Tableau algorithms defined naturally for pictures

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1 1 Introution Tblu lorithms in nturlly or piturs Mr A. A. vn Luwn CWI Kruisln SJ Amstrm Th Nthrlns mil: Abstrt W onsir piturs s in in [Zl1]. W lbort on th nrlistion o th Robinson-Shnst orrsponn to piturs in thr, n on th rsult in [FoGr] tht shows tht this orrsponn is nturl, i.., inpnnt o th pris rin orr o th squrs o skw irms tht is us in its inition. W iv simplii proo o this rsult by showin tht th nrlis Shnst insrtion prour n b in without usin this orr t ll. Our min rsults involv th oprtion o lissmnt in in [Shü2]. W show tht lissmnt n b nrlis to piturs, n is nturl. In t, w obtin two ul orms o lissmnt; onsquntly both tblux orrsponin to prmuttion in th Robinson-Shnst orrsponn n b obtin by lissmnt rom on pitur. W show tht th two orms o lissmnt ommut with h othr. From this t th min proprtis o lissmnt ollow in muh simplr wy thn thir oriinl rivtion in [Shü2]. 1. Introution. A pitur btwn skw irms is bijtion o thir squrs stisyin rtin onitions tht will b ivn blow. For spil hois o th omin n/or im irm, piturs r quivlnt to othr onpts, suh s stnr n smi-stnr (skw) tblux, Littlwoo-Rihrson illins, n prmuttions; morovr som wll known proprtis n onstrutions or ths spil ss n b nrlis to piturs. Zlvinsky hs shown in [Zl1] tht th numbr o piturs btwn ny pir o skw irms quls th intrtwinin numbr o th orrsponin rprsnttions o th symmtri roup, whih nrliss th Littlwoo-Rihrson rul, n tht th Robinson-Shnst n Shütznbrr orrsponns hv nrlistions to piturs. In th inition o ths orrsponns prtiulr totl orrin J on Z Z is us, tht lso ours in th inition o piturs thmslvs; usin this orrin on th ims o squrs, piturs n b viw s tblux, n thn th onstrution o ths orrsponns oinis with th usul onstrutions or th tblu s. Howvr, both in th inition o piturs n o th Robinson- Shnst orrsponn th us o J turns out to b inssntil: in [ClSt] it ws shown tht J n b rpl by th mor nturl prtil orrin, n in [FoGr] it ws shown tht in th inition o th Robinson-Shnst orrsponn or piturs J n b rpl by ny totl orrin omptibl with without tin th orrsponn. Followin [FoGr], lt us ll onstrution involvin piturs nturl nrlistion o similr onstrution or tblux, whn it rus to tht onstrution by totlly orrin th st o ims o pitur by som orrin omptibl with, n whn morovr th outom o th onstrution is inpnnt o th totl orrin us. W invstit th nturlity o th Robinson-Shnst n Shütznbrr orrsponns n th prours us to in thm, n whthr th oprtion o lissmnt in in [Shü2] hs nturl nrlistion to piturs; w in th ollowin rsults. Th Shnst insrtion n xtrtion prours n b in or piturs irtly in trms o, without hoosin totl orrin (lmm 3.3.2), whih irtly implis th nturlity o ths prours; thus w obtin simplr mor irt proo o th nturlity o th Robinson-Shnst orrsponn or piturs thn ws ivn in [FoGr] (thorm 3.2.1). Consirin th Robinson-Shnst orrsponn in rltion to symmtris o th pln tht prsrv th pitur proprty, n usin th wll known rltion btwn th Robinson-Shnst n Shütznbrr orrsponns, w in tht th Shütznbrr orrsponn or piturs is lso nturl (thorm 4.2.1); howvr th ltion (or vution) prour us to in th Shütznbrr orrsponn is not nturl. W lso obtin (non-obvious) bijtion btwn th sts o piturs with ivn omin n im n thos with th trnspos omin n im (thorm 4.3.1). W show tht lissmnt o skw tblux hs nturl nrlistion to piturs (thorm 5.1.1). In this s nturlity is in t nssry onition or hvin propr inition t ll; lik or th Shnst 1

2 2 Piturs insrtion prour th us o totl orrin n b voi ltothr. Th Robinson-Shnst n Shütznbrr orrsponns n both b xprss in trms o lissmnt (this hols or piturs in th sm wy s or tblux). Du to th t tht th invrs mp o pitur is in pitur, w obtin ul orm o lissmnt s wll, tht hns th shp o th im rthr thn tht o th omin. This itionl oprtion s powr n symmtry to th thory o lissmnt;.., whrs usin orinry lissmnt on o th two tblux ssoit to prmuttion unr th Robinson-Shnst orrsponn n b obtin rom th orrsponin skw tblu, on n obtin both ths tblux rom th pitur orrsponin to th prmuttion, usin th two orms o lissmnt (thorm 5.1.1). A ruil rsult is tht both orms o lissmnt ommut with h othr (thorm 5.3.1). This t shs liht on th unmntl proprtis o lissmnt: thy ollow sily rom it (thorm 5.4.1), without usin th rsults o [Shü2], or th proprtis o th Robinson-Shnst orrsponn ths r bs on. Thus lissmnts o piturs provi n inpnnt n lmntry pproh to th thory o th Robinson-Shnst n Shütznbrr orrsponns, both or piturs n or tblux. This ppr is ornis s ollows. In 2 w iv inition n bsi proprtis o piturs, n init onntions with othr ombintoril onpts n with th Littlwoo-Rihrson rul. In 3 w trt th Robinson-Shnst orrsponn or piturs, n isuss qustions o its nturlity. In 4 w ontinu by stuyin th rltion o th Robinson-Shnst orrsponn with symmtris o th st o piturs, n th Shütznbrr orrsponn. Ths two stions ontin rltivly w nw rsults; mphsis lis on sribin th orrsponns n thir proprtis, n th mnin o nturlity. In 5, th thory o lissmnt or piturs is vlop. For this 3 n 4 only srv to provi motivtion: thir rsults r not rquir or th thory, on th ontrry, it ivs n ltrntiv wy to obtin thos rsults. 2. Piturs Orrins on Z Z. Th strtin point or ll th objts tht w shll stuy is th intr ltti Z Z. Its lmnts will b pit, n otn rrr to, s squrs, n w shll lt th irst oorint inrs ownwrs n th son inrs to th riht, lik mtrix inis. W shll mploy two irnt prtil orrins on this st; on is th nturl oorintwis orrin tht will b not by, n is in by (i, j) (i, j ) i i j j n th othr is trnsvrs orrin not by n in by (i, j) (i, j ) i i j j. Th rrows tth to th sins r intn s rminr o th inition, n point in th irtion o th smllr lmnts (lik th < sin itsl). As usul, x < y mns x y n x y, n similrly or <. Rmrk. Both th hoi o th trnsvrs orrin n th symbols us to rprsnt th orrins r somwht rbitrry, n not lwys in rmnt with othr litrtur on th subjt; or instn in [FoGr] th opposit trnsvrs orrin is us but it is not by th sm symbol. W polois or ny onusion tht miht rsult, but sin it is impossibl to b in rmnt with ll litrtur, w hv hosn or onvntions tht r onsistnt n sy to rmmbr: movin rom lt to riht w inrs in both orrins. Bus o th us o irnt orrins, w shll not prtilly orr st (or post or short) xpliitly s pir (A, A ) o st with prtil orrin. Rll tht post morphism (A, A ) (B, B ) is mp : A B suh tht or ny 1, 2 A with 1 A 2 on hs ( 1 ) B ( 2 ). An orr il o prtilly orr st (S, ) is subst I o S suh tht or ll x S n y I with x y w hv x I; th omplmnt C = S \ I, whih hs th proprty tht or ll x C n y S with x y w hv y C, is ll n orr oil. For utur rrn w stt n ltrntiv hrtristion o post morphisms Proposition. A mp : A B is post morphism (A, A ) (B, B ) i n only i th invrs im o ny orr il o (B, B ) is n orr il o (A, A ). 2

3 2.2 Skw irms 2.2. Skw irms. A skw irm χ is init subst o Z Z tht is onvx with rspt to th nturl orrin, i.., i x, z χ n x < y < z thn y χ; not th st o ll skw irms by S. A typil skw irm n b pit s ollows: Lt P S b th st o Youn irms, i.., o init orr ils o (N N, ); ths orrspon bijtivly to prtitions. Th th non-mpty Youn irms r just th skw irms tht, viw s post by th nturl orrin, ontin th oriin (i.., th squr (0, 0)) s uniqu miniml lmnt. For h µ, ν P with µ ν th irn st ν \ µ is skw irm, n i skw irm is ontin in N N, it n lwys b writtn in this orm. Howvr, suh n xprssion is not nssrily uniqu; or instn, th skw irm pit bov, whr w ssum tht th oriin lis t th intrstion o its irst row n olumn, n b writtn s but lso s. For skw irm χ S in ornr to b squr s χ suh tht χ \ {s} is in skw irm, n oornr to b squr s χ suh tht χ {s} is in skw irm. A ornr s o χ is ll innr rsptivly outr i s is miniml rsptivly mximl in th post (χ, ), o whih t lst on is th s. Similrly oornr s o χ is ll innr or outr orin s s is miniml or mximl in (χ {s}, ) Dinition o piturs. Vrious initions hv bn ivn or piturs by irnt uthors. W shll onsir only th s tht omin n im r skw irms, whr ll ths initions (n tht o oo mps in [FoGr]) bom quivlnt, up to som trivil symmtris* Dinition. Lt χ, ψ S n : χ ψ bijtion; is ll pitur i it is morphism o prtilly orr sts (χ, ) (ψ, ), n 1 is morphism (ψ, ) (χ, ). To isply pitur, w my lbl h squr o χ n its im in ψ with uniqu lttr, ivin or instn b b χ ψ Lt Pi(χ, ψ) not th st o ll piturs rom χ to ψ. From th inition o piturs it is lr tht i is pitur, thn so is 1, so tht Pi(χ, ψ) is in bijtion with Pi(ψ, χ). For trnsltions t 1, t 2 o Z Z w lso hv n obvious bijtion btwn Pi(ψ, χ) n Pi(t 1 (ψ), t 2 (χ)). Th st S is los unr th oprtions o trnsposition (ivn by (i, j) (i, j) t = (j, i)) n ntrl symmtry (ivn by (i, j) (i, j) = ( i, j)). On sily vriis tht by pproprit omposition with ths rltions bijtions o Pi(χ, ψ) with Pi(χ t, ψ t ), Pi( χ t, ψ t ), n Pi( χ, ψ) r obtin. Hr r th rsults o pplyin ths symmtris to th pitur isply bov. b b b b b b * Our piturs r trnspos t omin n im si with rspt to thos o [Zl1] n [ClSt]. For th piturs o [JP], n th oo mps o [FoGr], on shoul pply rltion in horizontl xis t th im si (th im shp is thn not skw irm, but rthr onvx or ). 3

4 2.4 Enoins o piturs n spil ss: prmuttions n tblux Applyin trnsposition t both omin n im si os not prsrv th pitur onitions; nvrthlss bijtion btwn Pi(χ, ψ) n Pi(χ t, ψ t ) xists, n w shll onstrut suh bijtion ltr. Th pitur onition n b m mor xpliit by mkin tbl o llow rltiv positions o ims. To n orr pir o istint squrs w ssoit on o iht possibl rltiv positions, by trminin or both thir oorints whthr tht o th irst squr is lss thn, qul to, or rtr thn tht o th son; ths positions n b init by th iht ompss irtions. Th ollowin tbl xprsss th llow ombintions o th rltiv position o pir o squrs n o thir ims unr pitur. In rsonin bout piturs w shll otn us this tbl without xpliit mntion Enoins o piturs n spil ss: prmuttions n tblux. Thr r othr wys o rprsntin piturs thn shown bov. Th row noin (rsptivly olumn noin) o pitur : χ ψ is obtin by illin h squr s o χ with th numbr tht is th irst (rsptivly son) oorint o (s). For th pitur shown bov, ths r rsptivly (whr w hv ssum tht th oriin lis in th topmost row n ltmost olumn o ψ). Sin h row n olumn is totlly orr by, ithr th row or th olumn noin ully trmins, i ψ is ivn. Th post morphism proprty or implis tht in th row noin th rows r wkly rsin n th olumns stritly rsin, whil in th olumn noin rows r stritly inrsin n olumns wkly inrsin. To obtin tblux with wkly inrsin rows n stritly inrsin olumns (s smistnr tblux r usully in) on my us nt row noin (illin h squr with minus th row oorint o its im). In ition to ths monotoniity onitions on rows n olumns, th inition o piturs poss som lss obvious onitions. Howvr, or rtin kins o skw irms ths onitions simpliy, n thus w n t vrious kins o ombintoril objts s spil ss o (noins o) piturs. For instn, i ψ is n nti-hin or (i.., no two istint squrs r omprbl), thn th post morphism onition or 1 is trivilly stisi, n th post morphism onition or similrly boms trivil i χ is n nti-hin. Hn i both χ n ψ r nti-hins or, thn piturs r just rbitrry bijtions or, vi olumn noin, prmuttions. I only ψ is n nti-hin, w similrly t th notion o skw tblu, n i morovr χ is Youn irm, tht o (stnr) Youn tblu. I w intrhn χ n ψ, thn Youn tblu will b rprsnt by th nti-hin χ ill with numbrs suh tht, whn r rom bottom lt to top riht, thy orm ltti prmuttion or mot Ymnouhi. Hr is n xmpl o suh pitur, its olumn noin, n tht o its invrs. b b, ,

5 2.5 Altrntiv hrtristions o piturs I w tk or ψ horizontl strip, i.., skw irm with t most on squr in h olumn, thn w t s nt olumn noins tblux in whih intil ntris llow, subjt only to th mntion monotoniity onitions. Thus w t smistnr tblux (ll nrlis Youn tblux in [Knu1]) s spil ss o piturs, or instn is rprsnt by b j o h k i m n l b h i j k l m n o. I w lso tk or χ horizontl strip, thn pitur is ully spii by ivin or h row o χ how mny o its squrs mp to h row o ψ. Ths t prisly srib nrlis prmuttion in th sns o [Knu1], whih n b rprsnt by n intr mtrix or by two-lin rry. For instn, th nrlis prmuttion rprsnt by th mtrix orrspons to th pitur or by th two-lin rry ( ) l m i n h k b j o b h i j k l m n o. Finlly, i w tk or χ vrtil strip (no two squrs in on row) whil ψ rmins horizontl strip, thn piturs r mor rstrit, sin th im o ny olumn o χ n hv t most on squr in ommon with ny sinl row o ψ; ths piturs orrspon to th rstrit nrlis prmuttions in [Knu1], tht n b rprsnt by zro-on mtris Altrntiv hrtristions o piturs. Usin proposition 2.1.1, w n hrtris piturs : χ ψ s ollows: is bijtiv post morphism (χ, ) (ψ, ) tht mps h orr il o (χ, ) to n orr il o (ψ, ) (whih is skw irm). In viw o this it is sirbl in hkin th pitur onition to rpl by stronr orrin (wr inomprbl pirs); thn thr will b wr orr ils to tst. Th ollowin proposition stts tht, surprisinly, this n b on in n rbitrry wy without wknin th onition; it ws oun inpnntly by th uthor [vl1] n by Fomin n Grn [FoGr, Lmm 3.4] Proposition. Lt : χ ψ b bijtion btwn two skw irms, n ssum tht or ll pirs x, y χ th ollowin two onitions hol: (i) w o not simultnously hv x < y n (y) < (x), (ii) w o not simultnously hv (x) < (y) n y < x. Thn is pitur. Proo. Th proo is irly simpl, but it ssntilly uss th two inin onitions or skw irms, nmly initnss n onvxity with rspt to. Suppos stisis th onitions o th proposition 5

6 2.5 Altrntiv hrtristions o piturs but is not pitur. Thn possibly tr rplin by 1, w my ssum th xistn o pir x, y χ with x < y but (x) (y); morovr by onvxity w my ssum x to li ithr in th sm row or in th sm olumn s y. Th lttr s my b ru to th ormr by rplin by th orrsponin bijtion χ t ψ t, so ssum x n y li in th sm row. It thn ollows rom th ssumptions tht (x) < (y) n in t (y) lis stritly to th riht n blow (x). Thr my b svrl pirs (x, y) with ths proprtis, but by initnss o χ w my hoos (x, y) mon suh pirs to li in th irst (i.., hihst) possibl row o χ. Now lt p b th squr lyin in th sm olumn s (x) n in th sm row s (y) (s th illustrtion blow); by onvxity o ψ w hv p ψ. From th onitions ivn it ollows tht 1 (p) lis in som row bov tht o x (n y) n in som olumn to th lt o tht o y. Now lt q b th point in th sm row s 1 (p) n in th sm olumn s y; by onvxity o χ w hv q χ. By similr rsonin s or 1 (p) w ru tht (q) lis blow th row o (y) (n p) n to th riht o th olumn o p. 1 (x) (p) q χ ψ p (y) x y (q) But thn w hv p (q), whn ( 1 (p), q) is pir o points with th sm proprtis s (x, y), but in row bov thm, ontritin th hoi o (x, y). Thror th ssumption tht is not pitur must hv bn ls Corollry. Lt : χ ψ b bijtion btwn skw irms, n lt χ n ψ b prtil (or totl) orrins on χ n ψ rsptivly suh tht x y implis x χ y or x, y χ, n x ψ y or x, y ψ. Thn is pitur i n only i is post morphism (χ, ) (ψ, ψ ) n 1 is post morphism (ψ, ) (χ, χ ). Proo. Clrly th stt onitions r nssry. On th othr hn, i thy hol, thn th onitions o proposition will lso hol, n is pitur. As init bov, prtil pplition o this orollry is to ru th mount o work in tstin th post morphism onition or 1 in trms o orr ils. Tkin or χ totl orrin, n or ψ simply, on ins tht bijtion : χ ψ is pitur i n only i it is post morphism : (χ, ) (ψ, ), n th im o h orr il o (χ, χ ) is n orr il o (ψ, ). Th orr ils o (χ, χ ) n b numrt by strtin with th mpty st n sussivly joinin th squrs o χ in inrsin orr or χ ; th im o h nw squr must b n outr oornr o th skw irm orm by th ims o th squrs lry prsnt in th prvious orr il. Tstin th post morphism onition or n b on in th sm orr, by simply omprin th im o h nw squr with iniviul ims o prvious squrs; by th onvxity o χ it suis to onsir only th squrs irtly blow n to th lt o th nw squr, whnvr thy li in χ. For totl orrin omptibl with thr r two prtiulrly obvious nits, nmly th orrins r by rows n by olumns, in by (i, j) r (i, j ) i > i (i = i j j ) n (i, j) (i, j ) j < j (j = j i i ). Th totl orrin J tht is us inst o in th inition o piturs in [Zl1] n [ClSt] is th opposit o r, n thror not omptibl with our ; to mth thir piturs with ours vrythin must b trnspos, in whih s J orrspons to. Th spil s o orollry whr this orrin is tkn or χ n ψ ws lry prov by Clusn n Stötzr [ClSt, Stz 1.4]; th proo o proposition bov is similr to thir proo. By onstrution bs on ths onsirtions w n show tht in rtin sns thr xists n bunn o piturs. This is not so i w ix omin n im irms borhn, sin thr is no simpl ritrion or Pi(χ, ψ) to b non-mpty, but i w ix only th omin, thn piturs n b built up without obstrution. It will b onvnint to hv nottion or th squrs irtly bov, blow, lt n riht o ivn squr s; in x = x (1, 0), x = x + (1, 0), x = x (0, 1) n x = x + (0, 1) Proposition. Lt χ S b ivn, n totl orrin χ on χ suh tht x y with x, y χ implis x χ y; lt b bijtion rom n orr il χ o (χ, χ ) to skw irm ψ, suh tht is post morphism (χ, ) (ψ, ) n 1 is post morphism (ψ, ) (χ, χ ). Thn thr is t lst on wy to xtn to pitur χ ψ or som ψ S; in s ψ P th xtnsion n b m suh tht lso ψ P. 6

7 2.6 Piturs n th Littlwoo-Rihrson rul Proo. W rson by inution on χ\χ. Th s χ = χ is tkn r o by orollry 2.5.2, so it suis to show tht i χ χ, thn w n xtn χ by th squr x χ \ χ tht is miniml or χ, n in (x) suh tht th stt onitions rmin vli. Lt p = x, r = x, n q = p = r. As init bov, th onitions or (x) r tht it is n outr oornr o ψ, n tht (p) < (x) i p χ n (x) < (r) i r χ. For ny y χ w hv on o y p, y q or r y, whr y p n r y rsptivly imply p χ n r χ ; morovr i p, r χ thn lso q χ, n (p) < (q) < (r). It ollows tht i p χ thn (p) ψ, n i r χ thn (r) ψ, n i both hol, thn (p) (r). It is now sy to s tht in ll ss thr xists n outr oornr o ψ tht stisis ll onitions or (x); i ψ is Youn irm, it n b hosn insi N N, so tht th im rmins Youn irm Piturs n th Littlwoo-Rihrson rul. W n rphrs th prours ivn bov or hrtrisin n nrtin piturs in trms o row n olumn noins o piturs. For simpliity w irst onsir th s whr th im is Youn irm. Thn th row or olumn noin lon trmins th pitur: th lnth o row i (olumn i) o th im irm quls th numbr o tims i ours s ntry o th row (olumn) noin. Dinin th wiht o (prt o) irm ill with nturl numbrs s th squn ( 0, 1,...), whr i is th numbr o tims i ours s ntry, th wiht o th row or olumn noin must thror b prtition (i.., wkly rsin). Sin th im o ny orr il o (χ, χ ) is lso Youn irm, th wiht o th rstrition o th row or olumn noin to th orr il must lso b prtition. W n now hrtris row n olumn noins o piturs with Youn irm s im Proposition. Lt E b skw irm χ ill with nturl numbrs, n lt χ b totl orrin on χ suh tht x y with x, y χ implis x χ y. Thn E is th olumn noin (rsptivly row noin) o pitur : χ λ with λ P i n only i th ollowin onitions r stisi: (i) th ntris o E r stritly inrsin (rsptivly wkly rsin) lon h row, (ii) th ntris o E r wkly inrsin (rsptivly stritly rsin) lon h olumn, (iii) th wiht o th rstrition o E to ny orr il o (χ, χ ) is prtition. I so, is uniquly trmin, n λ t (rsptivly λ) is th Youn irm o th wiht o E. Furthrmor, ny prtil illin in on n orr il o (χ, χ ) tht stisis th ivn onitions or th in ntris n b xtn to omplt illin stisyin th onitions. Proo. It is lr tht th onitions r nssry. To ronstrut pitur rom its olumn or row noin, th missin oorint o th im o squr x shoul b tkn to b th numbr o squrs y < x in χ with th sm ntry s x. With this rul, th suiiny o th onitions ollows rom orollry 2.5.2, tkin (rsptivly r ) or ψ. From th proo o proposition it ollows tht pplyin this rul to prtil illin in on n orr il χ o (χ, χ ) will rsult in post morphism (χ, ) (λ, ); th xtnibility o suh illin thn ollows rom proposition Th rminin sttmnts r obvious. Rmrk. Conition (iii) is quivlnt to th rquirmnt tht rin th ntris o E in th inrsin orr or χ on obtins ltti prmuttion, i.., th wiht o ny initil subsqun is prtition. W shll omit n til sttmnt n proo o th nrlistion o this proposition or piturs whos im not Youn irm. I th im o pitur is λ \ µ, thn ivin µ in ition to th row or olumn noin o suis to trmin ; th only hn in th onitions or this s is tht in (iii) not th wihts thmslvs r rquir to b prtitions, but rthr th rsult o in µ to th wihts. Th illins srib in proposition r just Littlwoo-Rihrson illins. Mor prisly, in th tritionl ormultion o th Littlwoo-Rihrson rul (s or instn [M, I.9]), th llow illins r prisly th trnsposs o th illins llow by proposition or olumn noins, usin or χ. Th Littlwoo-Rihrson rul sribs th strutur oiints o th rin o symmtri untions on its Z-bsis o S-untions { s λ λ P }; w rr to [M] or pris initions. This rul n now b rstt in trms o piturs, s ollows Thorm [Littlwoo & Rihrson]. For λ, µ P on hs s µ s ν = λ P λ µ,νs λ, whr λ µ,ν = Pi(λ \ µ, ν). Althouh piturs λ \ µ ν orrspon to Littlwoo-Rihrson illins or λt µ t,ν, tht numbr quls t λ µ,ν sin s λ s λ t inus n utomorphism o th rin o symmtri untions. This symmtry is not (yt) obvious or piturs, but proposition os llow substntil vrition in onrt ormultions o th 7

8 2.6 Piturs n th Littlwoo-Rihrson rul Littlwoo-Rihrson rul, ll o whih r quivlnt, sin thy just srib th sm st o piturs in irnt wys: vrious orrins n b us or χ (suh s r or ), on my us row or olumn noin, n symmtris o piturs my b ppli, suh s 1, whih ls to illin th Youn irm ν inst o th skw irm λ \ µ. Enowin th rin o symmtri untions with th innr prout or whih th st o S-untions orms n orthonorml bsis, w hv λ µ,ν = s λ, s µ s ν. For λ, µ P with µ λ th skw S-untion s λ\µ is in by s λ\µ, s ν = λ µ,ν; this is wll in by th Littlwoo-Rihrson rul, n is invrint unr trnsltions o th skw irm λ \ µ. Th prout hs irt intrprttion in th orm o ionl ontntion o skw irms: or χ, ψ S w hv s χ s ψ = s χ ψ whr χ ψ is skw irm (in up to trnsltion) built rom χ n ψ s ollows χ ψ: ψ χ (s [M, I (5.7)]). From this it ollows tht λ µ,ν = π ρ,λ whr π, ρ P r suh tht µ ν = π \ ρ. Th orrsponin intity Pi(λ \ µ, ν) = Pi(λ, µ ν) n b unrstoo irtly. To s tht, w irst stt, n prov ombintorilly, n obvious onsqun o th Littlwoo-Rihrson rul, tht will lso b o us in th squl Proposition. For λ, µ P th st Pi(λ, µ) is mpty unlss λ = µ, in whih s it hs on lmnt. Proo. Consir pitur : λ µ, thn th irst olumn o λ is n orr il o (λ, ), so its im must b Youn irm ontin in µ; not hvin mor tht on squr in ny row, th im must b ontin in th irst olumn o µ. But sin w my ru similrly or th invrs im o tht olumn, it n only b tht mps th irst olumn o λ onto tht o µ. W n thn split o th irst olumns, n by inution in tht h olumn o λ is mpp onto th orrsponin olumn o µ, so λ = µ n is uniquly trmin. Th uniqu lmnt o Pi(λ, λ) will b not by 1 λ. W r now ry to monstrt th intity mntion bov, n in t slihtly mor nrl on Proposition. For ny λ, µ P with µ λ n ψ S, th st Pi(λ \ µ, ψ) is in bijtion with Pi(λ, µ ψ). Proo. Lt pitur : λ µ ψ b ivn. Sin µ is n orr il o (µ ψ, ), its invrs im is Youn irm µ ontin in λ; th rstrition o to µ is in pitur, whn µ = µ n th rstrition is qul to 1 µ. Th rstrition o to th omplmntry skw irm λ \ µ is lso pitur, n it is this pitur tht will orrspon to unr th bijtion o th proposition. On sily hks tht onvrsly th xtnsion o ny pitur λ \ µ ψ by 1 µ is pitur λ µ ψ. Th Littlwoo-Rihrson rul stts tht Pi(χ, ν) = s χ, s ν or ll χ S n ν P. This susts tht th sm miht b tru mor nrlly, with ν rpl by n rbitrry skw irm ψ. This is in th s, n n lry b u rom th ts prsnt o r Proposition [Zlvinsky]. For ll χ, ψ S on hs Pi(χ, ψ) = s χ, s ψ. Proo. It will sui to prov this or χ = λ \ µ with λ, µ P n µ λ. Thn by proposition w hv Pi(λ \ µ, ψ) = Pi(λ, µ ψ), whih by th Littlwoo-Rihrson rul is qul to s λ, s µ ψ = s λ, s µ s ψ = s λ\µ, s ψ, whr th inl qulity ollows by linrity rom s λ, s µ s ν = s λ\µ, s ν or ν P, sin (by th Littlwoo-Rihrson rul) s ψ n b writtn s linr ombintion o suh s ν. This t ws oriinlly stt by Zlvinsky [Zl1, Thorm 2], n prov by onstrutin bijtion, th Robinson-Shnst orrsponn or piturs tht w shll srib blow. As w hv init, th numrtiv intity n lry b riv without usin tht onstrution. 8

9 3 Th Robinson-Shnst orrsponn 3. Th Robinson-Shnst orrsponn Th Robinson-Shnst lorithm ppli to piturs. Sin { s λ λ P } is n orthonorml bsis o th rin o symmtri untions, proposition is quivlnt to Pi(χ, ψ) = λ P Pi(λ, χ) Pi(λ, ψ). Th Robinson-Shnst orrsponn or piturs is bijtion orrsponin to this intity Thorm [Zlvinsky]. For ll χ, ψ S thr is bijtion Pi(χ, ψ) λ P Pi(λ, ψ) Pi(λ, χ). Th bijtion is obtin by usin th (orinry) Robinson-Shnst lorithm. In on ormultion o tht lorithm, it ins bijtiv orrsponn btwn th st o bijtions : A B o two totlly orr sts o n lmnts, n pirs (P, Q) o post morphisms P : λ B n Q: λ A or som λ P. Hr orrspons to prmuttion o n n P n Q to Youn tblux o shp λ, but it is nturl to tk th lmnts o B s ntris or P, sin P is orm by insrtin th ims o into n initilly mpty tblu usin th Shnst insrtion prour; similrly it is nturl to tk th lmnts o A s th ntris o Q. Applyin th lorithm to ny bijtion χ ψ, whr χ n ψ r totlly orr by, on obtins pir o bijtions λ ψ n λ χ or som λ P. (As bor w hv trnspos vrythin with rspt to [Zl1]; thr th trnspos Robinson-Shnst lorithm is us.) Th ssntil point o th thorm is tht th bijtion χ ψ is pitur i n only i th sm is tru or th bijtions λ ψ n λ χ omput rom it. W omit proo o this thorm, sin w shll prov stronr sttmnt blow. Whil th numrtiv substrtum o this thorm ollows rom th Littlwoo-Rihrson rul, onvrs implition is prtilly n historilly muh mor rlvnt. Usin th thorm w n u th Littlwoo-Rihrson rul rom spil instn o th intity Pi(χ, ψ) = s χ, s ψ, nmly whr ψ is horizontl strip ψ µ or µ P, in by ψ µ = µ 0 µ 1, whr µ i is ( opy o) row i o µ. Th untion s ψµ is th prout h µ o th omplt symmtri untions h µi ssoit to th prts o µ; th lmnts o Pi(χ, ψ µ ) orrspon unr nt row noin to smistnr tblux o shp χ n wiht µ, n or this s th intity n b stblish irtly (s [M, I (5.14)]). Usin this t n thorm 3.1.1, w n prov thorm 2.6.2: Proo o th Littlwoo-Rihrson rul. W hv s χ, h µ = Pi(χ, ψ µ ) = λ P Pi(λ, χ) Pi(λ, ψ µ) = λ P Pi(λ, χ) s λ, h µ, whih, sin th untions h µ r known to b Z-bsis o th rin o symmtri untions, implis tht s χ = λ P Pi(λ, χ) s λ, n thror s λ, s χ = Pi(λ, χ). Rmrk. W hv ollow th proo o [M, I (9.2)], but its ruil lim (9.4) ws u rom thorm 3.1.1; this rus th 5-p proo to th w lins bov. Sin [M] prts th introution o piturs, its proo uss irnt lnu thn ours, but it is sy to intrprt th objts mnipult s piturs. Not tht Monl s proo is ronstrution n ompltion o th inomplt proo in [Rob] (whih ws rprou in [Litt]), whr th Robinson-Shnst orrsponn ws irst in. It pprs tht th min spt in whih Robinson s proo ws inomplt, is tht it ils to prov th prsrvtion o th proprtis tht orrspon, in thir isuis orm, to th pitur onitions. So on miht sy tht th orrsponn tht Robinson shoul hv in is not th on tht hs bom known s th (orinry) Robinson-Shnst orrsponn, but rthr Zlvinsky s nrlis vrsion! (This is not quit ir, sin th piturs or whih on ns th orrsponn in th proo r not ompltly nrl ons, but still th point is rmrkbl.) 3.2. Inpnn o hoi o totl orrins. In [FoGr] it ws shown tht in th onstrution o th bijtion o thorm on my rpl th orrin, us to mk χ n ψ into totlly orr sts, by othr totl orrins omptibl with (this is ll hoosin rins o χ n ψ), n still obtin th sm bijtion. This rsmbls wht w hv sn or th vrious wys to hrtris piturs, so w shll sy tht th orrsponn o thorm is nturl on (this trminoloy ws introu in [FoGr]). Nonthlss, this proprty is quit non-trivil ition to thorm 3.1.1, sin hnin th orrins on χ n ψ n hv siniint t on th prmuttion tht orrspons to th pitur, usin th insrtion pross to pro quit irntly. W shll now ormult stronr vrsion o thorm 3.1.1, tht mks both th nturlity n th rltion with th orinry Robinson-Shnst orrsponn xpliit; w irst n som initions. For n N lt [n] b th n-lmnt st { i N i < n }, n intiy th symmtri roup S n with th st o 9

10 3.3 Insrtion n xtrtion usin bijtions [n] [n]. For λ P lt T λ b th st o bijtiv post morphisms (λ, ) ([n], ); ths r th Youn tblux o shp λ. Put P n = { λ P λ = n }, n lt RS n : S n λ P n T λ T λ not th orinry Robinson-Shnst orrsponn (usin row-insrtion), s or instn [Sh], [Knu2], [vl3]. It will b onvnint to rprsnt totl orrin χ on skw irm χ by th uniqu post isomorphism α: (χ, χ ) ([n], ) (this is ssntilly rin o [FoGr]); omptibility o χ with is xprss by th t tht α is lso post morphism (χ, ) ([n], ) Thorm [Fomin & Grn]. Thr is bijtion RS χ,ψ : Pi(χ, ψ) λ P n Pi(λ, ψ) Pi(χ, λ) or ny χ, ψ S, suh tht i n = χ = ψ n RS χ,ψ () = (p, q), thn or ny pir o bijtiv post morphisms α: (χ, ) ([n], ) n β: (ψ, ) ([n], ) on hs RS n (β α 1 ) = (β p, α q 1 ). With rspt to thorm w hv invrt th son pitur (q), so tht χ lwys ours s omin, just s ψ lwys ours s im; this os not t th mnin o th thorm, but will mk it mth nir with lissmnts, tht will b isuss ltr. Th proo o th nturlity sttmnt ivn in [FoGr] is quit thnil. It shows tht on n trnsorm th rin α into stnr rin o χ (orrsponin to ) by smll stps, suh tht th orrsponin hns to th prmuttion β α 1 r riht Knuth trnsormtions ( subst o th lmntry trnsormtions o prmuttions ivn in [Knu1]), n tht or h suh stp orrsponn btwn piturs is unhn. Th uthor inpnntly obtin th nturlity rsult, usin th simplr n mor irt proo prsnt blow. W show tht Shnst insrtion n xtrtion prours or piturs n b srib irtly in trms o th orrin on ψ, without usin th rin β t ll, n tht thy prsrv th pitur onitions; thus th orrsponn in is utomtilly inpnnt o β. Lik in [FoGr] it suis to prov nturlity on on si, sin or th othr si it ollows by th wll known symmtry proprty o RS n Insrtion n xtrtion usin. Bor w n onstrut RS χ,ψ n prov th thorm, w n som simplr rsults. For λ P n k N, lt λ (k) = ({k} N) λ not th row k o λ, n put λ (>k) = λ ( k+1) = i>k λ (i) n λ (<k) = λ \ λ ( k) Lmm. Lt λ P, ψ S, p Pi(λ, ψ), n lt s b n outr oornr o ψ. Thn inus totl orrin on p(λ (0) ) {s}. I morovr s is not th mximum o this totlly orr st, thn its sussor min { y p(λ (0) ) s < y } is n outr oornr o p(λ (>0) ). Proo. Not irst tht λ (0) is n orr oil o (λ, ), so tht its im p(λ (0) ) is n orr oil o (ψ, ), whih is morovr (bin th im o row) totlly orr by, n in t horizontl strip. I s wr inomprbl with rspt to to ny squr x p(λ (0) ), thn x woul li stritly to th lt n bov s, so tht x ψ, n sin p(λ (0) ) is n orr oil, x p(λ (0) ); this woul ontrit th t tht p(λ (0) ) is horizontl strip. I s hs sussor, sy t, within th st p(λ (0) ) {s}, s mntion in th lmm, thn t n only li in row bov tht o s, n thror must b th ltmost lmnt o its row within p(λ (0) ). But thn t is miniml lmnt o th orr oil p(λ (0) ) o (ψ, ), n thror n outr oornr o its omplmntry orr il p(λ (>0) ). W now om to th Shnst insrtion n xtrtion prours or piturs Lmm. Thr is pir o mutully invrs prours tht trnsorm into h othr th ollowin sts o t: on on si pir (p, s) with p Pi(λ, ψ) or som λ P n ψ S, n with s n outr oornr o ψ; on th othr si pir (x, p ) with p Pi(λ, ψ ) or som λ P n ψ S n with x n outr ornr o λ. Th orrsponn is suh tht ψ = ψ {s} n λ = λ \ {x}. Morovr, or ny injtiv post morphism β: (ψ, ) (N, ) th Youn tblu β p is th rsult o insrtin th numbr β(s) into β p by th orinry Shnst row-insrtion prour. For ny hoi o β, th inl rquirmnt ompltly trmins th t o th prours; in or β orrsponin to th orrin, th onstrutions will xtly mth thos o [Zl1]. Nvrthlss w n to srib th prours xpliitly, in orr to show tht this n b on without rrrin to β. Our proo thn will onsist o two lmnts: th sription o th prours, n th proo tht thy prsrv th pitur onitions. Sin in th lttr prt inpnn o β is not importnt, w oul hv onin ourslvs to rrrin or it to th proo in [Zl1]. Thnks to proposition howvr, our proo is muh simplr n mor onis thn tht proo, whih is tully ontin in n ppnix o [Zl2], n is ivn only or th insrtion prour. 10

11 3.3 Insrtion n xtrtion usin Proo. Lt pir (p, s) s in th lmm b ivn. W onstrut squn x 0,..., x r or som r N, with x i λ or i < r n x r n outr oornr o λ (whih will in t b th squr x o th lmm), n orrsponin squn s 0,..., s r with s 0 = s n s i = p(x i 1 ) ψ or i > 0. W shll hv morovr tht h s i is n outr oornr o p(λ ( i) ). Th trms o th squns r trmin sussivly; ssum th w hv onstrut ll x i or i < k, n onsquntly ll s i or i k, n tht s k is n outr oornr o p(λ ( k) ). Thn by rstritin to λ ( k) n pplyin lmm w in tht p(λ (k) ) {s k } is totlly orr by. Put x k = (k, j k ), whr j k = { x λ (k) p(x) < s k } ; ithr x k is th ltmost squr x o λ (k) or whih s k < p(x), or, i thr r no suh squrs, it is th irst squr in row k byon th riht n o λ (k). In th lttr s w put r = k n x = x r, n th onstrution is omplt; othrwis w hv by lmm tht s k+1 = p(x k ) is n outr oornr o p(λ (>k) ), n w my pro to th nxt stp o th onstrution. Whn th onstrution is omplt w put λ = λ {x}, n in p : λ {s} ψ by p (x i ) = s i or 0 i r n p (y) = p(y) or y λ \ {x 0,..., x r 1 }. For ny β it is lr tht i w rpl th squrs o ψ by thir ims unr β, thn th onstrution rus to orinry Shnst insrtion. For th invrs prour w tr our stps bkwrs. Lt (p, x) s in th lmm b ivn, n lt x our in row r. W strt by sttin x r = x n s r = p (x r ); sin x r is mximl in (λ ( r), ), its im s r is n innr oornr o th orr oil p (λ (<r) ) o (ψ, ). Thn x r 1,..., x 0 n s r 1,..., s 0 r in s ollows, mnwhil showin tht h s i is n innr oornr o p (λ (<i)). Assumin this or i = k + 1, w hv nloously to lmm tht {s k+1 } p (λ (k) ) is totlly orr by ; morovr s k+1 is not its minimum, s p (x k+1 ) < p (x k+1 ) = s k+1. Put x k = (k, j k ), whr j k = { x λ (k) p (x) < s k+1 } 1, n s k = p (x k ); thn s k is th prssor o s k+1 in {s k+1 } p (λ (k) ) with rspt to, whih lis t th n o its row within p (λ (k) ), n thror is n innr oornr o p (λ (<k) ). At th n w put s = s 0, λ = λ \ {x}, ψ = ψ \ {s}, n in p: λ ψ by p(x i ) = s i+1 n p(y) = p (y) or y λ \ {x 0,..., x r 1 }. Lik bor, i or ny β w rpl th squrs o ψ by thir ims unr β, thn th onstrution rus to orinry Shnst xtrtion; in prtiulr, th two prours r h othrs invrss, provi tht w n show tht thy prsrv th pitur onitions. To prov tht th rsult o n insrtion or xtrtion is in pitur, w us proposition For ny hoi o β, th t tht β p n β p r Youn tblux obtin rom h othr by orinry Shnst insrtion n xtrtion implis tht onition 2.5.1(i) is stisi in both ss, n lso tht x r < < x 0. Now onsir th s o insrtion; suppos tht onition 2.5.1(ii) is not stisi or p, i.., thr r squrs y, z λ with p (y) < p (z) n z < y. Sin w know tht p is pitur, on sily ss tht this n only our i z {x 0,..., x r }, sy z = x k, n y {x 0,..., x r }. Thn p (z) = s k is n outr oornr o p(λ ( k) ), so tht p(λ ( k) ) {s k } = p (λ ( k) ) is n orr il o (ψ, ); sin p (y) < p (z) this orr il lso ontins p (y), n so y λ ( k). Now z < y implis y λ (k), so tht k < r, n p(y) lis in olumn to th riht o p(z) = s k+1 ; this ontrits p (y) < s k < s k+1. In th s o xtrtion, violtion p(y) < p(z) z < y o onition 2.5.1(ii) n only our i y = x k n z {x 0,..., x r 1 } Thn p(z) lis in th th orr oil {s k+1 } p (λ (<k+1) ) = p(λ (<k+1)) {s} o (ψ, ), whn z λ (k), lin to ontrition with p (y) = s k < s k+1 = p(y) < p(z) = p (z). Lt us iv n xmpl to illustrt th prours. I w pply th insrtion prour, tkin or p th pitur isply on th lt blow, n or s th outr oornr o its im mrk, thn th rsult will b tht p is th pitur isply on th riht, n x is th squr in its omin mrk m. b k i j h m l n p h i b n l m j k b k j h i l m n p h i b This rsult ws obtin by th ollowin stps; or onvnin w us x or th squr mrk x t th im si o th isply o p, n similrly x or th squr mrk x t th omin si o p, whih is p 1 (x). W strt with puttin s 0 =, n omprin it with p(λ (0) ), whih tothr orm th hin < b < < < < k. So s 1 =, th sussor o, n x 0 =. Thn s 1 is ompr with p(λ (1) ), ivin < < < i < j, so x 1 = i n s 2 = i. Similrly rom h < i < m w t x 2 = m n s 3 = m, n sin l < m th prour thn stops with x 3 = x = l. Sttin p (x i ) = s i or i = 0, 1, 2, 3, w t or p th pitur isply on th riht. 11 n l m j k

12 3.4 Nturlity o th ull Robinson-Shnst orrsponn 3.4. Nturlity o th ull Robinson-Shnst orrsponn. Proo o thorm W shll now in th nrlis Robinson-Shnst orrsponn RS χ,ψ o thorm W o so by inin RSχ,ψ α or ny hosn bijtiv post morphism α: (χ, ) ([n], ) suh tht it stisis th rquirmnts o th thorm or this α n ll β, n thn prov tht RSχ,ψ α is inpnnt o α. Th onstrution is irt trnsltion o th orinry Robinson-Shnst lorithm, usin th prours o lmm or insrtion n xtrtion. So lt χ, ψ n n b s in th thorm, n lt : χ ψ b bijtion suh tht α 1 is skw tblu, i.., post morphism (ψ, ) ([n], ) (vntully w shll rstrit to bin pitur). For i = 0, 1,..., n w sussivly omput piturs p i : λ (i) ψ (i), n t th sm tim in iniviul ims o mp q: χ N N; hr λ (i) r Youn irms, n ψ (i) = (α 1 ([i])), whih is n orr il o (ψ, ) sin [i] is n orr il o ([n], )). Strt with p 0 :, n tr p i is trmin, pply th insrtion prour o lmm to (p i, (α 1 (i))), rsultin in pir (x, p ); st λ (i+1) = λ (i) {x}, p i+1 = p, n q(α 1 (i)) = x. Whn p n is vntully trmin, put λ = λ (n), n RSχ,ψ α () = (p n, q), whr q th now ompltly in bijtion χ λ, or whih α q 1 T λ. Rvrsin th stps, n usin th xtrtion prour o lmm 3.3.2, in n invrs lorithm RSχ,ψ α 1, tht n b ppli to ny pir (p, q) o pitur p: λ ψ n bijtion q: χ λ with α q 1 T λ, or som λ P, n tht yils bijtion : χ ψ or whih α 1 is skw tblu. By onstrution w hv i RSχ,ψ α () = (p, q) tht RS n(β α 1 ) = (β p, α q 1 ) or ll β; lrly p n q r inpnnt o β. On th othr hn by th wll known t tht RS n (w) = (P, Q) implis RS n (w 1 ) = (Q, P ), w hv RS n (α 1 β 1 ) = (α q 1, β p). I w now ssum tht is pitur, thn RS β ψ,χ ( 1 ) = (q 1, p 1 ), implyin tht q 1 (n hn q) is pitur, n lso tht p n q r inpnnt o α. Convrsly, i q inst o is ssum to b pitur, thn rom RS β ψ,χ 1 (q 1, p 1 ) = 1 it ollows tht β is skw tblu; tothr with th oriinl ssumption tht α 1 is skw tblu, this implis by proposition tht is pitur. This omplts th proo o thorm Rmrk. Th subsripts χ, ψ tth to th oprtor RS n its invrs r us only to istinuish it rom RS n, n to srv s rminr o th omin n im o th pitur involv; in pplitions o ths oprtors ths subsripts my b supprss, lthouh w shll not o so. As n illustrtion o th lorithm, w shll pply it to th pitur tht w hv sn bor: b b. W hoos α: χ [7] orrsponin to r, or whih th squrs o χ in inrsin orr rry th lbls,,,,, b, (th only othr litimt hoi woul b to intrhn n ). W show hr th inl stps o th lorithm (th irst w stps r lss illustrtiv): th piturs p 4,..., p 7 r sussivly,, b b, b b. Th othr pitur omput is q = b b whr th orrsponin Youn tblu α q 1 is isplyin th orr in whih th ims wr trmin. I w h us th othr hoi or α, n hn th insrtion orr,,,,, b,, th intrmit pitur p 4 woul b irnt (n w woul hv λ (4) = (2, 1, 1) inst o λ (4) = (3, 1)); th ntris 3 n 4 woul b intrhn in α q 1, but th pitur q woul b unhn. Not tht th point-im pirs o q r trmin on by on, but th intrmit 12

13 4 Th Shütznbrr orrsponn prtil mps r not lwys piturs: tr p 5 is omput th pirs o q lbll,,,, r trmin, but th orrsponin subst o th omin χ is not skw irm. An intrstin spil s o this onstrution is whn χ n ψ r horizontl strips, i.., th pitur orrspons to nrlis prmuttion o [Knu1]. Thn thr is only on possibl hoi or th morphisms α n β, so tht th pris stps tkn by th lorithm r ompltly trmin. Inst o usin β n α to mk p n q into stnr Youn tblux, on n lso rprsnt thm s smistnr tblux by usin nt row noin; th insrtion n xtrtion prours us or p, n th inition o th othr pitur q 1, thn bom intil to thos in [Knu1]. Th ul orrsponn srib thr, whih oprts on zro-on mtris inst o nrlis prmuttions, n lso b obtin s spil s, by tkin or χ vrtil strip n or ψ horizontl strip, n usin olumn noin or p so tht it is trnspos smistnr tblu (or ul tblu in th trminoloy o [Knu1]), whil or q on kps nt row noin. Thror, th Robinson-Shnst lorithm or piturs n in t b sn s ommon nrlistion o both vrints o o Knuth s nrlis Robinson-Shnst lorithm. Unlik th orinry Robinson-Shnst lorithm, th Robinson-Shnst lorithm or piturs n b ppli to h o th omponnts o th pir it rturns. Suh itrtion os not prou ny intrstin nw piturs, howvr Proposition. For p Pi(λ, ψ) n q Pi(χ, λ) with λ P n χ, ψ S, on hs RS λ,ψ (p) = (p, 1 λ ) (1) RS χ,λ (q) = (1 λ, q) (2) Proo. Th irst s n b vrii irtly rom th inition o RSλ,ψ α, with (or simpliity) α orrsponin to or r ; h insrtion stp only involvs movs in sinl olumn o λ. Th vriition ssntilly oms own to th wll known t tht or ny Youn tblu P, i w pply th orinry Robinson-Shnst lorithm to th prmuttion obtin by rin th ntris in inrsin orr or or r, thn th lt tblu obtin will b P itsl; in, w s tht this is tru or ny orr omptibl with. Th son s ollows by symmtry. 4. Th Shütznbrr orrsponn Th Robinson-Shnst orrsponn in rltion to symmtris. As ws mntion bor, th st Pi(χ, ψ) is in bijtion with h o Pi( χ, ψ), Pi(χ t, ψ t ), n Pi( χ t, ψ t ), by omposin pitur with th init rltions in omin n im; w shll not th ountrprts o pitur so obtin by, t, n t (so w init th symmtry ppli to th omin, rthr thn tht ppli to th im). An obvious qustion is wht hppns to th pir o piturs omput by th Robinson-Shnst lorithm whn w pply ths symmtris to ; th nswr must b non-trivil, sin th lss o piturs llow or p n q is not ix by ths symmtris. Th nswr will involv th Shütznbrr orrsponn, n lorithmilly in shp prsrvin trnsormtion o Youn tblux; w shll not it by S n : T n T n, whr n N n T n = λ P n T λ. It ws irst in in [Shü1] (whr it is ll I); s lso [Knu2] (th oprtion P P S ) n [vl3]. It hs inition n som proprtis o typ similr to thos o th Robinson-Shnst lorithm, n thr is stron onntion btwn th orrsponns in by th two lorithms, tht w shll now ormult. Lt ñ S n b th uniqu prmuttion tht is n nti-isomorphism o ([n], ) to itsl, i.., ñ: i n 1 i Thorm [Knuth]. For σ S n n P, Q T n, th ollowin sttmnts r quivlnt: RS n (σ) = (P, Q) RS n (σ ñ) = (P t, S n (Q) t ) RS n (ñ σ) = (S n (P ) t, Q t ) RS n (ñ σ ñ) = (S n (P ), S n (Q)) In its ull orm th thorm irst pprs in [Knu2, Thorm D] (s lso [Shü2, 4.3], n [vl3, thorm 4.1.1]); importnt prtil rsults lry ppr in [Sh] n [Shü1]. W lso hv th intitis S n (P t ) = S n (P ) t n S n (S n (P )) = P, tht r in t impli by this thorm. 13

14 4.2 Th Shütznbrr orrsponn or piturs Viwin prmuttions s spil ss o piturs, this thorm prisly sribs th ts o th symmtris mntion bov on th tblux ssoit to prmuttions unr th Robinson-Shnst orrsponn: i is pitur orrsponin o prmuttion σ S n, thn t orrspons to σ ñ (th rvrs o σ), t to ñ σ (σ with ñ ppli to its ntris), n to ñ σ ñ Th Shütznbrr orrsponn or piturs. Ths sttmnts n b nrlis to rbitrry piturs, usin th Shütznbrr orrsponn or piturs tht is srib in [Zl1]; it is bs on th orrsponin lorithm or tblux in muh th sm wy s th Robinson-Shnst orrsponn or piturs is. W shll ll this oprtion S ψ or ψ S; it bijtivly mps Pi(λ, ψ) to Pi(λ, ψ) or ll λ P ψ. Th ntion o th im irm is quit nturl in viw o th inition o S n n thorm (in t it woul hv som vnts to lso in S n suh tht th ntris o S n (P ) r th ntivs o thos o P, s is on in [vl3]). Lik bor, w irst in n oprtion S β ψ usin bijtiv post morphism β: (ψ, ) ([n], ), n thn show tht it sns piturs to piturs n th outom os not pn on β. For β orrsponin to, th inition will mth th on in [Zl1]. W shll n post morphisms rom skw irms to [n] orrsponin to β, but in on ψ t, ψ, n ψ t ; ths will b ll β t, β n β t rsptivly, n r in by β t (s) = ñ(β(s t )), β(s) = ñ(β( s)), n β t (s) = β( s t ) (th omposition with ñ in th irst two ss is n to to obtin morphism). W in S β ψ by Sβ ψ (p) = ( β) 1 S n (β p) so tht ( β) S β ψ (p) = S n(β p); in othr wors, S β ψ is in in suh wy tht unr omposition with β n β to trnsorm piturs into tblux, it rus to th orinry Shütznbrr orrsponn Thorm. Thr is bijtion S ψ : λ P Pi(λ, ψ) λ P Pi(λ, ψ) or ny ψ S, suh tht i n = ψ thn or ny bijtiv post morphism β: (ψ, ) ([n], ) on hs ( β) S ψ (p) = S n (β p) or ll p Pi(λ, ψ), λ P. Morovr, i S χ: λ P Pi(χ, λ) λ P Pi( χ, λ) is orrsponinly in by S χ(q) = S χ (q 1 ) 1, thn th ollowin sttmnts r quivlnt: RS χ,ψ () = (p, q), (3) RS χ t, ψ t( t ) = ( p t, S χ t( qt ) ), (4) RS χ t,ψ t( t ) = ( S ψ t(p t ), q t), (5) RS χ, ψ ( ) = ( S ψ (p), S χ(q) ). (6) Furthrmor, S ψ t(p t ) = S ψ (p) t n S ψ (S ψ (p)) = p or ll p Pi(λ, ψ). This thorm ollows in strihtorwr wy rom thorms n Nvrthlss th nturlity sttmnt n th inorportion o qutions (4) n (5) ppr to b nw; th quivln o (3) n (6) is stt in [Zl1, proposition 9]. Proo. Lt Pi(χ, ψ) n RS χ,ψ () = (p, q). Choos morphisms α, β s in thorm 3.2.1, n put σ = β α 1 S n, thn ñ σ ñ = ( β) ( ) ( α) 1. Applyin RS n to this prmuttion, w t by thorm tht RS n (( β) ( ) ( α)) = (S n (β p), S n (α q 1 )) = (( β) S β ψ (p), ( α) Sα χ (q 1 )). It thn ollows rom thorm tht w must hv RS χ, ψ ( ) = (S β ψ (p), Sα χ (q 1 ) 1 ); thror, this is pir o piturs tht os not pn on α or β, whih stblishs th initil sttmnts bout S ψ n th quivln o (3) n (6). Th othr quivlns ollow by rsonin similrly or th prmuttions ( β t ) t (α t ) 1 n (β t ) ( t ) ( α t ) 1. Th rminin lims n b prov similrly, but lso ollow rom th stt quivlns. Not tht th nturlity is ssntil in obtinin th quivln o (3) with (4) or (5): i w woul only us oprtions o typ S β ψ with β orrsponin to, thn it woul or instn not b possibl to rlt S ψ t(p t ) to S n (( β t ) p t ), sin β t is not o th init typ Trnsposin omin n im simultnously. Th Robinson-Shnst n Shütznbrr orrsponns or piturs, in ombintion with th symmtris o piturs, provi svrl quivlnt wys to in th bijtion btwn Pi(χ, ψ) n Pi(χ t, ψ t ) tht ws nnoun rlir. 14

15 4.3 Trnsposin omin n im simultnously Thorm. Thr xists bijtiv mp T rom Pi(χ, ψ) to Pi(χ t, ψ t ) or ny χ, ψ S with th ollowin proprtis. For pitur : χ ψ with RS χ,ψ () = (p, q), on hs RS χ t,ψ t( T ) = ( S ψ t(p t ), S χ t( qt ) ), (7) RS χ, ψ (( T ) t ) = ( S ψ (p), q ), (8) RS χ,ψ ( ( T ) t ) = ( p, S χ(q) ), (9) RS χ t, ψ t( ( T )) = (p t, q t ). (10) For λ P n ny p Pi(λ, ψ), q Pi(χ, λ) on hs morovr p T = S ψ t(p t ) n q T = S χ t( qt ), (11) so tht (7) n b rstt s RS χ t,ψ t( T ) = (p T, q T ). (12) Finlly on hs T T =, n urthr ommuttion rltions ( 1 ) T = ( T ) 1, ( ) T = ( T ), ( t ) T = ( T ) t, n S ψ (p) T = S ψ t(p T ), S χ(q) T = S χ t(qt ). Proo. Eh o th qutions (7) (10) trmins uniqu vlu or T, n by thorm ths r ll qul. Applyin (7) with p or q or, n usin proposition on obtins (11). Th rminin intitis ollow by irt omputtion, usin th intitis lry stblish. I th omin or im o pitur is Youn irm, thn (11) shows tht T n b omput without usin th Robinson-Shnst lorithm. Suh piturs orrspon to Littlwoo-Rihrson illins, n or thos orrsponin oprtion hs bn srib lswhr, s or instn [HSu]. On th othr hn (10) shows tht T n lwys b omput without usin th Shütznbrr lorithm, so (11) lso implis tht S ψ n b xprss in trms o RS χ,ψ ; with 1 λ notin th uniqu pitur λ λ, w hv S ψ (p) = RS 1 λ,ψ (p, 1 λ), (13) n by intrhnin p n S ψ (p) this implis tht S ψ (p) is lso th irst omponnt o RS λ, ψ ( p). As n illustrtion o th rltion btwn n T or nrl piturs, w onsir in th pitur or whih w monstrt th Robinson-Shnst lorithm. W h = b b, p = b b, q = b b ; usin (11) n (12) w t p T = b b, q T = b b, T = t u v w x y z z y w x u v t. Not tht in omputin p T n q T by th Shütznbrr lorithm w hv hosn th intiyin lbls to mth thos o p n q on th im rsptivly on th omin. For T howvr th orrsponn with th iniviul point-im pirs o oul not b mintin in ny mninul wy, so w swith to irnt st o lbls. 15

16 4.4 Lk o nturlity o th ltion prour 4.4. Lk o nturlity o th ltion prour. So r w hv us thorm rthr thn th inition o S n (P ), but it is intrstin to s whthr th omputtion o S ψ (p) n b srib irtly in trms o piturs, s ws th s or RS χ,ψ (). Th omputtion o S n (P ) onsists o rpt pplition o ltion prour to P, whih rmovs n ntry, n rrrns th rminin ntris into smllr Youn tblu; th tblu S n (P ) rors th squn o shps o P, (P ), 2 (P ),..., n (P ). For ψ S n bijtiv post morphism β: (ψ, ) ([n], ), on n in n oprtion β suh tht or mps p: λ ψ or whih β p is Youn tblu on hs β β (p) = (β p); thn th tblu ( β) S β ψ (p) = S n(β p) will ror th squn o shps o p, β (p), 2 β (p),..., n β (p). Sin Sβ ψ os not pn on β on miht think tht th sm is tru or β. Howvr, this is not th s: th vry t tht S n (β p) = ( β) S ψ (p) shows tht S n (β p) vris with β, so th squn o shps i β (p) must vry s wll. So unlik th Shnst insrtion n xtrtion prours, nnot b in nturlly or piturs. In t, β os not vn prsrv th pitur onitions. An pplition o strts with rmovin th ntry t th oriin, rtin n mpty squr, n thn s lon s possibl slis ntris ltwrs or upwrs into th mpty squr; whnvr two ntris oul mov into th mpty squr, th smllr on tks prn, to kp th rows n olumns inrsin. In th omputtion o β (p), this omprison tks pl btwn ntris o β p. I th p-ims o th squrs in qustion r omprbl by thn this will trmin th omprison in β p, inpnntly o β. I thy r inomprbl howvr, thn β brks th ti, n th ntris ompr will n up ithr in th sm row or olumn; but this mns tht th pitur onition is stroy, sin or pitur th ims o squrs in on row or olumn must b omprbl by. Sin rpt pplition o rmovs th ntris rom th tblu in inrsin orr, omprison btwn ny pir o ntris is stblish t som point urin th pross, so tht unlss ψ is totlly orr by, som i β (p) will not b pitur. In t th pitur onitions will b violt in nothr wy s wll: th im shps o i β (p) will not ll b skw irms. Lt us iv onrt xmpl. Consir th ollowin pitur: b p: b, or whih S ψ (p) = b b. For onvnin w lt h o,..., not th squr in th im o p with tht lbl; this llows us to viw th pitur s Youn tblu ill with symbols inst o numbrs. A hoi o β ins n orrin o th symbols; lthouh thr r svrl possibilitis w will rstrit ourslvs to thos or whih < < b <, < <. Dpnin on whthr or not < w t th ollowin two squns or i β (p). < : b b b < : b b b W s tht lthouh irn is introu t th irst stp, th shps rmin th sm until th irst o {, } is rmov, n th tblux bom qul in tr both r rmov. This rmrkbl t is no oinin, sin th nturlity o S ψ implis tht th squr tht is r in th stp tht n ntry x is rmov, is th on tht mps to x unr S ψ (p), whih is inpnnt o th orrin us. An xtrm s is p = 1 λ whr λ is rtnulr irm: thn vry Youn tblu P o shp λ n b writtn s P = β p or som β, n ny hin o irms n our s shps o i β (p); in this s h two o β, P, n S n (P ) r link by simpl trnsormtion. A rlt t is tht or p Pi(λ, ψ) with λ rtnulr, S ψ (p) quls p, up to trnsltion o th omin; this ollows rom (13) n (1), but lso rom th nrl wy tht th Shütznbrr orrsponn n b xprss in trms o lissmnt, onstrution tht w shll onsir nxt. 16

17 5. Glissmnt. 5 Glissmnt In th prvious two stions th Robinson-Shnst n Shütznbrr orrsponns wr onsir s thy r in by thir trministi lorithms. Shütznbrr hs shown in [Shü2] tht ths orrsponns n lso b in by rwrit systm or skw tblux, whr th bsi rwrit stp is ll lissmnt. W shll now vlop similr thory or piturs; th onstrutions n rsults o 3 n 4 r not us, but thy o provi motivtion Dinition o omin-lissmnts n im-lissmnts o piturs. Lt us rll rom [Shü2] how lissmnt o skw tblu ϕ is orm. An innr oornr o th shp o ϕ is ppoint s initil position o n mpty squr, thn (s in th ltion prour) ntris r rptly sli ltwrs or upwrs into th mpty squr, th smllr ntry tkin prn i thr r two possibilitis. Whn no mor movs r possibl, th nw positions o th ntris in nw skw tblu ϕ, tht w shll ll th inwr lissmnt o ϕ into s. Givn th inl position s o th mpty squr (n outr oornr o th shp o ϕ ) th movs n b tr bk in similr shion, rovrin ϕ; w ll ϕ n outwr lissmnt o ϕ into s. In orr to in similr oprtion or pitur : χ ψ, on my tk bijtiv post morphism β: (ψ, ) ([n], ) n ll n inwr lissmnt o i β is n inwr lissmnt o β. It is howvr by no mns obvious tht suh will b pitur. A nssry onition or this is tht th ims unr o ny pir o squrs in th sm row or olumn r omprbl by. In prtiulr ny pir s, t ψ or whih th ntris β(s) n β(t) o β wr ompr in ormin th lissmnt must b omprbl by. But thn th rsultin pitur will not pn on β; in othr wors, th inition n only work i it is nturl. This turns out to b th s, whih is surprisin in viw o th ntiv rsults bout th ltion prour Thorm. Lt : χ ψ b pitur, n s n innr (rsptivly outr) oornr o χ. Thr xists uniqu pitur : χ ψ suh tht or ny bijtiv post morphism β: (ψ, ) ([n], ) th skw tblu β is th inwr (rsptivly outwr) lissmnt o β into s. Th pitur will b ll th inwr (rsptivly outwr) omin-lissmnt o into s. Anothr orm o lissmnt n b riv by th symmtry 1 : w shll ll ( ) 1 th inwr (outwr) im-lissmnt o 1 into s. Hr is n xmpl o ths oprtions: th lt pitur is n inwr ominlissmnt o th mil on, n th riht pitur is n outwr im-lissmnt o th mil on. b b DG b b IG b b Th omprisons m r < n < or th omin-lissmnt n < or th imlissmnt, whr ovrlins (unrlins) init lbll squrs in th im (omin) o th mil pitur. Rplin pitur by lissmnt o into squr onstituts on rwrit stp. Mor nrlly, w shll ll lissmnt o i thr is squn o piturs = 0, 1,..., l = whr h i+1 is (omin or im) lissmnt o i into som squr; i in ition ny o th quliitions omin, im, inwr, or outwr is us, thn tht quliition must pply to ll ths lissmnts into squr. To prov thorm 5.1.1, w only n to onsir th inwr s, by th symmtry. W strt with showin th nturlity; lik or th Shnst insrtion prour, th rsults o ll omprisons m r inpnnt o β Lmm. Lt n β b s in thorm I, urin th omputtion o th inwr lissmnt o β into s, th ntris β((x)) n β((y)) o two squrs x, y χ r ompr with h othr, thn (x) n (y) r omprbl by. Proo. Assum th ontrry, n lt (i, j) b th irst squr (i.., miniml or ) or whih th ntris o squrs x = (i + 1, j) n y = (i, j + 1) o χ r bin ompr, but (x) n (y) r inomprbl or. Thn (x) lis bov n to th lt o (y), i.., (x) = (k, l) n (y) = (k, l ) with k < k n l < l. Lt z b th squr (k, l) whih lis in th olumn o (x) n th row o (y); 17