SYMMETRIC FUNCTION GENERALIZATIONS OF THE q-BAKERFORRESTER EX-CONJECTURE AND SELBERG-TYPE INTEGRALS

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SYMMETRIC FUNCTION GENERALIZATIONS OF THE

q-BAKER–FORRESTER EX-CONJECTURE AND SELBERG-TYPE

INTEGRALS

GUOCE XIN AND YUE ZHOU*

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

constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-

Morris constant term identity. This conjecture was proved and extended by K´arolyi, Nagy,

Petrov and Volkov in 2015. In this paper, we obtain two symmetric function generalizations

of the q-Baker–Forrester ex-conjecture. These includes: (i) a q-Baker–Forrester type constant

term identity for a product of a complete symmetric function and a Macdonald polynomial;

(ii) a complete symmetric function generalization of KNPV’s result.

Keywords: Constant term identities, Selberg integrals, q-Morris identity, q-Baker–Forrester

conjecture, Macdonald polynomials, complete symmetric functions.

1. Introduction

In 1944, Atle Selberg [21] gave the following remarkable multiple integral:

· · · Z1

i=1

xα−1

i(1 −xi)β−1Y

1≤i<j≤n

|xi−xj|2γdx1· · · dxn

n−1

j=0

Γ(α+jγ)Γ(β+jγ)Γ(1 + (j+ 1)γ)

Γ(α+β+ (n+j−1)γ)Γ(1 + γ),

where α, β, γ are complex parameters such that

Re(α)>0,Re(β)>0,Re(γ)>−min{1/n, Re(α)/(n−1),Re(β)/(n−1)}.

When n= 1, the above Selberg integral reduces to the Euler beta integral. For the importance

of the Selberg integral, one can refer [6].

It is well-known that the Selberg integral is equivalent to the Morris constant term identity

[19]

(1.1) CT

i=1

(1 −x0/xi)a(1 −xi/x0)bY

1≤i6=j≤n

(1 −xi/xj)c=

n−1

i=0

(a+b+ic)!(i+ 1)c!

(a+ic)!(b+ic)!c!

Date: June 24, 2022.

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

*Corresponding author.

arXiv:2210.13469v1 [math.CO] 24 Oct 2022

2 GUOCE XIN AND YUE ZHOU*

for nonnegative integers a, b, c, where CT

xdenotes taking the constant term with respect to

x:= (x1, . . . , xn). Note that we can set x0= 1 in (1.1) without changing the constant term,

since we take the constant term in a homogeneous Laurent polynomial. The same principle

applies to all the homogeneous Laurent polynomials in this paper. In his Ph.D. thesis [19],

Morris also conjectured the following q-analogue constant term identity

(1.2) CT

i=1

(x0/xi)a(qxi/x0)bY

1≤i<j≤n

(xi/xj)c(qxj/xi)c=

n−1

i=0

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

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

where (z)c= (z;q)c:= (1 −z)(1 −zq)· · · (1 −zqc−1) is the q-factorial for a positive integer

cand (z)0:= 1. In 1988, Habsieger [10] and Kadell [12] independently proved Askey’s

conjectured q-analogue of the Selberg integral [3]. Expressing their q-analogue integral as a

constant term identity they thus proved Morris’ q-constant term conjecture (1.2). Now the

constant term identity (1.2) is called the Habsieger–Kadell q-Morris identity.

Recently, Albion, Rains and Warnaar obtained an AFLT type q-Selberg integral, which

is a q-Selberg integral over a pair of two Macdonald polynomials [2]. Their integral gives

aq-analogue of the AFLT Selberg integral [1], which arising from AGT conjecture. By

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

term identities [19], we obtain an equivalent constant term identity, see (1.4) below. For

nonnegative integers a, b, c and partitions λand µ, denote

(1.3) An(a, b, c, λ, µ) := CT

xx−|λ|−|µ|

0Pλ(x1, . . . , xn;q, qc)Pµhqc−b−1−qa

1−qcx0+

i=1

xii;q, qc

i=1

(x0/xi)a(qxi/x0)bY

1≤i<j≤n

(xi/xj)c(qxj/xi)c,

where Pλ(x;q, t) is the Macdonald polynomial, |λ|:= P∞

i=1 λiis the size of λ, and f[t+z] is

plethystic notation for the symmetric function f. Then, for `(λ)≤n

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

(1.4)

i=1

`(µ)

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 n(λ) := Pn

i=1(i−1)λiand `(λ) is the length of the partition λ. If b=c−a−1, the

constant term identity (1.4) reduces to [23, Theorem 1.7]. Note that if `(λ)> n then both

sides of (1.4) reduces to zero.

SYMMETRIC FUNCTION GENERALIZATIONS OF THE q-BAKER–FORRESTER EX-CONJECTURE 3

For nonnegative integers n, n0, a, b, m and a positive integer c, denote

(1.5)

Fn,n0(x;a, b, c, m) =

i=1

(x0/xi)a(qxi/x0)b+χ(i>n−m)Y

1≤i<j≤n

(xi/xj)c−χ(i≤n0)(qxj/xi)c−χ(i≤n0),

where χ(true) = 1 and χ(false) = 0. Intimately related to the theory of random matrices,

Baker and Forrester [4] conjectured the following extension of the Habsieger–Kadell q-Morris

identity:

(1.6) CT

xFn,n0(x;a, b, c, 0)

n−n0−1

i=1

(1 −q(i+1)c)

n−1

i=0

(qa+ic−n0−χ(i≤n0)(i−n0)+1)b(q)(i+1)c−n0−χ(i≤n0)(i−n0)−1

(q)b+ic−n0−χ(i≤n0)(i−n0)(q)c−χ(i≤n0)

Using Combinatorial Nullstellensatz, K´arolyi, Nagy, Petrov and Volkov [13] obtained a con-

stant term identity for Fn,n0(x;a, b, c, m) (see the l= 0 case of Theorem 1.2 below), which

extends the original q-Baker–Forrester conjecture (1.6) by adding the new parameter m.

Let X= (x0, x1, x2, . . . ) be an alphabet of countably many variables. Then for nonnegative

integer l, the l-th complete symmetric function hl(X) may be deﬁned in terms of its generating

function as

(1.7) X

l≥0

zlhl(X) = Y

i≥0

1−zxi

For la nonnegative integer and µa partition, let

(1.8) Bn,n0(a, b, c, l, µ) := CT

xx−l−|µ|

0Fn,n0(x;a, b, c, 0) ×hlhn

i=1

1−qc−χ(i≤n0)

1−qxii

×Pµhqc−b−1−qa

1−qcx0+

i=1

1−qc−χ(i≤n0)

1−qcxii;q, qc.

Our ﬁrst result is the next q-Baker–Forrester type constant term identity for a product of a

complete symmetric function and a Macdonald polynomial.

4 GUOCE XIN AND YUE ZHOU*

Theorem 1.1. Let Bn,n0(a, b, c, l, µ)be deﬁned in (1.8). Then for 0≤n0< n and `(µ)<

n−n0,

Bn,n0(a, b, c, l, µ) = (−1)l+|µ|q(l

2)+Pn

i=1 (µi

2)−cn(µ)hlh1−qnc−n0

1−qi×(qa−l+1)l

(qn0(c−1)+b+1)l

(1.9)

×Pµh1−qa+b+(n−1)c−n0+1

1−qci;q, qc

n−1

j=0

(qa+jc−n0−χ(j≤n0)(j−n0)+1)b(q)(j+1)c−n0−χ(j≤n0)(j−n0)−1

(q)b+jc−n0−χ(j≤n0)(j−n0)(q)c−χ(j≤n0)

n−n0−1

j=1

(1 −q(j+1)c)(qjc−b−µj)µj(q(n−j−1)c−n0+b+µj+1)l

(q(n−j)c+b+1−n0)uj+l

Note that if n0≥nthen this forces µ= 0 by the restriction `(µ)< n −n0. Then it is easy

to see that Bn,n0(a, b, c, l, 0) = Bn,0(a, b, c −1, l, 0) for n0≥n.

For la nonnegative integer, deﬁne

(1.10) Cn,n0(a, b, c, l, m) := CT

xx−l

0hlhn

i=1

1−qc−χ(i≤n0)

1−qxiiFn,n0(x;a, b, c, m).

Our second result is the next complete symmetric function generalization of the result of

K´arolyi et al.

Theorem 1.2. Let Cn,n0(a, b, c, l, m)be deﬁned in (1.10). For m≥n−n0,

(1.11) Cn,n0(a, b, c, l, m) = (−1)lq(l

2)hlh1−qnc−n0

1−qi(qa−l+1)l

(q(n−1)c−n0+b+2)l

n−n0

j=2

(1 −qjc)

n−1

j=0

(qa+j(c−1)+χ(j>n0)(j−n0)+1)b+χ(j≥n−m)(q)(j+1)(c−1)+χ(j>n0)(j−n0)

(q)b+j(c−1)+χ(j>n0)(j−n0)+χ(j≥n−m)(q)c−χ(j≤n0)

Note that by the deﬁnition of Cn,n0(a, b, c, l, m), for m≥nwe have Cn,n0(a, b, c, l, m) =

Cn,n0(a, b + 1, c, l, 0) = Bn,n0(a, b + 1, c, l, 0). One can also notice that Cn,n0(a, b, c, l, m) =

Cn,0(a, b, c −1, l, m) for n0≥n. The l= 0 case of Theorem 1.2 reduces to the result of

K´arolyi et al [13, Theorem 6.2].

The method employed to prove Theorems 1.1 and 1.2 is based on the Gessel–Xin method,

which ﬁrst appeared in [9] to prove the Zeilberger–Bressoud q-Dyson theorem [27]. We

managed to extend the Gessel–Xin method in the proof of the ﬁrst-layer formulas for the

q-Dyson product [16] and in dealing with the q-Dyson orthogonality problem [28]. The basic

idea of the Gessel–Xin method is 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 to prove our main results (1.9) and (1.11).

(1) Polynomiality. It is routine to show that the constant terms Bn,n0(a, b, c, l, µ) and

Cn,n0(a, b, c, l, m) (we refer these two constant terms as Band Cfor short respectively

SYMMETRIC FUNCTION GENERALIZATIONS OF THE q-BAKER–FORRESTER EX-CONJECTURE 5

in the following of this section) are polynomials in qa, assuming that all parameters

but aare ﬁxed. See Corollary 5.2. Then, we can extend the deﬁnitions of Band C

for negative a, especially negative integers.

(2) Rationality. By a rationality result, see Corollary 5.4 below, it suﬃces to prove (1.9)

and (1.11) for suﬃciently large c.

(3) Determination of roots. Let

B1=C1={−ic +n0−χ(i≤n0)(n0−i)−1,...,(1.12)

−ic +n0−χ(i≤n0)(n0−i)−b|i= 0,1, . . . , n −1},

B2=C2={l−1, l −2,...,0},

B3={−(n−j)c+n0−b−1,...,−(n−j)c+n0−b−µj|j= 1,2, . . . , `(µ)}.

Suppose c>b+µ1, then all the elements of B1∪B2∪B3are distinct. Let

(1.13) C3={−ic +n0−χ(i≤n0)(n0−i)−b−1|i=n−m, . . . , n −1}.

Suppose c > b + 1, then all the elements of C1∪C2∪C3are distinct.

For Band Cviewed as polynomials in qa, we will determine all their roots. Explic-

itly, Band Cvanish only when a∈B1∪B2∪B3and a∈C1∪C2∪C3respectively.

Note that B1, B3, C1, C3are sets of negative integers. For Band Cat a negative in-

teger a, we are actually concerned with constant terms of rational functions. Hence,

we need the theory of the ﬁeld of iterated Laurent series, which was developed by

Xin [24, 25].

(4) Value at an additional point. We obtain explicit expressions for Band Cat

a=−n0(c−1) −b−1 and a=−(n−m−1)(c−1) −b−1 respectively.

We can uniquely determine the closed-form expressions for Band Cby the above steps.

The steps (1) and (2) are routine but not trivial. The step (3) is quite lengthy but concep-

tually simple. In the step (4) we reduce Band Cto similar types. When carried out the

details, we made a breakthrough to the Gessel–Xin method by mixing the ideas of the orig-

inal Gessel–Xin method, plethystic substitutions and the splitting formula for the q-Dyson

product by Cai [5]. We will explain this below.

Let the degree of a rational function of xbe the degree of the numerator in xminus the

degree in xof the denominator. The original Gessel–Xin method [9] is a constant term

method to rational functions with negative degrees. We extended it to rational functions

with non-positive degrees to obtain the ﬁrst-layer formulas for the q-Dyson product in [16].

We attempted to prove the Forrester conjecture in [8]. However, the method failed in general.

The main obstacle is dealing with constant terms of rational functions with positive degrees.

In this paper, we overcome this diﬃculty.

For the rational functions with positive degrees in this paper, we ﬁnd that they reduce to

Laurent polynomials under certain conditions, see Lemma 9.1. Then, using Cai’s idea, we

obtain similar splitting formulas for the q-Baker–Forrester type constant terms, see Propo-

sition 3.5. By the splitting formulas, we can conﬁrm whether the constant terms of those

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SYMMETRICFUNCTIONGENERALIZATIONSOFTHEq-BAKER{FORRESTEREX-CONJECTUREANDSELBERG-TYPEINTEGRALSGUOCEXINANDYUEZHOU*Abstract.Itiswell-knownthatthefamousSelbergintegralisequivalenttotheMorrisconstanttermidentity.In1998,BakerandForresterconjecturedageneralizationoftheq-Morrisconstanttermidentity.Thisconject...

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SYMMETRIC FUNCTION GENERALIZATIONS OF THE q-BAKERFORRESTER EX-CONJECTURE AND SELBERG-TYPE INTEGRALS

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