nops(sig))]; end: preds:=proc(sig) locl n,out,i; n:=nops(sig); out:=null; for i from to n- do if sig[i]>sig[i+] then out:=out,[i,sigct(i,sig)]; fi; od; [out]; end coct:=proc(s,l) locl out,i,w; out:=null; for w in L do out:=out,[op(w),s]; od; out; end:

10 Topics in Algebric Combintorics LECTURE NOTES my, 00 7 This given, the following procedure with input permuttion σ returns ll the words corresponding to reduced fctoriztions of σ. It cn be esily checed tht it simply expresses in MAPLE lmost verbtim the identity in..0. REDS:=proc(sig) locl prevs,out,i,s,m,tu,te,med; prevs:=preds(sig); if prevs=[] then out:=[[]]; else te:=null; m:=nops(prevs); for i from to m do s:=prevs[i][]; tu:=prevs[i][]; med:=coct(s,reds(tu)); te:=te,med; od; out:=[te]; fi; out; end; Now cll of REDS([,,, ]) yielded reduced words s listed below... We need to introduce combintoril structure which will ply crucil role in our further developments. Given permuttion σ = σ σ σ σ n we ssocite to it n n n digrm with entries Ò Ó, ÒXÓ orò Ò, s follows. In column j nd row σ j we plce n X. This done, in ll the positions west or below this X we plce n Ò Ò. Finlly when ll the XÕs nd the Õs hve been plced we Þll the remining positions with Õs. The resulting Þgure will be referred to here nd fter s the ÒCircle DigrmÓ of the permuttion σ. The disply below gives the circle digrm of the permuttion σ =

11 Topics in Algebric Combintorics LECTURE NOTES my, 00 8 Remr.. We should note tht ech of the circles correspond to n inversion of σ. Indeed, from our construction of circle digrms we will hve Ò Ó in position (i, j) if nd only if the ÒXÓ in column j occurs below (i, j) nd the ÒXÓ inrowi occurs to the right of (i, j). This is equivlent to sying tht σ j >i nd j = σ i >j, Thus this Ò Ó corresponds to the inversion σ j >σ j.. The mtrix pproch Note tht the rerrngement X =(x,x,x,x,x,x,x 7,x 8 ) X =(x,x 8,x,x,x,x,x 7,x ) my simply be obtined by mtrix multipliction. In fct, if we must hve X = XM (interpreting X nd X s row vectors), then we re forced to te M = We clerly see tht the positions of the ones in this mtrix corresponds precisely to the positions of the X s in the circle digrm of 87. More generlly, the trnsition X =(x,x,x,...,x n ) X =(x σ,x σ,x σ,...,x σn ) cn obtined be obtined by right multipliction of X by the mtrix.. M(σ) = χ(i = σ j ) n i,j= We usully refer to M(σ) s the Òpermuttion mtrixó corresponding to σ. Note then tht the permuttion mtrix corresponding to the simple trnsposition s i =(i, i +) of S n my be schemticlly depicted s the n n mtrix M(s ) = i i i In other words, M(s i ) hs entries equl to one in positions (i, i +),(i+,i)nd (j, j) for j =,...,i nd j = i +,...,n, nd ll the remining entries equl to zero.

12 Topics in Algebric Combintorics LECTURE NOTES my, 00 9 This enbles us to view the line digrms in.. nd.. in completely different light. Indeed, note tht we my write the i, j-entry of the multipliction of + mtrices A (r) = (r) ij n i,j=,(r =,...,+) in the form (A () A () A () A (+) ) ij = n n n i = i = i = n i = () i,i () i,i () i,i (+) i,j... This expression hs very useful visuliztion. We depict sequence of +eqully spced columns, with nodes lbelled,,...,n nd view the sequence of indices i i i i j s pth successively hitting the lbels i, i,i,...,i,js indicted below for the cse n =,=nd the sequence,,,,,. We lso ssign to the edge joining lbel i of column r to lbel j of column r +the ÒweightÓ (r) i,j correspondingly ssign to ny pth weight equl to the product of the weights of its edges. This given, we cn then interpret the right hnd side of.. s the sum of the weights of ll the pths joining lbel i of column to lbel j of column +. nd, () () () () () We shll here nd fter brießy refer to these displys s Òmultipliction digrmsó. Clerly, the sum on the right hnd side of.. need only be crried out over the pths of weight 0. This given, to further simplify these digrms, we shll only depict edges i j of weight ij 0. In this mnner the multipliction digrm of M(s )M(s )M(s ) reduces to We cn thus visulize the identity M(s )M(s )M(s ) =

13 Topics in Algebric Combintorics LECTURE NOTES my, 00 0 by computing ech of the 9 i, j-entries in the product s sum of weights of pths. Zero i, j-entries corresponding to the cses when there is no pth joining i to j. Of course in this extremely simple cse for ny pir i, j either there is no pth or there is only one of weight. This ccounts for the right hnd side of... Although we my not see it from this exmple, we will soon pprecite how powerful this imgery cn be in understnding certin mtrix identities. At ny rte, we cn now visulize the displys in.. nd.. s instnces of multipliction digrms. In this mnner we cn use the disply in.. to obtin visul understnding of the identity M(s )M(s )M(s )M(s )M(s )M(s )M(s )M(s )M(s )M(s )M(s 7 ) = It develops tht Kssel, Lscoux nd Reutenuer [] discovered tht by dding single non-zero entry in ech of the mtrices M(s i ) we cn hve the resulting product retin full informtion s to ech of its fctors nd the order in which they occur. To be precise these uthors let P i (x) (for Þxed n)bethen nmtrix P(x) = i i i 0 x 0 0 This given, it is esy to see tht in the cse the product P (x)p (y)p (z) my be represented by the multipliction digrm x z y from which we derive tht P (x)p (y)p (z) = y +xz x z Here the y + xz entry ccounts for the fct tht there re two pths joining to. Nmely, nd of weights ÒxzÓ nd ÒyÓ respectively.

14 Topics in Algebric Combintorics LECTURE NOTES my, 00 Liewise from the digrm y x z we derive tht P (x)p (y)p (z) = y z x At this point it will be useful to note, for future reference, tht combining.. nd.. we obtin Similrly, in the n n cse, we derive tht P (x)p (y)p (z) = P (z)p (y +xz)p (x) P i (x)p i+ (y)p i (z) = P i+ (z)p i (y + xz)p i+ (x) (for i =,,...,n ).. More generlly, for given reduced word w = l Kssel et l. do set in [] P w (x,x,x,...,x l ) = P (x )P (x )P (x ) P l (x l )...7 Our gol here is to fully understnd the structure of this mtrix. We shll begin by showing tht in some cses its entries cn be written down without ny clcultion. To be precise we hve the following remrble fct. Theorem.. (Kssell, et l.) If w is the word of the cnonicl fctoriztion of permuttion σ, then the mtrix P w (x,x,x,...,x l ) is simply obtined from the circle digrm of σ by replcing every X by, every by0nd the s by the vribles x,x,x,...,x l successively up the columns strting from the left most column nd proceeding to the right. It will be good to strt with prticulr cse. For instnce, for the cnonicl fctoriztion of σ =, illustrted in.., this construction yields x x x x x x P (x,x,...,x 8 )= x 0 0 x

15 Topics in Algebric Combintorics LECTURE NOTES my, 00 To visulize the mechnism tht produces this result we resort to the multipliction digrm corresponding to the product tht yields P (x,x,...,x 8 ). Now it is not difþcult to see tht this digrm cn be simply obtined by dding edges with weights x,x,x,x,x,x,x 7,x 8 to the disply in.., s indicted below x x x x x x 7 x x 8 To clculte the, -entry in P (x,x,...,x n ) using this digrm we locte ll the pths tht join to. We see tht there is only one such pth. This is obtined by following the -line until it meets the edge lbled x then trverse this edge nd then follow the -line untill the end. This gives tht the, - entry is x. Now we should clerly see why the entries in positions (, ), (, ) nd (, ) turn out to be x,x,x respectively. This is simply becuse s we bring to Þrst position by the trnspositions s,s,s, in the product digrm corresponding to P (x )P (x )P (x ) the horizontl edges with weights x,x,x open up three pths respectively joining to, to nd to. Similrly in the portion of the digrm corresponding to the fctors P (x )P (x )P (x ) the horizontl edges with weights x,x,x open up three pths respectively joining to, to nd to. Tht ccounts for x,x,x lnding in positions (, ), (, ), (, ), of the resulting mtrix. Similr resoning ccounts for the positions of x 7 nd x 8. To estblish the result in the generl cse, we hve three crucil observtions: First, we note tht becuse in cnonicl fctoriztion, we bring the elements σ,σ,σ...to their positions successively one t the time, s we bring σ j to the j th in steps, +,+,...,+r the edges with weights x,x +,x +,...,x +r re ll bove the σ j -line. This given, when pth in the multipliction digrm trverses one of these edges it will then be forced to follow the σ j -line to its end nd therefore it will never be ble to trverse ny other x-weighted edge. This shows tht for ny pir (i, j) there is no pth joining i to j, or single pth. In the ltter cse the pth strts with the i-line nd either it never trverses one of the x-weighted edges thereby following the i-line ll the wy to the end (here i = σ j nd the i, j-entry is ÒÓ ) or it trverses n x-weighted edge nd then it must continue long the i = σ j -line ll the wy to the end (see Þgure below) i iõ iõ iõ j iõ i iõ i i i x iõ iõ i i i iõ i iõ iõ iõ i jõ i If the crossing occurs t step then the weight of the edge is x nd the i, j-entry will be x.

16 Topics in Algebric Combintorics LECTURE NOTES my, 00 Second, we note tht in the ltter cse, σ j = i (see Þgure bove) with j >jnd σ j = i >iimply tht the i, j-position is precisely Ò Ó-position in the circle digrm of σ. Finlly, if the weights of horizontl x-lbelled edges tht touch the σ j -line re successively x i,x i+,...,x i+r then these weights will necessrily lnd in the Ò Ó-positions of the j th column of the resulting mtrix. This completes our proof. Remr.. We hve shown bove tht if the th trnsposition in our reduced expression interchnges i with i = σ j then the vrible x will pper in the i, j-entry of the resulting mtrix. If we review the rgument we cn esily see tht this prticulr conclusion did not use the fct tht there we were deling with cnonicl fctoriztion. However, in the generl cse, s we shll see, there will lso be other pths joining i to j nd they will contribute further terms to the i, j-entry of the resulting mtrix. Keeping in mind this fct we cn prove the following remrble property of the mtrices P w (x,x,...,x l ). Theorem.. (Kssel, et l.) Let σ be permuttion of length l nd let J =(x i x j : i<j l)be the idel in the polynomil ring Q[x,x,...,x l ] generted by the products x i x j. Then for ny w RED(σ) the mtrix P w (x,x,...,x l ) modulo J my be obtined from the circle digrm of σ by replcing every X by, every by 0nd the s by permuttion of the vribles x,x,x,...,x l. More precisely, if w =,,..., l then the in position (i, j) is to be replced by x if the trnsposition s interchnges i with σ j. Recll tht we cn pss from w to the cnonicl fctoriztion w o of σ by succession of pplictions of the reltions ) nd ) of..7. Now from.. we deduce tht P i (x)p i+ (y)p i (z) = P i (z)p i+ (y)p i (x) (mod J ) for i =,...,l..9 nd we clerly hve P i (x)p j (y) =P j (y)p i (x) for j i...0 Thus if we use the sme reltions tht bring us from w to w o to the product P w (x,x,...,x l )=P (x )P (x )P (x ) P l (x l ), we see tht the reltions in..9 nd..0 will yield us n identity of the form P w (x,x,...,x l ) =P wo (x θ,x θ,...,x θl ) (mod J ) with θ,θ,...,θ l permuttion of,,...,n. This given, our ssertions follow from Theorem.. nd Remr... It will be worthwhile to illustrte this rgument by woring on speciþc exmple. For this we te σ = nd the word w = RED(σ). In the disply below we give the sequence of steps

17 Topics in Algebric Combintorics LECTURE NOTES my, 00 tht trnsform into the cnonicl fctoriztion of σ. On the right of the verticl line we hve indicted the trnsformtion we crried out from one step to the next. P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) 7 P(x ) 7 P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) 7 P(x ) 7 P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) 7 P(x ) 7 P(x ) P(x ) P(x ) P(x ) 9 P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) 9 P(x ) 9 P(x ) P(x ) 7 P(x ) 7 P(x ) 7 P(x ) P(x ) 9 P(x ) 9 P(x ) P(x ) P(x ) P(x ) 8 P(x ) 9 P(x ) 9 P(x ) 9 P(x ) P(x ) P(x ) P(x ) P(x ) P(x ) 9 P(x ) 8 P(x ) 8 P(x ) 8 P(x ) 8 P(x ) 8 P(x ) 8 P(x ) 8 P(x ) 8 This shows tht modulo the idel J = ( x i x j : i<j 9 ) we hve -> -> -> -> -> -> -> ->.. P (x,x,x,x,x,x,x 7,x 8,x 9 ) =P (x 7,x,x,x,x 9,x,x,x,x 8,).. Since s s s s s s s s s is the cnonicl fctoriztion of σ = we cn follow the recipe given by Theorem.. nd obtin x x P (x,x,x,x,x,x,x 7,x 8,x 9 ) 0 x x = 0 x 0 x 8 x 0 x 0 0 x Remr.. We should note tht the effect of woring in the quotient ring Q[x,x,...,x l ]/J is to ill ll contributions to the mtrix P w (x,x,...,x l ) coming from pths tht trverse more thn one of the x-weighted edges. In fct we cn esily see from the djoining product digrm tht the if we do not ill ll monomils of degree the resulting mtrix is x x x x x x x 7 x 8 x 9 P (x,x,...,x 9 ) = x x +x x 7 0 x x +x x 7 0 x 0 x 8 x +x x 7 0 x 0 0 x

18 Topics in Algebric Combintorics LECTURE NOTES my, 00 Our next gol is to show tht we cn produce equivlences such s in.. by woring directly with the Þnl mtrices, rther thn by cting on the fctors. To stte nd prove this result we need to me some deþnitions nd estblish some uxiliry propositions. To begin let us denote by Pw J (x,x,...,x l ) the mtrix we obtin when we compute the entries of P w (x,x,...,x l ) mod J. We shll lso refer to Pw J (x,x,...,x l ) s the Òliner prtó ofp w (x,x,...,x l ). For given indices j <j < <j,let us denote by Pw J [j,j,...,j ]the submtrix of Pw J (x,x,...,x l )contined in columns j,j,...,j nd rows σ j,σ j,...,σ j. Note tht if =nd σ j >σ j >σ j then the submtrix Pw J [j,j,j ]will be of the form Pw J [j,j,j ] = y z x This given, we shll cll Ò-Coxeter trnsition for ÓinPw(x J,x,...,x l ) replcement of the form x + x + x 0 x + x x More precisely, such trnsition consists in locting three indices j <j <j such tht the submtrix P J w [j,j,j ]is of one of the forms given in... This done, the -Coxeter trnsition consists in replcing one form by the other form in P J w (x,x,...,x l ). In the sme vein, Ò-Coxeter trnsition on Ó is the exchnge of x nd x + when x nd x + re not in the sme row or column. Thus this Coxeter trnsition crries out one of the following possible exchnges in the mtrix P J w (x,x,...,x l ): x x + x+ x x + x x x+ Proposition.. Let w =,, l be reduced word nd let = i, + = i +, + = i. Let w =,, l be the sme s w except in positions, +,+ where we hve = i +, + = i, + = i + Then the mtrix P J w (x,x,...,x l ) is simply obtined from P J w (x,x,...,x l ) by ming -Coxeter trnsition on. We hve P w (x,x,...,x l ) = P (x ) P i (x )P i+ (x + )P i (x + ) P l (x l ).

19 Topics in Algebric Combintorics LECTURE NOTES my, 00 Under this ssumption, the portion of the digrm tht contins the edges of weights x, x + nd x + will necessrily be of the form given below with the x, x + nd x + edges t heights i, i+nd i respectively. i i j i x i x + i i i i i x + i i i j i i i i i i j Indeed, if it is i -line nd the i -line tht cross t thew th step, nd if it is the i -line tht the i -line crosses t the + st step then the i nd i lines will necessrily cross t the + nd step. Since, in the line digrm of reduced decomposition, ny two lbelled lines cross only once, we will hve i <i <i nd the i, i nd i lines must respectively end up t levels j <j <j s indicted in the Þgure. Of course this mens tht σ j = i, σ j = i nd σ j = i Using this digrm nd the recipe given by Theorem.., we cn esily derive tht the submtrix P J w [j,j,j ]must be precisely s given below P J w [j,j,j ] = x + x x Note next tht if the portion of the product digrm of P (x )P (x ) P l (x l )given bove, is replced by the portion given below i i j i x i + i i i i x i i x + i i j i i i i i i j

20 Topics in Algebric Combintorics LECTURE NOTES my, 00 7 wht we get is precisely the multipliction digrm we cn use to compute the mtrix P w (x,x,...,x l ) = P (x ) P i+ (x )P i (x + )P i+ (x + ) P l (x l). On the other hnd, the reltion in..7 (modulo J ) gives This mens tht we lso hve P i+ (x )P i (x + )P i+ (x + ) = Pi (x + )P i+ (x + )P i (x ) (mod J ) P J w (x,x,...,x l ) = P (x ) P i (x + )P i+ (x + )P i (x ) P l (x l ) = P J w (x,,x +,x +,x,...,x l ) In other words P J w (x,x,...,x l )is obtined from P J w (x,x,...,x l )by interchnging x with x +. However, in view of.. this is precisely -Coxeter trnsition on. It is importnt to now t this point how the mtrix P J w (x,x,...,x l ) chnges s we increse or decrese the number of fctors. It develops tht these chnges cn be crried out by very simple recipe. More precisely we hve Proposition.. Let w = RED(σ), nd let σ j <σ j+ so tht w = j RED(σ s j ), then the trnsition P J w (x,x,...,x ) P J w (x,x,...,x + ) is simply obtined by interchnging columns j nd j + of Pw J (x,x,...,x )nd then chnging the (σ j,j)-entry of the resulting mtrix to x +. --> For convenience let M w nd M w denote the multiplictions digrms corresponding to w nd w nd let M w --> /w denote the the lst two columns we hve to dd to M w to get M w. Since by our ssumptions we hve --> P w (x,x,...,x + ) = P w (x,x,...,x ) P j (x + ), the digrm M w /w will necessrily be s depicted in the the djcent Þgure. We hve lso set there i = σ j nd i = σ j+. Now note tht, when s j or s j +, to compute n r, s entry in the mtrix P w (x,x,...,x + ) we simply follow the sme pths s for the computtion of the r, s entry of P w (x,x,...,x )up to the Þrst column of M w /w nd then proceed to the second column of M w /w trversing the horizontl edge t level s. This yields tht the s th columns of P w (x,x,...,x ) nd P w (x,x,...,x ) re identicl. Similrly we see tht, to compute n r, j +-entry of P w (x,x,...,x ), we must follow pth of M w tht goes from r to level j nd then drop down to level j +by following the lst step of the i-line in M w /w. This cuses the j + st column of P J w (x,x,...,x + ) to be identicl with the j th column of P J w (x,x,...,x ). To compute the r, j-entry of P J w (x,x,...,x + ) we hve two sets of pths. Those which in M w go from r to level j nd continue in M w /w horizontlly by trversing the x + -weighted edge (see Þgure), nd those which in M w go from r to level j +nd then climb up to level j by following the lst step of the i -line in M w /w. --> --> x + j --> i iõ j+ --> iõ i j+ --> n -->

21 Topics in Algebric Combintorics LECTURE NOTES my, 00 8 However from the Þrst set of pths only the i-line survives in computtion mod J. The reson for this is,except for the i-line, ll the other pths hve contributed n x-entry in Pw J (x,x,...,x )nd the continution cross the the x + -weighted edge will me their weight product of x s nd therefore equl to zero mod J. On the other hnd the i-line in M w followed by the x + -weighted edge will contribute n x + to the i, j-entry of P J w (x,x,...,x + ). Now pth in M w from second set tht goes from n r i to level j +, yields the r, j +-entry in Pw J (x,x,...,x )nd will cuse this entry to move to the r, j position in P J w (x,x,...,x + ) s it climbs to level j in M w /w. We hve now ccounted for ll but the i, j-entry in P J w (x,x,...,x + ). The possibility remins tht x + my not be the only term there becuse of some pth from second set tht went from i to level j +in M w. However note tht since σ j = i there is no Ò ÓorÒXÓin position i, j +in the circle digrm of σ so the i, j-entry in P w (x,x,...,x )is necessrily zero nd therefore there is no pth in M w tht joins i to j +. Thus the i, j-entry of P J w (x,x,...,x + ) must be x + precisely s sserted. In the disply below we illustrte the sequence of trnsitions corresponding to the reduced word w = RED() x x x x x x x x x 0 x x x x x x x x x 0 0 x x 0 x x x x 0 0 x x x x x x 0 x x 0 x 0 x x 0 x Since ech of these mtrix trnsitions cn be reversed, n immedite corollry of Proposition.. is tht the word w cn be reconstructed from P J w (x,x,...,x l ). To do this we simply crry out the illustrted process in reverse. In prticulr we obtin thus proof tht the mtrix P w (x,x,...,x l )is completely determined by its liner prt. Now it develops tht there is n even simpler wy, in fct recipe, for recovering w from P J w (x,x,...,x l ). This result cn be stted s follows.

22 Topics in Algebric Combintorics LECTURE NOTES my, 00 9 Theorem.. (C. Greene et Al []) Let w = l, nd for ech [,l] let c denote the number of x s with s> tht re directly NORTH or SOUTH of x in Pw J (x,x,...,x l ) nd let r be the number of x s with s>tht re directly WEST. This given, if x is in column j of Pw J (x,x,...,x l ) we necessrily hve = j + c r..8 It follows from Proposition.. nd it is esy to see from the process in..7 tht if = j then x lnds in column j t the moment it is inserted. However, s the process of construction of Pw J (x,x,...,x l ) continues, its column chnges. Nevertheless we cn esily eep trc of wht hppens. To begin we see tht every time n x s with s>gets inserted in the column of x the column number of x decreses by one. On the other hnd note tht if n x s with s>gets inserted in the row of x s this will necessrily te plce EAST of x, becuse to the right of x s we plce nd there is nothing but zeros in Pw J (x,x,...,x l )to the right of ny Õs. Now the only time when such n x s psses to the WEST of x is when x s is immeditely to the right of x nd their columns re interchnged. This cuses the column number of x to increse by one t tht time. Putting ll this together we derive tht when the trnsition process termintes we will Þnd x in column j with j = + r c. This proves..8. Proposition.. Let w =,, l be reduced word nd let = r nd + = s with r s...9 Let w =,, l be the sme s w except in positions, + where we hve = s nd + = r..0 Then the mtrix P J w (x,x,...,x l ) is simply obtined from Pw J (x,x,...,x l ) by ming -Coxeter trnsition on. Let M w (h) for moment denote the mtrix obtined fter h steps in the construction process tht yields Pw J (x,x,...,x l ). Liewise let M (h w be the mtrix obtined fter h steps in the construction process tht yields P J w (x,x,...,x l ). This given, from Proposition.. nd..9 it follows tht x nd x + will respectively be in columns r nd s of M w (+). It is lso cler tht x + is not inserted in the sme row s x becuse immeditely to the right of x in M w () there is. Now note tht since the two columns involved in the insertion of x do not overlp with the two columns involved in the insertion of x + we cn esily see tht M (+) w will necessrily be identicl with M (+) w except tht the positions of x nd x + re interchnged. Consequently, during the remining prt

23 Topics in Algebric Combintorics LECTURE NOTES my, 00 0 of the insertion processes yielding Pw J (x,x,...,x l ) nd P J w (x,x,...,x l ) we shll hve tht M w (h) will remin relted by -Coxeter trnsition nd it will be so the end s well proving our ssertion. M (h) w We re now Þnlly in position to estblish the following bsic result. Theorem.. For ny two reduced words w nd w of permuttion σ of length l we cn find sequence of Coxeter trnsitions tht trnsform Pw J (x,x,...,x l ) into Pw J (x,x,...,x l ). By Theorem.. we cn pss from w to w by sequence of pplictions of identities..7 ) nd ). But now from Propositions.. nd.. we derive tht..7 ) will cuse -Coxeter trnsition on the corresponding mtrix nd..7 ) will cuse -Coxeter trnsition. Thus the theorem is n immedite consequence of Theorem.. nd Propositions.. nd.... Blnced Lbeled Circle Digrms. From mtrices to tbloids The mtrix pproch of Kssel et. l. hs nturlly brought us to the generl notion of Blnced Lbeled Circle Digrm introduced in [] nd []. Although it will be good to eep in mind the mechnisms tht produce the mtrices Pw J (x,x,...,x l ) it will be more convenient to crry out ll our combintoril constructions nd mnipultions directly on these tbloids. Roughly speing, these tbloids re obtined by Þlling the circles in the digrm of σ with the lbels,,...,lso tht ÒÓ is in the sme position s Òx Ó is in Pw J (x,x,...,x l ). To be precise, in view of Theorem.., we hve the following DeÞnition. Given permuttion σ of length l, here nd fter we ssocite to ech word w = l RED(σ) the tbloid T (w) obtined by plcing in the tht is in position (i, j) if nd only if the trnsposition s interchnges i with σ j. Now it develops tht these tbloids hve very curious chrcteriztion. To stte it we need some nottion nd further deþnitions. To begin, it will be convenient to let ÒCD(σ)Ó denote the circle digrm of permuttion σ. Ifσhs length l then CD(σ) hs l circles nd Þlling of these circles with the lbels,,...,l will be clled n ÒinjectiveÓ lbeling of CD(σ) or brießy n Òinjective tbloidó. The lbel in position (i, j) in the resulting tbloid T will be denoted T ij. We shll of course use mtrix convention to denote loction nd thus i increses s we go SOUTH nd j increses s we go EAST. As we did for mtrices, if T is n injective lbeling of CD(σ), we shll denote by T (j,j,...,j )the subdigrm of T tht is contined in columns j,j,...,j nd rows σ j,σ j,...,σ j. We shll lso denote by T rs (j,j,...,j )the entry tht is in the r th row nd s th column of T (j,j,...,j ). For given cell (i, j) CD(σ) the collection of cells tht re directly EAST of (i, j) is clled the ÒrmÓ of(i, j). Liewise the collection of cells tht re directly SOUTH of (i, j) is clled the ÒlegÓ of(i, j). The collection consisting of the cell (i, j) together with its rm nd leg is usully referred to s the ÒhooÓ of (i, j), it will be denoted by ÒH ij Ó. A hoo H ij of n injective tbleu T is sid to be ÒblncedÓ ifnd nd

24 Topics in Algebric Combintorics LECTURE NOTES my, 00 only if the number of lbels in the rm of (i, j) tht re smller thn T ij is equl to the number of lbels in the leg tht re bigger thn T ij. In prticulr we see tht if the lbels in H ij re sorted in incresing order then plced bc in H ij strting from the bottom of the leg then NORTH up to (i, j) then Þnlly EAST long the rm, T ij will necessrily lnd right bc in its cell. We sy tht T itself is ÒblncedÓ if ll its hoos re blnced. The notions of ÒrmÓ, ÔlegÓ, ÒhooÓ nd Òblnced hooó nd Òblnced tbloidó re esily extended to subdigrms T (j,j,...,j ). For instnce we let the rm of T r,s (j,j,...,j ) be the collection of cells of T (j,j,...,j ) tht re EAST of T r,s (j,j,...,j ). The remining notions re nlogously deþned. In prticulr, we let H rs (j,j,...,j )denote the hoo of T r,s (j,j,...,j ). To be precise, H rs (j,j,...,j ) consists of T r,s (j,j,...,j ) together with its rm nd leg in T (j,j,...,j ). Liewise we sy tht T (j,j,...,j )is blnced if ll the hoos H rs (j,j,...,j )re blnced. It goes without sying tht ll the results we hve estblished for the mtrices P J (x,x,...,x l ) cn be trnsfered to the tbloids T (w). We shll use this fct here nd fter without necessrily spelling out in detil how this trnsfer should be crried out, since it only mounts to ming the replcements x, 0, X, In prticulr the -Coxeter nd -Coxeter trnsitions of section. now become s indicted below. Nmely, -Coxeter trnsitions re simply interchnges in T of -subdigrms T (j,j,j )of the form: while -Coxeter trnsitions re substitutions of the form In the sme vein Theorem.. my now be stted s Theorem.. For ny two reduced words w,w RED(σ) we cn find sequence of Coxeter trnsitions which trnsform T (w ) into T (w ). The notion of blnced tbloid rised quite erly [] in the study of reduced words. The wor of Kssel et. l. shows tht it hs nturl lgebric setting which beutifully explins its origin. We derive it here s corollry of Theorem... Proposition.. The tbloids T (w) re ll blnced.

25 Topics in Algebric Combintorics LECTURE NOTES my, 00 A view of the displys in.. nd.. should me it cler tht pplying or -Coxeter trnsition on blnced tbloid does not destroy blnce. At ny rte, note tht in the cse of the -Coxeter trnsition which goes from left to right in.. we see tht we re incresing by one the number of entries in the rm of +tht re less thn +but t the sme time we re incresing by one the number of entries in the leg of +tht re lrger thn +. Going from right to left in.. reverses this process nd cnnot ffect blnce of the hoo of +. All the other hoos H ij contin only, +or +,+nd their blnce is trivilly not ffected by either of the two chnges in... Liewise, the blnce of hoo is not ffected by ny of the two trnsition in the Þrst prt of.., for in this cse no hoo contins both nd +. As for the trnsitions in the second prt of.., note tht if T ij, +then T ij >if nd only if T ij >+nd the blnce of H ij cnnot be ffected by this trnsition. Similrly, if T ij = or T ij = +then replcing by +or vicevers cnnot ffect the blnce of H ij. To conclude, note tht the tbloid T (w o ) of ny cnonicl fctoriztion w o is necessrily blnced since, by the wy T (w o ) is constructed (cf. Theorem..), ll the lbels in the rm of hoo H i,j re lrger thn T ij nd ll the lbels in the leg re smller. Now when w nd w o re reduced words of the sme permuttion, by Theorem.., we cn pss from T (w o ) to T (w) by sequence of Coxeter trnsitions. Since when w o is cnonicl T (w o ) is blnced, T (w) must be blnced s well since, s we hve seen, ll these trnsitions preserve blnce. We should note tht Proposition.. yields us n lgorithm for constructing our tbloids T (w) without resorting to multipliction digrms. In fct, Proposition.., converted to tbloids, my be restted s Proposition.. Let w = RED(σ), nd let σ j <σ j+ so tht w = j RED(σ s j ), then the trnsition T (w) T (w ) is simply obtined by interchnging columns j nd j + of T (w) nd then chnging the (σ j,j)-entry of the resulting tbloid to +. This result s n immedite converse which my be stted s follows Proposition.. Let w = j RED(σ), nd let σ j >σ j+ so tht w = RED(σ s j ), then the trnsition T (w) T (w ) is simply obtined by interchnging columns j nd j + of T (w) nd then chnging the + to. At this point it is good to hve visul imge of these two trnsformtions. For convenience let ÒconstructÓ nd ÒdeconstructÓ denote the trnsformtions described in Propositions.. nd... More precisely when w RED(σ) nd σ j <σ j+ then construct [ T (w),j ] = T(wj)

26 Topics in Algebric Combintorics LECTURE NOTES my, 00 nd when w = + RED(σ) then deconstruct [ T ( + ) ] = ( T ( ),j). This given we cn schemticlly represent Propositions. nd.. by the following displys. + j j+ j j+ j j+ U V V U V U U V construct + deconstruct + W W W W Remr.. We should note tht to pply construct we need to give j nd then is the lrgest entry in T (w). To pply deconstruct we locte the lrgest entry, sy it is +nd it lies in the j th column. This given we operte s indicted in the Þgure nd return the resulting tbloid long with the index ÒjÓ. Now we see tht to construct tbloid T ( l ) we only need to crry l pplictions of construct. More precisely, we recursively set T( + ) = construct [ T( ), + ] (for =,,...,l ) with the initil step T( ) = construct [ T o ] where T o is the tbloid tht corresponds to the identity permuttion. In the following disply we hve crried out this lgorithm for w =. For moment let us sy tht injective lbeling T of the circle digrm CD(σ) is ÒconstructibleÓ if nd only if T = T ( l )for some l RED(σ).

27 Topics in Algebric Combintorics LECTURE NOTES my, 00 We hve the following remrble fct. Theorem.. An injective lbeling of the circle digrm of permuttion is constructible if nd only if it is blnced. In view of Proposition.., we need only prove tht every blnced tbloid is constructible. Let then T be blnced lbelling of the circle digrm of σ nd let N be the lrgest lbel in T. Suppose further tht N = T ij. We clim tht in position (i, j +)there necessrily is n X. To see this, note tht if this were not so then the j th nd j + st columns of T would hve one of the following forms: j j+ j j+ i N or i N b Indeed, if the X in column j +were bove the th row then immeditely to the left of it there would hve to be circle becuse there is no ÒXÓ to ill tht cell from the left or bove. Now the lbel in tht circle is necessrily number <Nbut tht would cuse the hoo of to be unblnced since there is lbel bigger thn SOUTH of nd no lbel less thn EAST. In fct no lbel t ll EAST of becuse of tht djcent X. This elimintes the Þrst lterntive. In cse the X in column j +is below the i th row then there would hve to be circle in column j +immeditely to the right of N becuse there is no ÒXÓ to ill tht cell from the left or from bove. Now gin in tht circle there would hve be lbel b<n, but then the hoo of N is unblnced be cuse there is lbel smller thn N EAST nd no lbel bigger thn N SOUTH. This elimintes the second possibility. This forces the j th nd j + st columns to be of the following form j j+ b.. i N where we clim tht every lbel b bove N,inthej th column, hs necessrily n djcent lbel <bin the j + st column. Clerly, there must be circle djcent to b in the j + st column becuse there is no ÒXÓto ill tht cell from the left or from bove. To show tht in tht circle there is lbel less thn b we proceed by contrddiction. Suppose tht the sitution is s indicted in.. with >b, nd tht pir is the lowest we cn Þnd. Let then p be the number of lbels, SOUTH of b, tht re lrger thn b. We hve p becuse N>b. But then, since T is blnced, there must lso be p lbels b,b,...,b p ll less thn b in the rm of b.

28 Topics in Algebric Combintorics LECTURE NOTES my, 00 Now, since >bll these lbels re less thn s well. But then gin, since T is blnced, there must be t lest p lbels u,u,...,u p lrger thn in the leg of. However ll these lbels must fll in circles of column j +tht re between the nd the X. Moreover, the presence of these circles in column j +forces circles djcent to them in column j. Let w,w,...,w p be the lbels tht fll in these circles, (indexed so tht w r is to the left of u r ). Since we chose b nd to form the lowest pir b<, we must hve w r >u r >>b(for r =,,...,p). In summry, these two columns would the be s depicted in the djcent Þgure. But this cnnot be since we now see p +lbels greter thn b in the leg of b, contrry to our initil choice of p. We hve now proved tht T is the form given in.. where every pir of djcent circles bove the pir N,X contin lbels b, with b>. We clim tht if we pply deconstruct to T the resulting tbloid T will be gin blnced injective lbelling of CD(σ). Indeed loo t the picture below should me it cler tht the only hoos whose collections of lbels hve been ffected in signiþcnt wy re those of nd b. Now only gins lbel greter thn it to the right, this does not ffect its rn mong the lbels in his hoo, so its hoo remins blnced. As for b we see tht it looses N>bin its leg but t the sme time it looses <bin its rm. These losses compenste ech other nd thus leve the hoo of b still blnced. j j+ b j j+ b j b w w w p N j < > > > + u u u p X deconstruct i N i We cn see now how the proof cn be completed. To begin the result is trivilly true for the circle digrm of the identity since there re no circles t ll to Þll. So we ssume by induction on the number of circles, tht ll blnced lbelings of CD(σ) tht hve less circles thn T re constructible. Now we see from the Þgure bove tht T is, in fct, lbelling of the circle digrm of the permuttion σ s j. The inductive hypothesis gives tht T is constructible. This given, we must hve tht T = T (,, l )with,, l RED(σ s j ) nd fortiori,, l j RED(σ). Since construct (T, j) = T We deduce tht T = T (,, l ) (with l = j) This shows tht T is constructible, completing the induction nd the proof. It develops tht constructibility, (nd now in prticulr blnce) forces whole fmily of restrictions on the lbeling.