Unimodality of generalized Gaussian coefficients.

Transcription

1 Uimodality of geeralized Gaussia coefficiets. Aatol N. Kirillov Steklov Mathematical Istitute, Fotaka 7, St.Petersburg, , Russia Jauary 1991 Abstract A combiatorial proof [ of] the uimodality of the geeralized N -Gaussia coefficiets based o the explicit formula for λ Kostka-Foulkes polyomials is give Let us metio that the proof of the uimodality of the geeralized Gaussia coefficiets based o theoretic-represetatio cosideratios was give by E.B. Dyki [1] (see also [], [10], [11]). Recetly K.O Hara [[6] gave a] costructive proof of the uimodality of the Gaussia coefficiet + k = s (k) (1,, k ), ad D. Zeilberger [1] derived some idetity which may be cosider as a algebraizatio of O Hara s costructio. By + k iductio this idetity immediately implies the uimodality of. Usig the [ observatio ] (see Lemma 1) that the geeralized Gaussia coefficiet λ may be idetified (up to degree ) with the Kostka-Foulkes polyomial () (see Lemma 1), the proof of the uimodality of K λ,µ λ is a simple coseuece of the exact formula for Kostka-Foulkes polyomials cotaied i []. Furthermore the expressio for () ithecaseλ =(k) K λ,µ coicides with idetity (KOH) from [8]. So we obtai a geeralizatio ad a combiatorial proof of (KOH) for arbitrary partitio λ. 1

2 0. Let us recall some well kow facts which will be used later. We base ourselves [9] ad [5]. Let λ =(λ 1 λ ) be a partitio, λ be the sum of its parts λ i, (λ) = i (i 1)λ i ad be the geeralized λ Gaussia coefficiet. Recall that s λ (1,, )= (λ) λ = 1 +c(x) 1 x λ h(x). (1) Here c(x) is the cotet ad h(x) is the hook legth correspodig to the box x λ, [5]. Lemma 1 Let λ be a partitio ad fix a positive iteger. Cosider ew partitios λ =( λ,λ) ad µ =( λ +1 ).The ( 1) λ +(λ) λ = (). () K λ,µ Proof. We use the descriptio of the polyomial λ +(λ) λ as a geeratig fuctio for the stadard Youg tableaux of the shape λ filled with umbers from the iterval [1,,]. Let us deote this set of Youg tableaux by STY (λ, ). The λ +(λ) λ = T. () T STY (λ, ) Here T is the sum of all umbers fillig the boxes of T. For ay tableau T (or diagram λ) let us deote by T [k] (orλ[k]) the part of T (or λ) cosistig of rows startig from the (k + 1)-st oe. Give tableau T STY (λ, ), the cosider tableau T STY ( λ, µ) such that T [1] = T + supp λ[1], ad we fill the first row of T with all remaiig umbers i icreasig order from left to right. Here for ay diagram λ we deote by suppλ the plae partitio of the shape λ ad cotet (1 λ ). It is easy to see that c( T )= T + so we obtai the idetity (). ( +1)( ) λ,

3 Let us cosider a explaatory example. Assume λ =(, 1), =. The λ =(9,, 1), µ =( ). It is easy to see that STY (λ, ) =8. T T T c( T ) Now we would like to use the formula for Kostka-Foulkes polyomials, obtaied i []. 1

4 0. First let us recall some defiitios from []. Give a partitio λ ad compositio µ, a cofiguratio {ν} of the type(λ, µ) is, by defiitio, a collectio of partitios ν (1),ν (), such that 1) ν (k) = j k+1 λ j; ) P (k) (λ, µ) :=Q (ν (k 1) ) Q (ν (k) )+Q (ν (k+1) ) 0 for all k, 1, where Q (λ) := j λ j,ad ν(0) = µ. Propositio 1 [] Let λ be a partitio ad µ be a compositio, the K λ,µ () = c(ν) [ P (k) ] (λ, µ)+m (ν (k) ) m {ν} k, (ν (k), () ) where the summatio i () is take over all cofiguratios of {ν} type (λ, µ), m (ν (k) )=(ν (k) ) (ν (k) ) +1. of the From Lemma 1 ad Propositio 1 we deduce Theorem 1 Let λ be a partitio. The N λ = c 0(ν) [ P (k) (λ, N)+m (ν (k) ) m {ν} k, (ν (k) ) ], (5) where the summatio i (5) is take over all collectios {ν} of partitios {ν} = {ν (1),ν (), } such that 1) ν (k) = j k λ j, k 1, ν (0) =0, ) P (k) (ν, N) :=(N +1) δ k,1 +Q (ν (k 1) ) Q (ν (k) )+Q (ν (k+1) ) 0, for all k, 1. Here c 0 (ν) =(ν (1) ) (λ)+ ad by defiitio ( α ( α (k) α (k+1) ) k, 1 ) := α(α 1) for ay α R., α (k) := (ν(k) ) (6) The idetity (5) may be cosided as a geeralizatio of the (KOH) - idetity (see [8]) for arbitrary partitio λ. Corollary 1 The geeralized -Gaussia coefficiet λ is a symmetric ad uimodal polyomial of degree (N 1) λ (λ).

5 Proof. First, it is well kow (e.g. [10],[11]) that the product of symmetric ad uimodal polyomials is agai symmetric ad uimodal. Secodly, [ we use a well ] kow fact (e.g.[10]), that the ordiary -Gaussia coefficiet m + is a symmetric ad uimodal polyomial of degree m. Soi order to prove Corollary 1, it is sufficiet to show that the sum c 0 (ν)+ k, m (ν (k) )P (k) (ν, N) (7) is the same for all collectios of partitios {ν} which satisfy the coditios 1) ad ) of the Theorem 1. I oder to compute the sum (7), we use the followig result (see []): Lemma Assume {ν} to be a cofiguratio of the type (λ, ν). The k, m (ν (k) )P (k) (ν, µ) =(µ) c(ν) µ α(1). (8) 1 Usig Lemma, it is easy to see that the sum (7) is eual to (N 1) λ (λ). This cocludes the proof. Note that i the proof of Corollary[ 1 we use ] symmetry ad uimodality m + of the ordiary -Gaussia coefficiet. However, we may prove m + the uimodality of by iductio usig the idetity (5) for the case λ =(1 ), N = m. Remark 1. The uimodality of geeralized -Gaussia coefficiets was also proved i the recet preprit [7]. The proof i [7] uses the result from []. However [7] does ot cotai the idetity (5). Remark. The proof of the idetity () give i [] is based o the costructio ad properties of the bijectio (see []) STY (λ, µ) QM(λ, µ). It is a iterestig task to obtai a aalytical proof of (5). I the case = 1 such a proof was obtaied i []. Ackowledgemets. The fial versio of this paper was writte durig the author s stay at RIMS. The author expresses his deep gratitude to RIMS for its hospitality. 5

6 Refereces [1] Dyki E.B., Some properties of the weight system of a liear represetatio of a semisimple Lie group (i Russia). Dokl. Akad. Nauk USSR, 1950, 71, 1-. [] Hughes J.W., Lie algebraic proofs of some theorems o partitios. I Number Theory ad Algebra, Ed. H. Zassehaus, Academic Press, NY, 1977, [] Kirillov A.N., Completeess of the Bethe vectors for geeralized Heiseberg maget. Zap. Nauch. Sem. LOMI (i Russia), 198, 1, [] Kirillov A.N., O the Kostka-Gree-Foulkes polyomials ad Clebsch- Gorda umbers. Jour. Geom. ad Phys., 1988, 5, [5] Macdoald I.G., Symmetric Fuctios ad Hall Polyomials. Oxford Uiversity Press, [6] O Hara K., Uimodality of Gaussia coefficiets: a costructive proof. Jour. Comb. Theory A, 1990, 5, 9-5. [7] Goodma F., O Hara K., Stato D., A uimodality idetity for a Schur fuctio. Preprit 1990/1991. [8] Stato D., Zeilberger D., The Odlyzko cojecture ad O Hara s uimodality proof. Bull. Amer. Math. Soc., 1989, 107, 9-. [9] Staley R., Theory ad applicatio of plae partitios I,II. Studies Appl. Math., 1971, 50, , [10] Staley R., Uimodal seueces arisig from Lie algebras. I Youg Day Proceedigs, Eds. J.V.Narayama, R.M.Mathse, ad J.G.Williams, Dekker, New York/Basel, 1980, [11] Staley R., Log-cocave ad uimodal seueces i algebra, combiatorics, ad geometry. Aals of the New York Academy of Scieces, 1989, 576, [1] Zeilberger D., Kathy O Hara s costructive proof of the uimodality of the Gaussia polyomials. Amer. Math. Mothly, 1989, 96,