Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn
We present explicit formulas for the Macdonald polynomials of types Cn and Dn in the one-row case. In view of the combinatorial structure, we call them ''tableau formulas''. For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeo...
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| Цитувати: | Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn / B. Feigin, A. Hoshino, M. Noumi, J. Shibahara, J. Shiraishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1471632019-02-14T01:25:02Z Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn Feigin, B. Hoshino, A. Noumi, M. Shibahara, J. Shiraishi, J. We present explicit formulas for the Macdonald polynomials of types Cn and Dn in the one-row case. In view of the combinatorial structure, we call them ''tableau formulas''. For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ₁₂W₁₁ series. We remark that the correlation functions of the deformed W algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables. 2015 Article Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn / B. Feigin, A. Hoshino, M. Noumi, J. Shibahara, J. Shiraishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D52; 33D80 DOI:10.3842/SIGMA.2015.100 http://dspace.nbuv.gov.ua/handle/123456789/147163 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We present explicit formulas for the Macdonald polynomials of types Cn and Dn in the one-row case. In view of the combinatorial structure, we call them ''tableau formulas''. For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ₁₂W₁₁ series. We remark that the correlation functions of the deformed W algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables. |
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Article |
| author |
Feigin, B. Hoshino, A. Noumi, M. Shibahara, J. Shiraishi, J. |
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Feigin, B. Hoshino, A. Noumi, M. Shibahara, J. Shiraishi, J. Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Feigin, B. Hoshino, A. Noumi, M. Shibahara, J. Shiraishi, J. |
| author_sort |
Feigin, B. |
| title |
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn |
| title_short |
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn |
| title_full |
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn |
| title_fullStr |
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn |
| title_full_unstemmed |
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn |
| title_sort |
tableau formulas for one-row macdonald polynomials of types cn and dn |
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Інститут математики НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/147163 |
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Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn / B. Feigin, A. Hoshino, M. Noumi, J. Shibahara, J. Shiraishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 14 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-11T01:30:27Z |
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2025-07-11T01:30:27Z |
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1837312175790620672 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 100, 21 pages
Tableau Formulas for One-Row Macdonald
Polynomials of Types Cn and Dn
?
Boris FEIGIN †1, Ayumu HOSHINO †2, Masatoshi NOUMI †
3
, Jun SHIBAHARA †4
and Jun’ichi SHIRAISHI †
5
†1 National Research University Higher School of Economics, International Laboratory
of Representation Theory and Mathematical Physics, Moscow, Russia
E-mail: borfeigin@gmail.com
†2 Kagawa National College of Technology, 355 Chokushi-cho, Takamatsu,
Kagawa 761-8058, Japan
E-mail: hoshino@t.kagawa-nct.ac.jp
†3 Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
E-mail: noumi@math.kobe-u.ac.jp
†4 Hamamatsu University School of Medicine, 1-20-1 Handayama, Higashi-ku,
Hamamatsu city, Shizuoka 431-3192, Japan
E-mail: s070521math@yahoo.co.jp
†5 Graduate School of Mathematical Sciences, University of Tokyo, Komaba,
Tokyo 153-8914, Japan
E-mail: shiraish@ms.u-tokyo.ac.jp
Received May 19, 2015, in final form November 26, 2015; Published online December 05, 2015
http://dx.doi.org/10.3842/SIGMA.2015.100
Abstract. We present explicit formulas for the Macdonald polynomials of types Cn and Dn
in the one-row case. In view of the combinatorial structure, we call them “tableau formulas”.
For the construction of the tableau formulas, we apply some transformation formulas for the
basic hypergeometric series involving very well-poised balanced 12W11 series. We remark
that the correlation functions of the deformed W algebra generators automatically give rise
to the tableau formulas when we principally specialize the coordinate variables.
Key words: Macdonald polynomials; deformed W algebras
2010 Mathematics Subject Classification: 33D52; 33D80
1 Introduction
I.G. Macdonald introduced the symmetric polynomials Pλ(x; q, t) as a (q, t)-deformation of the
Schur polynomials sλ(x). Then he extended this construction to the cases of the symmetric Lau-
rent polynomials invariant under the actions of the Weyl groups of simple root systems [10]. For
type An, he gave an explicit combinatorial formula for Pλ(x; q, t), usually called the “tableau
formula”. In [1] it was shown that the tableau formula for Pλ(x; q, t) of type An can be in-
terpreted as certain specialization of the correlation functions of the deformed W algebras of
type An. One of our motivations is to explore a little further the correspondence between the
Macdonald polynomials and the deformed W algebras associated with simple root systems.
More precisely, we calculate the correlation functions of W algebras of types Cn and Dn with
?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
mailto:borfeigin@gmail.com
mailto:hoshino@t.kagawa-nct.ac.jp
mailto:noumi@math.kobe-u.ac.jp
mailto:s070521math@yahoo.co.jp
mailto:shiraish@ms.u-tokyo.ac.jp
http://dx.doi.org/10.3842/SIGMA.2015.100
http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
principal specializations in coordinate variables (Definition 7.8). As a result, we obtain certain
combinatorial expressions, which we regard as tableau formulas. Then, on the basis of Lassalle’s
explicit formulas [8] (see [4] for a proof and their generalization), we prove that they are actually
the Macdonald polynomials of types Cn and Dn in the one-row case (Theorem 7.10).
We need to recall briefly the Kashiwara–Nakashima tableaux. In the study of quantum
algebras [5], they gave combinatorial descriptions of the crystal bases for the integrable highest
weight representations by using the “semi-standard tableaux” of respective types. For the crystal
bases of the symmetric tensor representations V (rΛ1) of types Cn and Dn, the semi-standard
tableau is defined to be a one-row diagram (r) of size r filled with entries in the ordered set
I = {1, 2, . . . , n, n, n− 1, . . . , 1} being arranged in the weakly increasing manner. The orderings
of I are defined by (6.1) for Cn and by (4.1) for Dn respectively. For a semi-standard tableau
of shape (r), denote by θi the number of the letter i ∈ I in the tableau. Then we have θ1 + θ2 +
· · ·+ θn + θn + θn−1 + · · ·+ θ1 = r. For type Dn, we have an aditional condition θnθn = 0 due
to the structure of the ordering.
We now present the main results of this paper (Theorems 4.3, 6.2 and 6.5). Let P
(Cn)
(r) (x; q, t, T )
and P
(Dn)
(r) (x; q, t) be the Macdonald polynomials of types Cn and Dn respectively attached to
a single row (r). As for the notation, see Appendix A.2.
Theorem 1.1. We have the following tableau formulas in the one-row cases:
P
(Cn)
(r)
(
x; q, t, t2/q
)
=
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ1=r
∏
k∈I
(t; q)θk
(q; q)θk
×
∏
1≤l≤n
(tn−l+1qθl+···+θl+1 ; q)θl(t
n−l+2qθl+1+···+θl+1−1; q)θl
(tn−l+2qθl+···+θl+1−1; q)θl(t
n−l+1qθl+1+···+θl+1 ; q)θl
× xθ1−θ11 x
θ2−θ2
2 · · ·xθn−θnn , (1.1)
P
(Dn)
(r) (x; q, t) =
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ
1
=r
θnθn=0
∏
k∈I
(t; q)θk
(q; q)θk
×
∏
1≤l≤n−1
(tn−l−1qθl+θl+1+···+θl+1+1; q)θl(t
n−lqθl+1+θl+2+···+θl+1 ; q)θl
(tn−lqθl+θl+1+···+θl+1 ; q)θl(t
n−l−1qθl+1+θl+2+···+θl+1+1; q)θl
× xθ1−θ11 x
θ2−θ2
2 · · ·xθn−θnn . (1.2)
Here and hereafter, we use the standard notation of q-shifted factorials
(z; q)∞ =
∞∏
k=0
(1− qkz), (z; q)k =
(z; q)∞
(qkz; q)∞
, k ∈ Z,
(a1, a2, . . . , ar; q)k = (a1; q)k(a2; q)k · · · (ar; q)k, k ∈ Z.
Remark 1.2. These tableau formulas for P
(Cn)
(r) (x; q, t, t2/q) and P
(Dn)
(r) (x; q, t) are obtained by
principally specializing the correlation functions of the deformedW algebras of types Cn and Dn
respectively. See Theorem 7.10.
We can extend the tableau formula of type Cn in (1.1) as follows to general q, t and T .
Theorem 1.3. Set θ := min(θn, θn) for simplicity of display. We have
P
(Cn)
(r) (x; q, t, T ) =
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ1=r
∏
k∈I\{n,n}
(t; q)θk
(q; q)θk
(t; q)|θn−θn|
(q; q)|θn−θn|
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 3
×
∏
1≤l≤n−1
(
(tn−l−1qθl+···+θn−1+|θn−θn|+θn−1+···+θl+1+1; q)θl
(tn−lqθl+···+θn−1+|θn−θn|+θn−1+···+θl+1 ; q)θl
×
(tn−lqθl+1+···θn−1+|θn−θn|+θn−1+···+θl+1 ; q)θl
(tn−l−1qθl+1+···+θn−1+|θn−θn|+θn−1+···+θl+1+1; q)θl
)
× (T ; q)θ(t
nqr−2θ; q)2θ
(q; q)θ(Ttn−1qr−θ; q)θ(tn−1qr−2θ+1; q)θ
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn . (1.3)
Remark 1.4. At present, we do not know any W algebra which explains the formula (1.3).
There are several combinatorial expressions for the Macdonald polynomials studied from
some different points of view. See [9, 13, 14] for example. It would be an intriguing problem to
find possible connections between those formulas and ours obtained in this paper.
This paper is organized as follows. In Sections 2, 3 and 5 we construct the transformation
formulas for the basic hypergeometric series for proving our tableau formulas. In Sections 4
and 6, we prove the tableau formulas for the one-row Macdonald polynomials of types Cn
and Dn respectively. In Section 7, we recall the deformed W algebras of types Cn and Dn,
and then prove that the correlation functions with principal specialization give us the tableau
formulas for the Macdonald polynomials of types Cn and Dn in the one-row case respectively.
In Appendix A, we recall briefly the Koornwinder polynomials, and the Macdonald polynomials
of types Cn and Dn as degenerations of the Koornwinder polynomials.
Throughout this paper, we use the standard notation for the basic hypergeometric series as
r+1φr
[
a1, a2, . . . , ar+1
b1, . . . , br
; q, z
]
=
∞∑
n=0
(a1, a2, . . . , ar+1; q)n
(q, b1, b2, . . . , br; q)n
zn,
r+1Wr(a1; a4, a5, . . . , ar+1; q, z) = r+1φr
[
a1, qa
1/2
1 ,−qa1/2
1 , a4, . . . , ar+1
a
1/2
1 ,−a1/2
1 , qa1/a4, . . . , qa1/ar+1
; q, z
]
.
We call the r+1Wr series very well-poised basic hypergeometric series. Moreover, we call a r+1Wr
series very well-poised balanced when it satisfies the balancing condition (a4a5 · · · ar+1)z =(
±(a1q)
1
2
)r−3
.
2 Transformation formula I
In this section, we give a transformation formula of basic hypergeometric series. We show that
a very well-poised balanced 12W11 series is transformed to a 4φ3 series which is neither balanced
nor well-poised. Recall the following proposition:
Proposition 2.1 ([12, Proposition 7.3]). We have for r, θ ∈ Z≥0
r+9Wr+8
(
a; q−θ, qθaf, a1, . . . , ar,
(
aq
f
) 1
2
,−
(
aq
f
) 1
2
,
(
aq2
f
) 1
2
,−
(
aq2
f
) 1
2
; q, z
)
(2.1)
=
(aq, f2/q; q)θ
(af, f ; q)θ
∑
m≥0
(q/f, q−θ, aq/f ; q)m
(q, q−θq2/f2, aq; q)m
qm r+5Wr+4
(
a; q−m, qmaq/f, a1, . . . , ar; q, z
)
.
The main result in this section is as follows:
4 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
Theorem 2.2. Assume af = a2a3, then
12W11
(
a; q−θ, qθaf, f, a2, a3,
(
aq
f
) 1
2
,−
(
aq
f
) 1
2
,
(
aq2
f
) 1
2
,−
(
aq2
f
) 1
2
; q, q/f
)
=
(aq, af/a2; q)θ
(af, aq/a2; q)θ
4φ3
[
q−θ, q−θa2/a, f, a2
q−θ+1a2/af, q−θ+1/f, aq/a3
; q, q2/f2
]
. (2.2)
Remark 2.3. By one of the anonymous referees, it was pointed out that (2.2) is a special case
of the transformation formula obtained by Langer, Schlosser and Warnaar [7, equation (4.2)].
Namely, we have (2.2) by letting d→ 0 (or d→∞) in the p = 0 case of [7, equation (4.2)]. The
authors thank the referee for informing them of this fact.
Remark 2.4. The 4φ3 series in the right hand side of (2.2) is neither balanced nor well-poised.
However it has the following structure: set for simplicity u1 := q−θ, u2 := q−θa2/a, u3 := f ,
u4 := a2, v1 := q, v2 := q−θ+1a2/af , v3 := q−θ+1/f and v4 := aq/a3. Then we have
(i) u1v1 = u3v3 = q−θ+1 and u2v4 = u4v2 = q−θ+1a2/a3,
(ii) u1v4 = u4v3 = aq−θ+1/a3 and u2v1 = u3v2 = q−θ+1a2/a.
Proof of Theorem 2.2. Applying (2.1) to the left hand side of (2.2), we have
l.h.s. of (2.2) =
(aq, f2/q; q)θ
(af, f ; q)θ
×
∑
m≥0
(q/f, q−θ, aq/f ; q)m
(q, q−θ+2/f2, aq; q)m
qm 8W7
(
a; q−m, aqm+1/f, f, a2, a3; q, q/f
)
.(2.3)
We apply Watson’s formula [3, p. 35, equation (2.5.1)]
8φ7
[
a, qa
1
2 ,−qa
1
2 , b, c, d, e, q−n
a
1
2 ,−a
1
2 , aq/b, aq/c, aq/d, aq/e, aqn+1
; q,
a2q2+n
bcde
]
=
(aq, aq/de; q)n
(aq/d, aq/e; q)n
4φ3
[
q−n, d, e, aq/bc
aq/b, aq/c, deq−n/a
; q, q
]
, (2.4)
to the right hand side of (2.3) with the substitutions b = aqm+1/f , c = a3, d = a2 and e = f .
Then we have
r.h.s. of (2.3) =
(aq, f2/q; q)θ
(af, f ; q)θ
∑
m≥0
(q−θ, aq/a2f, q/f ; q)m
(q−θ+2/f2, aq/a2, q; q)m
qm
×
m∑
j=0
(f, a2, q
−mf/a3, q
−m; q)j
(aq/a3, q−ma2f/a, q−mf, q; q)j
qj . (2.5)
Now we need to change the order of the summation. Setting s := m− j and using the condition
af = a2a3, we have
r.h.s. of (2.5) =
(aq, f2/q; q)θ
(af, f ; q)θ
θ∑
j=0
(q−θ, f, a2; q)j
(q−θ+2/f2, aq/a3, q; q)j
(q/f)2j
×
θ−j∑
s=0
(aq/a2f, q/f, q
−θ+j ; q)s
(aq/a2, q, q−θ+j+2/f2; q)s
qs. (2.6)
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 5
As a final step, we apply the q-Saalschütz transformation formula [3, p. 13, equation (1.7.2)] to
the summation with respect to s of (2.6). Then we have
r.h.s. of (2.6) =
(aq, q; q)θ
(af, f ; q)θ
θ∑
j=0
(af/a2, f ; q)θ−j
(aq/a2, q; q)θ−j
(f, a2; q)j
(q, aq/a3; q)j
=
(aq, af/a2; q)θ
(af, aq/a2; q)θ
4φ3
[
q−θ, q−θa2/a, f, a2
q−θ+1a2/af, q−θ+1/f, aq/a3
; q, q2/f2
]
.
This completes the proof of Theorem 2.2. �
In what follows, we use Theorem 2.2 in the form
12W11
(
a; q−θ, qθaf, f, a2, a3,
(
aq
f
) 1
2
,−
(
aq
f
) 1
2
,
(
aq2
f
) 1
2
,−
(
aq2
f
) 1
2
; q, q/f
)
=
(aq, q; q)θ
(af, f ; q)θ
θ∑
j=0
(af/a2, f ; q)θ−j
(aq/a2, q; q)θ−j
(f, a2; q)j
(q, aq/a3; q)j
. (2.7)
3 Transformation formula II
In this section, we present a transformation formula which will be used to describe the Macdonald
polynomials of types Dn and Cn.
Theorem 3.1. Let n ∈ Z≥2. Fix K,m1,m2, . . . ,mn ∈ Z≥0 arbitrarily. Set ml,n :=
n∑
k=l
mk,
φl,n :=
n∑
k=l
φk for simplicity of display. We have
∑
φ1,φ2,...,φn−1,i≥0
φ1+φ2+···+φn−1+i=K
( ∏
1≤l≤n−1
(t; q)φl(t; q)φl+ml
(q; q)φl(q; q)φl+ml
×
(tn−l−1qφl+2φl+1,n−1+ml,n+1; q)φl(t
n−lq2φl+1,n−1+ml+1,n ; q)φl
(tn−lqφl+2φl+1,n−1+ml,n ; q)φl(t
n−l−1q2φl+1,n−1+ml+1,n+1; q)φl
)
× (t; q)mn
(q; q)mn
(t; q)i(t
nq2K+m1,n−2i; q)i
(q; q)i(tn−1q2K+m1,n−2i+1; q)i
=
∑
φ1,φ2,...,φn−1,φn≥0
φ1+φ2+···+φn=K
∏
1≤j≤n
(t; q)φj (t; q)φj+mj
(q; q)φj (q; q)φj+mj
. (3.1)
We prove Theorem 3.1 by induction on n. In order to clarify the structure of our proof, we
first confirm the case n = 2 in Section 3.1, and then treat the general case in Section 3.2.
3.1 The case n = 2
Proposition 3.2. Fix K,m1,m2 ∈ Z≥0 arbitrarily. We have∑
φ1,i≥0
φ1+i=K
(t; q)m1+φ1(t; q)m2(t, qm1+m2+φ1+1, tqm2 ; q)φ1(t, t2q2K+m1+m2−2i; q)i
(q; q)m1+φ1(q; q)m2(q, tqm1+m2+φ1 , qm2+1; q)φ1(q, tq2K+m1+m2−2i+1; q)i
=
∑
φ1,φ2≥0
φ1+φ2=K
(t; q)m1+φ1(t; q)m2+φ2(t; q)φ1(t; q)φ2
(q; q)m1+φ1(q; q)m2+φ2(q; q)φ1(q; q)φ2
. (3.2)
6 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
Proof. One finds that the summation in l.h.s. of (3.2) with respect to φ1 is given by the
following 12W11 series
l.h.s. of (3.2) =
(t, t2qm1+m2 ; q)K(t; q)m1(t; q)m2
(q, tqm1+m2+1; q)K(q; q)m1(q; q)m2
× 12W11
(
a; q−K , qKaf, f, b, c,
(
aq
f
) 1
2
,−
(
aq
f
) 1
2
,
(
aq2
f
) 1
2
,−
(
aq2
f
) 1
2
; q, q/f
)
, (3.3)
where a = tqm1+m2 , b = tqm1 , c = tqm2 , f = t. Then applying formula (2.7), the r.h.s. of (3.3)
is rewritten as
(t; q)m1(t; q)m2
(q; q)m1(q; q)m2
K∑
φ1=0
(tqm2 , t; q)K−φ1
(qm2+1, q; q)K−φ1
(t, tqm1 ; q)φ1
(q, qm1+1; q)φ1
=
K∑
φ1=0
(t; q)m1+φ1(t; q)m2+K−φ1(t; q)φ1(t; q)K−φ1
(q; q)m1+φ1(q; q)m2+K−φ1(q; q)φ1(q; q)K−φ1
.
This completes the proof of Proposition 3.2. �
3.2 The general case
Assume the validity of the transformation formula (3.1) for n− 1. We have
l.h.s. of (3.1) =
K∑
φn−1=0
K−φn−1∑
φn−2=0
· · ·
K−φ3,n−1∑
φ2=0
∏
2≤l≤n−1
(t; q)φl(t; q)φl+ml
(q; q)φl(q; q)φl+ml
×
(tn−l−1qφl+2φl+1,n−1+ml,n+1; q)φl(t
n−lq2φl+1,n−1+ml+1,n ; q)φl
(tn−lqφl+2φl+1,n−1+ml,n ; q)φl(t
n−l−1q2φl+1,n−1+ml+1,n+1; q)φl
×
K−φ2,n−1∑
φ1=0
(t; q)φ1(t; q)φ1+m1(t; q)mn(t, tnq2φ1,n−1+m1,n ; q)K−φ1,n−1
(q; q)φ1(q; q)φ1+m1(q; q)mn(q, tn−1q2φ1,n−1+m1,n+1; q)K−φ1,n−1
×
(tn−2qφ1+2φ2,n−1+m1,n+1; q)φ1(tn−1q2φ2,n−1+m2,n ; q)φ1
(tn−1qφ1+2φ2,n−1+m1,n ; q)φ1(tn−2q2φ2,n−1+m2,n+1; q)φ1
. (3.4)
The summation with respect to φ1 in (3.4) can be written as follows
(t; q)m1(t; q)mn(t, tnq2φ2,n−1+m1,n ; q)K−φ2,n−1
(q; q)m1(q; q)mn(q, tn−1q2φ2,n−1+m1,n+1; q)K−φ2,n−1
× 12W11
(
a; q−θ, qθaf, f, b, c,
(
aq
f
) 1
2
,−
(
aq
f
) 1
2
,
(
aq2
f
) 1
2
,−
(
aq2
f
) 1
2
; q, q/f
)
, (3.5)
where a = tn−1q2φ2,n−1+m1,n , b = tqm1 , c = tn−1q2φ2,n−1+m2,n , f = t, θ = K − φ2,n−1. Applying
the formula (2.7), (3.5) is transformed into
(t; q)m1(t; q)mn
(q; q)m1(q; q)mn
×
K−φ2,n−1∑
j=0
(tn−1q2φ2,n−1+m2,n ; q)K−j−φ2,n−1(t; q)K−j−φ2,n−1(t; q)j(tq
m1 ; q)j
(tn−2q2φ2,n−1+m2,n+1; q)K−j−φ2,n−1(q; q)K−j−φ2,n−1(q; q)j(qm1+1; q)j
. (3.6)
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 7
Using (3.6) and changing the order of the summations, we can express the r.h.s. of (3.4) as
follows
K∑
j=0
(t; q)j(t; q)j+m1
(q; q)j(q; q)j+m1
K−j∑
φn−1=0
K−j−φn−1∑
φn−2=0
· · ·
K−j−φ4,n−1∑
φ3=0
∏
3≤l≤n−1
(t; q)φl(t; q)φl+ml
(q; q)φl(q; q)φl+ml
×
(tn−l−1qφl+2φl+1,n−1+ml,n+1; q)φl(t
n−lq2φl+1,n−1+ml+1,n ; q)φl
(tn−lqφl+2φl+1,n−1+φl,n ; q)φl(t
n−l−1q2φl+1,n−1+ml+1,n+1; q)φl
×
K−j−φ3,n−1∑
φ2=0
(t; q)φ2+m2(t; q)mn(t; q)φ2(tn−1q2φ2,n−1+m2,n−1 ; q)K−j−φ2,n−1(t; q)K−j−φ2,n−1
(q; q)φ2+m2(q; q)mn(q; q)φ2(tn−2q2φ2,n−1+m2,n+1; q)K−j−φ2,n−1(q; q)K−j−φ2,n−1
×
(tn−2q2φ3,n−1+m3,n ; q)φ2(tn−3qφ2+2φ3,n−1+m2,n+1; q)φ2
(tn−3q2φ3,n−1+m3,n+1; q)φ2(tn−2qφ2+2φ3,n−1+m2,n ; q)φ2
. (3.7)
By the induction hypothesis, (3.7) can be rewritten as
K∑
j=0
(t; q)j(t; q)j+m1
(q; q)j(q; q)j+m1
∑
φ2+φ3+···+φn−1+φ=K−j
(t; q)φ(t; q)φ+mn
(q; q)φ(q; q)φ+mn
∏
2≤l≤n−1
(t; q)φl(t; q)φl+ml
(q; q)φl(q; q)φl+ml
=
∑
j+φ2+φ3+···+φn−1+φ=K
(t; q)j(t; q)j+m1
(q; q)j(q; q)j+m1
(t; q)φ(t; q)φ+mn
(q; q)φ(q; q)φ+mn
∏
2≤l≤n−1
(t; q)φl(t; q)φl+ml
(q; q)φl(q; q)φl+ml
= r.h.s. of (3.1).
Hence we have completed the proof of Theorem 3.1.
4 Tableau formulas for Macdonald polynomials of type Dn
In this section, we investigate the tableau formula for the one-row Macdonald polynomials of
type Dn. Let I := {1, 2, . . . , n− 1, n, n, n− 1, . . . , 1} be the index set with the ordering
1 ≺ 2 ≺ · · · ≺ n− 1 ≺≺
n
n
≺
≺
n− 1 ≺ · · · ≺ 1. (4.1)
Denoting by Λ1 the first fundamental weight of type Dn, let P
(Dn)
(r) (x; q, t) be the Macdonald
polynomials of type Dn associated with the weights rΛ1 for r ∈ Z≥0.
We recall Lassalle’s formula for P
(Dn)
(r) (x; q, t). Lassalle introduced Gr(x; q, t) defined by the
generating function
n∏
i=1
(tuxi; q)∞
(uxi; q)∞
(tu/xi; q)∞
(u/xi; q)∞
=
∑
r≥0
Gr(x; q, t)ur. (4.2)
Comparing the coefficient of ur of the equation (4.2), we obtain
Gr(x; q, t) =
∑
θ1+θ2+···+θ1=r
∏
i∈I
(t; q)θi
(q; q)θi
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn , (4.3)
where θi, θi ∈ Z≥0, i = 1, 2, . . . , n.
The following theorem [4, Theorem 5.2] was conjectured by Lassalle [8].
8 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
Theorem 4.1 ([4, 8]). For any positive integer r we have
Gr(x; q, t) =
[r/2]∑
i=0
(t; q)r−2i
(q; q)r−2i
P
(Dn)
(r−2i)(x; q, t)
(t; q)i(t
nqr−2i; q)i
(q; q)i(tn−1qr−2i+1; q)i
. (4.4)
Conversely
P
(Dn)
(r) (x; q, t) =
(q; q)r
(t; q)r
[r/2]∑
i=0
Gr−2i(x; q, t)ti
(1/t; q)i(t
nqr−i; q)i
(q; q)i(tn−1qr−i; q)i
1− tnqr−2i
1− tnqr−i
. (4.5)
Remark 4.2. If we insert (4.3) in (4.5), we have an explicit combinatorial formula for
P
(Dn)
(r) (x; q, t). However, it is not clear how we can extract the combinatorics of the Kashiwara–
Nakashima tableaux of type D from this Lassalle’s version.
Here, we establish the tableau formula for P
(Dn)
(r) (x; q, t).
Theorem 4.3. We have
P
(Dn)
(r) (x; q, t) =
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ
1
=r
θnθn=0
∏
k∈I
(t; q)θk
(q; q)θk
(4.6)
×
∏
1≤l≤n−1
(tn−lqθl+1+θl+2+···+θl+1 ; q)θl(t
n−l−1qθl+θl+1+···+θl+1+1; q)θl
(tn−l−1qθl+1+θl+2+···+θl+1+1; q)θl(t
n−lqθl+θl+1+···+θl+1 ; q)θl
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn .
Remark 4.4. Set X := tn−lqθl+1+θl+2+···+θl+1 and Y := tn−l−1qθl+1+θl+2+···+θl+1 for simplicity.
The last product in (4.6) can be rewritten as
∏
1≤l≤n−1
(X; q)θl(q
θlY ; q)θl
(Y ; q)θl(q
θlX; q)θl
=
∏
1≤l≤n−1
(X; q)θl(X; q)θl(Y ; q)θl+θl
(Y ; q)θl(Y ; q)θl(X; q)θl+θl
.
This implies that r.h.s. of (4.6) has the symmetry (Z/2Z)n. Namely, it is invariant under the
exchange xl ↔ 1
xl
, 1 ≤ l ≤ n− 1.
Remark 4.5. It would be an intriguing problem to show the factorization of the Macdonald
polynomial P
(Dn)
(r) (x; q, t) from our formula (4.6) when we make the principal specialization:
P
(Dn)
(r)
(
tn−1, . . . , t, 1; q, t
)
= t−r(n−1) (tn; q)r(t
2(n−1); q)r
(t; q)r(t(n−1); q)r
.
Remark 4.6. Setting n = 1 in (4.6), we have
P
(D1)
(r) (x; q, t) = xr + x−r.
Setting n = 2 in (4.6), we have
P
(D2)
(r) (x; q, t) =
(q; q)r
(t; q)r
∑
θ1+θ2+θ2
+θ
1
=r
θ2θ2
=0
(t; q)θ1
(q; q)θ1
(t; q)θ2
(q; q)θ2
(t; q)θ2
(q; q)θ2
(t; q)θ1
(q; q)θ1
×
(tqθ2+θ2 ; q)θ1
(qqθ2+θ2 ; q)θ1
(qqθ1+θ2+θ2 ; q)θ1
(tqθ1+θ2+θ2 ; q)θ1
x
θ1−θ1
1 x
θ2−θ2
2
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 9
=
(
(q; q)r
(t; q)r
∑
µ1+µ2=r
(t; q)µ1
(q; q)µ1
(t; q)µ2
(q; q)µ2
x
(µ1−µ2)/2
1 x
−(µ1−µ2)/2
2
)
×
(
(q; q)r
(t; q)r
∑
ν1+ν2=r
(t; q)ν1
(q; q)ν1
(t; q)ν2
(q; q)ν2
x
(ν1−ν2)/2
1 x
(ν1−ν2)/2
2
)
,
which shows the symmetry D2 = A1 ×A1.
Proof of Theorem 4.3. Let
{
Ψ
(Dn)
(r) (x; q, t)
}
r∈Z≥0
be a certain collection of Laurent polyno-
mials. By using Lassalle’s formula (4.4), it is easily proved by induction that the infinite system
of equalities for Ψ
(Dn)
(r) (x; q, t)
Gr(x; q, t) =
[r/2]∑
i=0
(t; q)r−2i
(q; q)r−2i
Ψ
(Dn)
(r−2i)(x; q, t)
(t; q)i(t
nqr−2i; q)i
(q; q)i(tn−1qr−2i+1; q)i
, r ∈ Z≥0, (4.7)
gives us Ψ
(Dn)
(r) (x; q, t) = P
(Dn)
(r) (x; q, t), r ∈ Z≥0.
Set
Ψ
(Dn)
(r) (x; q, t) =
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ
1
=r
θnθn=0
∏
k∈I
(t; q)θk
(q; q)θk
×
∏
1≤l≤n−1
(tn−lqθl+1+θl+2+···+θl+1 ; q)θl(t
n−l−1qθl+θl+1+···+θl+1+1; q)θl
(tn−l−1qθl+1+θl+2+···+θl+1+1; q)θl(t
n−lqθl+θl+1+···+θl+1 ; q)θl
× xθ1−θ11 x
θ2−θ2
2 · · ·xθn−θnn .
We prove this family of Laurent polynomials satisfies (4.7). In view of the (Z/2Z)n symmetry
of Ψ
(Dn)
(r) (x; q, t), it is sufficient to consider in (4.7) the coefficients of the monomials xm1
1 · · ·xmnn
with nonnegative powers m1, . . . ,mn ∈ Z≥0 only. Let r ∈ Z≥0, and fix m1, . . . ,mn ∈ Z≥0
arbitrarily. Set K := 1
2(r −m1 −m2 − · · · −mn) for simplicity. Setting
θk = mk + φk, θk = φk, 1 ≤ k ≤ n− 1, θn = mn, θn = 0,
one finds that the coefficients of the monomials xm1
1 · · ·xmnn in (4.7) is exactly given by l.h.s.
of (3.1). On the other hand, the coefficients of the monomials xm1
1 · · ·xmnn in Gr(x; q, t) is clearly
r.h.s. of (3.1). Hence we have proved (4.7), which establishes the tableau formula P
(Dn)
(r) (x; q, t) =
Ψ
(Dn)
(r) (x; q, t). �
5 Transformation formula III
In this section, we present a transformation formula to describe the Macdonald polynomials of
type Cn.
Theorem 5.1. Let n ∈ Z≥2. Fix K,m1,m2, . . . ,mn ∈ Z≥0 arbitrarily. Set ml,n :=
n∑
k=l
mk,
φl,n :=
n∑
k=l
φk for simplicity of display. We have
∑
φ1,φ2,...,φn,i≥0
φ1+φ2+···+φn+i=K
∏
1≤k≤n
(t; q)φk(t; q)φk+mk
(q; q)φk(q; q)φk+mk
·
(
t2/q
)i (t−1q; q)i(t
nq2K+m1,n−2i; q)i
(q; q)i(tn+1q2K+m1,n−2i; q)i
10 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
×
∏
1≤l≤n
(tn−l+1qφl+φl+1,n+ml,n ; q)φl(t
n−l+2q2φl+1,n+ml+1,n−1; q)φl
(tn−l+2qφl+φl+1,n+ml,n−1; q)φl(t
n−l+1q2φl+1,n+ml+1,n ; q)φl
=
∑
φ1,φ2,...,φn≥0
φ1+φ2+···+φn=K
∏
1≤j≤n
(t; q)φj (t; q)φj+mj
(q; q)φj (q; q)φj+mj
. (5.1)
We prove Theorem 5.1 by induction on n. In Section 5.1 we show Theorem 5.1 in the case
of n = 2 and in Section 5.2 we treat the general case.
5.1 The case n = 2
Setting n = 2, we have
r.h.s. of (5.1) =
K∑
φ1=0
(t; q)m1+φ1(t; q)φ1(t; q)m2+K−φ1(t; q)K−φ1
(q; q)m1+φ1(q; q)φ1(q; q)m2+K−φ1(q; q)K−φ1
=
K∑
φ1=0
(t; q)m1(t; q)m2(tqm1 ; q)φ1(t; q)φ1(tqm2 ; q)K−φ1(t; q)K−φ1
(q; q)m1(q; q)m2(qm1+1; q)φ1(q; q)φ1(qm2+1; q)K−φ1(q; q)K−φ1
.
Then we have
l.h.s. of (5.1) =
K∑
φ2=0
K−φ2∑
φ1=0
(t; q)φ2(t; q)φ2+m2(t; q)φ1(t; q)φ1+m1
(q; q)φ2(q; q)φ2+m2(q; q)φ1(q; q)φ1+m1
×
(t2qm1+m2+2φ2+φ1 ; q)φ1(t3qm2+2φ2−1; q)φ1
(t3qm1+m2+2φ2+φ1−1; q)φ1(t2qm2+2φ2 ; q)φ1
×
(tqm2+φ2 ; q)φ2(t2q−1; q)φ2
(t2qm2−1+φ2 ; q)φ2(t; q)φ2
(t2/q)K−φ2−φ1
×
(q/t; q)K−φ2−φ1(t2qm1+m2+2φ2+2φ1 ; q)K−φ2−φ1
(q; q)K−φ2−φ1(t3qm1+m2+2φ2+2φ1 ; q)K−φ2−φ1
=
K∑
φ2=0
(t; q)m2+φ2(t; q)φ2(tqm2+φ2 ; q)φ2(t2q−1; q)φ2(q/t; q)K−φ2(t2qm1+m2+2φ2 ; q)K−φ2
(q; q)m2+φ2(q; q)φ2(tqm2−1+φ2 ; q)φ2(t; q)φ2(q; q)K−φ2(t3qm1+m2+2φ2 ; q)K−φ2
× 8φ7
[
a, qa
1
2 ,−qa
1
2 , b, c, d, e, qφ2−K
a
1
2 ,−a
1
2 , aq/b, aq/c, aq/d, aq/e, aqK−φ2+1
; q,
a2qK−φ2+2
bcde
]
, (5.2)
where a = t3qm1+m2−1+φ2 , b = tqm1 , c = t2qK+m1+m2+φ2 , d = t, e = t3qm2−1+2φ2 , a2qK−φ2+2
bcde =
q/t. Applying Watson’s transformation formula (2.4) to the 8φ7 series of (5.2), we have
8φ7 series of (5.2) =
(t3qm1+m2+2φ2 , t−1qm1+1; q)K−φ2
(t2qm1+m2+2φ2 , qm1+1; q)K−φ2
× 4φ3
[
qφ2−K−m2 , t, t3qm2−1+2φ2 , qφ2−K
t2qm2+2φ2 , tqφ2−K , tqφ2−K−m1
; q, q
]
. (5.3)
Using Sears’ transformation formulas [3, p. 242, Appendix III, equations (III.15) and (III.16)],
we rewrite the 4φ3 series in (5.3) as follows
4φ3
[
qφ2−K−m2 , t, t3qm2−1+2φ2 , qφ2−K
t2qm2+2φ2 , tqφ2−K , tqφ2−K−m1
; q, q
]
=
(qm1−K+φ2 , t−1q1−m2−K−φ2 ; q)K−φ2
(tq−m1−K−φ2 , t2q1−m2−K−φ2 ; q)K−φ2
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 11
× 4φ3
[
tqm1 , t, t−2q1−m2−K−φ2 , qφ2−K
tqφ2−K , t−1q1−m2−K−φ2 , qm1+1; q, q
]
. (5.4)
Sears’ transformation gives us
4φ3 series in (5.4) =
(t−2q1−m2−K−φ2 , q; q)K−φ2
(t−1q1−m2−K−φ2 , t−1q; q)K−φ2
× 4φ3
[
t, t2qm1+m2+k+φ2 , t−1q, qφ2−K
qm1+1, q, t2qm2
; q, q
]
, (5.5)
and
4φ3 series in (5.5) =
(t−1q, tqm2+2φ2 ; q)K−φ2
(q, t2qm2+2φ2 ; q)K−φ2
tK−φ2
× 4φ3
[
t, t−2q1−m2−K−φ2 , tqm1 , qφ2−K
qm1+1, tqφ2−K , t−1q1−m2−K−φ2 ; q, q
]
.
Then we have
l.h.s. of (5.2)
=
K∑
φ2=0
(t; q)m1(t)m2(t2qm2+φ2 ; q)φ2(t2q−1; q)φ2(t−1q; q)K−φ2(tqm2+2φ2 ; q)K−φ2
(q; q)m1(q; q)m2(q; q)φ2(qm2+1; q)φ2(t2qm2−1+φ2 ; q)φ2(q; q)K−φ2(t2qm2+2φ2 ; q)K−φ2
× 4φ3
[
t, t−2q1−m2−K−φ2 , tqm1 , qφ2−K
qm1+1, tqφ2−K , t−1q1−m2−K−φ2 ; q, q
]
=
K∑
φ1=0
(t; q)m1(t; q)m2(t−1q; q)K(tqm2 ; q)K(q−K , t−2q1−m2−K , t, tqm1 ; q)φ1
(q; q)m1(q; q)m2(q; q)K(t2qm2 ; q)K(tq−K , t−1q1−m2−K , q, qm1+1; q)φ1
×
K−φ1∑
φ2=0
(t2qm2−1, tq
m2+1
2 ,−tq
m2+1
2 , t2q−1, tqm2+K−φ1 , qφ1−K ; q)φ2
(q, tq
m2−1
2 ,−tq
m2−1
2 , qm2+1, tqφ1−K , t2qm2+K−φ1 ; q)φ2
(t2/q)K(q/t)φ2qφ1
=
K∑
φ1=0
(t; q)m1(t; q)m2(t−1q; q)K(tqm2 ; q)K(q−K , t−2q1−m2−K , t, tqm1 ; q)φ1
(q; q)m1(q; q)m2(q; q)K(t2qm2 ; q)K(tq−K , t−1q1−m2−K , q, qm1+1; q)φ1
(t2/q)Kqφ1
× 6φ5
[
t2qm2−1, tq
m2+1
2 ,−tq
m2+1
2 , t2q−1, tqm2+K−φ1 , qφ1−K
q, tq
m2−1
2 ,−tq
m2−1
2 , qm2+1, tqφ1−K , t2qm2+K−φ1
; q, q/t
]
=
k∑
φ1=0
(t; q)m1(t; q)m2(t−1q; q)K(tqm2 ; q)K(q−K , t−2q1−m2−K , t, tqm1 ; q)φ1
(q; q)m1(q; q)m2(q; q)K(t2qm2 ; q)K(tq−K , t−1q1−m2−K , q, qm1+1; q)φ1
×
(t2qm2 ; q)K−φ1(t−1q1−K+φ1 ; q)K−φ1
(qm2+1; q)K−φ1(tqφ1−K ; q)K−φ1
(t2/q)Kqφ1 = r.h.s. of (5.1).
5.2 The general case
We have
l.h.s. of (5.1) =
K∑
φ2,n=0
∏
2≤k≤n
(t; q)φk(t; q)φk+mk
(q; q)φk(q; q)φk+mk
×
∏
2≤l≤n
(tn−l+1qφl+φl+1,n+ml,n ; q)φl(t
n−l+2q2φl+1,n+ml+1,n−1; q)φl
(tn−l+2qφl+φl+1,n+ml,n−1; q)φl(t
n−l+1q2φl+1,n+ml+1,n ; q)φl
12 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
×
K−φ2,n∑
φ1=0
(tnqφ1+φ2,n+m1,n ; q)φ1(tn+1q2φ2,n+m2,n−1; q)φ1
(tn+1qφ1+φ2,n+m1,n−1; q)φ1(tnq2φ2,n+m2,n ; q)φ1
× (t2/q)K−φ2,n−φ1
(t−1q; q)K−φ2,n−φ1(tnq2φ2,n+m1,n+2φ1 ; q)K−φ2,n−φ1
(q; q)K−φ2,n−φ1(tn+1q2φ2,n+m1,n+2φ1 ; q)K−φ2,n−φ1
. (5.6)
Here, we can describe the summation with respect to φ1 in (5.6) as follows:
(t; q)m1(tnqm1,n+2φ2,n , t−1q; q)K−φ2,n
(q; q)m1(tn+1qm1,n+2φ2,n , q; q)K−φ2,n
(t2/q)K−φ2,n
× 8φ7
[
tn+1qm1,n+2φ2,n−1, t
n+1
2 q
m1,n+2φ2,n+1
2 ,
t
n+1
2 q
m1,n+2φ2,n−1
2 ,−t
n+1
2 q
m1,n+2φ2,n−1
2 ,
−t
n+1
2 q
m1,n+2φ2,n+1
2 , t, tqm1 , tn+1qm2,n+2φ2,n−1, tnqm1,n+2φ2,n+K , qφ2,n−K
tnqm1,n+2φ2,n , tnqm2,n+2φ2,n , qm1+1, tqφ2,n−K , tn+1qm1,n+2φ2,n+K
; q, q/t
]
=
(t; q)m1(tnqm1,n+2φ2,n , t−1q, t−nq1−K−m2,n−φ2,n ; q)K−φ2,n
(q; q)m1(qm1+1, q, tqφ2,n−K ; q)K−φ2,n
(t2/q)K−φ2,n
× 4φ3
[
tn−1qm2,n+2φ2,n , tn+1qm2,n+2φ2,n−1, tnqm1,n+φ2,n+K , qφ2,n−K
tnqm1,n+2φ2,n , tnqm2,n+2φ2,n , tnqm2,n+2φ2,n ; q, q
]
. (5.7)
Applying Sears’ 4φ3 transformation formula [3, p. 41, equation (2.10.4)] to the r.h.s. of (5.7), we
have
r.h.s. of (5.7) = (t2/q)K−φ2,n
(t; q)m1(t−1q; q)K−φ2,n(tn−1qm2,n+2φ2,n ; q)K−φ2,n
(q; q)m1(q; q)K−φ2,n(tnqm2,n+2φ2,n ; q)K−φ2,n
× 4φ3
[
tqm1 , t, t−nq1−K−m2,n−φ2,n , qφ2,n−K
qm1+1, tqφ2,n−K , t−n+1q1−K−m2,n−φ2,n ; q, q
]
. (5.8)
Using the following two formulas
(t−1q; q)K−φ2,n(qφ2,n−K ; q)φ1
(q; q)K−φ2,n(tqφ2,n−K ; q)φ1
=
(t−1q; q)K−φ2,n−φ1
(q; q)K−φ2,n−φ1
t−φ1 ,
(tn−1qm2,n+2φ2,n ; q)K−φ2,n(t−nq1−K−m2,n−φ2,n ; q)φ1
(tnqm2,n+2φ2,n ; q)K−φ2,n(t−n+1q1−K−m2,n−φ2,n ; q)φ1
=
(tn−1qm2,n+2φ2,n ; q)K−φ2,n−φ1
(tnqm2,n+2φ2,n ; q)K−φ2,n−φ1
t−φ1 ,
we have
r.h.s. of (5.8)
=
K−φ2,n∑
φ1=0
(t; q)m1(t; q)φ1(tqm1 ; q)φ1(t−1q; q)K−φ2,n−φ1(tn−1qm2,n+2φ2,n ; q)K−φ2,n−φ1
(q; q)m1(q; q)φ1(qm1+1; q)φ1(q; q)K−φ2,n−φ1(tnqm2,n+2φ2,n ; q)K−φ2,n−φ1
× (t2/q)K−φ2,n−φ1 .
Then we have
l.h.s. of (5.6) =
K∑
φ2,n=0
∏
2≤k≤n
(t; q)φk(t; q)φk+mk
(q; q)φk(q; q)φk+mk
×
∏
2≤l≤n
(tn−l+1qφl+φl+1,n+ml,n ; q)φl(t
n−l+2q2φl+1,n+ml+1,n−1; q)φl
(tn−l+2qφl+φl+1,n+ml,n−1; q)φl(t
n−l+1q2φl+1,n+ml+1,n ; q)φl
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 13
×
K−φ2,n∑
φ1=0
(t; q)m1(t; q)φ1(tqm1 ; q)φ1(t−1q; q)K−φ2,n−φ1(tn−1qm2,n+2φ2,n ; q)K−φ2,n−φ1
(q; q)m1(q; q)φ1(qm1+1; q)φ1(q; q)K−φ2,n−φ1(tnqm2,n+2φ2,n ; q)K−φ2,n−φ1
× (t2/q)K−φ2,n−φ1
=
K∑
φ1=0
(t; q)φ1(t; q)φ1+m1
(q; q)φ1(q; q)φ1+m1
K−φ1∑
φ2,n=0
∏
2≤k≤n
(t; q)φk(t; q)φk+mk
(q; q)φk(q; q)φk+mk
×
∏
2≤l≤n
(tn−l+1qφl+φl+1,n+ml,n ; q)φl(t
n−l+2q2φl+1,n+ml+1,n−1; q)φl
(tn−l+2qφl+φl+1,n+ml,n−1; q)φl(t
n−l+1q2φl+1,n+ml+1,n ; q)φl
×
(t−1q; q)K−φ1−φ2,n(tn−1qm2,n+2φ2,n ; q)K−φ1−φ2,n
(q; q)K−φ1−φ2,n(tnqm2,n+2φ2,n ; q)K−φ1−φ2,n
(t2/q)K−φ1−φ2,n
=
K∑
φ1=0
(t; q)φ1(t; q)φ1+m1
(q; q)φ1(q; q)φ1+m1
∑
φ2+···+φn+i=K−φ1
∏
2≤k≤n
(t; q)φk(t; q)φk+mk
(q; q)φk(q; q)φk+mk
× (t−1q; q)i(t
n−1qm2,n+2φ2,n ; q)i
(q; q)i(tnqm2,n+2φ2,n ; q)i
(t2/q)i
×
∏
2≤l≤n
(tn−l+1qφl+φl+1,n+ml,n ; q)φl(t
n−l+2q2φl+1,n+ml+1,n−1; q)φl
(tn−l+2qφl+φl+1,n+ml,n−1; q)φl(t
n−l+1q2φl+1,n+ml+1,n ; q)φl
. (5.9)
By the induction hypothesis, we obtain
r.h.s. of (5.9) =
K∑
φ1=0
(t; q)φ1(t; q)φ1+m1
(q; q)φ1(q; q)φ1+m1
∑
φ2+···+φn=K−φ1
∏
2≤k≤n
(t; q)φk(t; q)φk+mk
(q; q)φk(q; q)φk+mk
=
∑
φ1+φ2+···+φn=K
∏
1≤j≤n
(t; q)φj (t; q)φj+mj
(q; q)φj (q; q)φj+mj
= r.h.s. of (5.1).
6 Tableau formulas for Macdonald polynomials of type Cn
In this section, we establish the tableau formulas for the Macdonald polynomials of type Cn.
Let I := {1, 2, . . . , n− 1, n, n, n− 1, . . . , 1} be the index set with the ordering
1 ≺ 2 ≺ · · · ≺ n− 1 ≺ n ≺ n ≺ n− 1 ≺ · · · ≺ 1. (6.1)
Denoting by Λ1 the first fundamental weight of type Cn, let P
(Cn)
(r) (x; q, t, T ) be the Macdonald
polynomials of type Cn associated with the weights rΛ1 for r ∈ Z≥0.
The following theorem [4, Theorem 5.1] was conjectured by Lassalle [8].
Theorem 6.1 ([4, 8]). For any positive integer r we have
Gr(x; q, t) =
[r/2]∑
i=0
(t; q)r−2i
(q; q)r−2i
P
(Cn)
(r−2i)(x; q, t, T )T i
(t/T ; q)i(t
nqr−2i; q)i
(q; q)i(Ttn−1qr−2i+1; q)i
. (6.2)
Conversely
P
(Cn)
(r) (x; q, t, T ) =
(q; q)r
(t; q)r
[r/2]∑
i=0
Gr−2i(x; q, t)ti
(T/t; q)i(t
nqr−i; q)i
(q; q)i(Ttn−1qr−i; q)i
1− tnqr−2i
1− tnqr−i
.
First we prove the tableau formula for P
(Cn)
(r) (x; q, t, t2/q).
14 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
Theorem 6.2. We have
P
(Cn)
(r)
(
x; q, t, t2/q
)
=
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ1=r
∏
k∈I
(t; q)θk
(q; q)θk
(6.3)
×
∏
1≤l≤n
(tn−l+1qθl+···+θl+1 ; q)θl(t
n−l+2qθl+1+···+θl+1−1; q)θl
(tn−l+2qθl+···+θl+1−1; q)θl(t
n−l+1qθl+1+···+θl+1 ; q)θl
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn .
Remark 6.3. It would be an intriguing problem to show the factorization of the Macdonald
polynomial P
(Cn)
(r) (x; q, t, t2/q) from our formula (6.3) when we make the principal specialization
P
(Cn)
(r)
((
t2/q
)1/2
tn−1, . . . ,
(
t2/q
)1/2
t,
(
t2/q
)1/2
; q, t, t2/q
)
= qr/2t−rn
(tn; q)r(t
2(n+1)/q2; q)r
(t; q)r(tn+1/q; q)r
.
Remark 6.4. Setting n = 1 in (6.3) we have,
P
(C1)
(r)
(
x; q, t, t2/q
)
=
r∑
θ1=0
(t2/q, q−r; q)θ1
(q, t−2q2−r; q)θ1
(q/t)2θ1x−r+2θ1 = x−r 2φ1
[
t2/q, q−r
t−2q2−r ; q, (qx/t)2
]
.
Setting n = 2 in (6.3) we have
P
(C2)
(r)
(
x; q, t, t2/q
)
=
(q; q)r
(t; q)r
∑
θ1+θ2+θ2+θ1=r
(t; q)θ1(t; q)θ1(t; q)θ2(t; q)θ2
(q; q)θ1(q; q)θ1(q; q)θ2(q; q)θ2
×
(t2qθ1+θ2+θ2 , t3qθ2+θ2−1; q)θ1
(t3qθ1+θ2+θ2−1, t2qθ2+θ2 ; q)θ1
(tqθ2 , t2/q; q)θ2
(t2qθ2−1, t; q)θ2
x
θ1−θ1
1 x
θ2−θ2
2 .
Proof of Theorem 6.2. Let
{
Ψ
(Cn)
(r) (x; q, t, T )
}
r∈Z≥0
be a certain collection of Laurent poly-
nomials. By using Lassalle’s formula (6.2), it is proved by induction that the infinite system of
equalities for Ψ
(Cn)
(r) (x; q, t, T )
Gr(x; q, t) =
[r/2]∑
i=0
(t; q)r−2i
(q; q)r−2i
Ψ
(Cn)
(r−2i)(x; q, t, T )T i
(t/T ; q)i(t
nqr−2i; q)i
(q; q)i(Ttn−1qr−2i+1; q)i
(6.4)
gives us Ψ
(Cn)
(r) (x; q, t, T ) = P
(Cn)
(r) (x; q, t, T ), r ∈ Z≥0. We use this argument with the specializa-
tion of the parameter T = t2/q.
Set
Ψ
(Cn)
(r)
(
x; q, t, t2/q
)
=
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ1=r
∏
k∈I
(t; q)θk
(q; q)θk
×
∏
1≤l≤n
(tn−l+1qθl+···+θl+1 ; q)θl(t
n−l+2qθl+1+···+θl+1−1; q)θl
(tn−l+2qθl+···+θl+1−1; q)θl(t
n−l+1qθl+1+···+θl+1 ; q)θl
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn .
We prove this family of Laurent polynomials satisfies (6.4) with the specialization T = t2/q.
In view of the (Z/2Z)n symmetry of Ψ
(Cn)
(r) (x; q, t, t2/q), it is sufficient to consider in (6.4) the
coefficients of the monomials xm1
1 · · ·xmnn with nonnegative powers m1, . . . ,mn ∈ Z≥0 only. Let
r ∈ Z≥0, and fix m1, . . . ,mn ∈ Z≥0 arbitrarily. Set K := 1
2(r−m1−m2−· · ·−mn) for simplicity.
Setting
θk = mk + φk, θk = φk, 1 ≤ k ≤ n,
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 15
one finds that the coefficients of the monomials xm1
1 · · ·xmnn in (6.4) is exactly given by l.h.s.
of (5.1). On the other hand, the coefficients of the monomials xm1
1 · · ·xmnn in Gr(x; q, t) is r.h.s.
of (5.1). Hence we have proved (6.4) with T = t2/q, which establishes the tableau formula
P
(Cn)
(r) (x; q, t, t2/q) = Ψ
(Cn)
(r) (x; q, t, t2/q). �
Finally, we present a tableau formula for the one-row Macdonald polynomials P
(Cn)
(r) (x; q, t, T )
with general parameters (q, t, T ).
Theorem 6.5. Set θ := min(θn, θn). We have
P
(Cn)
(r) (x; q, t, T ) =
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ1=r
∏
k∈I\{n,n}
(t; q)θk
(q; q)θk
(t; q)|θn−θn|
(q; q)|θn−θn|
×
∏
1≤l≤n−1
(
(tn−l−1qθl+···+θn−1+|θn−θn|+θn−1+···+θl+1+1; q)θl
(tn−lqθl+···+θn−1+|θn−θn|+θn−1+···+θl+1 ; q)θl
×
(tn−lqθl+1+···θn−1+|θn−θn|+θn−1+···+θl+1 ; q)θl
(tn−l−1qθl+1+···+θn−1+|θn−θn|+θn−1+···+θl+1+1; q)θl
)
× (T ; q)θ(t
nqr−2θ; q)2θ
(q; q)θ(Ttn−1qr−θ; q)θ(tn−1qr−2θ+1; q)θ
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn . (6.5)
Remark 6.6. It would be an intriguing problem to show the factorization of the Macdonald
polynomial P
(Cn)
(r) (x; q, t, T ) from our formula (6.5) when we make the principal specialization:
P
(Cn)
(r)
(
T 1/2tn−1, . . . , T 1/2t, T 1/2; q, t, T
)
= T−r/2t−r(n−1) (tn; q)r(t
2(n−1)T ; q)r
(t; q)r(tnT ; q)r
.
Proof of Theorem 6.5. We prove that the system of equalities (6.4) is satisfied by the fol-
lowing Laurent polynomials
Ψ
(Cn)
(r) (x; q, t, T ) =
(q; q)r
(t; q)r
∑
θ1+θ2+···+θ1=r
∏
k∈I\{n,n}
(t; q)θk
(q; q)θk
(t; q)|θn−θn|
(q; q)|θn−θn|
×
∏
1≤l≤n−1
(
(tn−l−1qθl+···+θn−1+|θn−θn|+θn−1+···+θl+1+1; q)θl
(tn−lqθl+···+θn−1+|θn−θn|+θn−1+···+θl+1 ; q)θl
×
(tn−lqθl+1+···θn−1+|θn−θn|+θn−1+···+θl+1 ; q)θl
(tn−l−1qθl+1+···+θn−1+|θn−θn|+θn−1+···+θl+1+1; q)θl
)
× (T ; q)θ(t
nqr−2θ; q)2θ
(q; q)θ(Ttn−1qr−θ; q)θ(tn−1qr−2θ+1; q)θ
x
θ1−θ1
1 x
θ2−θ2
2 · · ·xθn−θnn .
Note that these Ψ
(Cn)
(r) (x; q, t, T ) have the (Z/2Z)n symmetry.
Let r ∈ Z≥0, and fix m1, . . . ,mn ∈ Z≥0 arbitrarily. We study the coefficient of the monomial
xm1
1 xm2
2 · · ·xmnn in r.h.s. of (6.4), namely we consider the case θk − θk = mk ≥ 0 (1 ≤ k ≤ n).
Set K := 1
2(r − m1 − m2 − · · · − mn), θl,m :=
m∑
k=l
θk, Ǐ := I\{n, n} for simplicity. Then the
coefficient of the monomial xm1
1 xm2
2 · · ·xmnn in r.h.s. of (6.4) is expressed as follows
∑
θ1+θ2+···+θ1+2i=r
∏
k∈Ǐ
(t; q)θk
(q; q)θk
(t; q)mn
(q; q)mn
16 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
×
∏
1≤l≤n−1
(tn−l−1qθl+2θl,n−1+ml,n+1; q)θl(t
n−lq2θl+1,n−1+ml+1,n ; q)θl
(tn−lqθl+2θl+1,n−1+ml,n ; q)θl(t
n−l−1q2θl+1,n−1+ml+1,n+1; q)θl
× (T ; q)θ(t
nqr−2θ; q)2θ
(q; q)θ(Ttn−1qr−θ; q)θ(tn−1qr−2θ+1; q)θ
(t/T ; q)i(t
nqr−2i; q)i
(q; q)i(Ttn−1qr−2i+1; q)i
T i
=
K∑
θ1,n−1=0
K−θ1,n−1∑
θn=0
∏
k∈Ǐ
(t; q)θk
(q; q)θk
(t; q)mn
(q; q)mn
×
∏
1≤l≤n−1
(tn−l−1qθl+2θl,n−1+ml,n+1; q)θl(t
n−lq2θl+1,n−1+ml+1,n ; q)θl
(tn−lqθl+2θl+1,n−1+ml,n ; q)θl(t
n−l−1q2θl+1,n−1+ml+1,n+1; q)θl
× (T ; q)θn(tnq2θ1,n−1+m1,n ; q)2θn
(q; q)θn(Ttn−1q2θ1,n−1+m1,n+θn ; q)θn(tn−1q2θ1,n−1+m1,n+1; q)θn
×
(t/T ; q)K−θ1,n−1−θn(tnq2θ1,n−1+m1,n+2θn ; q)K−θ1,n−1−θn
(q; q)K−θ1,n−1−θn(Ttn−1q2θ1,n−1+m1,n+2θn+1; q)K−θ1,n−1−θn
TK−θ1,n−1−θn
=
K∑
θ1,n−1=0
∏
k∈Ǐ
(t; q)θk
(q; q)θk
(t; q)mn
(q; q)mn
×
∏
1≤l≤n−1
(tn−l−1qθl+2θl,n−1+ml,n+1; q)θl(t
n−lq2θl+1,n−1+ml+1,n ; q)θl
(tn−lqθl+2θl+1,n−1+ml,n ; q)θl(t
n−l−1q2θl+1,n−1+ml+1,n+1; q)θl
×
(t/T ; q)K−θ1,n−1(tnq2θ1,n−1+m1,n ; q)K−θ1,n−1
(q; q)K−θ1,n−1(Ttn−1q2θ1,n−1+m1,n+1; q)K−θ1,n−1
TK−θ1,n−1
× 6W5
(
Ttn−1q2θ1,n−1+m1,n , T, tnqθ1,n−1+m1,n+K , q−K+θ1,n−1 ; q, q/t
)
. (6.6)
By the summation formula for 6φ5 series [3, p. 34, equation (2.4.2)], we have
6W5 series in (6.6) =
(Ttn−1q2θ1,n−1+m1,n+1; q)K−θ1,n−1(t−1q−K+θ1,n+1; q)K−θ1,n−1
(tn−1q2θ1,n−1+m1,n+1; q)K−θ1,n−1(Tt−1q−K+θ1,n−1−1; q)K−θ1,n−1
.
Note that the dependence on the parameter T in (6.6) disappears by the cancelation as
(t/T ; q)K−θ1,n−1(t−1q−K+θ1,n−1+1; q)K−θ1,n−1
(Tt−1q−K+θ1,n−1−1; q)K−θ1,n−1
TK−θ1,n−1 = (t; q)K−θ1,n−1 .
Hence we have recast the coefficient of the monomial xm1
1 xm2
2 · · ·xmnn in r.h.s. of (6.4) as
K∑
θ1,n−1=0
∏
k∈Ǐ
(t; q)θk
(q; q)θk
(t; q)mn
(q; q)mn
×
∏
1≤l≤n−1
(tn−l−1qθl+2θl,n−1+ml,n+1; q)θl(t
n−lq2θl+1,n−1+ml+1,n ; q)θl
(tn−lqθl+2θl+1,n−1+ml,n ; q)θl(t
n−l−1q2θl+1,n−1+ml+1,n+1; q)θl
×
(t; q)K−θ1,n−1(tnq2θ1,n−1+m1,n ; q)K−θ1,n−1
(q; q)K−θ1,n−1(tn−1q2θ1,n−1+m1,n+1; q)K−θ1,n−1
Changing the running indices as φk = θk, one finds that this is nothing but l.h.s. of (3.1). Then
Theorem 3.1 means that this is the coefficient of the monomial xm1
1 xm2
2 · · ·xmnn in Gr(x; q, t). �
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 17
7 Deformed W algebras of types Cl and Dl
In this section, we study a relation between the tableau formulas for Macdonald polynomials
of types Cl and Dl and the deformed W algebras of types Cl and Dl. We briefly recall the
definition of the deformed W algebras of types Cl and Dl [2].
Let {α1, α2, . . . , αl} and {ω1, ω2, . . . , ωl} be the sets of simple roots and of fundamental
weights of a simple Lie algebra g of rank l. Let (·, ·) be the invariant inner product on g and
C = (Ci,j)1≤i,j≤l the Cartan matrix where Ci,j = 2(αi, αj)/(αi, αi). Let r∨ be the maximal num-
ber of edges connecting two vertices of the Dynkin diagram of g and set D = diag(r1, r2, . . . , rl)
where ri = r∨(αi, αi)/2. Denote by I = (Ii,j)1≤i,j≤l the incidence matrix where Ii,j = 2δi,j−Ci,j .
Let B = (Bi,j)1≤i,j≤l = DC (i.e. Bi,j = r∨(αi, αi)). We define l × l matrices C(q, t), D(q, t),
B(q, t) and M(q, t) as follows
Ci,j(q, t) =
(
qrit−1 + q−rit
)
δi,j − [Ii,j ]q,
D(q, t) = diag([r1]q, [r2]q, . . . , [rl]q),
B(q, t) = D(q, t)C(q, t),
M(q, t) = D(q, t)C(q, t)−1 = D(q, t)B(q, t)−1D(q, t), (7.1)
where we use the standard notation [n]q = qn−q−n
q−q−1 .
Let Hq,t be the Heisenberg algebra with generators ai[n] and yi[n] (i = 1, 2, . . . , l; n ∈ Z)
with the following relations
[ai[n], aj [m]] =
1
n
(
qrin − q−rin
)(
tn − t−n
)
Bi,j
(
qn, tn
)
δn,−m,
[ai[n], yj [m]] =
1
n
(
qrin − q−rin
)(
tn − t−n
)
δi,jδn,−m,
[yi[n], yj [m]] =
1
n
(
qrin − q−rin
)(
tn − t−n
)
Mi,j
(
qn, tn
)
δn,−m. (7.2)
Note that we have
aj [n] =
l∑
i=1
Ci,j
(
qn, tn
)
yi[n].
Introduce the generating series:
Yi(z) := :exp
(∑
m6=0
yi[m]z−m
)
:.
Here we have used the standard notation for the normal ordering : · · · : for the Heisenberg
generators defined as follows. We call the negative modes ai[−n], yi[−n] (n > 0) creation
operators and positive modes ai[n], yi[n] (n > 0) annihilation operators. Then the normal
ordered product :O: of an operator O is obtained by moving all the creation operators to the
left and the annihilation operators to the right. For example we have
Yi(z) = exp
(∑
m>0
yi[−m]zm
)
exp
(∑
m>0
yi[m]z−m
)
.
Remark 7.1. In [2], suitable zero mode factors are included in the generating series Yi(z) to
ensure reasonable commutation relations with the screening operators. In this paper, however,
we omit writing them since we do not need any arguments based on the screening operators.
18 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
We define a set J and fields Λi(z) = Λ
(X)
i (z), i ∈ J , for X = Cl and Dl.
(i) The Cl series. J := {1, 2, . . . , l, l, l − 1, . . . , 1},
Λi(z) := :Yi
(
zq−i+1ti−1
)
Yi−1
(
zq−iti
)−1
:, i = 1, 2, . . . , l,
Λi(z) := :Yi−1
(
zq−2l+i−2t2l−i
)
Yi
(
zq−2l+i−3t2l−i+1
)−1
:, i = 1, 2, . . . , l. (7.3)
(ii) The Dl series. J := {1, 2, . . . , l, l, l − 1, . . . , 1},
Λi(z) := :Yi
(
zq−i+1ti−1
)
Yi−1
(
zq−iti
)−1
:, i = 1, 2, . . . , l − 2,
Λl−1(z) := :Yl
(
zq−l+2tl−2
)
Yl−1
(
zq−l+2tl−2
)
Yl−2
(
zq−l+1tl−1
)
:,
Λl(z) := :Yl
(
zq−l+2tl−2
)
Yl−1
(
zq−ltl
)−1
:,
Λl(z) := :Yl−1
(
zq−l+2tl−2
)
Yl
(
zq−ltl
)−1
:,
Λl−1(z) := :Yl−2
(
zq−l+1tl−1
)
Yl−1
(
zq−ltl
)−1
Yl
(
zq−ltl
)
:,
Λi(z) := :Yi−1
(
zq−2l+i+2t2l−i−2
)
Yi
(
zq−2l+i+1t2l−i−1
)−1
:, i = 1, 2, . . . , l − 2. (7.4)
Definition 7.2 ([2]). Define the first generating fields T (X)(x, z) of the deformed W algebras
of type X = Cl or Dl with the independent indeterminates x = (x1, x2, . . . , xl):
T (X)(x, z) := x1Λ1(z) + · · ·+ xlΛl(z) +
1
xl
Λl̄(z) + · · ·+ 1
x1
Λ1̄(z). (7.5)
Remark 7.3. It is not an easy task to define the deformed W algebras purely in terms of the
generators and relations, except for some simple cases such as the deformed Virasoro algebra.
One of the simplest bypass ways is to regard the deformed W as the algebra generated by
the T (X)(x, z) given in terms of the Heisenberg generators.
Remark 7.4. The xi’s (i = 1, 2, . . . , l) correspond to the zero mode factors, and they paramet-
rize the highest weight condition for the representation of the W algebras.
Lemma 7.5. For Cl and Dl series we obtain the following operator product expansions
f(w/z)Λi(z)Λj(w) = γi,j(z, w) :Λi(z)Λj(w):,
where
f(z) = f (X)(z) := exp
(
−
∞∑
n=1
(qn − q−n)(tn − t−n)M1,1(qn, tn)zn
)
,
and γi,j(z, w) = γ
(X)
i,j (z, w) for X = Cl or Dl given by
γ
(Cl)
i,j (z, w) =
1, i = j,
γ(w/z), i ≺ j, j 6= ī,
γ(z/w), i � j, j̄ 6= i,
γ(w/z)γ
(
q2i−2l−2t−2i+2lw/z
)
, i ≺ j, j = ī,
γ(z/w)γ
(
q−2i+2l+2t2i−2lz/w
)
, i � j, j̄ = i,
γ
(Dl)
i,j (z, w) =
1 i = j,
γ(w/z), i ≺ j, j 6= ī,
γ(z/w), i � j, j̄ 6= i,
γ(w/z)γ
(
q2i−2l+2t−2i+2l−2w/z
)
, i ≺ j, j = ī,
γ(z/w)γ
(
q−2i+2l−2t2i−2l+2z/w
)
, i � j, j̄ = i.
Here we have use the notation γ(z) = (1−t2z)(1−z/q2)
(1−z)(1−zt2/q2)
.
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 19
A proof of Lemma 7.5 can be obtained by straightforward but pretty lengthy calculations
using (7.1), (7.2), (7.3), (7.4). Therefore we safely can omit the detail.
Let |0〉 be the vacuum vector satisfying the annihilation conditions ai[n]|0〉 = 0 for all i and
n > 0. Let F be the Fock module obtained by inducing up the one dimensional representa-
tion C|0〉 of the algebra of annihilation operators to the whole Heisenberg algebra. Then one can
check that the Fourier modes of the generator T (X)(x, z) acting on F are well defined. Let 〈0|
to be the dual vacuum satisfying 〈0|ai[−n] = 0 for all i and n > 0.
Now we consider the correlation functions 〈0|T (x, z1)T (x, z2) · · ·T (x, zr)|0〉 of types Cl and Dl
with the normalization 〈0|Λi(z)|0〉 = 1.
Proposition 7.6. Let X = Cl or Dl. Set Il := {1, 2, . . . , l, l, l − 1, . . . , 1}. Let x1, . . . , xl be
indeterminates and set xi = 1/xi, 1 ≤ i ≤ l. We have∏
i<j
f (X)(zj/zi) · 〈0|T (X)(x, z1)T (X)(x, z2) · · ·T (X)(x, zr)|0〉
= F (X)(x1, . . . , xl|zi, . . . , zr|q, t),
where
F (X)(x1, . . . , xl|zi, . . . , zr|q, t) =
∑
ε1,ε2,...,εr∈Il
xε1xε2 · · ·xεr
∏
1≤i<j≤r
γεi,εj (zi, zj).
Proof of Proposition 7.6. In view of the expression (7.5) we only need to know the matrix
elements
〈0|Λ(X)
ε1 (z1)Λ(X)
ε2 (z2) · · ·Λ(X)
εr (zr)|0〉, (7.6)
for any fixed ε1, . . . , εr ∈ Il. Then Proposition 7.5 and the normal ordering rule imply that
(7.6) =
∏
i<j
(
f (X)(zj/zi)
)−1 ·
∏
1≤i<j≤r
γεi,εj (zi, zj). �
Remark 7.7. It is clearly seen from the definition that F (X)(x1, . . . , xl|zi, . . . , zr|q, t) is a sym-
metric rational function in zi’s. We conjecture that the F (X)(x1, . . . , xl|zi, . . . , zr|q, t) is a sym-
metric Laurent polynomial in xi’s associated with the Weyl group of corresponding type X
(Cl or Dl). For the case of type Al, we better understand the situation due to the theory of the
shuffle algebra (we refer the reader to [1]).
Definition 7.8. By principally specializing the zi’s, set
Φ(X)
r (x|q, t) = F (X)
(
x1, . . . , xl|qr−1, qr−2, . . . , 1|q1/2, q1/2t−1/2
)
.
Remark 7.9. Remark 7.7 implies that Φ
(X)
r (x|q, t) is a symmetric Laurent polynomial in xi’s
associated with the Weyl group of corresponding type X (Cl or Dl). It is easy to check that
the terms which do not vanish under the principal specialization correspond exactly to the set
of semi-standard tableaux of type X.
Theorem 7.10. We have
(i) Φ
(Cl)
r (x|q, t) = P
(Cl)
(r) (x; q, t, t2/q),
(ii) Φ
(Dl)
r (x|q, t) = P
(Dl)
(r) (x; q, t).
Proof. Straightforward calculation of Φ
(X)
r (x|q, t) gives us (1.1) for X = Cl and (1.2) for
X = Dl. �
20 B. Feigin, A. Hoshino, M. Noumi, J. Shibahara and J. Shiraishi
A Macdonald polynomials of types Cn and Dn
We recall briefly the Koornwinder polynomials, and the definitions of the Macdonald polyno-
mials of types Cn and Dn as degenerations of the Koornwinder polynomials.
A.1 Koornwinder polynomials
Let (a, b, c, d, q, t) be a set of complex parameters with |q| < 1. Set α = (abcdq−1)1/2 for
simplicity. Let x = (x1, . . . , xn) be a set of independent indeterminates. Koornwinder’s q-
difference operator Dx = Dx(a, b, c, d|q, t) is defined by [6]
Dx =
n∑
i=1
(1− axi)(1− bxi)(1− cxi)(1− dxi)
αtn−1(1− x2
i )(1− qx2
i )
∏
j 6=i
(1− txixj)(1− txi/xj)
(1− xixj)(1− xi/xj)
(Tq,xi − 1)
+
n∑
i=1
(1− a/xi)(1− b/xi)(1− c/xi)(1− d/xi)
αtn−1(1− 1/x2
i )(1− q/x2
i )
∏
j 6=i
(1− txj/xi)(1− t/xixj)
(1− xj/xi)(1− 1/xixj)
(Tq−1,xi − 1),
where Tq±1,xif(x1, . . . , xi, . . . , xn) = f(x1, . . . , q
±1xi, . . . , xn). The Koornwinder polynomial
Pλ(x) = Pλ(x; a, b, c, d, q, t) with partition λ = (λ1, . . . , λn) (i.e., λi ∈ Z≥0, λ1 ≥ · · · ≥ λn)
is uniquely characterized by the two conditions (a) Pλ(x) is a Sn n (Z/2Z)n invariant Lau-
rent polynomial having the triangular expansion in terms of the monomial basis (mλ(x)) as
Pλ(x) = mλ(x)+lower terms, (b) Pλ(x) satisfies DxPλ(x) = dλPλ(x). The eigenvalue is given by
dλ =
n∑
j=1
〈abcdq−1t2n−2jqλj 〉〈qλj 〉 =
n∑
j=1
〈αtn−jqλj ;αtn−j〉,
where we used the notation 〈x〉 = x1/2 − x−1/2 and 〈x; y〉 = 〈xy〉〈x/y〉 = x+ x−1 − y − y−1 for
simplicity of display.
A.2 Macdonald polynomials of types Cn and Dn
We consider some degeneration of the Koornwinder polynomials to the Macdonald polynomials.
As for the details, we refer the readers to [6] and [11]. Specializing the parameters in the
Koornwinder polynomial Pλ(x; a, b, c, d, q, t) as
(a, b, c, d, q, t)→
(
−b1/2, ab1/2,−q1/2b1/2, q1/2ab1/2, q, t
)
,
we obtain the Macdonald polynomial of type (BCn, Cn) [6]
P
(BCn,Cn)
λ (x; a, b, q, t) = Pλ
(
x;−b1/2, ab1/2,−q1/2b1/2, q1/2ab1/2, q, t
)
.
Namely, setting
D(BCn,Cn)
x =
∑
σ1,...,σn=±1
n∏
i=1
(1− ab1/2xσii )(1 + b1/2xσii )
1− x2σi
i
·
∏
1≤i<j≤n
1− txσii x
σj
i
1− xσii x
σj
i
·
n∏
i=1
Tqσi/2,xi ,
we have
P
(BCn,Cn)
λ (x) = mλ(x) + lower terms,
D(BCn,Cn)
x P
(BCn,Cn)
λ (x) = (ab)n/2tn(n−1)/4
∑
σ1,...,σn=±1
s
σ1/2
1 · · · sσn/2n
P
(BCn,Cn)
λ (x),
where si = abtn−iqλi .
Tableau Formulas for One-Row Macdonald Polynomials of Types Cn and Dn 21
Macdonald polynomials of type Cn and type Dn are obtained as follows
P
(Cn)
λ (x; b, q, t) = P
(BCn,Cn)
λ (x; 1, b, q, t),
P
(Dn)
λ (x; q, t) = P
(BCn,Cn)
λ (x; 1, 1, q, t).
Acknowledgements
This study was carried out within The National Research University Higher School of Economics
Academic Fund Program in 2013–2014, research grant No.12-01-0016k. Research of J.S. is
supported by the Grant-in-Aid for Scientific Research C-24540206. The financial support from
the Government of the Russian Federation within the framework of the implementation of the
5-100 Programme Roadmap of the National Research University Higher School of Economics is
acknowledged. The authors sincerely thank the anonymous referees for valuable comments and
informing them of the connection between one of our results (Theorem 2.2) and the known fact
obtained in [7].
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1 Introduction
2 Transformation formula I
3 Transformation formula II
3.1 The case n=2
3.2 The general case
4 Tableau formulas for Macdonald polynomials of type Dn
5 Transformation formula III
5.1 The case n=2
5.2 The general case
6 Tableau formulas for Macdonald polynomials of type Cn
7 Deformed W algebras of types Cl and Dl
A Macdonald polynomials of types Cn and Dn
A.1 Koornwinder polynomials
A.2 Macdonald polynomials of types Cn and Dn
References
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