Recurrence formulas for Macdonald polynomials of type A

DOI 10.1007/s10801-009-0207-y

Recurrence formulas for Macdonald polynomials of type A

Michel Lassalle·Michael J. Schlosser

Received: 12 February 2009 / Accepted: 9 October 2009 / Published online: 13 November 2009

© Springer Science+Business Media, LLC 2009

Abstract We consider products of two Macdonald polynomials of typeA, indexed by dominant weights which are respectively a multiple of the first fundamental weight and a weight having zero component on thekth fundamental weight. We give the explicit decomposition of any Macdonald polynomial of typeAin terms of this basis.

Keywords Macdonald polynomials·Pieri formula·Multidimensional matrix inverse

1 Introduction

In the 1980s, Macdonald [6–8] introduced a class of orthogonal polynomials which are Laurent polynomials in several variables and generalize the Weyl characters of compact simple Lie groups. In the simplest situation, given a root systemR, these polynomials are elements of the group algebra of the weight lattice ofR, indexed by the dominant weights and depending on two parameters(q, t ).

When R is of type An, these Macdonald polynomials are in bijective cor- respondence with the symmetric functions Pλ(q, t ) indexed by partitions, in- troduced by Macdonald some years before [4, 5]. In fact, they correspond to Pλ(q, t )(x1, . . . , xn+1), for a partitionλ=(λ1, . . . , λn)of lengthn, with the n+1 variables(x₁, . . . , x_n+1)linked by the conditionx₁· · ·x_n+1=1.

M. Lassalle (

Centre National de la Recherche Scientifique, Institut Gaspard-Monge, Université de Marne-la-Vallée, 77454 Marne-la-Vallée Cedex, France e-mail:[email protected]

url:http://igm.univ-mlv.fr/~lassalle

M.J. Schlosser

Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Vienna, Austria e-mail:[email protected]

url:http://www.mat.univie.ac.at/~schlosse

The purpose of this article is to extend the result of [3], given for the symmetric functionsPλ(q, t ), to the framework of the root systemA_n.

More precisely, in [3, Theorem 4.1] we obtained a recurrence formula giving the symmetric functionP(λ1,...,λn)(q, t )as a sum

P(λ1,...,λn)=

θ∈Nⁿ⁻¹

Cθ₁,...,θ_n−1P(λ1+θ1,...,λn−1+θn−1)Pλ_n−|θ| (1.1)

with|θ| =_n−1

i=1θ_i and Nthe set of nonnegative integers. This formula was ob- tained by inverting the “Pieri formula,” which conversely expresses the product P(λ1,...,λn−1)Pλnas a sum

P(λ₁,...,λ_n−1)Pλ_n=

θ∈Nⁿ⁻¹

cθ₁,...,θ_n−1P(λ₁+θ₁,...,λ_n−1+θ_n−1,λ_n−|θ|).

Both expansions are identities between symmetric functions, valid for any number of variables.

These identities may also be written in terms of Macdonald polynomials of typeAn. For this purpose, let{ωi,1≤i≤n}be the nfundamental weights of the root systemAn. LetPλdenote the Macdonald polynomial associated with the domi- nant weightλ=_n

i=1λ_iω_i. The recurrence formula (1.1), written forn+1 variables (x₁, . . . , x_n+1)linked byx₁· · ·x_n+1=1, yields

P_λ=

θ∈Nⁿ⁻¹

C_{θ1,...,θn−1}P_{(λn−|θ|)ω1}P_μ (1.2)

withμ=_n−2

i=1(λi+θi−θi+1)ωi+(λn−1+λn+θn−1)ωn−1. This alternative for- mulation is obvious and does not bring anything new.

However the method of [3], when applied in theAnroot system framework, allows us to get a much stronger result. Indeed, letkbe a fixed integer with 1≤k≤n. In this paper we shall write the Macdonald polynomialP_λin terms of productsP_rω1P_μ withμ=_n

i=1μ_iω_i andμ_k=0. There arensuch recurrence formulas, (1.2) being the particular casek=nof the latter.

This paper is organized as follows. In Sect.2we introduce our notation for the root systemA_n and recall general facts about the corresponding Macdonald polynomi- als. Their Pieri formula, which involves a specific infinite-multidimensional matrix, is studied in Sect.3, starting from the one given by Macdonald for the symmetric functionsPλ(q, t ) [5, p. 340]. In Sect.4 we invert the Pieri matrix by applying a particular multidimensional matrix inverse, given separately in theAppendix. This matrix inverse is equivalent to one previously obtained in [3, Sect. 2] by using opera- tor methods. As a result of inverting the Pieri formula, we obtain recurrence formulas forA_n Macdonald polynomials. Finally, in Sect.5we detail the examples of theA₂ andA₃cases and compare them to earlier results.

2 Macdonald polynomials of typeA

The standard references for Macdonald polynomials associated with root systems are [6–8].

Let us consider the spaceRⁿ⁺¹ endowed with the usual scalar product and the quotient spaceV =Rⁿ⁺¹/R(1, . . . ,1), whereR(1, . . . ,1)is the subspace spanned by the vector(1, . . . ,1). Letε1, . . . , εn+1 denote the images inV of the coordinate vectors ofRⁿ⁺¹, linked by_n+1

i=1εi=0.

The root system of typeA_nis formed by the vectors{ε_i−ε_j, i=j}. The positive roots are{ε_i−ε_j, i < j}, and the simple roots areε_i−ε_i+1for 1≤i≤n. The Weyl group is the symmetric groupW=S_n+1acting by permutation of the coordinates.

The weight latticeP is formed by integral linear combinations of the fundamental weights{ω_i,1≤i≤n}defined byω_i=ε₁+ · · · +ε_i. Letω_i=0 fori=0, n+1. We denote byP⁺ the set of dominant weights λ=_n

i=1λ_iω_i, which are nonnegative integral linear combinations of the fundamental weights.

There is the following correspondence between dominant weights and partitions.

Given a dominant weight, if we write it as

λ= n

i=1

λiωi=

n+1

i=1

μiεi,

the sequenceμ=(μ1, . . . , μn+1)is a partition with length≤n+1. We have

λ_i=μ_i−μ_i+1 and μ_i=μ_n+1+ n j=i

λ_j.

Thusμis defined up toμ_n+1, and two partitionsμ, νcorrespond to the same weight λif and only ifμ₁−ν₁= · · · =μ_n+1−ν_n+1. We denote byCλthe family of partitions thus defined.

LetA denote the group algebra overR of the free Abelian groupP. For each λ∈P, lete^λ denote the corresponding element ofA, subject to the multiplication rulee^λe^μ=e^λ+μ. The set{e^λ, λ∈P}forms anR-basis ofA.

The Weyl groupW=S_n+1acts onP and onA. LetW λdenote the orbit ofλ∈P andA^W the subspace ofW-invariants inA. There are two important bases ofA^W, both indexed by dominant weights. The first one is given by the orbit-sums

m_λ=

μ∈W λ

e^μ.

The second one is provided by the Weyl characters χ_λ=δ⁻¹

w∈W

det(w)e^w(λ+ρ)

withρ=_n

i=1(n−i+1)εi andδ=

w∈Wdet(w)e^w(ρ). The Macdonald polyno- mials{P_λ, λ∈P⁺}form another basis defined as the eigenvectors of a specific self- adjoint operator (which we do not describe here).

For 1≤ i≤ n+1, define x_i =e^εi, so that the variables x_i are linked by x₁· · ·x_n+1=1. Thenδ is the Vandermonde determinant

i<j(x_i −x_j). There is a correspondence betweenA^W and the symmetric polynomials restricted to n+1 variablesx=(x₁, . . . , x_n+1)linked by the previous condition.

In terms of bases this correspondence may be described as follows. Let λ be any dominant weight, and let x₁· · ·x_n+1=1. All monomial symmetric functions m_μ(x₁, . . . , x_n+1) with μ∈Cλ are equal, and their common value is the orbit- summ_λ. Similarly, the Weyl characterχ_λ is the common value of the Schur func- tionss_μ(x₁, . . . , x_n+1),μ∈Cλ, whereas the Macdonald polynomialP_λis the com- mon value of the symmetric polynomialsPμ(q, t )(x₁, . . . , x_n+1) withμ∈Cλ and Pμ(q, t )the symmetric function studied in Chap. 6 of [5].

Given a positive integerrand a dominant weightλ, the “Pieri formula” expands the product

Prω₁Pλ=

ρ

cρPλ+ρ

in terms of Macdonald polynomials, where the range ofρand the values of the coef- ficientscρ are to be determined.

LetQdenote the root lattice spanned by the simple roots. For any vectorτ, define (τ )=C(τ )∩(τ+Q)

withC(τ )the convex hull of the Weyl group orbit ofτ. Since the orbit ofω1=ε1is the set{ε_i =ω_i−ω_i−1,1≤i≤n+1}, it is clear that(rω1)is formed by vectors

n+1

i=1

θ_i(ω_i−ω_i−1)= n i=1

(θ_i−θ_i+1)ω_i

withθ=(θ1, . . . , θn+1)∈Nⁿ⁺¹and|θ| =n+1 i=1θi=r.

By general results [8, (5.3.8), p. 104], it is known that the sum on the right- hand side of the Pieri formula is restricted to vectorsρ such thatρ∈(rω1)and λ+ρ∈P⁺. In the next section we shall give a direct proof of this result and make the value of the coefficientcρexplicit.

3 Pieri formula

Let 0< q <1. For any integerr, the classicalq-shifted factorial(u;q)ris defined by (u;q)_∞=

j≥0

1−uq^j

, (u;q)_r=(u;q)_∞/ uq^r;q

∞.

Letu=(u₁, . . . , u_m)bemindeterminates, andθ=(θ₁, . . . , θ_m)∈N^m. For clar- ity of display, throughout this paper, any time such a pair(u, θ ) is given, we shall implicitly assumemauxiliary variablesv=(v₁, . . . , v_m)to be defined byv_i=q^θiu_i.

Macdonald polynomials of typeA_nsatisfy the following Pieri formula.

Theorem 3.1 Letλ=_n

i=1λ_iω_i be a dominant weight, andr∈N. For any 1≤i≤ n+1, define

u_i=q

n j=iλjt⁻ⁱ and forθ∈Nⁿ⁺¹,

d_θ(u₁, . . . , u_n+1;r)=(q;q)_r (t;q)_r

n+1 j=1

(t;q)_θj (q;q)_θj

1≤i<j≤n+1

(t v_i/v_j;q)_θj (qv_i/v_j;q)_θj

(qu_i/t v_j;q)_θj (u_i/v_j;q)_θj . We have

P_rω1P_λ=

θ∈Nⁿ⁺¹

|θ|=r

d_θ(u₁, . . . , u_n+1;r)P_λ+ρ

withρ=_n

i=1(θ_i−θ_i+1)ω_i.

Proof In a first step, we write the Pieri formula for arbitrary Pμ(q, t ) withμ= (μ₁, . . . , μ_n)being a partition having length≤n. We start from [5, p. 340, (6.24)(i)]

and [5, p. 342, Example 2(a)]. Replacingg_r by(t;q)_r/(q;q)_rP(r), we have P(r)Pμ=

κ⊃μ

ϕ_κ/μPκ,

where the skew-diagramκ−μis a horizontalr-strip, i.e., has at most one node in each column. The Pieri coefficientϕ_κ/μis given by

(t;q)_r

(q;q)_rϕ_κ/μ=

1≤i≤j≤l(κ)

f (q^κi−κjt^j−i) f (q^κi−μjt^j−i)

f (q^μi−μj+1t^j−i) f (q^μi−κj+1t^j−i)

=

1≤i≤j≤l(κ)

w_κj−μj(q^κi−κjt^j−i) wκ_j+1−μ_j+1(q^μi−κj+1t^j−i) withf (u)=(t u;q)_∞/(qu;q)_∞andws(u)=(t u;q)s/(qu;q)s.

Sinceκ−μis a horizontal strip, the lengthl(κ)ofκis at most equal ton+1, so we can writeκ=(μ1+θ1, . . . , μn+θn, θn+1)with|θ| =r. Then

(t;q)_r

(q;q)_r ϕκ/μ=

1≤i≤j≤l(κ)

wθ_j

q^κi−κjt^j−i

1≤i<j≤l(κ)+1

wθ_j

q^μi−κjt^j−i−1₋1

=

n+1 j=1

(t;q)θ_j

(q;q)_θj

1≤i<j≤n+1

(t vi/vj;q)θ_j

(qv_i/v_j;q)_θj

(qui/t vj;q)θ_j

(u_i/v_j;q)_θj , where for 1≤i≤n+1, we setu_i=q^μit⁻ⁱ andv_i=q^κit⁻ⁱ=q^θiu_i.

In a second step we translate this result in terms of A_n Macdonald polynomi- als. Given the dominant weightλ, we chooseμ=(μ₁, . . . , μ_n+1)to be the unique

element of Cλ such that μ_n+1=0, i.e., with length ≤n. For 1≤i≤n, we have μ_i =_n

j=iλ_j. As for the partitionκ (with length≤n+1), it belongs toCσ with σ=_n

k=1(κ_k−κ_k+1)ω_k=_n

k=1(λ_k+θ_k−θ_k+1)ω_k. Hence the statement.

Remark On the right-hand side of the Pieri formula, the conditionλ+ρ∈P⁺ is necessarily satisfied as soon asdθ(u1, . . . , un+1;r)=0. Using the correspondence between dominant weights and partitions, this may be verified on the Pieri formula

P(r)Pμ=

κ=(μ1+θ1,...,μn+θn,θn+1)

ϕ_κ/μPκ.

We only have to show that ϕ_κ/μ necessarily vanishes when the multiinteger κ is not a partition. But then there is an indexi such that κ_i < κ_i+1, so that the factor (qu_i/t v_i+1;q)_θi+1 inϕ_κ/μwrites out as

1−q^{1+μi−κi+1}

· · ·

1−q^μi−μi+1 .

Due toκi< κi+1, this product would be=0 only ifμi< μi+1, which is impossible sinceμis a partition.

From now on, we fix some integer 1≤k≤n. Substitutingr− |θ|forθ_k, the Pieri formula may be written in the more explicit form

P_rω1P_λ=

θ=(θ1,...,θk−1,0,θk+1,...,θn+1)∈Nⁿ

|θ|≤r

dˆ_θ(u₁, . . . , u_n+1;r)P_λ+ρ

with

ρ=

1≤i≤n i=k−1,k

(θi−θi+1)ωi+θk−1ωk−1+ r− |θ|

(ωk−ωk−1)−θk+1ωk

and

dˆ_θ(u₁, . . . , u_n+1;r)=(q;q)r

(t;q)r

(t;q)r−|θ|

(q;q)r−|θ| n+1

j=1 j=k

(t;q)_θj (q;q)θ_j

×

1≤i<j≤n+1 j=k

(t v_i/v_j;q)_θj (qvi/vj;q)θ_j

(qu_i/t v_j;q)_θj (ui/vj;q)θ_j

×

k−1 i=1

(t v_i/v_k;q)_r−|θ| (qv_i/v_k;q)_r−|θ|

(qu_i/t v_k;q)_r−|θ| (u_i/v_k;q)_r−|θ| .

Hereu_i, v_i (1≤i≤n+1)are as in Theorem3.1, exceptv_k=q^r−|θ|u_k. The sum is restricted to|θ| ≤rsince 1/(q;q)_s=0 fors <0.

In a second step, we concentrate on the situationλ_k=0. Then each term on the right-hand side vanishes unlessθk+1=0. Indeed, ifλk=0, one hasuk=t uk+1and v_k+1=q^θk+1u_k+1. Hence, fori=kandj =k+1, the factor(qu_i/t v_j;q)_θj evaluates as

(quk/t vk+1;q)θ_k+1=

q^1−θk+1;q

θk+1=δθ_k+1,0. Therefore, ifλk=0, the Pieri formula can be written as

P_rω1P_λ

=

θ=(θ1,...,θk−1,0,0,θk+2,...,θn+1)∈Nⁿ⁻¹

|θ|≤r

d˜_θ(u₁, . . . , u_k−1, u_k, u_k+2, . . . , u_n+1;k, r)P_λ+ρ

with

ρ=

1≤i≤n i=k−1,k,k+1

(θ_i−θ_i+1)ω_i+θ_k−1ω_k−1+ r− |θ|

(ω_k−ω_k−1)−θ_k+2ω_k+1

and

d˜_θ(u₁, . . . , u_k−1, u_k, u_k+2, . . . , u_n+1;k, r)

=(q;q)r

(t;q)r

(t;q)r−|θ|

(q;q)r−|θ| n+1

i=1 i=k,k+1

(t;q)θ_i

(q;q)θ_i

1≤i<j≤n+1 i=k,k+1 j=k,k+1

(t v_i/v_j;q)_θj (qvi/vj;q)θ_j

(qu_i/t v_j;q)_θj (ui/vj;q)θ_j

×

k−1 i=1

(t v_i/v_k;q)_r−|θ| (qv_i/v_k;q)_r−|θ|

(qu_i/t v_k;q)_r−|θ| (u_i/v_k;q)_r−|θ|

n+1 j=k+2

(t v_k/v_j;q)_θj (qv_k/v_j;q)_θj

(qu_k/t²v_j;q)_θj (u_k/t v_j;q)_θj . Here the notation is the same as before, includingv_k=q^r−|θ|u_k. Forj ≥k+2, we have used

(t v_k/v_j;q)_θj (qv_k/v_j;q)_θj

(qu_k/t v_j;q)_θj (u_k/v_j;q)_θj

(t v_k+1/v_j;q)_θj (qv_k+1/v_j;q)_θj

(qu_k+1/t v_j;q)_θj (u_k+1/v_j;q)_θj

= (t vk/vj;q)θ_j

(qv_k/v_j;q)_θj

(quk/t²vj;q)θ_j

(u_k/t v_j;q)_θj , which is a direct consequence ofv_k+1=u_k+1=u_k/t.

In a third step, we perform some relabeling in order to remove the two 0’s ap- pearing in θ. For that purpose, for n indeterminates (u0, u1, . . . , un−1) and θ = (θ₁, . . . , θ_n−1)∈Nⁿ⁻¹, we define

D_θ(u₀, u₁, . . . , u_n−1;k, r)

=(q/t )^|θ|(t²u0;q)_|θ|

(qt u0;q)_|θ|

n−1 i=1

(t;q)θ_i

(q;q)_θi

(q^|θ|+1ui;q)θ_i

(q^|θ|t u_i;q)_θi

×

1≤i<j≤n−1

(t vi/vj;q)θ_j

(qv_i/v_j;q)_θj

(qui/t vj;q)θ_j

(u_i/v_j;q)_θj

×

k−1 i=1

(u_i/u₀;q)_θi (qu_i/t u₀;q)_θi

(qu_i/t u₀;q)_θi−r+|θ| (u_i/u₀;q)_θi−r+|θ|

(u_i/t u₀;q)_θi−r+|θ| (qui/t²u0;q)θ_i−r+|θ|

×

n−1 i=k

(t u_i/u₀;q)_θi (qui/u0;q)θ_i

.

Lemma If we write w_i=

⎧⎨

⎩

q^−rt⁻², i=0, q^−ru_i/t u_k, 1≤i≤k−1, q^−ru_i+2/t u_k, k≤i≤n−1, we have

D_θ(w0, w1, . . . , w_n−1;k, r)

= ˜d(θ₁,...,θ_k−1,0,0,θ_k,...,θ_n−1)(u1, . . . , uk−1, uk, uk+2, . . . , un+1;k, r).

Proof Merely by substitution, and usingv_k=q^r−|θ|u_k, we only have to prove

(q/t )^|θ| (q^−r;q)_|θ| (q^1−r/t;q)_|θ|

n+1 j=k+2

(q^|θ|−r+1u_j/t u_k;q)_θj (q^|θ|−ruj/uk;q)θ_j

(t²u_j/u_k;q)_θj (qt uj/uk;q)θ_j

×

k−1 i=1

(q^|θ|−r+1ui/t uk;q)θ_i

(q^|θ|−rui/uk;q)θ_i

(t ui/uk;q)θ_i

(qui/uk;q)θ_i

×

k−1 i=1

(qui/uk;q)θ_i−r+|θ|

(t u_i/u_k;q)_θi−r+|θ|

(ui/uk;q)θ_i−r+|θ|

(qu_i/t u_k;q)_θi−r+|θ|

=(q;q)_r (t;q)_r

(t;q)r−|θ|

(q;q)_r−|θ|

k−1

i=1

(t vi/q^r−|θ|uk;q)r−|θ|

(qv_i/q^r−|θ|u_k;q)_r−|θ|

(qui/t q^r−|θ|uk;q)r−|θ|

(u_i/q^r−|θ|u_k;q)_r−|θ|

×

n+1 j=k+2

(t q^r−|θ|u_k/v_j;q)_θj (q^r−|θ|+1u_k/v_j;q)_θj

(qu_k/t²v_j;q)_θj (u_k/t v_j;q)_θj . We have obviously

(q^|θ|−r+1ui/t uk;q)θ_i

(q^|θ|−rui/uk;q)θ_i

(u_i/u_k;q)_θi−r+|θ|

(qui/t uk;q)θ_i−r+|θ|=(qu_i/t q^r−|θ|u_k;q)_r−|θ| (ui/q^r−|θ|uk;q)r−|θ| . Using the identities

(aq⁻ⁿ;q)n

(bq⁻ⁿ;q)n =(q/a;q)n

(q/b;q)n

(a/b)ⁿ,

(a;q)_n (b;q)_n

(b;q)_n−k

(a;q)_n−k =(q¹⁻ⁿ/a;q)_k (q¹⁻ⁿ/b;q)k

(a/b)^k, we get

(t u_i/u_k;q)_θi (qui/uk;q)θ_i

(qu_i/u_k;q)_θi−r+|θ|

(t ui/uk;q)θ_i−r+|θ| =(q^1−θiu_k/t u_i;q)_r−|θ|

(q^−θiuk/ui;q)r−|θ| (t /q)^r−|θ|

= (t v_i/q^r−|θ|u_k;q)_r−|θ| (qvi/q^r−|θ|uk;q)r−|θ|. Similarly, we obtain

(t /q)^θj (q^|θ|−r+1u_j/t u_k;q)_θj

(q^|θ|−ru_j/u_k;q)_θj = (t q^r−|θ|u_k/v_j;q)_θj (q^r−|θ|+1u_k/v_j;q)_θj, (q/t )^θj(t²u_j/u_k;q)_θj

(qt u_j/u_k;q)_θj =(qu_k/t²v_j;q)_θj (u_k/t v_j;q)_θj .

Finally, we have proved the following Pieri formula.

Theorem 3.2 Letλ=_n

i=1λiωi be a dominant weight, andr∈N. Assumeλk=0 for some fixed 1≤k≤n. Define

ui=

⎧⎪

⎪⎨

⎪⎪

⎩

q^−rt⁻², i=0, q^−r+

k−1

j=iλjt^k−i−1, 1≤i≤k−1, q^−r−

i+1

j=k+1λjt^k−i−3, k≤i≤n−1.

We have

Prω₁Pλ=

θ=(θ₁,...,θ_n−1)∈Nⁿ⁻¹

|θ|≤r

Dθ(u0, u1, . . . , un−1;k, r) Pλ+ρ

with

ρ=

k−2

i=1

(θi−θi+1)ωi+θk−1ωk−1+ r− |θ|

(ωk−ωk−1)−θkωk+1

+ n i=k+2

(θ_i−2−θ_i−1)ω_i.

Remark Fork=1,2 (resp.k=n,n−1), the first (resp. the last) sum in the above expression ofρmust be understood as zero. This convention will be kept in the next sections.