# Quantum Product of Symmetric Functions

Authors: Rafael Díaz; Eddy Pariguan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

---

## Abstract

We provide an explicit description of the quantum product of multisymmetric functions using the elementary multisymmetric functions introduced by Vaccarino.


---

## Body

## 1. Introduction

Fix a characteristic zero fieldK. The algebrogeometric duality allows us to identify affine algebraic varieties with the K-algebra of polynomial functions on it, and, reciprocally, a finitely generated algebra without nilpotent elements may be identified with its spectrum, provided with the Zarisky topology. Affine space K
n is thus identified with the algebra of polynomials in n-variables K
[
x
1
,
…
,
x
n
]. Consider the action of the symmetric group S
n on K
n by permutation of vector entries. The quotient space K
n
/
S
n is the configuration space of n unlabeled points with repetitions in K. Polynomial functions on K
n
/
S
n may be identified with the algebra K
[
x
1
,
…
,
x
n
]
S
n of S
n-invariant polynomials in K
[
x
1
,
…
,
x
n
]. A remarkable classical fact is that K
n
/
S
n is again a n-dimensional affine space [1]; indeed we have an isomorphism of algebras (1)
K
x
1
,
…
,
x
n
S
n
≃
K
e
1
,
…
,
e
n
,where, for α
∈
[
n
]
=
{
1
,
…
,
n
}, e
α is the elementary symmetric polynomial given by (2)
e
α
x
1
,
…
,
x
n
=
∑
a
=
α
‍
x
α
=
∑
a
⊆
n
,
a
=
α
∏
j
∈
a
x
j
.The elementary symmetric polynomials are determined by the identity (3)
∏
i
=
1
n
1
+
x
i
t
=
∑
α
=
0
n
e
α
x
1
,
…
,
x
n
t
α
.Using characteristic functions one shows for α
1
,
…
,
α
m
∈
[
n
] that (4)
e
α
1
⋯
e
α
m
=
∑
a
∈
N
n
c
α
1
,
…
,
α
m
,
a
x
a
,where c
(
α
1
,
…
,
α
m
,
a
) is the cardinality of the subset of matrices of format n
×
m with entries in {
0,1
} such that (5)
∑
j
=
1
n
A
i
j
=
α
i
for
i
∈
m
,
∑
i
=
1
m
A
i
j
=
a
j
for
j
∈
n
.A subtler situation arises when one considers the configuration space(6)
K
d
n
S
nof n unlabeled points with repetitions in K
d, for d
≥
2. In this case (
K
d
)
n
/
S
n is no longer an affine space; instead it is an affine algebraic variety. Polynomial functions on (
K
d
)
n
/
S
n are the so-called multisymmetric functions, also known as vector symmetric functions or MacMahon symmetric functions [1, 2], and they coincide with the algebra of invariant polynomials (7)
K
x
11
,
…
,
x
1
d
,
…
,
x
n
1
,
…
,
x
n
d
S
n
,which admits a presentation of the following form: (8)
K
e
α
∣
α
∈
n
I
n
,
d
,where the elementary multisymmetric functions e
α, for α
=
(
α
1
,
…
,
α
d
)
∈
N
d a vector such that α
=
α
1
+
⋯
+
α
d
≤
n, are defined by the identity (9)
∏
i
=
1
n
1
+
x
i
1
t
1
+
⋯
+
x
i
d
t
d
=
∑
α
∈
N
d
,
α
≤
n
e
α
x
11
,
…
,
x
1
d
,
…
,
x
n
1
,
…
,
x
n
d
t
1
α
1
⋯
t
d
α
d
.Explicitly, the multisymmetric function e
α is given by (10)
e
α
x
11
,
…
,
x
1
d
,
…
,
x
n
1
,
…
,
x
n
d
=
∑
a
x
a
=
∑
a
=
α
∏
j
∈
d
∏
i
∈
a
j
x
i
j
,where in the middle term we regard a
∈
M
n
×
d
(
{
0,1
}
) as a matrix such that (11)
∑
i
=
1
n
a
i
j
=
α
j
for
j
∈
d
,
∑
j
=
1
d
a
i
j
≤
1
for
i
∈
n
,
x
a
=
∏
i
=
1
n
∏
j
=
1
d
x
i
j
a
i
j
;and in the right-hand side term we let a
=
(
a
1
,
…
,
a
d
) be a d-tuple of disjoint sets a
j
⊆
[
n
] such that (12)
a
=
a
1
,
…
,
a
d
=
α
1
,
…
,
α
d
.It is not difficult to check that any multisymmetric function can be written (not uniquely) as a linear combination of products of elementary multisymmetric functions. The nonuniqueness is controlled by the idealI
n
,
d. For an explicit description of I
n
,
d the reader may consult Dalbec [3] and Vaccarino [4].One checks forα
1
,
…
,
α
m
∈
N
d that (13)
e
α
1
⋯
e
α
m
=
∑
a
∈
M
n
×
d
N
c
α
1
,
…
,
α
m
,
a
x
a
,where c
(
α
1
,
…
,
α
m
,
a
) counts the number of cubical matrices (14)
A
=
A
i
j
l
∈
Map
m
×
n
×
d
,
0,1such that (15)
∑
i
=
1
m
A
i
j
l
=
a
j
l
for
j
∈
n
,
l
∈
d
,
∑
l
=
1
d
A
i
j
l
≤
1
for
i
∈
m
,
j
∈
n
,
∑
j
=
1
n
A
i
j
l
=
α
i
l
for
i
∈
m
,
l
∈
d
.Recall that an algebra may be analyzed by describing it by generators and relations or alternatively, as emphasized by Rota and his collaborators, by finding a suitable basis such that the structural coefficients are positive integers with preferably a nice combinatorial interpretation. The second approach for the case of multisymmetric functions was undertaken by Vaccarino [4] and his results will be reviewed in Section 2. The main goal of this work (see Section 5) is to generalize this combinatorial approach to multisymmetric functions from the classical to the quantum setting.Quantum mechanics, the century old leading small distances physical theory, is still not quite fully understood by mathematicians. The transition from classical to quantum mechanics has been particularly difficult to grasp. An appealing approach to this problem is to characterize the process of quantization as a process of deformation of a commutative Poisson algebra into a noncommutative algebra [5]. In this approach classical phase space is replaced by quantum phase space, where an extra dimension parametrized by a formal variable ħ is added.The classical phase space of a Lagrangian theory is naturally endowed with a closed two-form. In the nondegenerated case (i.e., in the symplectic case) this two-form can be inverted given rise to a Poisson bracket on the algebra of smooth functions on phase space. In a sense, the Poisson bracket may be regarded as a tangent vector in the space of deformations of the algebra of functions on phase space, that is, as an infinitesimal deformation. That this infinitesimal deformation can be integrated into a formal deformation is a result according to Fedosov [6] for the symplectic case and according to Kontsevich [7] for arbitrary Poisson manifolds.Many Lagrangian physical theories are invariant under a continuous group of transformations; in that case the two-form on phase space is necessarily degenerated. Nevertheless, a Lagrangian theory might be invariant under a finite group and still retain its nondegenerated character. In the latter scenario all the relevant constructions leading to the quantum algebra of functions on phase space are equivariant and thus give rise to quantum algebra of invariant functions under the finite group. We follow this path along this work, being as explicit and calculative as possible. Our main aim is thus to provide foundations as well as practical tools for dealing with quantum symmetric functions.


## 2. Multisymmetric Functions

In this section we introduce Vaccarino’s multisymmetric functionse
α
(
p
) which are defined in analogy with the elementary multisymmetric functions of Introduction, as yet the definition is general enough to account for the symmetrization of arbitrary polynomial functions [4].Fixa
,
n
,
d
∈
N
+. Let y
1
,
…
,
y
d and t
1
,
…
,
t
a be a pair of sets of commuting independent variables over K. For α
=
(
α
1
,
…
,
α
a
)
∈
N
a we set (16)
α
=
∑
i
=
1
a
α
i
,
t
α
=
∏
i
=
1
a
t
i
α
i
.For q
∈
K
[
y
1
,
…
,
y
d
] and i
∈
[
n
] we let q
(
i
)
=
q
(
x
i
1
,
…
,
x
i
d
) be the polynomial obtained by replacing each appearance of y
j in q by x
i
j, for j
∈
[
d
]. For example, for n
=
2 and q
=
y
1
y
2
y
3
∈
R
[
y
1
,
y
2
,
y
3
] we have(17)
q
1
=
x
11
x
12
x
13
,
q
2
=
x
21
x
22
x
23
.Definition 1.
Considerα
∈
N
a such that |
α
|
≤
n and p
=
(
p
1
,
…
,
p
a
)
∈
K
[
y
1
,
…
,
y
d
]
a. The multisymmetric functions e
α
(
p
)
∈
K
[
(
K
d
)
n
]
S
n are determined by the identity(18)
∏
i
=
1
n
1
+
p
1
i
t
1
+
⋯
+
p
a
i
t
a
=
∑
α
≤
n
e
α
p
t
α
.Example 2.
Forn
=
3 and p
=
(
y
1
y
2
,
y
3
y
4
), we have that e
(
1,1
)
(
y
1
y
2
,
y
3
y
4
) is equal to x
13
x
14
x
21
x
22
+
x
21
x
22
x
33
x
34
+
x
23
x
24
x
31
x
32
+
x
11
x
12
x
33
x
34
+
x
13
x
14
x
31
x
32
+
x
11
x
12
x
23
x
24 and e
(
2,1
)
(
y
1
y
2
,
y
3
y
4
)
=
x
11
x
12
x
21
x
22
x
33
x
34
+
x
11
x
12
x
23
x
24
x
31
x
32
+
x
13
x
14
x
21
x
22
x
31
x
32.Example 3.
Forp
=
(
y
1
,
…
,
y
d
) and α
∈
N
d with |
α
|
∈
[
n
], the multisymmetric functions e
α
(
y
1
,
…
,
y
d
) are the elementary multisymmetric functions defined in Introduction.Next couple of lemmas follow directly from Definition1.Lemma 4.
Letα
∈
N
a be such that |
α
|
≤
n and let p
=
(
p
1
,
…
,
p
a
)
∈
K
[
y
1
,
…
,
y
d
]
a. The multisymmetric function e
α
(
p
) is given by the combinatorial identity(19)
e
α
p
=
∑
c
∏
l
=
1
a
∏
i
∈
c
l
p
l
i
,where c
=
(
c
1
,
…
,
c
a
) is a tuple of disjoint subsets of [
n
], with |
c
|
=
(
|
c
1
|
,
…
,
|
c
a
|
)
=
α.Recall that the symmetrization mapK
[
(
K
d
)
n
]
→
K
[
(
K
d
)
n
]
S
n sends f to (20)
∑
σ
∈
S
n
f
∘
σ
,where one regards σ
∈
S
n as a map σ
:
(
K
d
)
n
→
(
K
d
)
n.Lemma 5.
Letα
∈
N
a be such that |
α
|
≤
n and p
=
(
p
1
,
…
,
p
a
)
∈
K
[
y
1
,
…
,
y
d
]
a. The multisymmetric function e
α
(
p
) is the symmetrization of the polynomial (21)
p
1
1
⋯
p
1
α
1
⋯
p
i
∑
l
=
1
i
-
1
α
l
+
1
⋯
p
i
∑
l
=
1
i
α
l
⋯
p
a
∑
l
=
1
a
-
1
‍
α
l
+
1
⋯
p
a
∑
l
=
1
a
‍
α
l
.Lemma 6.
Letp
=
(
p
1
,
…
,
p
a
)
∈
K
[
y
1
,
…
,
y
d
]
a be expanded in monomials as(22)
p
1
=
∑
j
1
∈
[
k
1
]
c
1
j
1
m
1
j
1
,
…
,
p
a
=
∑
j
a
∈
[
k
a
]
c
a
j
a
m
a
j
a
.Then (23)
e
α
p
=
∑
β
∈
N
k
,
r
β
=
α
e
β
m
c
β
,where m
=
(
m
11
,
…
,
m
1
k
1
,
…
,
m
a
1
,
…
,
m
a
j
a
)
,
c
=
(
c
11
,
…
,
c
1
k
1
,
…
,
c
a
1
,
…
,
c
a
k
a
), and for β
=
(
β
11
,
…
,
β
1
k
1
,
…
,
β
a
1
,
…
,
β
a
k
a
)
∈
N
|
k
| we set (24)
r
(
β
)
=
β
11
+
⋯
+
β
1
k
1
,
…
,
β
a
1
+
⋯
+
β
a
k
a
.Proof.
Let|
k
|
=
k
1
+
⋯
+
k
a and c
t
=
(
c
11
t
1
,
…
,
c
1
k
1
t
1
,
…
,
c
a
1
t
1
,
…
,
c
a
k
a
t
a
). Then (25)
∑
α
∈
N
a
,
α
≤
n
e
α
p
t
α
=
∏
i
=
1
n
1
+
∑
j
1
∈
[
k
1
]
c
1
j
1
m
1
j
1
i
t
1
k
k
k
k
k
k
i
+
⋯
+
∑
j
a
∈
[
k
a
]
c
a
j
a
m
a
j
a
i
t
a
=
∑
β
∈
N
k
,
β
≤
n
e
β
m
c
t
β
=
∑
β
∈
N
k
,
β
≤
n
‍
e
β
m
c
β
t
r
(
β
)
.Thus we get(26)
e
α
p
=
∑
β
∈
N
k
,
r
β
=
α
e
β
m
c
β
.The following result according to Vaccarino [4] provides an explicit formula for the product of multisymmetric functions. We include the proof since the same technique carries over to the more involved quantum case.Theorem 7.
Fixa
,
b
,
n
∈
N
+
,
p
∈
K
[
y
1
,
…
,
y
d
]
a, and q
∈
K
[
y
1
,
…
,
y
d
]
b. Let α
∈
N
a and β
∈
N
b be such that |
α
|
,
|
β
|
≤
n. Then one has (27)
e
α
p
e
β
q
=
∑
γ
∈
L
α
,
β
,
n
e
γ
p
,
q
,
p
q
,
w
h
e
r
e
:
(i)
(
p
,
q
,
p
q
)
=
(
p
1
,
…
,
p
a
,
q
1
,
…
,
q
b
,
p
1
q
1
,
…
,
p
1
q
b
,
…
,
p
a
q
1
,
…
,
p
a
q
b
),
(ii)
L
(
α
,
β
,
n
) is the set of matrices γ
∈
Map
(
[
0
,
a
]
×
[
0
,
b
]
,
N
) such that (28)
γ
00
=
0
,
γ
=
∑
l
=
0
a
∑
r
=
0
b
γ
l
r
≤
n
,
∑
r
=
0
b
γ
l
r
=
α
l
f
o
r
l
∈
a
,
∑
l
=
0
a
γ
l
r
=
β
r
f
o
r
r
∈
b
.Proof.
Identify the matrixγ with the vector (29)
γ
=
0
,
γ
01
,
…
,
γ
0
b
,
γ
10
,
…
,
γ
a
0
,
γ
20
,
…
,
γ
2
b
,
…
,
γ
a
0
,
…
,
γ
a
b
.We have that(30)
∑
α
,
β
≤
n
e
α
p
e
β
q
t
α
s
βis equal to (31)
=
∑
α
≤
n
e
α
p
t
α
∑
β
≤
n
e
β
q
s
β
=
∏
i
=
1
n
‍
1
+
∑
l
=
1
a
‍
p
l
i
t
l
∏
i
=
1
n
‍
1
+
∑
r
=
1
b
‍
q
r
i
s
r
=
∏
i
=
1
n
‍
1
+
∑
l
=
1
a
‍
p
l
(
i
)
t
l
1
+
∑
r
=
1
b
‍
q
r
(
i
)
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
i
t
l
+
∑
r
=
1
b
‍
q
r
i
s
r
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
p
l
i
q
r
i
t
l
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
i
w
l
0
+
∑
r
=
1
b
‍
q
r
j
w
0
r
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
p
l
i
q
r
i
w
l
r
=
∑
γ
‍
e
γ
p
,
q
,
p
q
w
γ
,where for γ
∈
L
(
α
,
β
,
n
) we set (32)
w
γ
=
∏
l
=
0
a
∏
r
=
0
b
w
l
r
γ
l
r
=
∏
l
=
0
a
∏
r
=
0
b
t
l
s
r
γ
l
r
=
∏
l
=
0
a
∏
r
=
0
b
t
l
γ
l
r
s
r
γ
l
r
,using the conventions (33)
t
0
=
s
0
=
1
,
w
l
r
=
t
l
s
r
for
l
,
r
≥
0
.For w
γ to be equal to t
α
s
β we must have(34)
w
γ
=
∏
l
=
0
a
∏
r
=
0
b
t
l
γ
l
r
s
r
γ
l
r
=
∏
l
=
1
a
‍
t
l
∑
r
=
0
b
γ
l
r
∏
r
=
1
b
‍
s
r
∑
l
=
0
a
γ
l
r
=
∏
l
=
1
a
‍
t
l
α
l
∏
k
=
1
b
‍
s
r
β
r
.Thus we conclude that (35)
∑
r
=
0
b
‍
γ
l
r
=
α
l
for
l
∈
a
,
∑
l
=
0
a
‍
γ
l
r
=
β
r
for
r
∈
b
.Graphically, a matrixγ
∈
L
(
α
,
β
,
n
) is represented as(36)
0
γ
01
γ
02
γ
03
⋯
γ
0
b
γ
10
γ
11
γ
12
γ
13
⋯
γ
1
b
⟶
α
1
γ
20
γ
21
γ
22
γ
23
⋯
γ
2
b
⟶
α
2
⋮
⋮
⋮
⋮
⋮
γ
a
0
γ
a
1
γ
a
2
γ
a
3
⋯
γ
a
b
⟶
α
a
↓
↓
↓
↓
β
1
β
2
β
3
⋯
β
b
,where the horizontal and vertical arrows represent, respectively, row and column sums.Example 8.
Forn
=
3
,
p
=
(
y
1
y
2
,
y
1
), and q
=
(
y
1
y
2
,
y
3
) we have(37)
e
1,1
y
1
y
2
,
y
1
e
2,1
y
1
y
2
,
y
3
=
∑
γ
‍
e
γ
y
1
y
2
,
y
1
,
y
1
y
2
,
y
3
,
y
1
2
y
2
2
,
y
1
y
2
y
3
,
y
1
2
y
2
,
y
1
y
3
,where γ
=
(
γ
10
,
γ
20
,
γ
01
,
γ
02
,
γ
11
,
γ
12
,
γ
21
,
γ
22
)
∈
N
8 is such that γ
≤
3 and (38)
γ
10
+
γ
11
+
γ
12
=
1
,
γ
20
+
γ
21
+
γ
22
=
1
,
γ
01
+
γ
11
+
γ
21
=
2
,
γ
02
+
γ
12
+
γ
22
=
1
.Looking at the solutions in N of the system of linear equations above we obtain (39)
e
1,1
y
1
y
2
,
y
1
e
2,1
y
1
y
2
,
y
3
=
e
1,1
,
1
y
3
,
y
1
2
y
2
2
,
y
1
2
y
2
+
e
1,1
,
1
y
1
y
2
,
y
1
y
2
y
3
,
y
1
2
y
2
+
e
1,1
,
1
y
1
y
2
,
y
1
2
y
2
2
,
y
1
y
3
.Example 9.
Forn
=
4
,
p
=
(
y
1
2
y
2
,
y
2
3
y
3
,
y
1
y
2
y
3
), and q
=
(
y
1
3
y
2
2
y
3
,
y
1
2
y
3
,
y
2
y
3
) we have (40)
e
(
1,1
,
1
)
(
y
1
2
y
2
,
y
2
3
y
3
,
y
1
y
2
y
3
)
e
(
1,2
,
1
)
(
y
1
3
y
2
2
y
3
,
y
1
2
y
3
,
y
2
y
3
)
=
∑
γ
‍
e
γ
y
1
2
y
2
,
y
2
3
y
3
,
y
1
y
2
y
3
,
y
1
3
y
2
2
y
3
,
y
1
2
y
3
,
y
2
y
3
,
k
k
k
k
k
k
k
k
k
k
y
1
5
y
2
3
,
y
1
4
y
2
y
3
,
y
1
2
y
2
2
y
3
,
y
1
3
y
2
5
y
3
2
,
y
2
4
y
3
2
,
y
1
y
2
4
y
3
,
k
k
k
k
k
k
k
k
k
k
y
1
4
y
2
3
y
3
2
,
y
1
3
y
2
y
3
2
,
y
1
2
y
2
2
y
3
,where γ
=
(
γ
10, γ
20, γ
30, γ
01, γ
02, γ
03, γ
11, γ
12, γ
13, γ
21
,
γ
22, γ
23, γ
31, γ
32, γ
33
)
∈
N
15 is such that |
γ
|
≤
4 and (41)
γ
10
+
γ
11
+
γ
12
+
γ
13
=
1
,
γ
20
+
γ
21
+
γ
22
+
γ
23
=
1
,
γ
30
+
γ
31
+
γ
32
+
γ
33
=
1
,
γ
01
+
γ
11
+
γ
21
+
γ
31
=
1
,
γ
02
+
γ
12
+
γ
22
+
γ
32
=
2
,
γ
03
+
γ
13
+
γ
23
+
γ
33
=
1
.Looking at the solutions in N of the system of linear equations above we obtain (42)
e
1,1
,
1
y
1
2
y
2
,
y
2
3
y
3
,
y
1
y
2
y
3
e
1,2
,
1
y
1
3
y
2
2
y
3
,
y
1
2
y
3
,
y
2
y
3
=
e
(
1,1
,
1,1
)
(
y
2
y
3
,
y
1
4
y
2
y
3
,
y
2
4
y
3
2
,
y
1
y
2
y
3
)
+
e
(
1,1
,
1,1
)
(
y
2
y
3
,
y
1
4
y
2
y
3
,
y
1
3
y
2
5
y
3
2
,
y
1
3
y
2
y
3
2
)
+
e
1,1
,
1,1
y
2
y
3
,
y
1
5
y
2
3
,
y
2
4
y
3
2
,
y
1
3
y
2
y
3
2
+
e
1,1
,
1,1
y
1
2
y
3
,
y
1
2
y
2
2
y
3
,
y
2
4
y
3
2
,
y
1
4
y
2
3
y
3
2
+
e
1,1
,
1,1
y
1
2
y
3
,
y
1
2
y
2
2
y
3
,
y
1
3
y
2
5
y
3
2
,
y
1
3
y
2
y
3
2
+
e
1,1
,
1,1
y
1
2
y
3
,
y
1
4
y
2
y
3
,
y
1
y
2
4
y
3
,
y
1
4
y
2
3
y
3
2
+
e
1,1
,
1,1
y
1
2
y
3
,
y
1
4
y
2
y
3
,
y
1
3
y
2
5
y
3
2
,
y
1
2
y
2
2
y
3
+
e
1,1
,
1,1
y
1
2
y
3
,
y
1
5
y
2
3
,
y
1
y
2
4
y
3
,
y
1
3
y
2
y
3
2
+
e
1,1
,
1,1
y
1
2
y
3
,
y
1
5
y
2
3
,
y
2
4
y
3
2
,
y
1
2
y
2
2
y
3
+
e
1,1
,
1,1
y
1
3
y
2
2
y
3
,
y
1
2
y
2
2
y
3
,
y
2
4
y
3
2
,
y
1
3
y
2
y
3
2
+
e
1,1
,
1,1
y
1
3
y
2
2
y
3
,
y
1
4
y
2
y
3
,
y
1
y
2
3
y
3
2
,
y
1
2
y
2
2
y
3
+
e
1,1
,
1,1
y
1
3
y
2
2
y
3
,
y
1
4
y
2
y
3
,
y
2
3
y
3
2
,
y
1
2
y
2
2
y
3
.


## 3. Review of Deformation Quantization

In this section we review a few needed notions on deformation quantization. We assume the reader to be somewhat familiar with Kontsevich’s work [7], although that level of generality is not necessary to understand the applications to the quantization of canonical phase space. A Poisson bracket [8, 9] on a smooth manifold M is a R-bilinear antisymmetric map (43)
,
:
C
∞
M
×
C
∞
M
⟶
C
∞
M
,where C
∞
(
M
) is the space of real-valued smooth functions on M, and for f
,
g
,
h
∈
C
∞
(
M
) the following identities hold: (44)
f
,
g
h
=
f
,
g
h
+
g
f
,
h
,
f
,
g
,
h
=
f
,
g
,
h
+
g
,
f
,
h
.A manifold equipped with a Poisson bracket is called a Poisson manifold. The Poisson bracket {
,
} is determined by an antisymmetric bilinear form α on T
*
M, that is, by the Poisson bivector α
∈
⋀
2
T
M given in local coordinates (
x
1
,
x
2
,
…
,
x
d
) on M by (45)
α
i
j
=
x
i
,
x
j
.The bivector α determines the Poisson bracket as follows: (46)
f
,
g
=
α
d
f
,
d
g
=
∑
i
,
j
∈
d
α
i
j
∂
f
∂
x
i
∂
g
∂
x
j
,
for
f
,
g
∈
C
∞
M
.If the Poisson bivector α
i
j is nondegenerated (i.e., det
⁡
(
α
i
,
j
)
≠
0) the Poisson manifold M is called symplectic.Example 10.
The spaceR
2
d is a symplectic Poisson manifold with Poisson bracket given in the linear coordinates (
x
1
,
…
,
x
d
,
y
1
,
…
,
y
d
) by (47)
f
,
g
=
∑
i
=
1
d
‍
∂
f
∂
x
i
∂
g
∂
y
i
-
∂
f
∂
y
i
∂
g
∂
x
i
,
for
f
,
g
∈
C
∞
R
2
d
.Equivalently, the Poisson bracket {
,
} on C
∞
(
R
2
d
) is determined by the identities (48)
x
i
,
x
j
=
0
,
y
i
,
y
j
=
0
,
x
i
,
y
j
=
δ
i
j
,
for
i
,
j
∈
d
.This example is the so-called canonical phase space with n degrees of freedom.Example 11.
Let(
g
,
[
,
]
) be Lie algebra over R of dimension d. The dual vector space g
* is a Poisson manifold with Poisson bracket given on f
,
g
∈
C
∞
(
g
*
) by (49)
f
,
g
α
=
α
,
d
α
f
,
d
α
g
,where α
∈
g
* and the differentials d
α
f and d
α
g are regarded as elements of g via the identifications T
α
*
g
*
=
g
*
*
=
g. Choose a linear basis e
1
,
…
,
e
d for g. The structural coefficients c
i
j
k of g are given, for i
,
j
,
k
∈
[
d
], by (50)
e
i
,
e
j
=
∑
k
=
1
d
‍
c
i
j
k
e
k
.Let (
x
1
,
…
,
x
d
) be the linear system of coordinates on g
* relative to the basis e
1
,
…
,
e
d of g. The Poisson bracket is determined by continuity and the identities (51)
x
i
,
x
j
=
∑
k
=
1
d
‍
c
i
j
k
x
k
.A formal deformation, or deformation quantization, of a Poisson manifoldM is an associative product, called the star product, (52)
⋆
:
C
∞
M
ħ
⊗
R
ħ
C
∞
M
ħ
⟶
C
∞
M
ħ
,defined on the space C
∞
(
M
)
[
[
ħ
]
] of formal power series in ħ with coefficients in C
∞
(
M
) such that the following conditions hold for f
,
g
∈
C
∞
(
M
):(i)
f
⋆
g
=
∑
n
=
0
∞
B
n
(
f
,
g
)
ħ
n, where the maps (53)
B
n
,
:
C
∞
M
×
C
∞
M
⟶
C
∞
M
are bidifferential operators.
(ii)
f
⋆
g
=
f
g
+
(
1
/
2
)
{
f
,
g
}
ħ
+
O
(
ħ
2
), where O
(
ħ
2
) stand for terms of order 2 and higher in the variable ħ.Kontsevich in [7] constructed a ⋆-product for any finite dimensional Poisson manifold. For linear Poisson manifolds the Kontsevich ⋆-product goes as follows. Fix a Poisson manifold (
R
d
,
α
); the Kontsevich ⋆-product is given on f
,
g
∈
C
∞
(
M
) by (54)
f
⋆
g
=
∑
n
=
0
∞
‍
B
n
f
,
g
ħ
n
n
!
=
∑
n
=
0
∞
‍
∑
Γ
∈
G
n
‍
ω
Γ
B
Γ
f
,
g
ħ
n
n
!
,
where
(i)
G
n is a collection of graphs, called admissible graphs, each with 2
n edges;
(ii)
for each graphΓ
∈
G
n, the constant ω
Γ
∈
R is independent of d and α and it is computed through an integral in an appropriated configuration space;
(iii)
B
Γ
(
,
)
:
C
∞
(
R
d
)
×
C
∞
(
R
d
)
→
C
∞
(
R
d
) is a bidifferential operator associated with the graph Γ
∈
G
n and the Poisson bivector α. The definition of the operators B
Γ
(
,
) is quite explicit and fairly combinatorial in nature.Remark 12.
Kontsevich himself has highlighted the fact that explicitly computing the integrals defining the constantsω
Γ is a daunting task currently beyond reach. One can however use the symbols ω
Γ as variables, and they will define a deformation quantization (with an extended ring of constants) as soon as these variables satisfy a certain system of quadratic equations [10].We are going to use the Kontsevich⋆-product in a slightly modified form(55)
Let
G
=
∐
n
=
0
∞
‍
G
n
,
for
Γ
∈
G
,
set
Γ
¯
=
n
iff
Γ
∈
G
n
.With this notation the Kontsevich ⋆-product is given on functions f
,
g
∈
C
∞
(
R
d
) by (56)
f
⋆
g
=
∑
Γ
∈
G
‍
ω
Γ
Γ
¯
!
B
Γ
f
,
g
ħ
Γ
¯
.Remark 13.
The Kontsevich⋆-product is defined over R since c
Γ
∈
R. If α is a regular Poisson bivector, that is, the entries α
i
j of the Poisson bivector are polynomial functions, then the ⋆-product on C
∞
(
R
d
) is restricted to a well-defined ⋆-product on the space R
[
x
1
,
…
,
x
d
] of polynomial functions on R
d. We are interested in the quantization of symmetric polynomial functions; thus we assume that α is a regular Poisson bivector and work with quantum algebra (
R
[
x
1
,
…
,
x
d
]
,
⋆
).


## 4. Quantum Symmetric Functions

LetS
n be the symmetric group on n letters. For each subgroup K
⊆
S
n, consider the Polya functor P
K
:
R
-alg
→
R
-alg from the category of associative R-algebras to itself, defined on objects as follows [10]. Let A be R-algebra; the underlying vector space of P
K
A is given by (57)
P
K
A
=
A
⊗
n
K
=
A
⊗
n
a
1
⊗
⋯
⊗
a
n
-
a
σ
1
⊗
⋯
⊗
a
σ
n
∣
a
i
∈
A
,
σ
∈
K
.Elements of P
K
A are written as a
1
⊗
⋯
⊗
a
n
¯. For a
i
j
∈
A, the following identity determines the product on P
K
A: (58)
K
m
-
1
∏
i
=
1
m
‍
⨂
j
=
1
n
a
i
j
¯
=
∑
σ
∈
{
1
}
×
K
m
-
1
⨂
j
=
1
n
∏
i
=
1
m
‍
a
i
σ
i
-
1
(
j
)
¯
.The Polya functor P
K is also known as the coinvariants functor. The invariants functor (59)
I
K
:
R
-alg
⟶
R
-alg
,
I
i
K
:
A
⟶
A
⊗
n
Kis given on objects by (60)
A
⊗
n
K
=
a
∈
A
⊗
n
∣
g
a
=
a
for
g
∈
K
.The product on (
A
⊗
n
)
K comes from the inclusion (
A
⊗
n
)
K
⊂
A
⊗
n.The functorsI
K and P
K are naturally isomorphic to each other [10].Suppose a finite groupK acts on a Poisson manifold M and that the induced action of K on (
C
∞
(
M
)
[
[
ħ
]
]
,
⋆
) is by algebra automorphisms; then we define the algebra of quantum K-symmetric functions on M as (61)
C
∞
M
ħ
,
⋆
K
≃
C
∞
M
ħ
,
⋆
K
.Let(
R
d
,
α
) be a regular Poisson manifold. The Cartesian product of Poisson manifolds is naturally endowed with the structure of a Poisson manifold; thus we get a regular Poisson manifold structure on (
R
d
)
n. We use the following coordinates on the n-fold Cartesian product of R
d with itself: (62)
R
d
n
=
x
1
,
…
,
x
n
∣
x
i
=
x
i
1
,
…
,
x
i
d
∈
R
d
,
k
k
x
1
,
…
,
x
n
k
k
x
i
j
∈
R
,
i
,
j
∈
n
×
d
.The ring of regular functions on (
R
d
)
n is the ring of polynomials on d
n commutative variables: (63)
R
R
d
n
=
R
x
11
,
…
,
x
1
d
,
…
,
x
n
1
,
…
,
x
n
d
.Consider another set of commutative variables y
1
,
…
,
y
d. Recall from Section 2 that for f
∈
R
[
y
1
,
…
,
y
d
] and i
∈
[
n
] we set f
(
i
)
=
f
(
x
i
1
,
…
,
x
i
d
)
∈
R
[
x
i
1
,
…
,
x
i
d
]
⊆
R
[
(
R
d
)
n
].The Poisson bracket on(
R
d
)
n is determined by the following identities:(64)
x
k
i
,
x
l
j
=
δ
k
l
α
i
j
k
,
for
i
,
j
∈
d
,
k
,
l
∈
n
,where the coordinates α
i
j of the Poisson bivector α
=
∑
‍
α
i
j
∂
i
∧
∂
j are regarded as polynomials in R
[
y
1
,
…
,
y
n
]. The Poisson bracket on (
R
d
)
n is S
n-invariant; indeed for σ
∈
S
n we have (65)
x
σ
k
i
,
x
σ
l
j
=
δ
σ
k
,
σ
l
α
i
j
σ
k
=
σ
δ
k
l
α
i
j
k
.Next results [10] provide a natural construction of groups acting as algebra automorphisms on the algebras (
R
[
R
d
]
[
[
ħ
]
]
,
⋆
).Theorem 14.
Let(
R
d
,
{
,
}
) be a regular Poisson manifold and let K be a subgroup of S
d such that the Poisson bracket {
,
} is K-equivariant. Then the action of K on (
R
[
R
d
]
[
[
ħ
]
]
,
⋆
) is by algebra automorphisms.Corollary 15.
Let(
R
d
,
{
,
}
) be a regular Poisson manifold and consider a subgroup K
⊆
S
n. Then K acts by algebra automorphisms on (
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
).Definition 16.
Let(
R
d
,
{
,
}
) be a regular Poisson manifold. The algebra of quantum symmetric functions on (
R
d
)
n is given by (66)
R
R
d
n
ħ
,
⋆
S
n
≃
R
R
d
n
ħ
,
⋆
S
n
.Example 17.
ConsiderR
2
d with its canonical symplectic Poisson structure; then (
R
2
d
)
n is also a symplectic Poisson manifold. Choose coordinates on (
R
2
d
)
n as follows: (67)
R
2
d
n
=
x
1
,
y
1
,
…
,
x
n
,
y
n
∣
x
i
=
x
i
1
,
…
,
x
i
d
,
x
i
j
,
y
i
j
∈
R
k
k
k
x
1
,
y
1
,
…
,
x
n
,
y
n
k
k
k
x
1
,
y
1
,
…
,
x
n
,
y
n
k
y
i
=
y
i
1
,
…
,
y
i
d
,
k
k
k
x
1
,
y
1
,
…
,
x
n
,
y
n
k
x
i
j
,
y
i
j
∈
R
.The S
n-invariant Poisson bracket on (
R
2
d
)
n is given for i
,
j
∈
[
d
] and k
,
l
∈
[
n
] by (68)
x
k
i
,
x
l
j
=
0
,
y
k
i
,
y
l
j
=
0
,
x
k
i
,
y
l
j
=
δ
k
l
δ
i
j
.Example 18.
Letg be d-dimensional Lie algebra over R and let g
* be its dual vector space. Then g
* is a Poisson manifold, and therefore (
g
*
)
n is also a Poisson manifold. The S
n-invariant Poisson bracket on (
g
*
)
n is given, for i
,
j
∈
[
d
] and k
,
l
∈
[
n
], by (69)
x
k
i
,
x
l
j
=
δ
k
l
∑
m
=
0
d
c
i
j
m
x
k
m
,where c
i
j
m are the structural coefficients of g.Specializing Definition16 we obtain the following natural notions. The algebra of quantum symmetric functions on (
R
2
d
)
n is given by (70)
R
R
2
d
n
ħ
,
⋆
S
n
≃
R
R
2
d
n
ħ
,
⋆
S
n
.More generally, the algebra of quantum symmetric functions on (
g
*
)
n is given by (71)
R
g
*
n
ħ
,
⋆
S
n
≃
R
g
*
n
ħ
,
⋆
S
n
.


## 5.⋆-Product of Multisymmetric Functions

We are ready to state and proof the main result of this work which extends Theorem7 from the classical to the quantum case: we provide an explicit formula for the ⋆-product of multisymmetric functions.Recall that the⋆-product can be expanded as a formal power series in ħ as (72)
f
⋆
g
=
∑
n
=
0
∞
B
n
f
,
g
ħ
n
n
!
.Theorem 19.
Let(
R
d
,
{
,
}
) be a regular Poisson manifold and let (
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n be the algebra of quantum symmetric functions on (
R
d
)
n. Fix a
,
b
,
n
∈
N
+
,
p
∈
R
[
y
1
,
…
,
y
d
]
a, and q
∈
R
[
y
1
,
…
,
y
d
]
b. Let α
∈
N
a and β
∈
N
b be such that |
α
|
,
|
β
|
≤
n. The ⋆-product of e
α
(
p
) and e
β
(
q
) is given by (73)
e
α
p
⋆
e
β
q
=
∑
m
=
0
∞
‍
∑
γ
∈
Q
(
α
,
β
,
n
,
m
)
e
γ
B
p
,
q
ħ
m
,
w
h
e
r
e
(i)
B
(
p
,
q
)
=
(
p
,
q
,
…
,
B
k
(
p
,
q
)
,
…
) and (74)
B
k
p
,
q
=
B
k
p
1
,
q
1
,
…
,
B
k
p
1
,
q
b
,
…
,
k
k
k
k
B
k
p
a
,
q
1
,
…
,
B
k
p
a
,
q
b
,
(ii)
Q
(
α
,
β
,
n
,
m
) is the subset of Map
(
[
0
,
a
]
×
[
0
,
b
]
×
N
,
N
) consisting of cubical matrices (75)
γ
:
0
,
a
×
0
,
b
×
N
⟶
N
s
u
c
h
t
h
a
t
(a)
γ
00
k
=
0 for k
≥
0; if either l
=
0 or r
=
0, then γ
l
r
k
=
0 for k
≥
1,
(b)
|
γ
|
=
∑
l
=
0
a
∑
r
=
0
b
∑
k
=
0
∞
γ
l
r
k
≤
n, and ∑
l
=
0
a
∑
r
=
0
b
∑
k
=
0
∞
k
γ
l
r
k
=
m,
(c)
∑
r
=
0
b
∑
k
=
0
∞
γ
l
r
k
=
α
l for l
∈
[
a
], and ∑
l
=
0
a
∑
k
=
0
∞
γ
l
r
k
=
β
r for j
∈
[
b
].Proof.
We have(76)
∑
α
,
β
≤
n
∑
m
=
0
∞
B
m
e
α
p
,
e
β
q
t
α
s
β
ħ
m
=
∑
|
α
|
,
|
β
|
≤
n
e
α
p
⋆
e
β
q
t
α
s
β
=
∑
α
≤
n
‍
e
α
p
t
α
⋆
∑
β
≤
n
‍
e
β
q
s
β
=
∏
i
=
1
n
‍
1
+
∑
l
=
1
a
‍
p
l
i
t
l
⋆
∏
i
=
1
n
‍
1
+
∑
r
=
1
b
‍
q
r
i
s
r
=
∏
i
=
1
n
‍
1
+
∑
l
=
1
a
‍
p
l
i
t
l
⋆
1
+
∑
r
=
1
b
‍
q
r
i
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
(
i
)
t
l
+
∑
r
=
1
b
‍
q
r
(
i
)
s
r
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
p
l
(
i
)
⋆
q
r
(
i
)
t
l
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
(
i
)
t
l
+
∑
r
=
1
b
‍
q
r
(
i
)
s
r
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
∑
k
=
0
∞
B
k
p
l
i
,
q
r
i
t
l
s
r
ħ
k
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
(
i
)
w
l
00
+
∑
r
=
1
b
‍
q
r
(
i
)
w
0
r
0
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
∑
k
=
0
∞
B
k
p
l
i
,
q
r
i
w
l
r
k
=
∑
γ
∈
Q
α
,
β
,
n
,
m
‍
e
γ
B
p
,
q
w
γ
,
where:
w
γ
=
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
w
l
r
k
γ
l
r
k
=
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
t
l
s
r
ħ
k
γ
l
r
k
w
γ
=
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
t
l
γ
l
r
k
s
r
γ
l
r
k
ħ
k
γ
l
r
k
,and we are using the conventions (77)
t
0
=
s
0
=
1
,
w
r
u
k
=
t
r
s
u
ħ
k
for
r
,
u
,
m
≥
0
.For w
γ to be equal to t
α
s
β
ħ
m we must have (78)
∏
l
=
0
a
‍
t
l
α
l
∏
r
=
0
b
‍
s
r
β
r
ħ
m
=
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
t
l
γ
l
r
k
s
r
γ
l
r
k
ħ
k
γ
l
r
k
=
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
t
l
γ
l
r
k
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
s
r
γ
l
r
k
·
∏
l
=
0
a
∏
r
=
0
b
∏
k
=
0
∞
ħ
k
γ
l
r
k
=
∏
l
=
1
a
‍
t
l
∑
r
=
0
b
∑
k
=
0
∞
γ
l
r
k
∏
r
=
1
b
‍
s
r
∑
l
=
0
a
∑
k
=
0
∞
γ
l
r
k
·
ħ
∑
l
=
0
a
∑
r
=
0
b
∑
k
=
0
∞
k
γ
l
r
kand thus we conclude that (79)
∑
r
=
0
b
∑
k
=
0
∞
γ
l
r
k
=
α
l
for
l
∈
[
a
]
,
∑
l
=
0
a
∑
k
=
0
∞
γ
l
r
k
=
β
r
for
j
∈
[
b
]
,
∑
l
=
0
a
∑
r
=
0
b
∑
k
=
0
∞
k
γ
l
r
k
=
m
.Corollary 20.
With the assumptions of Theorem19, the Poisson bracket of the multisymmetric functions e
α
(
p
) and e
β
(
q
) is given by (80)
e
α
p
,
e
β
q
=
2
∑
γ
∈
Q
α
,
β
,
n
,
1
e
γ
B
p
,
q
.Proof.
It follows from Theorem19 and the identity (81)
e
α
p
,
e
β
q
=
2
∂
∂
ħ
e
α
p
⋆
e
α
q
e
α
p
⋆
e
α
q
ħ
=
0
.For our next result we regard(
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n as topological algebra with topology induced by the inclusion (82)
R
R
d
n
ħ
,
⋆
S
n
⊆
R
R
d
n
ħ
,
⋆
,where a fundamental system of neighborhoods of 0
∈
R
[
(
R
d
)
n
]
[
[
ħ
]
] is given by the decreasing family of subalgebras (83)
R
R
d
n
ħ
⊇
ħ
R
R
d
n
ħ
⊇
⋯
⊇
ħ
n
R
R
d
n
ħ
⊇
⋯
.Recall from Introduction that the elementary multisymmetric functionse
k, for k
∈
N
d with |
k
|
≤
n, are defined by the identity (84)
∏
i
=
1
n
‍
1
+
x
i
1
t
1
+
⋯
+
x
i
d
t
d
=
∑
k
∈
N
d
,
k
≤
n
‍
e
k
t
k
.Similarly, the homogeneous multisymmetric functionsh
k, for k
=
(
k
1
,
…
,
k
d
)
∈
N
d, are defined by the identity (85)
∏
i
=
1
n
‍
1
1
-
x
i
1
t
1
-
⋯
-
x
i
d
t
d
=
∑
k
∈
N
d
‍
h
k
t
k
.LetM
d be the set of (nontrivial) monomials in the variables y
1
,
…
,
y
d. The power sum symmetric function e
1
(
m
) is given, for m
∈
M
d, by (86)
e
1
m
=
m
1
+
⋯
+
m
d
.Theorem 21.
The elementary multisymmetric functionse
k  for |
k
|
≤
n, the homogeneous multisymmetric functions h
k for |
k
|
≤
n, and the power sum multisymmetric functions e
1
(
m
) with m
∈
M
d a monomial of degree less than or equal to n, together with ħ generate, respectively, the topological algebra (
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n.Proof.
It is known [1, 3, 4, 11, 12] that each of the aforementioned sets of multisymmetric functions generate the algebra of classical multisymmetric functions R
[
(
R
d
)
n
]
S
n.
To go the quantum case the same argument is applied in each case, so we only consider the elementary symmetric functions. Takef
∈
(
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n and expand it as formal power series (87)
f
=
∑
k
=
0
∞
‍
f
k
ħ
k
with
f
k
∈
R
R
d
n
S
n
.We can write f
0 as a linear combination of a product of elementary symmetric functions. For simplicity assume that f
0
=
e
k
1
⋯
e
k
m, then (88)
f
-
e
k
1
⋆
⋯
⋆
e
k
m
∈
O
ħ
.Assume next that (
f
-
e
k
1
⋆
⋯
⋆
e
k
m
)
1 can be written as e
l
1
⋯
e
l
r, then (89)
f
-
e
k
1
⋆
⋯
⋆
e
k
m
-
e
l
1
⋆
⋯
⋆
e
l
r
ħ
∈
O
ħ
2
.Proceeding by induction we see that f can be written as a formal power series in ħ with coefficients equal to the sum of the ⋆-product of elementary multisymmetric functions.Choose a variablet
m for each m
∈
M
d, the set of monomials in the variables y
1
,
…
,
y
d, and set (90)
∑
|
α
|
≤
n
‍
e
α
t
α
=
∏
i
=
1
n
‍
1
+
∑
m
∈
M
d
‍
m
(
i
)
t
m
,where α
:
M
d
→
N of finite support and t
α
=
∏
m
∈
M
d
t
m
α
(
m
).Theorem 22.
The set{
e
α
ħ
k
∣
|
α
|
≤
n
,
k
≥
0
} is a topological basis for the topological algebra (
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n. The product of basic elements is given by Lemma 6 and Theorem 19.Proof.
It is well known that the symmetrization of monomials yields a basis forR
[
(
R
d
)
n
]
S
n; thus from Lemma 5 we see that the set {
e
α
∣
|
α
|
≤
n
} is a basis for R
[
(
R
d
)
n
]
S
n as well. Thus forming the products e
α
ħ
k we obtain a topological basis for (
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n.Next result describes the product of multisymmetric functions using the Kontsevich’s⋆-product. In this case one can give a more precise formula for the computation of the quantum higher corrections, that is, the coefficients that accompany the higher order powers in ħ.Theorem 23.
Let(
R
d
,
{
,
}
) be a regular Poisson manifold and let (
R
[
(
R
d
)
n
]
[
[
ħ
]
]
,
⋆
)
S
n be the algebra of quantum symmetric functions on (
R
d
)
n with the Kontsevich ⋆-product. Fix a
,
b
,
n
∈
N
+
,
p
∈
R
[
y
1
,
…
,
y
d
]
a, and q
∈
R
[
y
1
,
…
,
y
d
]
b. Let α
∈
N
a and β
∈
N
b be such that |
α
|
,
|
β
|
≤
n. The ⋆-product of e
α
(
p
) and e
β
(
q
) is given by (91)
e
α
p
⋆
e
β
q
=
∑
m
=
0
∞
‍
∑
γ
∈
K
α
,
β
,
n
,
m
‍
e
γ
B
p
,
q
ħ
m
,
w
h
e
r
e
(i)
B
(
p
,
q
)
=
(
p
,
q
,
…
,
(
ω
Γ
/
Γ
¯
!
)
B
Γ
(
p
,
q
)
,
…
) and (92)
B
Γ
p
,
q
=
B
Γ
p
1
,
q
1
,
…
,
B
Γ
p
1
,
q
b
,
…
,
k
k
k
B
Γ
p
a
,
q
1
,
…
,
B
Γ
p
a
,
q
b
,
the polynomialB
Γ
(
p
i
,
q
j
) results of applying Kontsevich’s bidifferential operator B
Γ to the pair (
p
i
,
q
j
);
(ii)
K
(
α
,
β
,
n
,
m
) is the subset of Map
(
[
0
,
a
]
×
[
0
,
b
]
×
G
,
N
) consisting of maps (93)
γ
:
0
,
a
×
0
,
b
×
G
⟶
N
s
u
c
h
t
h
a
t
(a)
γ
00
Γ
=
0; if either l
=
0 or r
=
0, then γ
l
r
Γ
=
0 for Γ
¯
≥
1;
(b)
|
γ
|
=
∑
l
=
0
a
∑
r
=
0
b
∑
Γ
∈
G
γ
l
r
Γ
≤
n, and ∑
l
=
0
a
∑
r
=
0
b
∑
Γ
∈
G
Γ
¯
γ
l
r
Γ
=
m,
(c)
∑
r
=
0
b
∑
Γ
∈
G
γ
l
r
Γ
=
α
l for l
∈
[
a
], and ∑
l
=
0
a
∑
Γ
∈
G
γ
l
r
Γ
=
β
r for j
∈
[
b
].Proof.
We have(94)
∑
|
α
|
,
|
β
|
≤
n
∑
m
=
0
∞
B
m
e
α
p
,
e
β
q
t
α
s
β
ħ
m
=
∑
|
α
|
,
|
β
|
≤
n
‍
e
α
(
p
)
⋆
e
β
(
q
)
t
α
s
β
=
∑
|
α
|
≤
n
‍
e
α
(
p
)
t
α
⋆
∑
|
β
|
≤
n
‍
e
β
(
q
)
s
β
=
∏
i
=
1
n
‍
1
+
∑
l
=
1
a
‍
p
l
(
i
)
t
l
⋆
1
+
∑
r
=
1
b
‍
q
r
(
i
)
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
(
i
)
t
l
+
∑
r
=
1
b
‍
q
r
(
i
)
s
r
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
p
l
(
i
)
⋆
q
r
(
i
)
t
l
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
(
i
)
t
l
+
∑
r
=
1
b
‍
q
r
(
i
)
s
r
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
∑
Γ
∈
G
ω
Γ
Γ
¯
!
B
Γ
p
l
i
,
q
r
i
t
l
s
r
ħ
Γ
¯
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
p
l
(
i
)
w
l
0
∅
+
∑
r
=
1
b
‍
q
r
(
i
)
w
0
r
∅
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
∑
Γ
∈
G
ω
Γ
Γ
¯
!
B
Γ
p
l
i
,
q
r
i
w
r
k
Γ
=
∑
γ
∈
K
α
,
β
,
n
,
m
‍
e
γ
B
p
,
q
w
γ
,
where:
w
γ
=
∏
l
=
0
a
∏
r
=
0
b
∏
Γ
∈
G
w
l
r
Γ
γ
l
r
Γ
=
∏
l
=
0
a
∏
r
=
0
b
∏
Γ
∈
G
∞
t
l
γ
l
r
k
s
r
γ
l
r
k
ħ
Γ
¯
γ
l
r
Γ
,and by convention ∅ stands for the unique graph in G with no edges (representing the classical product), and (95)
t
0
=
s
0
=
1
,
w
r
u
Γ
=
t
r
s
u
ħ
Γ
¯
for
r
,
u
≥
0
,
Γ
∈
G
.For t
α
s
β
ħ
m
=
w
γ we must have (96)
∏
l
=
0
a
‍
t
l
α
l
∏
r
=
0
b
‍
s
r
β
r
ħ
m
=
∏
l
=
0
a
∏
r
=
0
b
∏
Γ
∈
G
t
l
γ
l
r
k
s
r
γ
l
r
k
ħ
Γ
¯
γ
l
r
Γ
=
∏
l
=
1
a
‍
t
l
∑
r
=
0
b
∑
Γ
∈
G
γ
l
r
Γ
∏
r
=
1
b
‍
s
r
∑
l
=
0
a
∑
Γ
∈
G
γ
l
r
Γ
·
ħ
∑
l
=
0
a
∑
r
=
0
b
∑
Γ
∈
G
Γ
¯
γ
l
r
Γ
.Thus we conclude that (97)
∑
r
=
0
b
∑
Γ
∈
G
γ
l
r
Γ
=
α
l
for
l
∈
a
,
∑
l
=
0
a
∑
Γ
∈
G
γ
l
r
Γ
=
β
r
for
j
∈
b
,
∑
l
=
0
a
∑
r
=
0
b
∑
Γ
∈
G
Γ
¯
γ
l
r
Γ
=
m
.


## 6. Symmetric Powers of the Weyl Algebras

In this section we study the case of two-dimensional canonical phase space, that is, the symplectic manifoldR
2 with the canonical Poisson bracket given as follows: (98)
f
,
g
=
∂
f
∂
x
∂
g
∂
y
-
∂
f
∂
y
∂
g
∂
x
.Definition 24.
The Weyl algebra is defined by generators and relations by(99)
W
=
R
x
,
y
ħ
y
x
-
x
y
-
ħ
.The deformation quantization of(
R
2
,
{
,
}
) is well known to be given by the Moyal product [13]. Moreover, one has the following result.Theorem 25.
The Weyl algebra is isomorphic to the deformation quantization of polynomial functions onR
2 with the canonical Poisson structure.Our goal in this section is to study the deformation quantization of the space(
R
2
)
n
/
S
n, which can be identified with the algebra of quantum symmetric functions (100)
R
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
ħ
,
⋆
S
n
,or, equivalently, with the symmetric powers of the Weyl algebra (101)
W
⊗
n
S
n
.One shows by induction [10] that the following identity holds in the Weyl algebra: (102)
x
c
y
d
⋆
x
f
y
g
=
∑
k
=
0
min
⁡
‍
B
k
x
c
y
d
,
x
f
y
g
ħ
k
x
c
y
d
⋆
x
f
y
g
=
∑
k
=
0
min
⁡
d
k
f
k
x
c
+
f
-
k
y
d
+
g
-
k
ħ
k
,
where
min
⁡
=
min
⁡
d
,
f
,
f
k
=
f
f
-
1
f
-
2
f
-
k
+
1
.Theorem 26.
ConsiderR
2 with its canonical Poisson structure. Fix a
,
b
∈
N
+ and let (103)
x
c
1
y
d
1
,
…
,
x
c
a
y
d
a
∈
R
x
,
y
a
,
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
∈
R
x
,
y
b
.For α
∈
N
a and β
∈
N
b the following identity holds: (104)
e
α
x
c
1
y
d
1
,
…
,
x
c
a
y
d
a
⋆
e
β
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
=
∑
m
=
0
∞
∑
γ
∈
Q
α
,
β
,
n
,
m
e
γ
B
x
c
1
y
d
1
,
…
,
x
c
a
y
d
a
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
i
k
k
k
k
k
k
∑
γ
∈
Q
α
,
β
,
n
,
m
e
γ
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
ħ
m
,where (105)
B
x
c
1
y
d
1
,
…
,
x
c
a
,
y
d
a
,
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
=
x
c
1
y
d
1
,
…
,
x
c
a
,
y
d
a
,
x
f
1
y
g
1
,
…
,
d
r
k
k
k
k
k
k
k
x
f
b
y
g
b
,
…
,
d
r
k
f
r
k
x
c
l
+
f
r
-
k
y
d
l
+
g
r
-
k
,
…
.Proof.
We have(106)
∑
|
α
|
,
|
β
|
≤
n
∑
m
=
0
∞
B
m
e
α
x
c
1
y
d
1
,
…
,
x
c
a
y
d
a
,
k
k
k
k
k
k
k
k
k
k
k
k
k
e
β
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
t
α
s
β
ħ
m
=
∑
α
,
β
≤
n
e
α
x
c
1
y
d
1
,
…
,
x
c
a
y
d
a
k
k
k
k
k
k
k
k
k
k
k
⋆
e
β
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
t
α
s
β
=
∑
α
≤
n
‍
e
α
x
c
1
y
d
1
,
…
,
x
c
a
y
d
a
t
α
⋆
∑
β
≤
n
‍
e
β
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
s
β
=
∏
i
=
1
n
‍
1
+
∑
l
=
1
a
‍
x
i
c
l
y
i
d
l
t
l
⋆
1
+
∑
r
=
1
b
‍
x
i
f
r
y
i
g
r
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
x
i
c
l
y
i
d
l
t
l
+
∑
r
=
1
b
‍
x
i
f
r
y
i
g
r
s
r
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
x
i
c
l
y
i
d
l
⋆
x
i
f
r
y
i
g
r
t
l
s
r
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
x
i
c
l
y
i
d
l
t
l
+
∑
r
=
1
b
‍
x
i
f
r
y
i
g
r
s
r
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
∑
k
=
0
min
⁡
d
r
k
f
r
k
x
i
c
l
+
f
r
-
k
k
k
k
k
k
k
k
k
k
·
∑
r
=
1
b
‍
x
i
f
r
y
i
d
l
+
g
r
-
k
t
l
s
r
ħ
k
=
∏
i
=
1
n
1
+
∑
l
=
1
a
‍
x
i
c
l
y
i
d
l
w
l
00
+
∑
r
=
1
b
‍
x
i
f
r
y
i
g
r
w
0
r
0
k
k
k
k
k
k
k
k
k
+
∑
l
=
1
a
∑
r
=
1
b
∑
k
=
0
min
⁡
d
r
k
f
r
k
x
i
c
l
+
f
r
-
k
k
k
k
k
k
k
k
k
k
i
·
∑
r
=
1
b
‍
y
i
d
l
+
g
r
-
k
w
l
r
k
=
∑
γ
∈
Q
(
α
,
β
,
n
,
m
)
e
γ
B
x
c
1
y
d
1
,
…
,
x
c
a
,
y
d
a
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
w
γ
,where min
⁡
=
min
⁡
{
d
l
,
f
r
} and w
l
r
k
=
t
l
s
r
ħ
k.Corollary 27.
With the assumptions of Theorem26, the Poisson bracket of the multisymmetric functions e
α
(
p
) and e
β
(
q
) is given by (107)
e
α
p
,
e
β
q
=
2
∑
γ
∈
Q
α
,
β
,
n
,
1
‍
e
γ
B
x
c
1
y
d
1
,
…
,
x
c
a
,
y
d
a
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
x
f
1
y
g
1
,
…
,
x
f
b
y
g
b
.Example 28.
Letp
=
y
,
q
=
x
∈
R
[
x
,
y
]. We have(108)
e
α
y
⋆
e
β
x
=
∑
m
≥
0
∑
γ
∈
Q
(
α
,
β
,
n
,
m
)
e
γ
y
,
x
,
x
y
,
1
ħ
m
,where the vectors γ
=
(
γ
100
,
γ
010
,
γ
110
,
γ
111
)
∈
N
4 is such that |
γ
|
≤
n and (109)
γ
100
+
γ
110
+
γ
111
=
α
,
γ
010
+
γ
110
+
γ
111
=
β
,
γ
111
=
m
.For example, for n
=
3
,
α
=
2
,
β
=
3, we have (110)
e
2
y
⋆
e
3
x
=
e
1,2
x
,
x
y
+
e
1,1
,
1
x
,
x
y
,
1
ħ
+
e
1,2
x
,
1
ħ
2since in this case γ
=
(
γ
100
,
γ
010
,
γ
110
,
γ
111
)
∈
N
4 is such that (111)
γ
100
+
γ
010
+
γ
110
+
γ
111
≤
3
,
γ
100
+
γ
110
+
γ
111
=
2
,
γ
010
+
γ
110
+
γ
111
=
3
,
γ
111
=
m
.Solving this equation for m
=
0,1
,
2 we, respectively, obtain (112)
γ
=
0,1
,
2,0
,
γ
=
0,1
,
1,1
,
γ
=
0,1
,
0,2
,yielding the desired result.On the other hand, from Definition1 we get (113)
e
2
(
y
)
=
y
1
y
2
+
y
1
y
3
+
y
2
y
3
,
e
3
(
x
)
=
x
1
x
2
x
3
,and thus(114)
e
2
y
⋆
e
3
x
=
y
1
y
2
+
y
1
y
3
+
y
2
y
3
⋆
x
1
x
2
x
3
=
x
1
x
2
x
3
y
1
y
2
+
x
1
x
2
x
3
y
1
y
3
+
x
1
x
2
x
3
y
2
y
3
+
x
1
x
3
y
1
+
x
1
x
2
y
1
+
x
2
x
3
y
2
+
x
1
x
2
y
2
k
k
k
k
k
k
k
+
x
2
x
3
y
3
+
x
1
x
3
y
3
ħ
+
x
1
+
x
2
+
x
3
ħ
2
,which indeed is equal to (115)
e
1,2
x
,
x
y
+
e
1,1
,
1
x
,
x
y
,
1
ħ
+
e
1,2
x
,
1
ħ
2
.Note that{
e
2
(
y
)
,
e
3
(
x
)
}
=
2
e
(
1,1
,
1
)
(
x
,
x
y
,
1
).Example 29.
Letp
=
q
=
x
y
∈
R
[
x
,
y
], then we have (116)
e
α
x
y
⋆
e
β
x
y
=
∑
m
≥
0
∑
γ
∈
Q
L
(
α
,
β
,
n
,
m
)
e
γ
x
y
,
x
y
,
x
2
y
2
,
x
y
ħ
m
,where the vector γ
=
(
γ
100
,
γ
010
,
γ
110
,
γ
111
)
∈
N
4 is such that |
γ
|
≤
n and (117)
γ
100
+
γ
110
+
γ
111
=
α
,
γ
010
+
γ
110
+
γ
111
=
β
,
γ
111
=
m
.Thus for n
=
2
,
α
=
2
,
β
=
1, we get (118)
e
2
x
y
⋆
e
1
x
y
=
e
1,1
x
y
,
x
2
y
2
+
e
1,1
x
y
,
x
y
ħsince in this case γ
=
(
γ
100
,
γ
010
,
γ
110
,
γ
111
)
∈
N
4 is such that (119)
γ
100
+
γ
010
+
γ
110
+
γ
111
≤
2
,
γ
100
+
γ
110
+
γ
111
=
2
,
γ
010
+
γ
110
+
γ
111
=
1
,
γ
111
=
m
.Solving this equation for m
=
0,1 we, respectively, obtain (120)
γ
=
1,0
,
1,0
,
γ
=
1,0
,
0,1
,yielding the desired result.From Definition1 we have e
2
(
x
y
)
=
x
1
y
1
x
2
y
2
,
e
1
(
x
y
)
=
x
1
y
1
+
x
2
y
2, and thus (121)
e
2
x
y
⋆
e
1
x
y
=
x
1
y
1
x
2
y
2
⋆
x
1
y
1
+
x
2
y
2
=
x
1
2
y
1
2
x
2
y
2
+
x
1
y
1
x
2
2
y
2
2
+
2
x
1
y
1
x
2
y
2
ħ
,which indeed is equal to e
(
1,1
)
(
x
y
,
x
2
y
2
)
+
e
(
1,1
)
(
x
y
,
x
y
)
ħ.Note that{
e
2
(
x
y
)
,
e
1
(
x
y
)
}
=
2
e
(
1,1
)
(
x
y
,
x
y
).Example 30.
Letn
=
2
,
α
=
β
=
2; then (122)
e
2
x
a
y
⋆
e
2
x
y
b
=
e
2
x
a
+
1
y
b
+
1
+
e
1,1
x
a
+
1
y
b
+
1
,
x
a
y
b
ħ
+
e
2
x
a
y
b
ħ
2
.Using Theorem 19 we have (123)
e
2
x
a
y
⋆
e
2
x
y
b
=
∑
γ
∈
Q
L
2,2
,
2
,
m
‍
e
γ
x
a
y
,
x
y
b
,
x
a
+
1
y
b
+
1
,
x
a
y
b
ħ
m
,where γ
=
(
γ
100
,
γ
010
,
γ
110
,
γ
111
)
∈
N
4 is such that (124)
γ
100
+
γ
010
+
γ
110
+
γ
111
≤
2
,
γ
100
+
γ
110
+
γ
111
=
2
,
γ
010
+
γ
110
+
γ
111
=
2
,
γ
111
=
m
.Solving this equation for m
=
0,1
,
2 we, respectively, obtain (125)
γ
=
0,0
,
2,0
,
γ
=
0,0
,
1,1
,
γ
=
0,0
,
0,2
.Thus we get (126)
e
2
x
a
y
⋆
e
2
x
y
b
=
e
2
x
a
+
1
y
b
+
1
+
e
1,1
x
a
+
1
y
b
+
1
,
x
a
y
b
ħ
+
e
2
x
a
y
b
ħ
2
.On the other hand, from Definition 1 we have (127)
e
2
x
a
y
=
x
1
a
y
1
x
2
a
y
2
,
e
2
x
y
b
=
x
1
y
1
b
x
2
y
2
b
.Computing directly the ⋆-product we obtain (128)
e
2
x
a
y
⋆
e
2
x
y
b
=
x
1
a
y
1
x
2
a
y
2
⋆
x
1
y
1
b
x
2
y
2
b
=
x
1
a
+
1
y
1
b
+
1
x
2
a
+
1
y
2
b
+
1
+
x
1
a
y
1
b
x
2
a
+
1
y
2
b
+
1
+
x
1
a
+
1
y
1
b
+
1
x
2
a
y
2
b
ħ
+
x
1
a
y
1
b
x
2
a
y
2
b
ħ
2
=
e
2
x
a
+
1
y
b
+
1
+
e
1,1
x
a
+
1
y
b
+
1
,
x
a
y
b
ħ
+
e
2
x
a
y
b
ħ
2
.
Note that{
e
2
(
x
a
y
)
,
e
2
(
x
y
b
)
}
=
2
e
(
1,1
)
(
x
a
+
1
y
b
+
1
,
x
a
y
b
).We close this work stating the main problem that our research opens.Problem 31.
Describe the relations in the algebra of quantum symmetric functions.



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*Source: 476926-2015-03-11.xml*