On the Number of Rim Hook Tableaux. Sergey Fomin* and. Nathan Lulov. Department of Mathematics. Harvard University

Transcription

1 Zapiski Nauchn. Seminaov POMI, to appea On the Numbe of Rim Hook Tableaux Segey Fomin* Depatment of Mathematics, Massachusetts Institute of Technology Cambidge, MA 0239 Theoy of Algoithms Laboatoy SPIIRAN, Russia and Nathan Lulov Depatment of Mathematics Havad Univesity Cambidge, MA 0238 Abstact A hooklength fomula fo the numbe of im hook tableaux is used to obtain an inequality elating the numbe of im hook tableaux of a given shape to the numbe of standad oung tableaux of the same shape. This povides an uppe bound fo a cetain family of chaactes of the symmetic goup. The analogues fo shifted shapes and ooted tees ae also given.. Intoduction and Main Result Let be a Fees shape of n boxes. Fo a box t 2, the hook H (t) with the vetex t is the set of all boxes of lying below t in the same column o to the ight of t in the same ow, including t itself. The im hook R (t) is the contiguous bode stip of connecting the ightmost and the lowemost boxes of the hook H (t). The hook and the im hook with the vetex t contain the same numbe of boxes called the hooklength h (t) (o just h(t)). Fo the est of the pape a positive intege is xed; we will only conside -im hooks, i.e., im hooks of length. A shape that can be tansfomed to the empty shape by *Patially suppoted by the NSF (DMS ). 99 Mathematics Subject Classication. 05E0, 05A7, 05A20, 20B30. Key wods and phases. oung's lattice, im hook tableau, chaactes of the symmetic goup. Typeset by AMS-TEX

2 2 consecutive emovals of -im hooks is called -decomposable. The poset RH of all - decomposable shapes odeed by im hook emoval is called the im hook lattice. This lattice is isomophic to the catesian powe of the oung lattice RH (cf. [FS92]). Removing an -im hook fom an -decomposable shape coesponds to emoving a box fom one of the shapes elated to unde this isomophism. Let f denote the numbe of satuated chains in RH connecting an -decomposable shape with the empty shape. In othe wods, f is the numbe of standad -im hook tableaux of shape (cf. [SW85]). In paticula, f = f is the numbe of standad oung tableaux of shape. An example of a standad -im hook tableau, = 5, is given below The main esult of the pape uses explicit expessions fo f and f in tems of hooklengths of (the hooklength fomulae) to obtain an inequality elating these numbes to each othe.. Theoem. Let be an -decomposable shape of n = m boxes. Then f m! m (n!) = (f ) Note that, when is xed, this can be ewitten as f Stiling fomula. = O(n 2? 2 (f ) ), using the Let be an ieducible chaacte of the symmetic goup S n. The value of on the conjugacy class [ m ] (i.e., m cycles of length ) can be computed using the Munaghan- Nakayama ule (see, e.g., [JK8]), which in this case yields j ([ m ])j = f Thus Theoem. gives an uppe bound fo the evaluation of an ieducible chaacte on pemutations with equal cycle lengths. In the case = 2 (xed point fee involutions), this becomes j ([2 n 2 ])j = O(n 4 pf ) ; an estimate that plays a cucial ole in the study [L94] by the second autho of an associated andom walk on the symmetic goup. Infomally speaking, [L94] shows that a andom walk on the altenating goup geneated by the set of xed point fee involutions appoaches

3 3 the unifom distibution afte thee steps. The poof of this esult is a combination of Theoem., the \uppe bound lemma" of Diaconis and Shahshahani [DS8], and the X asymptotics lim n! f = 2. `n Note that in the case = 2 Theoem. and Coollay 2.2 (see below) efe to the numbe of standad domino tableaux. This numbe is also known to count standad oung tableaux of a given shape which ae invaiant unde the Schutzenbege involution (see, e.g., [St94]). 2. Quotient Shapes. Poof of the Main Theoem. Unde the canonical isomophism RH =, an -decomposable shape coesponds to an -tuple of shapes ( 0 ; ; ;? ) whee j 0 j + j j + + j? j = m, jj = m. The k ae called quotient shapes. Thee ae seveal equivalent constuctions poducing these shapes, including the bead conguations, o the abacus of [JK8], the faiy sequences [FS92], and the sta diagams [R6, JK8]. Let us descibe the sta constuction. Fo each box x = (i; j) of, we wite the emainde (i? j) modulo into x. To constuct the shape k, take the ows whose ightmost box contains k and the columns with (k + ) modulo at the bottom. The boxes of lying on the intesections of these ows and columns fom the quotient shape k (cf. [JK8], p.85). 2. Example. Let = 3 and = [74 2 2]. Then ( 0 ; ; 2 ) = ([3]; [ 2 ]; []). In the sta constuction, the boxes contibuting to the quotient shapes ae exactly those whose hooklengths ae divisible by. Let t be one of these boxes. It is well-known that h k (t) = h (t) (see [FRT54, R6]). A hooklength fomula fo the numbe of im hook tableaux follows immediately. 2.2 Coollay. Fo an -decomposable shape, f = m! Poof. Combine the last obsevation, the hooklength fomula and the fomula f = m f 0 f f? j 0 jj j j? j 2.3 Theoem. Fo any -decomposable shape, t2 h(t) h(t) t2 h(t) In paticula, in any 2-decomposable shape the poduct of even hooklengths is geate o equal than the poduct of odd hooklengths. To pove this theoem, we will need some well-known facts about the -coes.

4 4 2.4 Denition. A shape without hooks of length is called an -coe shape. 2.5 Lemma (cf. [JK8, p.86]). An -coe shape has no hooklengths divisible by. 2.6 Lemma (cf. [St84, p.7]). Removing an -im hook fom a shape deceases the total numbe of -divisible hooklengths exactly by one. 2.7 Coollay. In a shape of n boxes, the numbe of hooklengths divisible by is less than o equal to n. It is equal to n if and only if is -decomposable. Poof of Theoem 2.3. Dene a function p! Zby? + ; if h(t) 0 mod ; p(t) = ; if h(t) 6 0 mod Dene a patial ode on by (k; l) (i; j) if k i and l j. (This ode is dual to the usual patial ode on ). Then Coollay 2.7 implies X st p(s) 0 fo any t 2 wheeas Theoem 2.3 claims that u2 h(u) p(u) Let be the Mobius function of the poset (; ). Then (t; u) = 8 >< > By Mobius invesion, u h(u) p(u) = u ; if (t; u) = ((k; l); (k; l)) o (t; u) = ((k; l); (k? ; l? ));?; if (t; u) = ((k; l); (k? ; l)) o (t; u) = ((k; l); (k; l? )); 0; othewise. h(u) X t (t; u) X st p(s) = t To pove Theoem 2.3, it suces to show that, fo any t 6= (; ), u h(u) (t;u) ; since we aleady know that X st p(s) 0 (note that, fo t = (; );! X h(u) (t;u) st X p(s) = st s2 by Coollay 2.7). The inequality in question essentially amounts to which follows fom and h(k? ; l? ) h(k; l) h(k? ; l) h(k; l? ) ; u X h(k? ; l? ) + h(k; l) = h(k? ; l) + h(k; l? ) p(s) p(s) = 0, h(k? ; l? ) > h(k? ; l) > h(k; l) ; h(k? ; l? ) > h(k; l? ) > h(k; l) Poof of Theoem.. Combine Coollay 2.2, Theoem 2.3, and the hooklength fomula.

5 5 3. Shifted Shapes In this section, we follow the usual teminology elated to the shifted shapes (see, e.g., [Sa90]). Let be a shifted shape of n boxes. The symmetized shape () = [ T is the disjoint union of and its eection T. The shifted hooklengths h (t) and the shifted im hooks R (t) ae dened as in [M86]. Namely, emove R(t) and then R(t T ) fom (); the emaining shape is the symmetization of n R (t). As befoe, the size of R (t) is h (t). Notice that the multiset fh () (t)g t2() is a disjoint union of two copies of the multiset fh (t)g t2. The poset SRH 0 of -decomposable shifted shapes is a sublattice of the shifted im hook lattice SRH dened in [FS92]; in the constuction of SRH, othe types of im hooks ae also allowed. In this pape, we will only deal with SRH 0. The lattice isomophism SRH 0 = S b? 2 c is implicit in [M86] and [FS92] whee Sis the distibutive lattice of shifted shapes. Let g denote the numbe of satuated chains in SRH connecting an -decomposable shifted shape with the empty shape, i.e., the numbe of standad shifted -im hook tableaux of shape. In paticula, g = g is the numbe of standad shifted oung tableaux of shape. The main esult fo shifted shapes is an analogue of Theoem.. 3. Theoem. Let 2 SRH 0 Then be an -decomposable shifted shape of n = m boxes. g m! m (n!) = (g ) Fo a xed, this can be ewitten as g = O(n 2? 2 (g ) ). To pove this theoem, we will need to ecall the constuction of the quotient shapes in the shifted case [M86]. Let 2 SRH 0. Denote P? q = b 2 c. The quotient shapes ( 0 ; ; ; q ) of ae dened as follows (as befoe, j k j = m). Let ( 0 ; ;? ) be the quotient shapes fo the symmetized shape () (see Sec. 2). It tuns out that (i) i = 0?i, i = ; ; q; (ii) =2 = ; if is even; (iii) 0 is a symmetized shape. Now dene 0 by ( 0 ) = 0 and, fo i = ; ; q, let i = i. With this denition, g = = m! m j 0 jj j j q j i=0 t2 i h i (t) g 0 f f q = m! q!? 2 = implying the following analogue of Coollay 2.2. i=0 h () (t)0mod t2 i h i (t) h () (t)!? A? 2

6 6 3.2 Coollay. Fo an -decomposable shifted shape 2 SRH 0, g = m! h(t) Poof of Theoem 3.. Apply Theoem 2.3 to the symmetized shape () g t m! m! = = h 2 () (t) t m! m! = = h (t) m! m (n!) = (g ) 4. Rooted Tees Anothe class of combinatoial objects aoding a hooklength fomula ae ooted tees [Sa88, p.3]. The numbe of linea extensions of a ooted tee is expessed by a hooklength fomula, the hooks being dened as the subtees of the tee. Let be a ooted tee with n nodes. Fo a node t 2, dene the hook H (t) to be the set of all successos of t, i.e., the subtee of with oot t. The notion of a im hook coincides hee with that of a hook. The hooklength h (t) is the cadinality of H (t). The notions of an -decomposable tee and a standad -im hook labeling of a tee (a tableau) ae exact analogs of the coesponding notions fo the Fees and shifted shapes. Let l be the numbe of standad -labelings fo a tee. In paticula, l = l is the numbe of linea extensions of. 4. Theoem. Let be an -decomposable tee with n = m nodes. Then l m! m (n!) = (l ) The poof is based on the following fact, which is peculia to the case of ooted tees. 4.2 Lemma. Fo an -decomposable ooted tee, a decomposition into a union of -im hooks is unique. Poof. Conside any -hook H of the tee and pove that any decomposition would contain it. Let s be any leaf of H. In any decomposition, the im hook containing s has to coincide with H. Let us dene the quotient tee by contacting all edges within each of the -im hooks of the unique decomposition of. Notice that hooklengths of ae exactly the -divisible hooklengths of divided by ; theefoe applying the hooklength fomula to yields an analogue of Coollay 2.2.

7 7 4.3 Poposition. l = m! In ode to get the desied uppe bound on l, we need to pove an analogue of Theoem Lemma. Fo any -decomposable tee, t2 h(t) h(t) t2 h(t) Poof. Fo t 2, let t 0 be the oot of the im hook containing t in the decomposition of. Then h(t 0 ) h(t), and the lemma follows. Refeences [DS8] [FS92] [FRT54] [JK8] [L94] [M79] [M86] [R6] [Sa90] [Sa9] [St84] [St94] [SW85] P. Diaconis, M. Shahshahani, Geneating a andom pemutation with andom tanspositions. Z. Wahscheinlichkeitstheoie vew. Gebiete, 57 (98), S. Fomin, D. Stanton, Rim hook lattices, Repot No.23 (99/92), Institut Mittag- Lee, 992. J. S. Fame, G. de B. Robinson, R. M. Thall, The hook gaphs of the symmetic goup. Canad. J. of Math. 6 (954), G. James, A. Kebe, The epesentation theoy of the symmetic goup, Encyclopedia of Mathematics and its Applications, Vol. 6, G.-C. Rota, Ed., Addison-Wesley, Reading, Mass., 98. N. Lulov, Geneating a andom pemutation by andom involutions, in pepaation. I. G. Macdonald, Symmetic functions and Hall polynomials, Oxfod Univesity Pess, 979. A. O. Mois, A. K. asseen, Some combinatoial esults involving shifted oung diagams, Math. Poc. Camb. Phil. Soc. 99 (986), G. de B. Robinson, Repesentation theoy of the symmetic goup, Univesity of Toonto Pess, 96. B. E. Sagan, The ubiquitous oung tableaux, Invaiant theoy and oung tableaux, D. Stanton, Ed., Spinge-Velag, 990, B. E. Sagan, The symmetic goup epesentations, combinatoial algoithms and symmetic functions, Wadswoth and Books/Cole, 99. R. P. Stanley, The stable behaviou of some chaactes of SL(n; C ), Lin. and Multilin. Algeba 6 (984), J. Stembidge, Canonical bases and self-evacuating tableaux, pepint. D. W. Stanton, D. E. White, A Schensted algoithm fo im hook tableaux, J. Combin. Theoy, Se. A 40 (985),