the electronic journal of combinatorics 4 (997), #R22 2 x. Introuction an the main results Asymptotic calculations are applie to stuy the egrees of ce

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1 Asymptotics of oung Diagrams an Hook Numbers Amitai Regev Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot 7600, Israel an Department of Mathematics The Pennsylvania State University University Park, PA 6802, U.S.A. Anatoly Vershik y St. Petersburg branch of the Mathematics Institute of the Russian Acaemy of Science Fontanka 27 St. Petersburg, 90 Russia an The Institute for Avance Stuies of the Hebrew University Givat Ram Jerusalem, Israel Submitte: August 22, 997; Accepte: September 2, 997 Abstract: Asymptotic calculations are applie to stuy the egrees of certain sequences of characters of symmetric groups. Starting with a given partition, we euce several skew iagrams which are relate to. To each such skew iagram there correspons the prouct of its hook numbers. By asymptotic methos we obtain some unexpecte arithmetic properties between these proucts. The authors o not know "nite", nonasymptotic proofs of these results. The problem appeare in the stuy of the hook formula for various kins of oung iagrams. The proofs are base on properties of shifte Schur functions, ue to Okounkov an Olshanski. The theory of these functions arose from the asymptotic theory of Vershik an Kerov of the representations of the symmetric groups. Work partially supporte by N.S.F. Grant No.DMS y Partially supporte by Grant INTAS an Russian Fun

2 the electronic journal of combinatorics 4 (997), #R22 2 x. Introuction an the main results Asymptotic calculations are applie to stuy the egrees of certain sequences of characters of symmetric groups S n ; n!. We obtain some unexpecte arithmetic properties of the set of the hook numbers for some special families of (xe) skew-oung iagrams (Theorem.2). The problem appeare in the stuy of the hook formula for various kins of oung iagrams. The proof of.2 is base on the properties of shifte Schur functions [Ok.Ol] which appeare in the asymptotic theory of the representation theory of the symmetric groups in [Ver.Ker]. The authors o not know a \nite" proof of the theorem. Given a partition, we escribe in. - a construction of certain skew iagrams which are erive from : these are S(); SR(); SR( 0 ); R an D below. Next, one lls these skew iagrams with their corresponing hook numbers [Mac, page 0]. Theorem.2, which is the main result here, gives some ivisibility properties of the proucts of these hook numbers. We remark again that even though the statement of theorem.2 has nothing to o with asymptotics, its proof oes use asymptotic methos. It shoul be interesting to n an \asymptotic free" proof of theorem.2. We start with.: A Construction: Given a partition ( iagram), let D enote the ouble reection of. For example, if (4; 2; ) then D x x x x x x x an D x x x : x x x x Recall that 0 `() is the number of nonzero parts of. Complete D to the 0 rectangle R(), then raw D on top an on the left of R. Finally, erase the rst D. Denote the resulting skew iagram by S(). For example, with (4; 2; ) we get { 2 {

3 the electronic journal of combinatorics 4 (997), #R22 3 S(4; 2; ) A A 2 & x x x x x x x & x x x x x x x x x x x x - A We subivie S() into the three areas A; A an A 2 : A R? D ; A is the D on the left of R an A 2 is the D on top of R. Denote SR() A [ A, the \shifte rectangle". Clearly, ja [ A j ja [ A 2 j jrj; ja j ja 2 j jj, so js()j jrj + jj. Now, ll S(); SR(); R an with their hook numbers. For example, when (4; 2; ) S(4; 2; ) : SR(4; 2; ) : R(4; 2; ) : { 3 {

4 the electronic journal of combinatorics 4 (997), #R22 4 an (4; 2; ) : Thus, for example, x2(4;2;) Note that the hook numbers in SR() are the same as those in the area A [A of S(). As usual, 0 `() is the number of nonzero parts of. Recall that s (x ; x 2 ; ) is the corresponing Schur function, an s (; ; ) is the number 0 of (semi-stanar, i.e. rows weakly an column strictly increasing) tableaux of shape, lle with elements from f; 2; ; 0 g [Mac]. Similarly for s 0 (; ; )..2 Theorem: Let be a partition. With the above construction of S() A [ A [ A 2 an R, we have () x2r!, x2a [A A s (; ; ): 0 In particular, x2a [A ivies x2r. [Note that A[A S(), an for x 2 A [ A; is the corresponing hook number in x 2 S()]. (') Similarly, x2r!, x2a 2 [A A s 0(; ; ): (2) x2r! x2 A : x2s() { 4 {

5 the electronic journal of combinatorics 4 (997), #R22 5 We conjecture that a statement much stronger than.2.2 hols, namely: the two multisets f j x 2 S()g an f j x 2 Rg [ f j x 2 g are equal. Theorem.2. is an obvious consequence of the following \asymptotic" theorem..3. Theorem: Let ( ; ; k ), be a partition. Let n k`, `!, an enote (`) (`k). Then (a) lim `! jj s (; ; ) k k an (b) lim `! jj k x2r( ; 0 ) x2a [A A : Theorem.2.' follows from.2. by conjugation. Theorem.2.2 is a consequence of the following \asymptotic" theorem.4. Theorem: Let be a xe partition. Let `!, 0 m! ; n `m an (`; m) (`m). Then (a) lim `;m! x2 : (b) lim `;m! x2r!, x2s() A : In this note we apply the following main tools: a) The theory of symmetric functions [Mac]. In particular, we apply the hook formula jj! x2 { 5 {

6 the electronic journal of combinatorics 4 (997), #R22 6 an I.3, Example 4, page 45 in [Mac]. b) The Okounkov-Olshanski [Ok.Ol] theory of \shifte symmetric functions". In particular, we apply formula (0.4) of [Ok.Ol]: Let ` k; ` n; k n;, then s () n(n? ) (n? k + ) : Here s (x) is the \shifte Schur function" [Ok.Ol]; one of its key properties is that s (x) s (x)+ lower terms, where s (x) is the orinary Schur function. We remark that the paper [Ok.Ol] was inuence by the work of Vershik an Kerov on the asymptotic theory of the representations of the symmetric groups. See for example [Ver.Ker], in which the characters of the innite symmetric group are foun from limits involving orinary Schur functions. See also the introuction of [Ok.Ol]..2.'). x2. Here we prove theorem.3 which, as note before, implies.2. (an 2.. The proof of theorem.3. (`) (`) s ( (`); ; k (`)) n(n? ) (n? jj + ) ; where n jj k`. Since `! ; n(n? ) (n? jj + ) ' (k`) jj. Also, s () s () + (lower terms in n); hence k s () ' s () s (`; ; `) : { 6 {

7 the electronic journal of combinatorics 4 (997), #R22 7 Recall that for two sequences a n, b n of real numbers, a n ' b n means that lim n! a n bn. Since s (x) is homogeneous of egree jj, The proof now follows easily. s () `jj s (; ; ) : k 2.2. The proof of theorem.3.b: Since is a rectangle, hence, where is the ouble reection of. Denote by ~ D the ouble reection of. Thus ` D i To calculate an by the hook formula, ll (`) an with their respective hook numbers. In both, examine the i th row from the bottom - with their respective hook numbers. Divie into B an B 2 as follows: Notice that B SR() of.. Note also that the hook numbers in B are those in SR(), an they are inepenent of `. Examine the hook numbers in B 2. In the i th row (from bottom), these are + i; + i + ; ; ` + i?? i, consecutive integers. We also ivie (`) into two rectangles: { 7 {

8 the electronic journal of combinatorics 4 (997), #R22 8 ` B 2 B D ` (`) R 2 R Again, the hook numbers in R are inepenent of `, an those in the i th row (from bottom) of R 2 are + i; + i + ; ; ` + i?, again consecutive integers. By the \hook" formula, the left han sie of.3.b is " #," # (`) (n? jj)! n! (`) (`) x2 x2(`) where n k`. Since `!, (n? jj)! n! " # x2(`) x2 (n? jj)! n! ' jj n jj : k` { 8 {

9 the electronic journal of combinatorics 4 (997), #R22 9 Now x2(`) x2 " x2r x2b # " x2r 2 x2b 2 # : Note that the right han sie of.3.b is ( k )jj. We calculate : thus (since `! ). x2r 2 x2b 2 0 i 0 i 0 i [( + i)( + i + ) (` + i? )]; [( + i)( + i + ) (` + i?? i )]; [(` + i? i )(` + i? i + ) (` + i? )] ' `jj ; Hence, lim `! (`) (`) jj k an the proof is complete. x3. Here we prove theorem.4 which, as note before, implies theorem The proof of.4.a: Let (`; m) (`m); `; m!. We show rst that s () ' s (), as follows: By [Ok.Ol.(0.9)], X e r() (` + r? )(` + r? 2) ` ii <<i r m m `rm r (` + r? )(` + r? 2) ` ' : r r! Similarly, e r () ' `rm r r!. { 9 {

10 the electronic journal of combinatorics 4 (997), #R22 0 Let ; be given as in [Ok.Ol.x3]. By [Ok.Ol.(3.8)] it easily follows that for any u an r, ;?u e r() ' e r() ' e r (): Applying the Jacobi Trui formulas for s () (Mac. I, (3.5), page 4] an for s () [Ok.Ol.(3.0)], it clearly follows that s () ' s (). Now in 2., here (`;m) (`;m) s ( (`; m); ; m+k (`; m)) n(n? ) (n? jj + ) where n `m: Here Thus m s ((`; m)) ' s ((`; m)) `jj s (; ; ): (`;m) (`;m) ' jj s (; ; ) n m m jj x2 m + c(x) ; ([Mac, pg. 45, Ex 4]) where c(x) is the content of x 2. Since m! ; m+c(x) ' m for all x 2, an the proof follows The proof of.4.b: Choose `; m large so that (`; m). Let be the ouble reection of (`; m), so (`;m), then calculate by the hook formula. To analyze the hook numbers in, we subivie into the areas A ; ; ; A 4; as shown below: i.e., D is rawn at the bottom-right of the ` m rectangle. We then follow. an construct A 4; S(). Now A ; is the (`? ) (m? 0 ) rectangle, an this etermines A 2; an A 3;. We also split the ` m rectangle (`; m) accoringly: { 0 {

11 the electronic journal of combinatorics 4 (997), #R22 ` : m A ; A 3; A 4; A 2; {z} 0 D A ;(`;m) A 3;(`;m) A 2;(`;m) 0 A 4;(`;m) Since (`; m) ` `m an ` `m? jj, ' (`;m) `m jj x2(`;m) h (`;m)(x) x2 h : (x) Now, h (`;m) (x) h (x) for x 2 A ; A ;(`;m). As in 2.3 x2a 2;(`;m) h (`;m) (x) x2a 2; h (x) { { ' `jj :

12 the electronic journal of combinatorics 4 (997), #R22 2 Similarly (or, by conjugation), x2a 3; h (`;m) (x) x2a 3; h (x) m jj : After cancellations we have ' an the proof is complete. x2a 4; h (`;m) (x) x2a 4; h (x) x2r( ; 0 ) x2s() References [Ok.Ol] Okounkov A. an Olshanski G., Shifte Schur functions, preprint. [Mac] Maconal I.G., Symmetric functions an Hall polynomials, Oxfor University Press, 2n eition 995. [Ver.Ker] Vershik A.M. an Kerov, S.V., Asymptotic Theory of characters of the symmetric group, Funct. Anal. Appl. 5 (98) aresses: regev@wisom.weizmann.ac.il, vershik@pmi.ras.ru { 2 {