A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics, Massachusetts Istitute of Techology Cambridge, MA 0139, USA rsta@mathmitedu Submitted: Jauary 7, 000; Accepted: February 1, 000 Abstract I this paper, we prove that the dimesio of the space spaed by the characters of the symmetric powers of the stadard -dimesioal represetatio of S is asymptotic to / This is proved by usig geeratig fuctios to obtai formulas for upper ad lower bouds, both asymptotic to /, for this dimesio I particular, for 7, these characters do ot spa the full space of class fuctios o S Primary AMS subect classificatio: 05E10 Secodary: 05A15, 05A16, 05E05 Notatio Let P deote the umber of uordered partitios of ito positive itegers, ad let φ deote the Euler totiet fuctio Let V be the stadard -dimesioal represetatio of S,sothatV = Ce 1 Ce with σe i =e σi for σ S Let S N V deote the N th symmetric power of V,adletχ N : S Z deote its character Fially, let D deote the dimesio of the space of class fuctios o S spaed by all the χ N, N 0 1 Supported by a NSERC PGS-B fellowship Partially supported by NSF grat DMS-9500714 1
the electroic oural of combiatorics 7 000, #R6 1 Prelimiaries Our aim i this paper is to ivestigate the umbers D It is a fudametal problem of ivariat theory to decompose the character of the symmetric powers of a irreducible represetatio of a fiite group or more geerally a reductive group A special case with a ice theory is the reflectio represetatio of a fiite Coxeter group This is essetially what we are lookig at The defiig represetatio of S cosists of the direct sum of the reflectio represetatio ad the trivial represetatio This trivial summad has o sigificat effect o the theory I this cotext it seems atural to ask: what is the dimesio of the space spaed by the symmetric powers? Moreover, decomposig the symmetric powers of the character of a irreducible represetatio of S is a example of the operatio of ier plethysm [1, Exer 774], so we are also obtaiig some ew iformatio related to this operatio We begi with: Lemma 11 Let λ =λ 1,,λ k be a partitio of which we deote by λ, ad suppose σ S is a λ-cycle The χ N σ is equal to the umber of solutios x 1,,x k i oegative itegers to the equatio λ 1 x 1 + + λ k x k = N Proof Suppose without loss of geerality that σ =1 λ 1 λ 1 +1 λ 1 + λ λ 1 + + λ k 1 +1 Cosider a basis vector e c 1 1 e c of S N V, so that c 1 + + c = N with each c i 0 This vector is fixed by σ if ad oly if c 1 = = c λ1, c λ1 +1 = = c λ1 +λ ad so o Sice χ N σ equals the umber of basisvectorsfixedbyσ, the lemma follows It seems difficult to work directly with the χ N s; fortuately, it is ot too hard to restate the problem i more cocrete terms Give a partitio λ =λ 1,,λ k of, defie 1 f λ q = 1 1 q λ 1 1 q λ k Next, defie F C[[q]] to be the complex vector space spaed by all of these f λ q s We have: Propositio 1 dim F = D Proof Cosider the table of the characters χ N ; we are iterested i the dimesio of the row-spa of this table Sice the dimesio of the row-spa of a matrix is equal to the dimesio of its colum-spa, we ca equally well study the dimesio of the space spaed by the colums of the table By the preceedig lemma, the N th etry of the colum correspodig to the λ-cycles is equal to the umber of oegative iteger solutios to the equatio λ 1 x 1 + + λ k x k = N Cosequetly, oe easily verifies that f λ q is the geeratig fuctio for the etries of the colum correspodig to the λ-cycles The dimesio of the colum-spa of our table is therefore equal to dim F, ad the propositio is proved
the electroic oural of combiatorics 7 000, #R6 3 Upper Bouds o D Our basic strategy for computig upper bouds for dim F is to write all of the geeratig fuctios f λ q as ratioal fuctios over a commo deomiator; the the dimesio of their spa is bouded above by 1 plus the degree of their umerators For example, oe ca see without much difficulty that 1 q1 q 1 q isthe least commo multiple of the deomiators of the f λ q s Puttig all of the f λ q s over this commo deomiator, their umerators the have degree + 1/, which proves D +1 By modifyig this strategy carefully, it is possible to fid a somewhat better boud Observe that the deomiator of each of our f λ s is up to sig chage a product of cyclotomic polyomials I fact, the power of the th cyclotomic polyomial Φ q dividig the deomiator of f λ q is precisely equal to the umber of λ i s which are divisible by It follows that Φ q divides the deomiator of f λ q at most times, ad the partitios λ for which this upper boud is achieved are precisely the P partitios of which cotai copies of LetS be the collectio of f λ s correspodig to these P partitios Oe sees immediately that the dimesio of the space spaed by the fuctios i S is ust D : i fact, the fuctios i this space are exactly 1/1 q times the fuctios i F Now the power of Φ q i the least commo multiple of the deomiators of all of the f λ q s excludig those i S is oly 1, so the degree of this commo deomiator is oly +1/ φ Therefore, as i the first paragraph of this sectio, the dimesio of the space spaed by all of the f λ s except those i S is at most /+1 φ; sice the dimesio spaed by the fuctios i S is D, we have proved the upper boud D +1 φ+d If it happes that D <φ, the this upper boud is a improvemet o our origial upper boud If we repeat this process, this time simultaeously excludig the sets S for all of the s which gave us a improved upper boud i the above argumet, we fid that we have proved:
the electroic oural of combiatorics 7 000, #R6 4 Propositio 1 D +1 =1 max 0,φ D Fially, we obtai a upper boud for D which does ot deped o other values of D : Corollary Recursively defie U0 = 1 ad U = +1 =1 max 0,φ U The D U Proof We proceed by iductio o Equality certaily holds for =0 Forlarger, the iductive hypothesis shows that D U whe >0, ad so D +1 max 0,φ D =1 +1 max 0,φ U = U =1 Below is a table of values of D adu for 1 34, calculated for 1 3 usig Maple ad for 4 34 usig a Pytho program For cotrast, P ad our first estimate 1 + 1 are provided for 4, but are omitted due to space cosideratios for 5 Note that i the rage 1 34, we have D =U except for =19, 0, 5, 7, 8, 31, whe U D = 1, ad =3, 33, whe U D =, 3 respectively What is the behaviour of D+ +1 =1 max 0,φ D as? Example 3 The first dimesio where D <P is = 7, ad it is easy the to show that D <P for all 7 The differece P 7 D7 = arises from the followig two relatios: 4 1 x 1 x 3 = 3 1 x 3 1 x 4 + 1 1 x 3 1 x
the electroic oural of combiatorics 7 000, #R6 5 1 3 4 5 6 7 8 9 10 11 1 13 14 D 1 3 5 7 11 13 19 3 9 35 45 51 6 U 1 3 5 7 11 13 19 3 9 35 45 51 6 /+1 1 4 7 11 16 9 37 46 56 67 79 9 P 1 3 5 7 11 15 30 4 56 77 101 135 15 16 17 18 19 0 1 3 4 D 69 79 90 106 118 134 146 161 176 195 U 69 79 90 106 119 135 146 161 176 195 /+1 106 11 137 154 17 191 11 3 76 300 P 176 31 97 385 490 67 79 100 155 1575 5 6 7 8 9 30 31 3 33 34 D 1 33 55 77 93 315 337 370 395 41 U 13 33 56 78 93 315 338 37 398 41 Table 1: Values of D, U, /+1,P for small ad 3 1 x 3 1 x 1 x = 1 x 4 1 x + 1 3 1 x 4 1 x 3 The first relatio, for example, says that if χ is a liear combiatio of χ N s, the 4 χ, -cycle = 3 χ3-cycle + χ3,, -cycle Alterately, it tells us that for ay N 0, four times the umber of oegative itegral solutios to x 1 +x + x 3 + x 4 + x 5 = N is equal to three times the umber of such solutios to 3x 1 + x + x 3 + x 4 + x 5 = N plus the umber of such solutios to 3x 1 +x +x 3 = N 3 Lower Bouds o D Let λ =λ 1,,λ k The ratioal fuctio f λ q of equatio 1 ca be writte as f λ q =p λ 1,q,q,, where p λ deotes a power sum symmetric fuctio See [1, Ch 7] for the ecessary backgroud o symmetric fuctios Sice the p λ for λ form a basis for the vector space say over C Λ of all homogeeous symmetric fuctios of degree [1, Cor 77], it follows that if {u λ } λ is ay basis for Λ the D = dim spa C {u λ 1,q,q,:λ }
the electroic oural of combiatorics 7 000, #R6 6 I particular, let u λ = e λ, the elemetary symmetric fuctio idexed by λ Defie dλ = λi i Accordig to [1, Prop 783], we have e λ 1,q,q,= q dλ i 1 q1 q 1 q λ i Sice power series of differet degrees where the degree of a power series is the expoet of its first ozero term are liearly idepedet, we obtai from Propositio 1 the followig result Propositio 31 Let E deote the umber of distict itegers dλ, where λ rages over all partitios of The D E Note We could also use the basis s λ of Schur fuctios istead of e λ, sice by [1, Cor 713] the degree of the power series s λ 1,q,q,isdλ, where λ deotes the cougate partitio to λ Defie G + 1 to be the least positive iteger that caot be writte i the form λi i,whereλ Thus all itegers 1,,,G ca be so represeted, so D E G We ca obtai a relatively tractable lower boud for G, as follows For a positive iteger m, write uiquely m = k1 + k + + kr, 3 where k 1 k k r adk 1,k, are chose successively as large as possible so that k1 k ki m 0 for all 1 i r For istace, 6 = 7 + 3 + + Defie νm =k1 + k + + k r Suppose that νm for all m N The if m N we ca write m = k 1 + + kr so that k1 + + k r Hece if λ = k 1,,k r, 1 P k i where 1 s deotes s parts equal to 1, the λ is a partitio of for which λi i = m It follows that if νm for all m N the G N Hece if we defie H to be the largest iteger N for which νm wheever m N, thewehave established the strig of iequalities D E G H 4 Here is a table of values of these umbers for 1 3 Note that D appears to be close to E + 1 We do t have ay theoretical explaatio of this observatio
the electroic oural of combiatorics 7 000, #R6 7 1 3 4 5 6 7 8 9 10 11 1 13 14 D 1 3 5 7 11 13 19 3 9 35 45 51 6 E 1 3 5 7 9 13 18 1 7 34 39 46 54 G 0 1 1 3 4 4 7 13 13 18 5 3 3 3 H 0 1 1 3 4 4 7 11 13 18 19 19 5 3 15 16 17 18 19 0 1 3 D 69 79 90 106 118 134 146 161 176 E 61 7 83 9 106 118 130 145 16 G 40 49 5 6 73 85 10 11 17 H 40 43 5 6 73 85 89 10 116 Table : Values of D, E, G, H for small Propositio 3 We have νm m +3m 1/4 5 for all m 405 Proof The proof is by iductio o m It ca be checked with a computer that equatio 5 is true for 405 m 50000 Now assume that M>50000 ad that 5 holds for 405 m<mletp = p M be the uique positive iteger satisfyig p p +1 M< Thus p is ust the iteger k 1 of equatio 3 Explicitly we have 1+ 8M +1 p M = By the defiitio of νm wehave νm =p M + ν M pm Itcabecheckedthatthemaximumvalueofνm form<405 is ν404 = 4 Set q M =1+ 8M +1/ Sice M p M pm q M, by the iductio hypothesis we have νm q M + max4, q M +3q 1/4 M It is routie to check that whe M>50000 the right had side is less tha M + 3M 1/4, ad the proof follows
the electroic oural of combiatorics 7 000, #R6 8 Propositio 33 There exists a costat c>0 such that H c3/ for all 1 Proof From the defiitio of H ad Propositio 3 ad the fact that the right-had side of equatio 5 is icreasig, alog with the iquality νm 4 = 405 + 3 405 1/4 for m 404, it follows that H m +3m 1/4 m for m>404 For sufficietly large, we ca evidetly choose m such that = m +3m 1/4,soH m Sice m +3m 1/4 +1>, a applicatio of the quadratic formula agai for sufficietly large shows m 1/4 3+ 9+4 1, from which the result follows without difficulty Sice we have established both upper bouds equatio ad lower bouds equatio 4 ad Propositio 33 for D asymptotic to /, we obtai the followig corollary Corollary 34 There holds the asymptotic formula D 1 Ackowledgemets The first author thaks Mark Dickiso for his help i computig values of D Refereces [1] R Staley, Eumerative Combiatorics, vol, Cambridge Uiversity Press, New York/Cambridge, 1999