International Journal of Mathematics and Mathematical Sciences
Volume 2015 (2015), Article ID 476926, 13 pages
http://dx.doi.org/10.1155/2015/476926

Research Article

Quantum Product of Symmetric Functions

Rafael Díaz¹ and Eddy Pariguan²

¹Instituto de Matemáticas y sus Aplicaciones, Universidad Sergio Arboleda, Bogotá, Colombia
²Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia

Received 22 September 2014; Accepted 25 February 2015

Academic Editor: Hernando Quevedo

Copyright © 2015 Rafael Díaz and Eddy Pariguan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

Fix a characteristic zero field . The algebrogeometric duality allows us to identify affine algebraic varieties with the -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 is thus identified with the algebra of polynomials in -variables . Consider the action of the symmetric group on by permutation of vector entries. The quotient space is the configuration space of unlabeled points with repetitions in . Polynomial functions on may be identified with the algebra of -invariant polynomials in . A remarkable classical fact is that is again a -dimensional affine space [1]; indeed we have an isomorphism of algebras

where, for , is the elementary symmetric polynomial given by

The elementary symmetric polynomials are determined by the identity

Using characteristic functions one shows for that

where is the cardinality of the subset of matrices of format with entries in such that

A subtler situation arises when one considers the configuration space

of unlabeled points with repetitions in , for . In this case is no longer an affine space; instead it is an affine algebraic variety. Polynomial functions on 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

which admits a presentation of the following form:

where the elementary multisymmetric functions , for a vector such that , are defined by the identity

Explicitly, the multisymmetric function is given by

where in the middle term we regard as a matrix such that

and in the right-hand side term we let be a -tuple of disjoint sets such that

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 ideal . For an explicit description of the reader may consult Dalbec [3] and Vaccarino [4].

One checks for that

where counts the number of cubical matrices

such that

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 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 functions 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].

Fix . Let and be a pair of sets of commuting independent variables over . For we set

For and we let be the polynomial obtained by replacing each appearance of in by , for . For example, for and we have

Definition 1. Consider such that and . The multisymmetric functions are determined by the identity

Example 2. For and , we have that is equal to and .

Example 3. For and with , the multisymmetric functions are the elementary multisymmetric functions defined in Introduction.

Next couple of lemmas follow directly from Definition 1.

Lemma 4. Let be such that and let . The multisymmetric function is given by the combinatorial identity

where is a tuple of disjoint subsets of , with .

Recall that the symmetrization map sends to

where one regards as a map .

Lemma 5. Let be such that and . The multisymmetric function is the symmetrization of the polynomial

Lemma 6. Let be expanded in monomials as

Then

where , and for we set

Proof. Let and . Then

Thus we get

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. Fix , and . Let and be such that . Then one has

(i),(ii) is the set of matrices such that

Proof. Identify the matrix with the vector

We have that

is equal to

where for we set

using the conventions

For to be equal to we must have

Thus we conclude that

Graphically, a matrix is represented as

where the horizontal and vertical arrows represent, respectively, row and column sums.

Example 8. For , and we have

where is such that and

Looking at the solutions in of the system of linear equations above we obtain

Example 9. For , and we have

where , , , , , , , , , , , , , is such that and

Looking at the solutions in of the system of linear equations above we obtain

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 is a -bilinear antisymmetric map

where is the space of real-valued smooth functions on , and for the following identities hold:

A manifold equipped with a Poisson bracket is called a Poisson manifold. The Poisson bracket is determined by an antisymmetric bilinear form on , that is, by the Poisson bivector given in local coordinates on by

The bivector determines the Poisson bracket as follows:

If the Poisson bivector is nondegenerated (i.e., ) the Poisson manifold is called symplectic.

Example 10. The space is a symplectic Poisson manifold with Poisson bracket given in the linear coordinates by

Equivalently, the Poisson bracket on is determined by the identities

This example is the so-called canonical phase space with degrees of freedom.

Example 11. Let be Lie algebra over of dimension . The dual vector space is a Poisson manifold with Poisson bracket given on by

where and the differentials and are regarded as elements of via the identifications . Choose a linear basis for . The structural coefficients of are given, for , by

Let be the linear system of coordinates on relative to the basis of . The Poisson bracket is determined by continuity and the identities

A formal deformation, or deformation quantization, of a Poisson manifold is an associative product, called the star product,

defined on the space of formal power series in with coefficients in such that the following conditions hold for :

(i), where the maps

 are bidifferential operators.(ii), where stand for terms of order and higher in the variable .

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 ; the Kontsevich -product is given on by

(i) is a collection of graphs, called admissible graphs, each with edges;(ii)for each graph , the constant is independent of and and it is computed through an integral in an appropriated configuration space;(iii) is a bidifferential operator associated with the graph and the Poisson bivector . The definition of the operators is quite explicit and fairly combinatorial in nature.

Remark 12. Kontsevich himself has highlighted the fact that explicitly computing the integrals defining the constants is a daunting task currently beyond reach. One can however use the symbols 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

With this notation the Kontsevich -product is given on functions by

Remark 13. The Kontsevich -product is defined over since . If is a regular Poisson bivector, that is, the entries of the Poisson bivector are polynomial functions, then the -product on is restricted to a well-defined -product on the space of polynomial functions on . We are interested in the quantization of symmetric polynomial functions; thus we assume that is a regular Poisson bivector and work with quantum algebra .

4. Quantum Symmetric Functions

Let be the symmetric group on letters. For each subgroup , consider the Polya functor from the category of associative -algebras to itself, defined on objects as follows [10]. Let be -algebra; the underlying vector space of is given by

Elements of are written as . For , the following identity determines the product on :

The Polya functor is also known as the coinvariants functor. The invariants functor

is given on objects by

The product on comes from the inclusion .

The functors and are naturally isomorphic to each other [10].

Suppose a finite group acts on a Poisson manifold and that the induced action of on is by algebra automorphisms; then we define the algebra of quantum -symmetric functions on as

Let 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 . We use the following coordinates on the -fold Cartesian product of with itself:

The ring of regular functions on is the ring of polynomials on commutative variables:

Consider another set of commutative variables . Recall from Section 2 that for and we set .

The Poisson bracket on is determined by the following identities:

where the coordinates of the Poisson bivector are regarded as polynomials in . The Poisson bracket on is -invariant; indeed for we have

Next results [10] provide a natural construction of groups acting as algebra automorphisms on the algebras .

Theorem 14. Let be a regular Poisson manifold and let be a subgroup of such that the Poisson bracket is -equivariant. Then the action of on is by algebra automorphisms.

Corollary 15. Let be a regular Poisson manifold and consider a subgroup . Then acts by algebra automorphisms on .

Definition 16. Let be a regular Poisson manifold. The algebra of quantum symmetric functions on is given by

Example 17. Consider with its canonical symplectic Poisson structure; then is also a symplectic Poisson manifold. Choose coordinates on as follows:

The -invariant Poisson bracket on is given for and by

Example 18. Let be -dimensional Lie algebra over and let be its dual vector space. Then is a Poisson manifold, and therefore is also a Poisson manifold. The -invariant Poisson bracket on is given, for and , by

where are the structural coefficients of .

Specializing Definition 16 we obtain the following natural notions. The algebra of quantum symmetric functions on is given by

More generally, the algebra of quantum symmetric functions on is given by

5. -Product of Multisymmetric Functions

We are ready to state and proof the main result of this work which extends Theorem 7 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 as

Theorem 19. Let be a regular Poisson manifold and let be the algebra of quantum symmetric functions on . Fix , and . Let and be such that . The -product of and is given by

(i) and

(ii) is the subset of consisting of cubical matrices

(a) for ; if either or , then for ,(b), and ,(c) for , and for .

Proof. We have

and we are using the conventions

For to be equal to we must have

and thus we conclude that

Corollary 20. With the assumptions of Theorem 19, the Poisson bracket of the multisymmetric functions and is given by

Proof. It follows from Theorem 19 and the identity

For our next result we regard as topological algebra with topology induced by the inclusion

where a fundamental system of neighborhoods of is given by the decreasing family of subalgebras

Recall from Introduction that the elementary multisymmetric functions , for with , are defined by the identity

Similarly, the homogeneous multisymmetric functions , for , are defined by the identity

Let be the set of (nontrivial) monomials in the variables . The power sum symmetric function is given, for , by

Theorem 21. The elementary multisymmetric functions   for , the homogeneous multisymmetric functions for , and the power sum multisymmetric functions with a monomial of degree less than or equal to , together with generate, respectively, the topological algebra .

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 .
To go the quantum case the same argument is applied in each case, so we only consider the elementary symmetric functions. Take and expand it as formal power series

We can write as a linear combination of a product of elementary symmetric functions. For simplicity assume that , then

Assume next that can be written as , then

Proceeding by induction we see that can be written as a formal power series in with coefficients equal to the sum of the -product of elementary multisymmetric functions.

Choose a variable for each , the set of monomials in the variables , and set

where of finite support and .

Theorem 22. The set is a topological basis for the topological algebra . 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 for ; thus from Lemma 5 we see that the set is a basis for as well. Thus forming the products we obtain a topological basis for .

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 .

Theorem 23. Let be a regular Poisson manifold and let be the algebra of quantum symmetric functions on with the Kontsevich -product. Fix , and . Let and be such that . The -product of and is given by

(i) and

 the polynomial results of applying Kontsevich’s bidifferential operator to the pair ;(ii) is the subset of consisting of maps

(a); if either or , then for ;(b), and ,(c) for , and for .

Proof. We have

and by convention stands for the unique graph in with no edges (representing the classical product), and

For we must have

Thus we conclude that

6. Symmetric Powers of the Weyl Algebras

In this section we study the case of two-dimensional canonical phase space, that is, the symplectic manifold with the canonical Poisson bracket given as follows:

Definition 24. The Weyl algebra is defined by generators and relations by

The deformation quantization of 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 on with the canonical Poisson structure.

Our goal in this section is to study the deformation quantization of the space , which can be identified with the algebra of quantum symmetric functions

or, equivalently, with the symmetric powers of the Weyl algebra

One shows by induction [10] that the following identity holds in the Weyl algebra:

Theorem 26. Consider with its canonical Poisson structure. Fix and let

For and the following identity holds:

where

Proof. We have

where and .

Corollary 27. With the assumptions of Theorem 26, the Poisson bracket of the multisymmetric functions and is given by

Example 28. Let . We have

where the vectors is such that and

For example, for , we have

since in this case is such that

Solving this equation for we, respectively, obtain

yielding the desired result.

On the other hand, from Definition 1 we get

and thus

which indeed is equal to

Note that .

Example 29. Let , then we have

where the vector is such that and

Thus for , we get

since in this case is such that

Solving this equation for we, respectively, obtain

yielding the desired result.

From Definition 1 we have , and thus

which indeed is equal to .

Note that .

Example 30. Let ; then

Using Theorem 19 we have

where is such that

Solving this equation for we, respectively, obtain

Thus we get

On the other hand, from Definition 1 we have

Computing directly the -product we obtain

Note that .

We close this work stating the main problem that our research opens.

Problem 31. Describe the relations in the algebra of quantum symmetric functions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Camilo Ortiz and Fernando Novoa for helping them to develop the software used in the examples. Eddy Pariguan thanks “Pontificia Universidad Javeriana” for providing funds for this research through Project ID-00006185 titled Descripción de Producto Cuántico para Funciones Multisimétricas.

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