Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson [5] is a emakable combinatoial coespondence associated with the theoy of symmetic functions. Schützenbege s jeu de taquin[10] can be used to give altenative desciptions of both P - and Q- tableaux of the Schensted algoithm as well as the odinay and dual Knuth elations. In this pape we descibe the jeu de taquin on shifted im hook tableaux using the switching ule, which shows that the sum of the weights of the shifted im hook tableaux of a given shape and content does not depend on the ode of the content if content pats ae all odd. 1. Intoduction Thee is a emakable combinatoial coespondence associated with the theoy of symmetic functions, called the Schensted algoithm. Theoem 1.1. (Schensted algoithm) Let S n be the symmetic goup of degee n. Then thee is a bijection π (P, Q) between pemutations π of S n and the set of all pais (P λ, Q λ ) of standad Young tableaux of the same shape λ, whee λ n. Received July 22, 2015. Revised Septembe 6, 2015. Accepted Septembe 9, 2015. 2010 Mathematics Subject Classification: 05E10. Key wods and phases: patition, shifted im hook tableau, Schensted algoithm, switching ule, jeu de taquin. c The Kangwon-Kyungki Mathematical Society, 2015. This is an Open Access aticle distibuted unde the tems of the Ceative commons Attibution Non-Commecial License (http://ceativecommons.og/licenses/by -nc/3.0/) which pemits unesticted non-commecial use, distibution and epoduction in any medium, povided the oiginal wok is popely cited.
428 Jaejin Lee It was fist descibed in 1938 by Robinson [5], in a pape dealing with an attempt to pove the coectness of the Littlewood-Richadson ule. Schensted algoithm was ediscoveed independently by Schensted [7] in 1961, whose main objective was counting pemutations with given lengths of thei longest inceasing and deceasing subsequences. Schensted coespondence about inceasing and deceasing subsequences is extended by C. Geene [3], to give a diect intepetation of the shape of the standad Young tableaux coesponding to a pemutation. Knuth [4] gave a genealization of the Schensted algoitm, whee standad Young tableaux ae eplaced by column stict tableaux, and pemutations ae eplaced by multi-pemutations. And he descibed conditions fo two pemutation to have the same P -tableaux unde Schensted algoithm. In [11] Viennot gave a geometic intepetation fo Schensted algoithm. The combinatoial significance of Schensted algoithm was indicated by Schützenbege [10], who intoduced the evacuation algoithm using the jeu de taquin. The jeu de taquin can be used to give altenative desciptions of both P - and Q-tableaux of the Schensted algoithm as well as the odinay and dual Knuth elations. Afte Schützenbege intoduced the jeu de taquin on standad Young tableaux, vaious analogs of the jeu de taquin came: vesions fo im hook tableaux [9] and shifted tableaux[6]. In this pape we descibe the jeu de taquin on shifted im hook tableaux using the switching ule, which shows that the sum of the weights of the shifted im hook tableaux of a given shape and content does not depend on the ode of the content if content pats ae all odd. In section 2, we outline the definitions and notation used in this pape. In Section 3, we pove the switching ule fo the shifted im hook tableaux. The jeu de taquin on shifted im hook tableaux is given in Section 4. 2. Definitions We use standad notation P fo the set of all positive integes. Let λ = (λ 1, λ 2,..., λ l ) be a patition of the nonnegative intege n, denoted λ n o λ = n, so λ is a weakly deceasing sequence of positive integes summing to n. We say each tem λ i is a pat of λ and n is the weight of λ. The numbe of nonzeo pats is called the length of λ and is witten l = l(λ). Let P n be the set of all patitions of n and P be the
The jeu de taquin on the shifted im hook tableaux 429 set of all patitions. We also denote DP = {µ P µ has all distinct pats}, DP n = {µ P n µ has all distinct pats}. We sometimes abbeviate the patition λ with the notation 1 j 1 2 j 2..., whee j i is the numbe of pats of size i. Sizes which do not appea ae omitted and if j i = 1, then it is not witten. Thus, a patition (5, 3, 2, 2, 2, 1) 15 can be witten 12 3 35. Fo each λ DP, a shifted diagam D λ of shape λ is defined by D λ = {(i, j) i j λ j + i 1, 1 i l(λ)}. And fo λ, µ DP with D µ D λ, a shifted skew diagam D λ/µ is defined as the set-theoetic diffeence D λ \ D µ. Figue 2.1 and Figue 2.2 show D λ and D λ/µ espectively when λ = (9, 7, 4, 2) and µ = (5, 3). Figue 2.1 Figue 2.2 Figue 2.3 A shifted skew diagam θ is called a single im hook if θ is connected and contains no 2 2 block of cells. If θ is a single im hook, then its head is the uppe ightmost cell in θ and its tail is the lowe leftmost cell in θ. See Figue 2.3. A double im hook is a shifted skew diagam θ fomed by the union of two single im hooks both of whose tails ae on the main diagonal. If θ is a double im hook, we denote by A[θ] (esp., α 1 [θ]) the set of diagonals of length two (esp., one). Also let β 1 [θ] (esp., γ 1 [θ]) be a single im hook in θ which stats on the uppe (esp., lowe ) of the two main diagonal cells and ends at the head of α 1 [θ]. The tail of β 1 [θ] (esp., γ 1 [θ]) is called the fist tail (esp., second tail) of θ and the head of β 1 [θ] o γ 1 [θ] (esp., γ 2 [θ], β 2 [θ], whee β 2 [θ] = θ \ β 1 [θ] and γ 2 [θ] = θ \ γ 1 [θ]) is called the fist head (esp., second head, thid head) of θ. Hence we have
430 Jaejin Lee the following desciptions fo a double im hook θ: θ = A[θ] α 1 [θ] = β 1 [θ] β 2 [θ] = γ 1 [θ] γ 2 [θ]. A double im hook is illustated in Figue 2.4. We wite A, α 1, etc. fo A[θ], α 1 [θ], etc. when thee is no confusion. θ A α 1 β 2 β 1 γ 2 γ 1 Figue 2.4 We will use the tem im hook to mean a single im hook o a double im hook. A shifted im hook tableau of shape λ DP and content ρ = (ρ 1,..., ρ m ) is defined ecusively. If m = 1, a im hook with all 1 s and shape λ is a shifted im hook tableau. Suppose P of shape λ has content ρ = (ρ 1, ρ 2,..., ρ m ) and the cells containing the m s fom a im hook inside λ. If the emoval of the m s leaves a shifted im hook tableau, then P is a shifted im hook tableau. We define a shifted skew im hook tableau in a simila way. If P is a shifted im hook tableau, we wite κ P (o just κ ) fo a im hook of P containing. If θ is a single im hook then the ank (θ) is one less than the numbe of ows it occupies and the weight w(θ) = ( 1) (θ) ; if θ is a double im hook then the ank (θ) is A[θ] /2 + (α 1 [θ]) and the weight w(θ) is 2( 1) (θ). The weight of a shifted im hook tableau P, w(p ), is the poduct of the weights of its im hooks. The weight of a shifted skew im hook tableau is defined in a simila way. 1 1 1 1 2 2 1 3 3 3 3 1 1 1 1 2 2 1 3 3 3 3 1 1 1 1 2 2 1 3 3 3 3 Figue 2.5 Figue 2.6
The jeu de taquin on the shifted im hook tableaux 431 Figue 2.5 shows an example of a shifted im hook tableau P of shape (6, 4, 1) and content (5, 2, 4). Hee (κ 1 ) = 1, (κ 2 ) = 0 and (κ 3 ) = 1. Also w(κ 1 ) = 2, w(κ 2 ) = 1 and w(κ 3 ) = 1. Hence w(p ) = ( 2) (1) ( 1) = 2. Suppose P is a shifted im hook tableau. Then we denote by one of the tableaux obtained fom P by cicling o not cicling the second tail of each double im hook in P. The is called a second tail cicled im hook tableau. We use the notation to efe to the uncicled vesion; e.g., = P. See Figue 2.6 fo examples of second tail cicled im hook tableaux. We now define a new weight function w fo second tail cicled im hook tableaux. If τ is a im hook of, we define w (τ) = ( 1) (τ). The weight w ( ) is the poduct of the weights of im hooks in. Fo each double im hook τ of a im hook tableau P, thee ae two second cicled im hooks τ 1, τ 2 such that w(τ) = w (τ 1 ) + w (τ 2 ). This fact implies the following: Poposition 2.1. Let γ OP. Then we have w(p ) = w ( ), P whee the left-hand sum is ove all shifted im hook tableaux P of shape λ/µ and content γ, while the ight-hand sum is ove all shifted second tail cicled im hook tableaux of shape λ/µ and content γ. 3. Switching ule on the shifted im hook tableaux In [9] Stanton and White give the switching ule fo im hook tableau. It shows that the sum of the signs of im hook tableaux of a given shape and content is independent of the ode of the content. In this section we give a shifted im hook analog of this switching pocedue. Ou switching algoithm shows that the sum of the weights of the shifted im hook tableaux of a given shape and content does not depend on the ode of the content if content pats ae all odd. This gives us the combinatoial poof fo the invaiance of spin chaactes of Sn. See [8] fo detail.
432 Jaejin Lee is said to be a shifted second tail cicled -im hook tableau if is a shifted second tail cicled im hook tableau whose enties include and ae fom the set {1, 2,..., m, }, whee 1 < < fo some intege. We intoduce the symbol to make it clea that no established ode elationship govens. We say that is coveed by (denoted by ) if is the next intege lage than in. Fom now on, unless we explicitly specify to the contay, we assume is a shifted second tail cicled -im hook tableau of shape λ and contents all odd and in. The cicling of the second tail is necessay to compensate fo the weight of 2 on double im hooks. If κ κ is disconnected in, we call and disconnected. We say that and is a single ( esp.,double) im hook union if κ κ is a single (esp., double) im hook. If κ κ is neithe disconnected no any im hook union, we call and ovelapping. We define an assignment X( ) that sends into anothe shifted second tail cicled -im hook tableau ˆ of shape λ as follows: 1. If and ae disconnected in, then X( ) = ˆ =, but with. 2. If and is a single im hook union, then X( ) moves all of the symbols at the head of τ = κ κ to the tail of τ, and vice vesa. The numbe of s and s is peseved. In this case, eithe in ˆ o in ˆ. Figue 3.1 gives us an example fo case 2 with in ˆ and Figue 3.2 shows case 2 with in ˆ. X( )( ) X( )( ) Figue 3.1 Figue 3.2 3. If and is a double im hook union, let τ = κ κ. Recall that we can wite τ as follows: τ = β 1 β 2 = γ 1 γ 2 = A α 1.
The jeu de taquin on the shifted im hook tableaux 433 Let a = κ, b = κ and c = β 1 = γ 1. Then we have β 2 = γ 2 = a + b c, α 1 = 2c a b and A = 2(a + b c). We say we fill τ fom β 1 if the wod with a s followed by b s is inseted in τ, stating at the head of β 1, unning down β 1 to the diagonal, then up β 2. Similaly, define filling τ fom β 2, fom γ 1 and fom γ 2. It is not had to veify the following two lemmas. Fo examples, see Figue 3.3 and Figue 3.4. Lemma 3.1. If a, b A /2 then thee ae exactly two shifted skew im hook tableaux of shape τ with a s and b s. One of these (say T 1 ) fills τ fom β 1 o fom γ 1. The othe (say T 2 ) fills τ fom β 2 o fom γ 2. If < in T 1 and T 2 o if < in T 1 and T 2, then w(t 1 ) = w(t 2 ). Othewise, w(t 1 ) = w(t 2 ). Lemma 3.2. If a = A /2 (esp., b = A /2), then thee ae exactly thee shifted skew im hook tableaux of shape τ with a s and b s. In one of these (say T 4 ), β 2 will contain the s (esp., s). In the second (say T 5 ), γ 2 will contain the s (esp., s). The thid (say T 6 ) fills τ fom β 1 o fom γ 1 (esp., fom β 2 o fom γ 2 ). Also, w(t 4 ) = w(t 5 ) and if < in T 6 then w(t 6 ) = w(t 4 ) w(t 5 ) (esp., w(t 6 ) = w(t 5 ) w(t 4 )) while if < in T 6 then w(t 6 ) = w(t 5 ) w(t 4 ) (esp., w(t 6 ) = w(t 4 ) w(t 5 )). (a) T 1 T 2 (b) T 1 T 2 Figue 3.3
434 Jaejin Lee (a) T 4 (b) T 4 T 5 T 5 Figue 3.4 T 6 T 6 We now descibe an assignment X( ) when and is a double im hook union in. Suppose fist a, b A /2. If contains T 1, then X( ) = ˆ contains T 2, and vice vesa. See Figue 3.5. (a) (b) X( )( ) X( )( ) Figue 3.5 Suppose now a = A /2 o b = A /2. Say a = A /2. Since b = c and in, cannot contain T 4. If contains T 5, then ˆ contains T 6 with no cicle on the second tail of τ; if contains T 6 with no cicle on the second tail of τ, then ˆ contains T 5 ; if contains T 6 with a cicle on the second tail of τ, then ˆ contains T 4. See Figue 3.6. (a) (b) (c) X( )( ) X( )( ) X( )( )
The jeu de taquin on the shifted im hook tableaux 435 Figue 3.6 4. If and is ovelapping, then X( ) exchanges and along diagonals of. See Figue 3.7. X( )( ) Figue 3.7 Poposition 3.3. We have in ˆ = X( ) if and only if w (κ P2 )w (κ P2 ) = w (κ ˆP2 )w (κ ˆP2 ), and in ˆ = X( ) if and only if w (κ P2 )w (κ P2 ) = w (κ ˆP2 )w (κ ˆP2 ). Poof. It is easy to veify the above statements with a case-by-case agument. Fom Poposition 3.3 we have the following theoem: Theoem 3.4. Let λ be a patition with all distinct pats and ρ OP n and ρ be any eodeing of ρ. Then w ( ) = w (), whee the left-hand sum is ove all shifted second cicled im hook tableaux of shape λ and content ρ, and the ight-hand sum is ove all shifted second cicled im hook tableaux P 2 of shape λ and content ρ. Poof. If ρ and ρ diffe by an adjacent tansposition, X defined above establishes this identity. The theoem follows because any eodeing can be witten as a sequence of adjacent tanspositions. The signed bijection in the geneal case is given by the involution pinciple of Gasia and Milne [2]. See [9] fo details. Theoem 3.4 and Poposition 2.1 imply the following coollaies: P 2
436 Jaejin Lee Coollay 3.5. Let λ DP n. Let ρ have all odd pats and ρ be any eodeing of ρ. Then w(p ) = w(p ), whee the left-hand sum is ove all shifted im hook tableaux P of shape λ and content ρ, and the ight-hand sum is ove all shifted im hook tableaux P of shape λ and content ρ. 4. The jeu de taquin on the shifted im hook tableau Related in seveal ways to the Schensted coespondence is the jeu de taquin of Schützenbege[10]. It is an algoithm defined on column stict tableaux. An essentially simila pocedue of switching values is descibed by Bende and Knuth[1]. In this section we will descibe the jeu de taquin on the shifted im hook tableau using the shifted im hook switching pocedue descibed in the pevious section. We need some assumption and notation in ode to define the shifted im hook jeu de taquin fom the opeato X defined in Section 3. Let k be a fixed odd positive intege. We now assume that all shifted im hooks ae of length k and all shifted im hook tableaux(and shifted -im hook tableaux) ae shifted k-im hook tableaux. If is a shifted second tail cicled im hook tableau containing a shifted im hook of s s, then Change(s, )( ) is the shifted second tail cicled -im hook tableau obtained by eplacing evey s in with. The obvious convention s 1 < < s + 1 is also assumed. If is a shifted second tail cicled im hook tableau whose lagest value is m, then Ease(m)( ) is the shifted second tail cicled im hook tableau with the m s eased. Note that can be a shited second tail cicled -im hook tableau with being the lagest value. Then Ease( )( ) eases the s in. Finally we come to the shifted im hook analog of the Schützenbege evacuation pocedue. Suppose is a shifted second tail cicled im hook tableau which contains a shifted im hook of s s and thee ae l values in lage than s. Let E s ( ) = Ease( ) X l ( ) Change(s, )( ). The opeato E s essentially evacuates s fom by easing s and successively sliding lage valued shifted im hooks into the vacated positions. When the vacated positions each the oute im, the sliding stops.
The jeu de taquin on the shifted im hook tableaux 437 Figue 4.1 gives an example of E s. = 1 1 1 1 2 2 4 4 4 1 2 2 2 3 4 3 3 3 3 4 5 5 5 5 5 E 1 ( ) = 2 2 2 2 3 4 4 4 4 2 3 3 3 4 5 3 5 5 5 5 Figue 4.1 We conclude this section by descibing the shifted im hook analog of the Schützenbege evacuation tableau. Suppose is a shifted second tail cicled im hook tableau with enties 1 s 1 < s 2 < < s l m. Since E s ( ) diffes in shape fom by a shifted im hook outside the shape of E s ( ), we can constuct a shifted second tail cicled im hook tableau by successively evacuating s 1, s 2,, s l fom ; and, afte each evacuation, inseting m + 1 s 1, m + 1 s 2,, m + 1 s l espectively into these shited im hooks. This new tableau has the same shape as and is called ( ). If l = m so that the enties of ae {1, 2,..., m}, then the enties of ( ) ae also {1, 2,..., m}. Moe pecisely, we can define ( ) inductively as follows. Suppose m is given and is a shited second tail cicled im hook tableau whose content is {s 1, s 2,..., s l } with 1 s 1 < s 2 < < s l m. We define (P2 ) as the shifted im hook tableau with the same shape as, and with content {m + 1 s l,..., m + 1 s 2, m + 1 s 1 }, such that E m+1 s1 ( ) = (E s1 ( )). Note that since m + 1 s 1 is the lagest enty of ( ), in this case E m+1 s1 = Ease(m + 1 s 1 ). In Figue 4.2 we give an example of a pai and ( ). = 1 1 1 2 2 4 4 4 4 1 1 2 3 4 5 2 2 3 5 5 3 3 3 5 5 1 1 1 1 1 4 5 5 5 2 2 2 3 4 5 2 2 3 4 5 3 3 3 4 4 (P2 ) = Figue 4.2
438 Jaejin Lee In subsequent pape we will show that all connections between the Schensted coespondence fo shifted tableaux and shifted jeu de taquin have shifted im hook vesions. Refeences [1] E. A. Bende and D. E. Knuth, Enumeation of plane patitions, J. Combin. Theoy (A) 13 (1972), 40 54. [2] A. Gasia and S. Milne, A Roges-Ramanujan bijection, J. Combin. Theoy (A) 31 (1981), 289 339. [3] C. Geene, An extension of Schensted s theoem, Adv. in Math., 14 (1974), 254 265. [4] D. E. Knuth, Pemutations, matices and genealized Young tableaux, Pacific J. Math., 34 (1970), 709 727. [5] G. de B. Robinson, On the epesentations of the symmetic goup, Ame. J. Math., 60, (1938), 745 760. [6] B. E. Sagan, Shifted tableaux, Schu Q-functions and a conjectue of R. Stanley, J. Combin. Theoy Se. A 45 (1987), 62 103. [7] C. Schensted, Longest inceasing and deceasing subsequences, Canad. J. Math., 13 (1961), 179 191. [8] J. R. Stembidge, Shifted tableaux and pojective epesentations of symmetic goups, Advances in Math., 74 (1989), 87 134. [9] D. W. Stanton and D. E. White, A Schensted algoithm fo im hook tableaux, J. Combin. Theoy Se. A 40 (1985), 211 247. [10] M. P. Schützenbege, Quelques emaques su une constuction de Schensted, Math., Scand. 12 (1963), 117 128. [11] G. Viennot, Une fome géométique de la coespondance de Robinson- Schensted, in Combiatoie et Repésentation du Goupe Symétique, D. Foata ed., Lectue Notes in Math., Vol. 579, Spinge-Velag, New Yok, NY, 1977, 29 58. Jaejin Lee Depatment of Mathematics Hallym Univesity Chunchon, Koea 200-702 E-mail: jjlee@hallym.ac.k