math.co/ v2 28 Aug 1998

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1 ELEMENTAR PROOFS OF IDENTITIES FOR SCHUR FUNCTIONS AND PLANE PARTITIONS Davd M. Bressoud math.co/ v2 28 Aug 998 August 24, 998 To George Andrews on the occason of hs 60th brthday Abstract. We use elementary methods to prove product formulas for sums of restrcted classes of Schur functons. These mply nown denttes for the generatng functon for symmetrc plane parttons wth even column heght and for the generatng functon for symmetrc plane parttons wth an even number of angles at each level.. Introducton By a plane partton, we mean a nte set, P, of lattce ponts wth postve nteger coecents, f( j )g N 3, wth the property that f (r s t) 2 P and r j s t, then ( j ) must also be n P. A plane partton s symmetrc f ( j ) 2 P f and only f (j ) 2 P. The heght of stac ( j) s the largest value of for whch there exsts a pont ( j ) n the plane partton. A plane partton s column strct f the heght of stac ( j) s strctly less than the heght of stac (? j) whenever 2 and ( j ) s n the plane partton. Symmetrc plane parttons were studed by P. A. MacMahon [2] who conjectured n 898 that the generatng functon for symmetrc plane parttons wth j n and m s n? q m+2?? q 2? <jn? q 2(m++j?)? q 2(+j?) Ths was proven ndependently by Andrews [] and Macdonald []. As shown by Andrews [2], ths s equvalent to the Bender-Knuth conjecture [3], that the generatng functon for column strct plane parttons wth n, m s? q m++j?? q +j? jn Both of these generatng functons are consequences of the followng theorem of Macdonald [], the rst when we set x q 2n?2+ and the second when we set x q n+?. Copyrght to ths wor s retaned by the author. Permsson s granted for the noncommercal reproducton of the complete wor for educatonal or research purposes, and for the use of gures, tables and short quotes from ths wor n other boos or journals, provded a full bblographc ctaton s gven to the orgnal source of the materal. Typeset by AMS-TE

2 2 DAVID M. BRESSOUD Theorem I (Macdonald). For postve ntegers m and n, (.) fm n g s (x x n ) det(x j?? x m+2n?j ) Q Q n (? x ) <jn (x x j? )(x? x j ) The sum s over all parttons nto at most n parts, each of whch s less than or equal to m. I gave an elementary proof of Macdonald's dentty n [4]. Desarmenen [7] and Stembrdge [5] found a smlar theorem where the sum on the left s over parttons nto even parts. Desarmenen [8] has also found the generalzaton n whch any number of odd parts are speced. Theorem II (Desarmenen-Stembrdge). For postve even nteger m and postve nteger n, (.2) fmn g even s (x x n ) det(x j?? x m+2n+?j ) Q n (? )Q x2 <jn (x x j? )(x? x j ) Ths theorem has two corollares that were found by Desarmenen and Stembrdge and, ndependently, Proctor [4]. The q case was rst dscovered by DeSante-Catherne and Vennot [6]. The generatng functon for symmetrc plane parttons wth j n and m where m s even and every stac has even heght s gven by n? q m+2? q 2 <jn? q 2(m++j)? q 2(+j) The generatng functon for column strct plane parttons wth n, j m (m even), and all rows of even length s n? q m+2? q 2 <jn? q m++j? q +j Oada [3] has proven the followng companon usng hs mnor summaton formula. Krattenthaler [0] has used the specal orthogonal tableaux of Lashmba, Musl, and Seshadr to generalze ths result to one n whch the number of columns of odd length s speced. Theorem III (Oada). For postve nteger m and postve even nteger n, (.3) s (x x n ) det(x j?? x m+2n??j ) + det(x Q j? 2 <jn (x x j? )(x? x j ) fmn g 0 even + x m+2n??j ) where 0 s the partton conjugate to. In other words the sum on the left s over parttons wth even column lengths. Ths has the followng corollary when x q 2n?2+.

3 ELEMENTAR PROOFS OF IDENTITIES FOR SCHUR FUNCTIONS AND PLANE PARTITIONS3 Corollary. The generatng functon for symmetrc plane parttons, j n (n even) and m, such that for each there are an even number of lattce ponts of the form ( ) s gven by 2 n? 0 (? q m+2 ) + n? 0 ( + q m+2 )! <jn? q 2(m++j?2)? q 2(+j?) The generatng functons that are derved from Theorems I, II, and III have partcularly nce formulatons. We dene B(n n m) f( j ) j j n mg and B(n n m)s 2 to be the set of orbts of B(n n m) under transposton of the rst two coordnates. For 2 B(n n m)s 2, we dene Ht() + j +? 2 where ( j ) s any one element of. An orbt countng generatng functon s the sum n whch each plane partton s weghted by q to the number of orbts. The generatng functon for symmetrc plane parttons n B(n n m) s gven by jj(+ht())? q? q jj Ht() 2B(nnm)S 2 The orbt countng generatng functon for symmetrc plane parttons n B(n n m) s gven by? q +Ht( )? q Ht() 2B(nnm)S 2 The generatng functon for symmetrc plane parttons wth even stac heght n B(n n m) (m even) s gven by jj(2+ht())? q? q jj(+ht()) 2B(nnm)S 2 The orbt countng generatng functon for symmetrc plane parttons wth even stac heght n B(n n m) (m even) s gven by 2+Ht( )? q? q +Ht() 2B(nnm)S 2 The generatng functon for symmetrc plane parttons n B(n n m) (n even) such that for each, m, there are an even number of ponts of the form ( ) s gven by jj(+ht())? q? q jj q Ht() Ht() 2B(nnm?)S 2 2S Sf(m) j ng There s a formula gven by Krattenthaler (equaton(7.5) n [0]) for the correspondng orbt countng generatng functon. It s not as readly stated n terms of orbts. In secton 2, we shall warm up to the proof of Theorems II and III wth a general result that ncludes the lmtng cases of Theorems I, II, and III. It was rst proved by Ishawa and Waayama [9] usng Oada's mnor-summaton formula of Pfaans.

4 4 DAVID M. BRESSOUD Theorem IV (Ishawa and Waayama). For any postve nteger n, we have that (.4) f (t v)s (x x n ) n (? tx )(? vx )? x x j <jn where we let a j be the number of columns of length j n (equvalently, the number of parts of sze j n 0 ) and We note that f (t v) j odd v aj +? t aj+ v? t j even aj +? (tv)? tv f(0 ) 0 f any aj s odd f(?) otherwse 0 f any aj s postve for any odd j f(0 0) otherwse Theorem IV mples the followng Lttlewood formulas ([], examples 4 and 5 n secton I.5) (.5) (.6) (.7) even 0 even s (x x n ) s (x x n ) s (x x n ) n n? x? x 2? x x j <jn? x x j <jn? x x j <jn In secton 3, we shall gve the proof of Theorem III as well as a new proof of Theorem II. Secton 4 wll show the dervaton of the generatng functon for symmetrc plane parttons wth an even number of lattce ponts of the form ( ) for each. Wth the exceptons of Lemmas and 2, the results presented n ths paper are not new. The proofs, however, are consderably smpler than those that have been gven before. 2. Proof of Theorem IV Lemma. For any postve nteger n we have that (2.) x x n n x? (? tx )(? vx ) n 6? x x x? x ( (? tx x n )(? vx x n ) f n s odd (? x x n )(? tvx x n ) f n s even

5 ELEMENTAR PROOFS OF IDENTITIES FOR SCHUR FUNCTIONS AND PLANE PARTITIONS5 Proof Ths lemma s correct for n. We assume that t s correct wth n? varables Q and proceed by nducton. If we multply both sdes of equaton (2.) by <jn (x? x j ), each sde s an alternatng polynomal n x x n. It follows that both sdes of equaton (2.) are symmetrc polynomals that are quadratc n each of the varables x through x n. We only need to show that they agree at three values of x. Both polynomals are when x 0. When x t?, the polynomal on the left s equal to (2.2) t? x 2 x n n 2 x? (? tx )(? vx )? t? x t?? x n 2 6? x x x? x n x 2 x n x? (?? x x t? x )(? vx ) x 2 2? x 6 ( (? x2 x n )(? t? vx 2 x n ) f n s odd (? t? x 2 x n )(? vx 2 x n ) f n s even n? t? x t?? x Smlarly, the two polynomals agree at x v?. Lemma 2. For even postve nteger n we have that (2.3) (x x n ) 2 n n l l6 x?2 x? l n 6? x x x? x n 6l? x x l x? x l? x x n Proof Ths follows from lemma wth t v 0, summng rst over l and then over. Proof of Theorem IV When n, the left sde of equaton (.4) s v +? t + x v? t 0 (? vx)(? tx) We proceed by nducton and assume that the equaton s vald for n? varables. We rewrte equaton (.4) as (2.3) (?) I() f (t v) n x ()+n?() (? tx )(? vx ) (? x x j ) (x? x j ) <jn <jn where I() s the nverson number of. We shall prove that the left sde s equal to the Vandermonde determnant. We tae the double summaton and rst sum over the possble values of n and? (n). We let be the restrcton of to f ngnfg. If we subtract n from each of the parts n, we are left wth a partton,, nto at most n? parts.

6 6 DAVID M. BRESSOUD We have that f (t v) c n f (t v) where c n s (v n+? t n+ )(v? t) f n s odd, (? (vt) n+ )(? vt) f n s even. The left sde of equaton (2.3) s equal to n n0 (?) n? x? (? tx )(? vx )c n (x x n ) n+ (?) I() f (t v) n 6 n 6 x ()+n??() (? tx )(? vx ) We use our nducton hypothess to rewrte ths as <jn (x? x j ) n n0 By Lemma, the double sum s equal to. (? x x ) <jn j6 x? (? tx )(? vx )c n (x x n ) n+ 3. Proof of Theorems II and III (? x x j ) n 6? x x x? x The proofs of Theorems II and III are smlar n structure to the proof of Theorem IV, just more complcated n detal. Proof of Theorem II We verfy that ths theorem s correct for n and proceed by nducton on the number of varables. We shall prove ths theorem n the form (3.) fmn g even det(x j+n?j ) n (? x 2 ) <jn (x x j? ) (?) I()+jSj 2S x m+2n+?() 2S x ()? where m s an even nteger. As n the proof of Theorem IV, we expand the left sde as a sum over parttons,, and permutatons,. We then sum separately over n whch must now be even, n 2t, and over? (n), leavng, the partton obtaned from when n s subtracted from each part, and, the restrcton of to f ngnfg. We then apply our nducton hypothess. The left sde of equaton (3.) becomes m2 n t0 m2 (?) n? x? (? x2 )(x x n ) 2t+ (?) I() n 6 t0 n? n 6 x ()+n??() (? x 2 ) (?) n? x? (? x2 )(x x n ) 2t+ nfg(?) I()+jSj 2S (x x? ) <jn j6 n 6 (x x j? ) (x x? ) x m?2t+2(n?)+?() 2S x ()?

7 ELEMENTAR PROOFS OF IDENTITIES FOR SCHUR FUNCTIONS AND PLANE PARTITIONS7 where S n? s the set of { mappngs from f ngnfg to f n? g and S s the complement of S n f ngnfg. We smplfy ths summaton and then sum over 2 S n? and over t. The left sde of equaton (3.) s equal to m2 n t0? nfg 2S n x m+2n?() 2S nfg 2S x 2S x ()? (?) n?+i()+jsj x? (? x2 ) 2S Q (?) n?+jsj x? (? x2 )? Q 2S xm+2? 2S x2 x m+2n? (?) ( n? 2 ) <jn j6 (x? x j j ) x 2t+ n 6 n 6 (x x? ) (x x? ) where s? f 2 S and + f 2 S. We reverse the order of summaton so that we rst sum over all proper subsets of f ng and then over all 2 S. For each 2 S, we rewrte x x? as?x (x equaton (3.) has become? x (?) ( n 2 ) ) f <, and rewrte t as x (x 2S (?) jsj 2S x x? (? x2 ) 2S 2S 6 x m+2n? x? Q 2S xm+2? Q 2S x2? x x x? x ) f >. The left sde of (x <jn? x j j ) Q By Lemma, the second lne s equal to? 2S x2 whch cancels wth the factor n the denomnator. We now expand the Vandermonde product. The left sde of equaton (3.) s equal to? We use the fact that (?) I()+jSj 2S (?) I()+jSj 2S (?) I()+jSj 2S x m+2n+?() x m+2n+?() 2S x m+2n+?() 2S det x ()? 2S x m+()+ x m+j+ x m+()+? x m+2n+?j 0 to replace? (?) I()+jSj 2S x m+2n+?() 2S x m+()+

8 8 DAVID M. BRESSOUD by (?) I()+n n x m+2n+?() Ths puts the left sde of equaton (3.) n the desred form. Proof of Theorem III We begn by rewrtng the dentty to be proven as (3.2) fmn g 0 even (?) I() n x ()+n?() <jn (?) I() 2S (x x j? ) x m+2n??() 2S x ()? We agan proceed by nducton. For ths theorem, we need to dentfy both? (n) and? (n? ). We form by subtractng n from each part. Snce each column has even length, has at most n? 2 parts. The left sde of equaton (3.2) s equal to m n0 <ln (x x l? ) (?) n?+n?l (x?2 n 6l (?) I() n 6l x? l? x? x?2 l (x x? )(x x l? ) x ()+n?2?() <jn j6l )(x x n ) n+2 (x x j? ) We apply our nducton hypothess to the nner sum and then sum over n and 2 S n?2, the set of { mappngs from f ngnf lg to f n? 2g. The left sde of equaton (3.2) becomes <ln (x x l? ) nflg x 2 2S 2S n 6l (?) +l (x?2 x? l? x? (x x? )(x x l? ) x m+2n?4 (?) ( n?2 2 ) <jn j6l x?2 l (x Agan we have? f 2 S and + f 2 S. For 2 S, we rewrte (x x? )(x x l? ) as x 2 (x Q )? Q 2S xm+? 2S x? x j j )? x? x l l ). We then )(x nterchange the sum on S, proper subsets of f ng wth even cardnalty, and the sum on and l whch now must le n the complement of S. The left sde of

9 ELEMENTAR PROOFS OF IDENTITIES FOR SCHUR FUNCTIONS AND PLANE PARTITIONS9 equaton (3.2) s equal to (?) ( n 2 ) 2S Sfng x 2 <ln l2s 2S x m+2n?2? Q 2S xm+? Q 2S x (x?2 x? l? x? x?2 l ) x x l? x? x l The second lne of ths expresson s equal to? x x x 2 2S 2S l2s l6 x?2 x? l 2S 6 x? x (x <jn 2S 6l 2S 6l? x j j ) (? x x )(? x x ) (x? x )(x? x l )? x x l x? x l Q By Lemma 2, ths s equal to? 2S x whch cancels wth the factor n the denomnator. As n the proof of Theorem II, we replace the Vandermonde product by the sum over permutatons. The left sde of equaton (3.2) becomes? We now observe that (?) I() 2S (?) I() 2S (?) I() 2S x m+2n?()? 2S x m+2n?()? x m+2n?()? 2S x ()? 2S x m+() x m+() 0 Ths s true because f we nterchange the nverse mages of n and n? and change whether or not each nverse mage s n S, then we change the sgn of the nverson number but do not change the monomal. As a result, we have that? (?) I() 2S x m+2n?()? 2S x m+() (?) I() The left sde of equaton (3.2) s equal to the desred sum. 4. Consequence and Queston n x m+2n?()? If we set x q 2n?2+ n Theorem III, the left sde of equaton (.3) becomes the generatng functon for symmetrc plane parttons wth j n, m, such that for each there are an even number of lattce ponts of the form ( ). The rght sde of equaton (.3) becomes n q (2n?2+)(m+2n?2)2 <jn (?) I() n (q 2n?2+? q 2n?2j+ )? (q 4n?2?2j+? )? q (2n+m?2())(2?2n?)2

10 0 DAVID M. BRESSOUD where? f 2 S and + f 2 S. We combne the B n form of the Weyl denomnator formula, (?) I()+jSj n x (2?2n?)2 () n x (?2n)2 (? x ) <jn and the dentty obtaned when each x s replace by?x (?) I() n x (2?2n?)2 () to derve the result that we need (4.) 2 n x (?2n)2 (?) I() n n n x (?2n)2 ( + x ) x (2?2n?)2 () (? x ) + n ( + x ) <jn! <jn The corollary now follows drectly. It would be of nterest to nd the analogous formula for fm n g f (t v)s (x x n ) (x? x j )(x x j? ) (x? x j )(x x j? ) (x? x j )(x x j? ) although the form of t wll certanly be much more complcated than anythng presented here. References. George Andrews, Plane parttons (I) the MacMahon conjecture, Studes n Foundatons and Combnatorcs, Advances n Mathematcs Supplementary Studes (978), 3{50. 2., Plane parttons (II) the equvalence of Bender-Knuth and MacMahon conjectures, Pacc J. Math. 72 (977), 283{ E. A. Bender and D. Knuth, Enumeraton of plane parttons, J. Combnatoral Th. 3 (972), 40{ Davd M. Bressoud, Elementary proof of MacMahon's conjecture, J. Algebrac Comb. 7 (998), no. 3, 253{ , Proofs and Conrmatons the Story of Alternatng Sgn Matrx Conjecture, Cambrdge Unversty Press and the MAA, expected Myram DeSante-Catherne and Gerard. Vennot, Enumeraton of certan oung tableaux wth bounded heghts, Combnatore enumeratve (G. Labelle and P. Leroux, eds.), Lecture Notes n Mathematcs, vol. 234, Sprnger-Verlag, 986, pp. 58{ Jacques Desarmenen, La d monstraton des dentts de Gordon et MacMahon et de deux denttes nouvelles, Strasbourg, Publ. I.R.M.A., Actes du 5 e Semnare Lotharngen de Combnatore 340/S-5 (987), 39{49.

11 ELEMENTAR PROOFS OF IDENTITIES FOR SCHUR FUNCTIONS AND PLANE PARTITIONS 8., Une generalsaton des formules de Gordon et de MacMahon, C. R. Acad. Sc. Pars Seres I, Math. 309 (989), no. 6, 269{ Masao Ishawa and Masato Waayama, Applcatons of mnor-summaton formula II, Pfaf- ans and Schur polynomals, preprnt. 0. C. Krattenthaler, Identtes for classcal group characters of nearly rectangular shape, J. Algebra (to appear).. I. G. Macdonald, Symmetrc Functons and Hall Polynomals, second edton, Oxford Unversty Press, P. A. MacMahon, Parttons of numbers whose graphs possess symmetry, Trans. Cambrdge Phl. Soc. 7 (898{99), 49{ Soch Oada, Applcatons of mnor summaton formulas to rectangular-shaped representatons of classcal groups, J. Algebra 205 (998), 337{ R. A. Proctor, New symmetrc plane partton denttes from nvarant theory wor of De Concn and Proces, European J. Combn. (990), no. 3, 289{ John R. Stembrdge, Hall-Lttlewood functons, plane parttons, and the Rogers-Ramanujan denttes, Trans. AMS 39 (990), no. 2, 469{498. Dept. of Mathematcs & Computer Scence, Macalester College, St. Paul, MN 5505, USA E-mal address bressoud@macalester.edu