THE AFLT q-MORRIS CONSTANT TERM IDENTITY YUE ZHOU Abstract. It is well-known that the Selberg integral is equivalent to the Morris constant

2025-05-06 0 0 586.9KB 39 页 10玖币

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THE AFLT q-MORRIS CONSTANT TERM IDENTITY

YUE ZHOU

Abstract. It is well-known that the Selberg integral is equivalent to the Morris constant

term identity. More generally, Selberg type integrals can be turned into constant term

identities for Laurent polynomials. In this paper, by extending the Gessel–Xin method

of the Laurent series proof of constant term identities, we obtain an AFLT type q-Morris

constant term identity. That is a q-Morris type constant term identity for a product of two

Macdonald polynomials.

Keywords: AFLT Selberg integral, Macdonald polynomials, constant term identities, q-

Morris identity, symmetric functions.

1. Introduction

Deﬁne the n-dimensional Jackson-integral or q-integral over [0,1]n:

Z[0,1]n

f(x1, . . . , xn)dqx1· · · dqxn:= (1 −q)nX

v1,...,vn≥0

f(qv1, . . . , qvn)qv1+···+vn,

where 0 < q < 1 and f:Rn→Cis a function such that the sum on the right is absolutely

convergent. In 1980, Askey [3] conjectured the next q-analogue of the Selberg integral [6,26].

For α, β ∈C\ {0,−1,−2,...,}such that Re(α)>0 and γa positive integer,

(1.1) Z[0,1]n

i=1

xα−1

i(qxi)β−1Y

1≤i<j≤n

x2γ

i(q1−γxj/xi)2γdqx1· · · dqxn

=qγα(n

2)+2γ2(n

3)n−1

i=0

Γq(α+iγ)Γq(β+iγ)Γq1+(i+ 1)γ

Γqα+β+ (n−1 + i)γΓq(1 + γ),

where for z∈C

(a)z= (a;q)z:= (a)∞

(aqz)∞

is the q-shifted factorial and Γq(z) = (q)z−1/(1 −q)z−1is the q-gamma function. Here

(a)∞= (a;q)∞:= (1 −a)(1 −aq)· · · . Askey’s conjecture was proved independently by

Habsieger [10] and Kadell [13]. Hence, we refer the integral (1.1) as the Askey–Habsieger–

Kadell q-Selberg integral. Expressing the integral as a constant term identity, Habsieger and

Date: June 30, 2022.

2010 Mathematics Subject Classiﬁcation. 05A30, 33D70, 05E05.

arXiv:2210.13245v1 [math.CO] 24 Oct 2022

2 YUE ZHOU

Kadell obtained the following equivalent result [31]. For nonnegative integers a, b, c,

(1.2) CT

i=1 x0

xiaqxi

x0bY

1≤i<j≤nxi

xjcxj

qc=

n−1

i=0

(q)a+b+ic(q)(i+1)c

(q)a+ic(q)b+ic(q)c

where x:= (x1, . . . , xn) and CT

xf(x) means to take the constant term of the Laurent polyno-

mial (series) f(x). The q= 1 case of (1.2) was proved by Morris [24] in his Ph.D thesis. He

also showed that the Morris identity is equivalent to the Selberg integral and conjectured the

above q-analogue identity. Since (1.2) was conjectured by Morris and proved by Habsieger

and Kadell, this identity is usually referred as the Habsieger–Kadell q-Morris identity. Note

that the Laurent polynomial in the left-hand side of (1.2) is homogeneous in x0, x1, . . . , xn,

we can take x0= 1 without aﬀecting the constant term. Hence, the operator CT

xin (1.2) is

equivalent to CT

x0,...,xn

. The same trick also applies to An(a, b, c, λ, µ) below.

Recently, Albion, Rains and Warnaar obtained an elliptic ALFT-Selberg integeral [2]. As

a corollary, they got a q-Selberg type integral for a product of two Macdonald polynomials [2,

Corollary 1.5], see (1.3) below. Let λand µbe two partitions such that the length of µis l.

For α, β, q, t ∈Csuch that |β|,|q|,|t|<1,

n!(2πi)nZTn

Pλ(x;q, t)Pµhx+t−β

1−ti;q, t

(1.3)

i=1

(α/xi, qxi/α)∞

(β/xi, xi)∞Y

1≤i<j≤n

(xi/xj, xj/xi)∞

(txi/xj, txj/xi)∞

dx1

· · · dxn

=β|λ|t|µ|Pλh1−tn

1−ti;q, tPµh1−βtn−1

1−ti;q, t

i=1

(t, αtn−m−iqλi, αt1−i/β, qti−1β/α)∞

(q, ti, βti−1, αt1−iqλi/β)∞

i=1

j=1

(αtn−i−j+1qλi+µj)

(αtn−i−jqλi+µj)∞

where Tis a positively oriented unit circle, (a1, . . . , ak)∞=Qk

i=1(ai)∞,mis an arbitrary

integer such that m≥l,Pλ(x;q, t) is the Macdonald polynomial, and f[t+z] is plethystic

notation for the symmetric function f. Using the Cauchy residue theorem and take t=

qc, α =qb+1, β =qa+b+1, xi7→ xiqb+1 for all the i, it is not hard to transform the above

integral into a constant term identity, i.e., Theorem 1.1 below.

For nonnegative integers a, b, c, and partitions λand µsuch that the length of µis l, denote

An(a, b, c, λ, µ) := CT

xx−|λ|−|µ|

0Pλ(x;q, qc)Pµhqc−b−1−qa

1−qcx0+

i=1

xii;q, qc

(1.4)

i=1 x0

xiaqxi

x0bY

1≤i<j≤nxi

xjcxj

qc.

Our main result in this paper is a direct constant term proof of the next AFLT type q-Morris

constant term identity.

q-AFLT IDENTITY 3

Theorem 1.1. Let An(a, b, c, λ, µ)be deﬁned in (1.4). Then

(1.5)

An(a, b, c, λ, µ) = (−1)|λ|qPn

i=1 (λi

2)−cn(λ)Pλh1−qnc

1−qci;q, qcPµhqc−b−1−qa+nc

1−qci;q, qc

i=1

j=1

(qb+(n−i−j)c+λi+µj+1+1)µj−µj+1

i=1

(qa+(i−1)c−λi+1)b+λi(q)ic

(q)b+(n−i)c+λi+µ1(q)c

where |λ|:= Pn

i=1 λiis the size of λ, and n(λ) := Pn

i=1(i−1)λi.

Note the right-hand side of (1.5) can be written as a compact product of q-factorials by

(6.13) below. Many specializations of (1.5) have rich history. When λ=µ= 0, (1.5) is the

Habsieger–Kadell q-Morris identity (1.2) above. When t=qγand take q→1, the Macdonald

polynomial Pλ(x;q, t) reduces to the Jack polynomial P(1/γ)

λ(x). Kadell [15] generalized the

Morris identity by adding a Jack polynomial, i.e., the q= 1 and µ= 0 case of (1.5). Then,

he raised an open problem to ﬁnd the q-analogue. The question was solved by Macdonald

by developing the theory of Macdonald polynomials [21, 22]. Macdonald gave an explicit

q-analogue integral of Kadell’s open problem in his book [22, Page 374]. By the standard

transformation between the q-Selberg type integrals and the q-Morris type constant term

identities [4, 13], it is not hard to obtain a constant term identity equivalent to Macdonald’s

result. That is the µ= 0 case of (1.5). In 1993, Kadell [14] gave a two Jack symmetric

function generalization of the Morris identity. Kadell’s result in 1993 corresponds to the

q= 1 and c=a+b+ 1 case of (1.5). If c=a+b+ 1, the constant term identity (1.5)

reduces to [28, Theorem 1.7]. It should be noted that the q= 1 case of (1.5) is equivalent to

the next ALFT type Selberg integral [1].

Z[0,1]n

P(1/γ)

λ(x)P(1/γ)

µ[x+β/γ −1]

i=1

xα−1

i(1 −xi)β−1Y

1≤i<j≤n

|xi−xj|2γdx1· · · dxn

(1.6)

=P(1/γ)

λ[n]P(1/γ)

µ[n+β/γ −1]

i=1

Γ(β+ (i−1)γ)Γ(α+ (n−i)γ+λi)Γ(1 + iγ)

Γ(α+β+ (2n−l−i−1)γ+λi)Γ(1 + γ)

i=1

j=1

Γ(α+β+ (2n−i−j−1)γ+λi+µj)

Γ(α+β+ (2n−i−j)γ+λi+µj),

where

Re(α)>−λn,Re(β)>0,Re(γ)>−min

1≤i≤n−1n1

n,Re(α) + λi

n−i,Re(β)

n−1o.

Since Hua [12] discovered the γ= 1 and β=γcase of the above integral, and Kadell [14]

obtained the β=γcase for general γ, the β=γcase of (1.6) is called the Kadell–Hua

integral.

4 YUE ZHOU

The idea to prove Theorem 1.1, is based on the well-known fact that to prove the equality

of two polynomials of degree at most d, it is suﬃcient to prove that they agree at d+ 1

distinct points. We brieﬂy outline the key steps.

(1) Polynomiality.

It is routine to show that the constant term An(a, b, c, λ, µ) (we refer the constant

term as Afor short in this section) is a polynomial in qa, assuming that all parameters

but aare ﬁxed. Then, we can extend the deﬁnition of Afor negative a, especially

negative integers.

(2) Rationality By a rationality result, see Corollary 7.4 below, it suﬃces to prove (1.5)

for c≥l. Here lcan be any nonnegative integer independent of c.

(3) Determination of roots

Let

A1={−ic −1,−ic −2,...,−ic −b|i= 0,1, . . . , n −1},

(1.7)

A2={−(i−1)c+λi−1,−(i−1)c+λi−2,...,−(i−1)c|i= 1, . . . , `(λ)},

A3={−(n−j)c−b−1,−(n−j)c−b−2,...,−(n−j)c−b−µj|j= 1, . . . , `(µ)},

where `(λ) is the length of the partition λ. Suppose c>b+λ1+µ1in (1.7), then all

the elements of A1∪A2∪A3are distinct.

For Aviewed as a polynomial in qa, we will determine all its roots under the

assumption that cis a suﬃciently large integer (so that all the roots of Aare distinct).

To be precise, Avanishes only when aequals one of the values in A1∪A2∪A3.

(4) Value at an additional point.

We characterize the expressions for Aat a=−b−1 if `(µ)< n and at a=

−(n−`(µ)−1)c−b−1 if `(µ)≥nrespectively.

The steps (1) and (2) are routine. A similar argument appeared in [9] and [31]. The Step

(3) is lengthy. To carry out the details, we need to mix three tools: the iterated Laurent

series, plethystic notation and substitution, and Cai’s splitting idea for Laurent polynomials.

We have combined the ﬁrst two tools in [32] to prove and extend Kadell’s orthogonality

conjecture. In Step (4), we reduce the expression for Aat the additional point to a similar

type. We can uniquely determine a closed-form expression for Aafter completing the above

four steps.

This paper is organised as follows. In the next section, we introduce the basic notation

used in this paper. In Section 3, we give a brief introduction to a commonly used tool in

the ring of symmetric functions — the plethystic notation. In Section 4, we present the

essential material in the ﬁeld of iterated Laurent series. In Section 5, we ﬁnd that a family of

constant terms vanish. In Section 6, we obtain necessary results of Macdonald polynomials.

In Section 7, we show the polynomiality and rationality of An(a, b, c, λ, µ). In Section 8, we

prepare the preliminary for the determination of the roots of An(a, b, c, λ, µ). In the last

section, we prove Theorem 1.1.

q-AFLT IDENTITY 5

2. Basic notation

In this section we introduce some basic notation used throughout this paper.

A partition λ= (λ1, λ2, . . . ) is a sequence of decreasing nonnegative integers such that

only ﬁnitely-many λiare positive. The length of a partition λ, denoted `(λ) is the number of

nonzero λi(such nonzero λiare called parts of λ). The tails of zeros of a partition is usually

not displayed. If there are exactly miof the parts of λare equal to i, we can also denote a

partition by

λ= (1m12m2· · · rmr· · · ).

The Young diagram of a partition is a collection of left-aligned rows of squares such that the

ith row contains λisquares. For example, the partition (7,4,3,1) corresponds to

If λ, µ are partitions, we shall write µ⊂λif µi≤λifor all i≥1. We can construct a skew

diagram λ/µ whenever µ⊂λby removing the squares of µfrom those of λ. For example, if

λ= (7,4,3,1) and µ= (4,3,1), then µ⊂λand the skew diagram λ/µ is the following:

A skew diagram θis said to be a horizontal r-strip if |θ|=rand it contains at most one

square in every column. For example, the above diagram for (7,4,3,1)/(4,3,1) is a horizontal

7-strip. The size of the partition λis |λ|=λ1+λ2+· · · . If λ, µ are partitions of the same

size then we write µ≤λif µ1+· · · +µi≤λ1+· · · +λifor all i≥1. This partial order on

the set of partitions of the same size is called the dominance order. As usual, we write µ<λ

if µ≤λbut µ6=λ.

The inﬁnite q-shifted factorial is deﬁned as

(z)∞= (z;q)∞:=

∞

i=0

(1 −zqi)

where, typically, we suppress the base q. Then, for kan integer,

(z)k= (z;q)k:= (z;q)∞

(zqk;q)∞

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THEAFLTq-MORRISCONSTANTTERMIDENTITYYUEZHOUAbstract.Itiswell-knownthattheSelbergintegralisequivalenttotheMorrisconstanttermidentity.Moregenerally,SelbergtypeintegralscanbeturnedintoconstanttermidentitiesforLaurentpolynomials.Inthispaper,byextendingtheGessel{XinmethodoftheLaurentseriesproofofconstantt...

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THE AFLT q-MORRIS CONSTANT TERM IDENTITY YUE ZHOU Abstract. It is well-known that the Selberg integral is equivalent to the Morris constant

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