Unimodality of generalized Gaussian coefficients.

arXiv:hep-th/9212152v1 26 Dec 1992Unimodality of generalized Gaussian coefficients.Anatol N. KirillovSteklov Mathematical Institute,Fontanka 27, St.Petersburg, 191011, RussiaJanuary 1991AbstractA combinatorial proof of the unimodality of the generalizedq-Gaussian coefficientsNλqbased on the explicit formula forKostka-Foulkes polynomials is given.10.Let us mention that the proof of the unimodality of the general-ized Gaussian coefficients based on theoretic-representation considerationswas given by E.B. Dynkin [1] (see also [2], [10], [11]).

Recently K.O’Hara[6] gave a constructive proof of the unimodality of the Gaussian coefficient"n + kn#q= s(k)(1, · · · , qk), and D. Zeilberger [12] derived some identitywhich may be consider as an “algebraization” of O’Hara’s construction. Byinduction this identity immediately implies the unimodality of"n + kn#q.Using the observation (see Lemma 1) that the generalized Gaussian coef-ficient"nλ′#qmay be identified (up to degree q ) with the Kostka-Foulkespolynomial Keλ,µ(q) (see Lemma 1), the proof of the unimodality of"nλ′#qis a simple consequence of the exact formula for Kostka-Foulkes polynomialscontained in [4].

Furthermore the expression for Keλ,µ(q) in the case λ = (k)coincides with identity (KOH) from [8]. So we obtain a generalization anda combinatorial proof of (KOH) for arbitrary partition λ.1

20.Let us recall some well known facts which will be used later. Webase ourselves [9] and [5].

Let λ = (λ1 ≥λ2 ≥· · ·) be a partition, |λ| bethe sum of its parts λi, n(λ) = Pi(i −1)λi and"nλ#qbe the generalizedGaussian coefficient.Recall thatsλ(1, · · · , qn) = qn(λ)"nλ′#q=Yx∈λ1 −qn+c(x)1 −qh(x) . (1)Here c(x) is the content and h(x) is the hook length corresponding to thebox x ∈λ, [5].Lemma 1Let λ be a partition and fix a positive integer n. Consider newpartitions eλ = (n · |λ|, λ) and µ = (|λ|n+1).

Thenqn(n−1)2·|λ|+n(λ)"nλ′#q= Keλ,µ(q). (2)Proof.We use the description of the polynomial q|λ|+n(λ) ·"nλ′#qas agenerating function for the standard Young tableaux of the shape λ filledwith numbers from the interval [1, · · · , n].

Let us denote this set of Youngtableaux by STY (λ, ≤n). Thenq|λ|+n(λ)"nλ′#q=XT∈STY (λ,≤n)q|T| .

(3)Here |T| is the sum of all numbers filling the boxes of T . For any tableau T(or diagram λ) let us denote by T[k] (or λ[k]) the part of T (or λ) consistingof rows starting from the (k + 1)-st one.

Given tableau T ∈STY (λ, ≤n),then consider tableau eT ∈STY (eλ, µ) such that eT[1] = T + supp λ[1], andwe fill the first row of ˜T with all remaining numbers in increasing order fromleft to right. Here for any diagram λ we denote by suppλ the plane partitionof the shape λ and content (1|λ|).

It is easy to see thatc( eT ) = |T| + (n + 1)(n −2)2· |λ| ,so we obtain the identity (2).2

Let us consider an explanatory example.Assume λ = (2, 1), n = 3.Then eλ = (9, 2, 1), µ = (34). It is easy to see that |STY (λ, ≤3)| = 8.T|T|eTc( eT )112411123344422310113511123334422411122511122344423311123611122334423412132611122334424312133711122333424413223711122234433413233811122233434414Now we would like to use the formula for Kostka-Foulkes polynomials,obtained in [4].3

30.First let us recall some definitions from [4]. Given a partition λand composition µ, a configuration {ν} of the type (λ, µ) is, by definition,a collection of partitions ν(1), ν(2), · · · such that1) |ν(k)| = Pj≥k+1 λj;2) P (k)n (λ, µ) := Qn(ν(k−1))−2Qn(ν(k))+Qn(ν(k+1)) ≥0 for all k, n ≥1,where Qn(λ) := Pj≤n λ′j ,and ν(0) = µ.Proposition 1 [4] Let λ be a partition and µ be a composition, thenKλ,µ(q) =X{ν}qc(ν) Yk,n"P (k)n (λ, µ) + mn(ν(k))mn(ν(k))#q,(4)where the summation in (4) is taken over all configurations of {ν} of thetype (λ, µ), mn(ν(k)) = (ν(k))′n −(ν(k))′n+1.From Lemma 1 and Proposition 1 we deduceTheorem 1 Let λ be a partition.

Then"Nλ′#q=X{ν}qc0(ν) Yk,n"P (k)n (λ, N) + mn(ν(k))mn(ν(k))#q,(5)where the summation in (5) is taken over all collections {ν} of partitions{ν} = {ν(1), ν(2), · · ·} such that1) |ν(k)| = Pj≥k λj, k ≥1, |ν(0)| = 0,2) P (k)n (ν, N) := n(N +1)·δk,1+Qn(ν(k−1))−2Qn(ν(k))+Qn(ν(k+1)) ≥0,for all k, n ≥1. Herec0(ν) = n(ν(1)) −n(λ) +Xk,n≥1 α(k)n−α(k+1)n2!, α(k)n:= (ν(k))′n(6)and by definition α2!

:= α(α−1)2for any α ∈R.The identity (5) may be consided as a generalization of the (KOH) -identity (see [8]) for arbitrary partition λ.Corollary 1 The generalized q-Gaussian coefficient"nλ′#qis a symmetricand unimodal polynomial of degree (N −1)|λ| −n(λ).4

Proof. First, it is well known (e.g.

[10],[11]) that the product of symmet-ric and unimodal polynomials is again symmetric and unimodal. Secondly,we use a well known fact (e.g.

[10]), that the ordinary q-Gaussian coefficient"m + nn#qis a symmetric and unimodal polynomial of degree mn. So inorder to prove Corollary 1, it is sufficient to show that the sum2c0(ν) +Xk,nmn(ν(k))P (k)n (ν, N)(7)is the same for all collections of partitions {ν} which satisfy the conditions1) and 2) of the Theorem 1.

In oder to compute the sum (7), we use thefollowing result (see [4]):Lemma 2 Assume {ν} to be a configuration of the type (λ, ν). ThenXk,nmn(ν(k))P (k)n (ν, µ) = 2n(µ) −2c(ν) −Xn≥1µ′n · α(1)n .

(8)Using Lemma 2, it is easy to see that the sum (7) is equal to (N −1)|λ|−n(λ). This concludes the proof.Note that in the proof of Corollary 1 we use symmetry and unimodalityof the ordinary q-Gaussian coefficient"m + nn#q.

However, we may provethe unimodality of"m + nn#qby induction using the identity (5) for thecase λ = (1n), N = m.Remark 1. The unimodality of generalized q-Gaussian coefficients wasalso proved in the recent preprint [7].

The proof in [7] uses the result from[4]. However [7] does not contain the identity (5).Remark 2.

The proof of the identity (4) given in [4] is based on theconstruction and properties of the bijection (see [4])STY (λ, µ) ⇀↽QM(λ, µ).It is an interesting task to obtain an analytical proof of (5). In the caseq = 1 such a proof was obtained in [3].Acknowledgements.

The final version of this paper was written duringthe author’s stay at RIMS. The author expresses his deep gratitude to RIMSfor its hospitality.5

References[1]Dynkin E.B., Some properties of the weight system of a linear repre-sentation of a semisimple Lie group (in Russian). Dokl.

Akad. NaukUSSR, 1950, 71, 221-224.

[2]Hughes J.W., Lie algebraic proofs of some theorems on partitions. InNumber Theory and Algebra, Ed.

H. Zassenhaus, Academic Press, NY,1977, 135-155. [3]Kirillov A.N., Completeness of the Bethe vectors for generalized Heisen-berg magnet.

Zap. Nauch.

Sem. LOMI (in Russian), 1984, 134, 169-189.

[4]Kirillov A.N., On the Kostka-Green-Foulkes polynomials and Clebsch-Gordan numbers. Journ.

Geom. and Phys., 1988, 5, 365-389.

[5]Macdonald I.G., Symmetric Functions and Hall Polynomials. OxfordUniversity Press, 1979.

[6]O’Hara K., Unimodality of Gaussian coefficients: a constructive proof.Jour. Comb.

Theory A, 1990, 53, 29-52. [7]Goodman F., O’Hara K., Stanton D., A unimodality identity for aSchur function.

Preprint 1990/1991. [8]Stanton D., Zeilberger D., The Odlyzko conjecture and O’Hara’s uni-modality proof.

Bull. Amer.

Math. Soc., 1989, 107, 39-42.

[9]Stanley R., Theory and application of plane partitions I,II. StudiesAppl.

Math., 1971, 50, 167-188, 259-279. [10] StanleyR.,UnimodalsequencesarisingfromLiealgebras.InYoung Day Proceedings, Eds.

J.V.Narayama,R.M.Mathsen, andJ.G.Williams, Dekker, New York/Basel, 1980, 127-136. [11] Stanley R., Log-concave and unimodal sequences in algebra, combina-torics, and geometry.

Annals of the New York Academy of Sciences,1989, 576, 500-535. [12] Zeilberger D., Kathy O’Hara’s constructive proof of the unimodality ofthe Gaussian polynomials.

Amer. Math.

Monthly, 1989, 96, 590-602.6

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