Quantum Symmetric Functions

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Quantum Symmetric FunctionsRafael Díaz a & Eddy Pariguan ba Departamento de Matematicas , Instituto Venezolano deInvestigaciones Científicas , Caracas, Venezuelab Departamento de Matematicas , Universidad Central deVenezuela , Caracas, VenezuelaPublished online: 01 Feb 2007.

To cite this article: Rafael Díaz & Eddy Pariguan (2005) Quantum Symmetric Functions,Communications in Algebra, 33:6, 1947-1978, DOI: 10.1081/AGB-200063327

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Communications in Algebra®, 33: 1947–1978, 2005Copyright © Taylor & Francis, Inc.ISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1081/AGB-200063327

QUANTUM SYMMETRIC FUNCTIONS#

Rafael DíazDepartamento de Matematicas, Instituto Venezolano de InvestigacionesCientíficas, Caracas, Venezuela

Eddy PariguanDepartamento de Matematicas, Universidad Central de Venezuela,Caracas, Venezuela

We study quantum deformations of Poisson orbivarieties. Given a Poisson manifold��m� �� we consider the Poisson orbivariety ��m�n/Sn. The Kontsevich star producton functions on ��m�n induces a star product on functions on ��m�n/Sn. We provideexplicit formulae for the case ��× ��/�� , where �� is the Cartan subalgebra of a classicalLie algebra �� and �� is the Weyl group of ��. We approach our problem from a fairlygeneral point of view, introducing Polya functors for categories over nonsymmetricHopf operads.

Key Words: Deformation; Quantization; Symmetric functions.

Mathematics Subject Classification: Primary 53D55, 05E05.

1. INTRODUCTION

Let k be a field of characteristic 0, M be a set and G be a subgroup of thepermutation group on n-letters Sn. A function f � Mn → k is said to be G-symmetricif f�x��1�� x��2�� x��n�� = f�x1� x2� � � � � xn� for all � ∈ G ⊆ Sn and all x1� � � � � xn ∈ M .The k-space of G-symmetric functions Func�Mn� k�G is a subalgebra of the k-algebraFunc�Mn� k� of all functions from Mn to k. One of the goals of this article is tofind explicit formula for the product on the algebra Func�Mn� k�G in a variety ofcontexts. Our approach is based on the following observations:

• It is often easier to work with coinvariant functions Func�Mn� k�G instead ofworking with invariant functions.

• Symmetric functions arise as an instance of a general construction which assignsto any k-algebra A its nth symmetric power algebra Symn�A�. This insight ledus to introduce the notion of Polya functors, which we present in the context ofcategories over nonsymmetric Hopf operads.

Received January 2004; Accepted December 2004#Communicated by J. Alev.Address correspondence to Rafael Díaz, Departamento de Matematicas, Instituto Venezolano

de Investigaciones Científicas (IVIC), Caracas, Venezuela; E-mail: [email protected]

1947

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1948 DIAZ AND PARIGUAN

Our main interest is to study formal deformations of the algebra Func�Mn� k�G.We take the real numbers � as the ground field, and let ��m� �� be a Poissonmanifold. Under this conditions Kontsevich (2003) have shown the existence ofa canonical formal deformation �C��m�� of the algebra �C��m�� ·� ofsmooth functions on �m. We prove that if the Poisson bracket on ��m� �� is G-equivariant for G ⊂ Sm, then the �-product on C��m�� induces a �-producton the algebra of symmetric functions C��m�G�, which we call the algebra ofquantum symmetric functions. We regard this algebra as the deformation quantizationof the Poisson orbifold �m/G. We remark that in a recent paper, Dolgushev (2005),has proven the existence of a quantum product on the algebra of invariant functionsC��M�G� for an arbitrary Poisson manifold M acted upon by a finite group G.His result is based on an alternative proof of the Kontsevich formality theoremwhich is manifestly covariant.

We present a general description of the quantum product on �m/G usingthe Kontsevich star product. We give explicit formulae for the product rule in thefollowing three cases:

• symplectic orbifold �× �/� , where � is a Cartan subalgebra of a classical Liealgebra �, and � is the Weyl group associated to �;

• symplectic orbifold �n/�nm � Sn;• symplectic orbifold �n/�nm � Sn, where �m is the dihedral group of 2m elements.

Our motivation to consider these orbifolds came from the study ofnoncommutative solitons in orbifolds (Gopakumar et al., 2000; Martinec andMoore, 2001) and the quantization of the moduli space of vacua in M-theoryas considered in the matrix model approach. Our results will be raised to thecategorical context in Diaz and Pariguan (2005). In a different direction, they maybe extended to include the q-Weyl and h-Weyl algebras as it is done in Diaz andPariguan (2004). We would like to mention that these orbifolds have also beenstudied from a different point of view in Alev et al. (1990), and more recently inEtingof and Ginzburg (2002).

We consider the quantum symmetric functions of type An and uncoverits relation with the Schur�� n� algebras. The latter algebras are naturalgeneralizations of the Schur algebras as defined in Green (1980). We also study thesymmetric powers of the M-Weyl algebra, which we define as the algebra generatedby x−1 and �

�x. We provided explicit formulae for the normal coordinates for the

M-Weyl algebra as well as for its symmetric powers. Similarly, we make clearthe relation between quantum symmetric odd functions and the Schur algebras ofvarious dimensions. Finally, we give a cohomological interpretation of the algebraof supersymmetric functions.

2. INVARIANTS vs. COINVARIANTS

In this section, we introduce the notion of Polya functors for categoriesover nonsymmetric Hopf operads, and provide a list of applications of the Polyafunctors. We will consider invariant theory for finite groups as well as for compacttopological groups. To avoid duplication, we will consider only the latter casein the proofs.

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QUANTUM SYMMETRIC FUNCTIONS 1949

Let k be a field of characteristic 0 and consider �Vectk�⊗� k� the monoidalcategory of vector spaces with linear transformations as morphisms. For any setI , consider the category of I-graded vector spaces VectI , it has as objects I-gradedvector spaces,

V =⊕i∈I

Vi� Vi ∈ Ob�Vectk��

Morphisms between objects V�W ∈ Ob�VectI � are given by

Mor�V�W� =∏i∈I

Hom�Vi�Wi��

The category �VectI �⊗I � kI� has a monoidal structure compatible with directsums induced by the corresponding structures on �Vectk�⊗� k�. Explicitly, givenV�W ∈ Ob�VectI �, we have �V ⊕W�i = Vi ⊕Wi, �V ⊗W�i = Vi ⊗Wi, and �kI�i =k. For a finite group G, we denote by VectI �G� the category of I-graded vectorspaces provided with grading preserving G actions. Morphisms in VectI �G� areintertwiners, i.e., maps � V −→ W such that �gv� = g �v�, for all v ∈ V� g ∈G. Abusing notation, for an infinite compact topological group provided witha biinvariant Haar measure dg, we denote by VectI �G� the category of finitedimensional vector spaces over � provided with a G action. We define thesymmetrization map sV � V −→ VG as the map given by

sV �v� =1

vol�G�

∫G�gv�dg� if G is infinite and k = ��

sV �v� =1

��G�

∑g∈G

gv� if G is finite and k is a field of characteristic zero,

where ��G� denotes the cardinality of G. The sequence 0 −→ Ker�sV � −→ V −→VG −→ 0� is exact and we obtain the corresponding commutative triangle

V � VG

��

��

��

V/Ker�SV �

sV

We denote the space V/Ker�sV � by VG, and thus we have an isomorphismsV � VG −→ VG. We define two functors

Inv � VectI �G� −→ VectI

V �−→ VG

Coinv � VectI �G� −→ VectI

V �−→ VG

For any v ∈ V , v denotes the equivalence class of v in VG. We have the following.

Proposition 1. The maps sV above define a natural isomorphism s � Coinv −→ Inv.

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Proof. For each V ∈ VectI �G� the construction above provides an isomorphismsV � VG −→VG. For a given morphism V

�→W , we have

� sV �v� = �

(1

vol�G�

∫G�gv�dg

)= 1

vol�G�

∫G��gv�dg

= 1vol�G�

∫Gg��v�dg = sW ��v� = sW��v�� for all v ∈ V

thus proving that for each arrow V�→W , the diagram

VG

sV−−−−→ VG

Coinv��

� �Inv��WG

sW−−−−→ WG

is commutative. �

Notice that VG may also be defined as VG = V/v− gv � v ∈ V� g ∈ G�. Con-structions above can be generalized to the category Catk of all k-linear categories.We make a more general construction in the next section in order to include linearcategories over non-symmetric Hopf operads.

2.1. Categories Over Nonsymmetric Operads

We review the notion of operads and the notion of algebras over operads(Kriz and May, 1995). For finite groups G ⊂ H such that G acts on a k-vectorspace V , the induced representation is defined by IndH

G�V� = �kG⊗ V�H wherekG denotes the group algebra of G. Let us define two k-linear categories � and S:

• Ob�� = �0� 1� 2� � � � � n� � � � �

• Mor��n�m� ={k� if m = n

0� if m �= n�

• Ob�S� = �0� 1� 2� � � � � n� � � � �

• MorS�n�m� ={Sn� if m = n

0� if m �= n

The category Funct�Sop�Vectk� of contravariant functors from S to Vectk possessesthree important monoidal structures given on objects by

• �V +W��n� = V�n�⊕W�n�.

• V ⊗W�n� =⊕i+j=n Ind

SnSi×Sj

�V�i�⊗k W�j��.

• V W�n� =⊕p≥0

⊕a1+···+ap=n Ind

SnSa1×···×Sap

V�p�⊗spW�a1�⊗k · · · ⊗k W�ap�.

The category Funct��Vectk� admits similar monoidal structures by forgetting theSn actions.

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Definition 2.

• An operad is a monoid in the monoidal category �Funct�Sop�Vectk�� 1�, where1�n� = 0� n �= 1 and 1�1� = k.

• A nonsymmetric operad is a monoid in the monoidal category�Funct��Vectk�� 1�.

Explicitly, a (nonsymmetric) operad is given by mp � �p�⊗ �a1�⊗ · · · ⊗�ap� −→ �a1 + · · · + ap� satisfying the list of axioms given for example inKriz and May (1995). If no confusion arises we write m instead of mp.

Definition 3. Let V ∈ Vectk, we define the endomorphisms operad by EndV �n� =Hom�V⊗n�V�, for all n ∈ �. Composition are given by arrows

Hom�V⊗p� V �⊗Hom�V⊗a1� V �⊗ · · · ⊗Hom�V⊗ap � V ��Hom�V⊗p� V �⊗Hom�V⊗�a1+···+ap�� V⊗p��

Hom�V⊗n� V �

for integers n� a1� � � � � ap such that a1 + · · · + ap = n. For more details, see Kriz andMay (1995).

Let us introduce the category Pre-Catk of small precategories. Objects ofPre-Catk are called small precategories. A small precategory consists of thefollowing data:

• A set of objects Ob��.• A vector space Mor�x� y� associated to each pair of objects x� y ∈ Ob��.

A morphisms F ∈ MorPre-Cat�� from precategory to precategory D consistsof a map F � Ob�� −→ Ob�� and a family of maps Fx�y � Mor�x� y� −→Mor��F�x�� F�y��, for x� y ∈ Ob��. Pre-Catk has a natural partial monoidalstructure. Given precategories and � such that Ob�� = Ob�� = X, we definethe product precategory � as follows:

• Ob�� = X.• Mor��x� z� =

⊕y∈X Mor�x� y�⊗Mor��y� z�.

The partial units are the precategories kX , defined as follows:

• Ob�kX� = X.

• MorkX �a� b� ={k� if a = b

0� otherwise�

Given a precategory , we define the nonsymmetric operad End�n� =MorPre-Cat�

n��, n ∈ �. We used the convention 0 = kOb��.

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�

�

�

�m(t;(t1,t2,t3))

⇓

�

�

�

�

�

p(t;p(t1;a1,a2,a3),p(t2;a4,a5),p(t3;a6,a7,a8))

a1

a4

a3

a5

a7

a6

a8

a2

�

�

�

� �

�

tt1

t2

t3

a1

a4

a3

a5

a7

a6

�

a8

a2

�

p(t1;a1,a2,a3)

p(m(t;t1,t2,t3),a1,a2,...,a8)

t

⇓

��

p(t2;a4,a5)

p(t3;a6,a7,a8)

�

⇒

⇓

⇐

Figure 1 Pictorial representation of the associative axiom of Definition 4.

Definition 4. Let be a nonsymmetric k-linear operad. An -category �� isa precategory together with a nonsymmetric operad morphism � � −→ End.Explicitly, a k-linear -category consist of the following data:

• Objects Ob��.• Morphisms: Hom�x� y� ∈ Ob�Vectk�, for each pair x� y ∈ Ob��.• For each, k ∈ � and objects x0� x1� � � � � xk ∈ Ob�� maps

px0��xk� �k�⊗Hom�x0� x1�⊗ · · · ⊗Hom�xk−1� xk� −→ Hom�x0� xk�

We usually write p instead of px0��xk.

These data should satisfy the following associativity axiom: Given objectsx0� � � � � xn1+···+nk

, and morphisms ai ∈ Hom�xi−1� xi�� i ∈ n1 + · · · + nk, t ∈ �k�,ti ∈ �ni�, then

p�m�t� t1� � � � � tk�� a1� � � � � an1+···+nk�

= p�t� p�t1� a1� � � � � an1�� p�tk� an1+···+n�k−1�+1� � � � � an1+···+nk

��

For example, if we are given objects xi ∈ Ob��, for i = 0� 1� � � � � 5, morphismsai ∈ Mor�xi−1� xi� for i = 1� � � � � 5 and t ∈ �5�, then the morphism p�t� a1� � � � � a5�

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from object x0 to object x5 is represented by the following diagram.

��

��

�

�a1

a 2

a3a4

a5⇓t

p(t ; a1,…,a5)

Given objects xi ∈ Ob��, for i = 0� 1� � � � � 8, morphisms ai ∈ Mor�xi−1� xi�,i = 1� 2� � � � � 8, t1� t3 ∈ �3�, t2 ∈ �2� and t ∈ �4�, then the axiom from Definition 4is represented by the following commutative diagram (Figure 1).

Given a nonsymmetric operad , we define the category OCatk as follows:

• Ob�OCatk� = small -categories.• Morphism Mor�� from -category to -category � are functors from to � such that

F�p�t� a1� � � � � an�� = p�t� F�a1�� F�an��

for given objects x0� x1� � � � � xn, morphisms ai ∈ Hom�xi−1� x1�, F�ai� ∈Hom��F�xi−1�� F�xi�� and t ∈ �n�. We call such a functor F an -functor.

3. POLYA FUNCTORS

Given an -category , Aut1�� ⊂ Funct�� denotes the collection ofinvertible -functors identical on objects. Let G be a compact topological group.A G-action on an -category is a representation � � G −→ Aut1��. It is definedby a collection of actions �x�y � G −→ GL�Hom�x� y�� such that

�x0�xm�g��p�t� a1� � � � � am�� = p�t� �x0�x1

�g��a1�� xn−1�xm�g��am��

for objects x0� x1� � � � � xn ∈ Ob��, ai ∈ Hom�xi−1� xi�, for all i ∈ m and g ∈ G.Abusing notation we shall write ga instead of �x�y�g��a� where x� y ∈ Ob�� anda ∈ Hom�x� y�. We define OCatk�G� to be the category of all linear -categoriesprovided with G actions. Morphisms F from -category to -category � areG-equivariant -functors F from into �, i.e., F�ga� = gF�a�, for all a ∈ Mor�x� y�,g ∈ G, where x� y ∈ Ob��. We define

Inv � OCatk�G� −→ OCatk �−→ G

as follows: Ob�G� = Ob��, MorG�x� y� = Mor�x� y�G. G is a -category since

gp�t� a1� � � � � am� = p�t� ga1� � � � � gam� = p�t� a1� � � � � am��

We define

Coinv � OCatk�G� −→ OCatk �−→ G

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as follows: Ob�G� = Ob��, MorG�x� y� = Mor�x� y�G and

p�t� a1� a2� � � � � am� =1

vol�G�m−1

∫G�n−1�

p�t� a1� g2a2� g3a3� � � � � gmam�dg2dg3 � � � dgm

(1)

Theorem 5. There is a natural isomorphism s � Coinv −→ Inv i.e., for ∈Ob�OCatk�G�� we are given an isomorphism s � Coinv�� −→ Inv�� such thatInv�F� s = s� Coinv�F�, for all functors F � −→ � in the category OCatk�G�.

Proof. Given a category and objects x� y ∈ Ob��, let sx�y � Hom�x� y� −→Hom�x� y�

G the symmetrization map defined in the section 2. By Proposition 1, itinduces an isomorphism of vector spaces

sx�y � Hom�x� y�G −→ Hom�x� y�G�

It remains to check that the following diagram is commutative:

where s = sx0�x1 ⊗ · · · ⊗ sxm−1�xm. Clearly,

p�t�−�id⊗ S = 1vol�G�m

∫Gm

p�t� g1a1� � � � � gmam�dg1dg2 � � � dgm�

On the other hand,

sx0�xmp�t� a1� � � � � am� =1

vol�G�

∫Gg1p�t� a1� � � � � am�dg1

= 1vol�G�m

∫Gm

g1p�t� a1� h2a2� � � � � hmam�dg1dh2 � � � dhm

= 1vol�G�m

∫Gm

p�t� g1a1� g1h2a2� � � � � g1hmam�dg1dh2 � � � dhm

= 1vol�G�m

∫Gm

p�t� g1a1� � � � � gmam�dg1dg2 � � � dgm

making the change of variables g2 = g1h2� � � � � gm = g1hm. �

Notice that if is an operad then ⊗ is naturally an operad with � ⊗��n� = �n�⊗ �n�.

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Definition 6. A Hopf operad is an operad together with an operad morphism � � −→ ⊗ .

We have the following lemma.

Lemma 7. If is a Hopf operad then the category of -algebras is monoidal.

Proof. If A and B are -algebras then A⊗ B is also an -algebra, as the followingdiagram shows:

�n�⊗ �A⊗ B�⊗n −→ ��n�⊗ A⊗n�⊗ ��n�⊗ B⊗n� −→ A⊗ B� �

We now construct a partial monoidal structure on OCat.

Definition 8. Let be a Hopf operad, given -categories and � such thatOb�� = Ob�� = X, we define the tensor product category ⊗� as follows:

• Ob� ⊗�� = X.• Mor⊗��x� y� = Mor�x� y�⊗Mor��x� y�.• p�t� a1 ⊗ b1� � � � � an ⊗ bn�=

∑p�t�1�� a1� � � � � an�⊗� p�t�2�� b1� � � � � bn�, where ��t� =∑

t�1� ⊗ t�2� using Swedler notation.

Recall the well-known definition. Given a pair of groups G and K ⊂ Snthe semidirect product Gn

� K is the set Gn� K = ��g� a� � g ∈ Gn� a ∈ K� provided

with the product �g� a��h� b� = �ga�h�� ab� where g� h ∈ Gn, a� b ∈ K and ifh = �h1� � � � � hn� then a�h1� � � � � hn� = �ha−1�1�� ha−1�n��. The following result isobvious.

Lemma 9.

a. Ob�⊗n� = Ob��, Mor⊗n �x� y� = Mor�x� y�⊗n.

b. If G acts on and K ⊂ Sn then Gn� K acts on ⊗n.

Assume that is a category over a nonsymmetric Hopf operad . Let G bea compact topological group, and K a subgroup of Sn. We construct functor PG�K

which we call the Polya functor of type G�K

PG�K � OCatk�G� −→ OCatk

as follows:

• On objects: Given ∈ Ob�OCatk�G��, then PG�K�� ∈ Ob�OCatk� is the category

PG�K�� = ⊗n/Gn� K�

• Explicitly: Ob�PG�K�� = Ob�⊗n� = Ob��, and for given objects x� y ∈Ob�PG�K��

MorPG�K��x� y� = �Mor�x� y�⊗n�/Gn

� K�

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• Identity: idx ∈ MorPG�K��x� x� = id⊗nx ∈ �Mor�x� x�

⊗n�/Gn� K.

• Composition: Given x0� � � � � xm ∈ Ob�PG�K�� and morphisms ai ∈MorPG�K��xi−1� xi�, for i = 1� � � � � m, we have the following:

p�t� a1� � � � � am� =1

��G�n��K��m−1

∑�g�s�∈�Gn�K�m−1

p�t� a1� �g2� s2�a2� � � � � �gm� sm�am�

if G is finite,

p�t� a1� � � � � am�

= 1�vol�Gn��K��m−1

∑s∈Km−1

∫g∈�Gn�m−1

p�t� a1� �g2� s2�a2� � � � � �gm� sm�am�dg2 � � � dgm�

if G is compact and gi ∈ Gn.

• On morphisms: each functor �−→� induces a functor ⊗n

�⊗n−→�⊗n. Thisfunctor descends to a well-defined functor:

PG�K�� = ⊗n/Gn� K −→ �⊗n/Gn

� K = PG�K��

Example 10. Consider the nonsymmetric operad �� given by �� n� = k, forall n ≥ 0. �� -categories are categories in the usual sense. This example will beapplied in section 4 to introduce explicit formulae for the composition of morphismsin the Schur categories of various types.

Definition 11. Let be an operad. An -algebra is a pair �A� ��, where A is avector space and � � −→ EndA is an operad morphism.

Notice that an -algebra may be regarded as an -category with one objectby setting A = Mor�1� 1�. We denote by OAlgk the category of -algebras andby OAlgk�G� the category of -algebras provided with a G action. We have twonaturally isomorphic functors:

Inv � OAlgk�G� −→ OAlgk

A �−→ AG

Coinv � OAlgk�G� −→ OAlgk

A �−→ AG�

Let be a Hopf operad. Assume that A is an -algebra. Let G be a compacttopological group and K ⊂ Sn. We have a functor:

PG�K � OAlgk�G� −→ OAlgk

A �−→ A⊗n/Gn� K

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The -algebra structure on A⊗n/Gn� K is given for any a1� � � � � am ∈ A,

t ∈ �m� by

p�t� a1� � � � � am� =1

��G�n��K��m−1

∑�g�s�∈�Gn�K�m−1

p�t� a1� �g2� s2�a2� � � � � �gm� sm�am�

(2)

if G is finite. If G is compact taking gi ∈ Gn, we have that

p�t� a1� � � � � am�

= 1�vol�Gn��K��m−1

∑s∈Km−1

∫g∈�Gn�m−1

p�t� a1� �g2� s2�a2� � � � � �gm� sm�am�dg2 � � � dgm�

Corollary 12. Let A be an -algebra, G a finite group acting by algebraautomorphism on A. Take K = �id�. The following identity hold in PG�A�:

a b = 1��G�

∑g∈G

a�gb� (3)

where a� b ∈ A.

We remark that Joyal theory of analytic functors (see Joyal, 1981, 1986), maybe extended from the context of k-vector spaces to -algebras by defining a functorF which sends a family A = �An�n≥0 of -algebras provided with right Sn actions,into the functor FA from the OAlg into OAlg given as follows:

F � Funct�Sop�OAlg� −→ Funct�OAlg�OAlg�

A = �An�n≥0 �−→ FA�B� =⊕n≥0

�An ⊗ B⊗n�Sn for all B ∈ Ob�OAlg��

Example 13. Consider the operad Ass given by Ass�n� = kSn, for n ≥ 0. Ass-algebras are the same as associative algebras. If we take the family k = �k�n, n ≥ 0,provided with the trivial Sn action, then Fk�A� = Sym�A�

The following remarks justify our choice of name for the Polya functors. LetK ⊂ Sn be a permutation group. For k ∈ K, let bs�k� be the number of cycles of kof length s. The cycle index polynomial of K ⊂ Sn, is the polynomial in n variablesx1� � � � � xn

PK�x1� � � � � xn� =1

��K�

∑k∈K

xb1�k�1 x

b2�k�2 · · · xbn�k�n �

Let A be a finite dimensional -algebra and X a basis for A. Using Polyatheory (Wehrhahn, 1990), we can compute the dimension of the -algebra �A⊗n�K

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as follows:

dim��A⊗n�K� = ��Xn/K�

= PK��X�� X�� X��

= PK�dimA� dimA� � � � � dimA��

3.1. General Multiplication Rule

For each K ⊂ Sn consider the Polya functor PK � k-alg −→ k-alg from thecategory of associative k-algebras into itself defined on objects as follows: if A is ak-algebra, then PK�A� denotes the algebra whose underlying vector space is

PK�A� = �A⊗n�/a1 ⊗ · · · ⊗ an − a�−1�1� ⊗ · · · ⊗ a�−1�n� � ai ∈ A� � ∈ K��Our next theorem provides the rule for the product of m elements in PK�A�.

Theorem 14. For any aij ∈ A, the following identities holds in PK�A�:

��K�m−1m∏i=1

(n⊗

j=1

aij

)= ∑

�∈�id�×Km−1

n⊗j=1

( m∏i=1

ai�−1i �j�

)� (4)

Proof. It follows from formula (2), taking Ass as the underlying operad and settingG = �id�. �

Theorem 14 implies the following.

Proposition 15. Let A be an algebra provided with a basis �es � s ∈ r�. Assume thateset = c�k� s� t�ek (sum over k), for all s� t ∈ r. For any given a = �aij� ∈ Mn×m�r�the following identity holds in PK�A�

�n!�m−1m∏i=1

(n⊗

j=1

eaij

)=∑

��

( n∏j=1

c��j� a� ��

) n⊗j=1

e�jm−1�

where the sum runs over all � ∈ �id�× Km−1, � = ��ji � ∈ M�m−1�×n�r�, and

c��j� a� �� = c��j1� a1�−1

1 �j�� a2�−12 �j��c��

j2� �

j1� a3�−1

3 �j�� · · · c��jm−1� �jm−2� am�−1

m �j��.

Proof.

�n!�m−1m∏i=1

(n⊗

j=1

eaij

)= ∑

�∈�id�×Gm−1

n⊗j=1

(m∏i=1

eai�−1i �j�

)

= ∑��

n⊗j=1

c�� a� ��e�m−1

= ∑��

(n∏

j=1

c��j� a� ��

)n⊗

j=1

e�jm−1 �

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Let us consider a nonsymmetric Hopf operad . Assume that a basis ptm,

t ∈ km, for �m� is given for each m ∈ �. Moreover, let us assume that ��ptm� =

ptm ⊗ · · · ⊗ pt

m, then the next proposition follows from formula �4�.

Proposition 16. Let A be an -algebra provided with a basis �es � s ∈ r�, andlet �n� = pt

m � t ∈ rm�. Assume that ptm�es1� � � � � esm� = c�t� k�m� s1� � � � � sn�ek, (sum

over k). For any given a = �aij� ∈ Mn×m�r�, the following identity holds in PK�A�:

ptm

(n⊗

j=1

ea1j � � � � �n⊗

j=1

eamj

)=∑

��u

n∏j=1

c�t� uj�m� a1�−11 �j�� am�−1

m �j��n⊗

j=1

euj

where the sum runs over all � ∈ �id�×Km−1 and u ∈ rn.

4. SYMMETRIC POWER OF A SUPERCATEGORY

Let us consider Polya functor PG�K � Catk�G� −→ Catk for the case G = id,K = Sn, i.e, we consider for each n ∈ � the functor

Symn � Catk −→ Catk�

Recall that a supercategory is a category over the category Supervect of �2-graded vector spaces with the Koszul rule of signs. Functor Symn may beapplied to supercategories as well. The next result provides formula for thecomposition of morphisms in the symmetric powers of a supercategory. We use thenotation a1 · · · an = a1 ⊗ · · · ⊗ an ∈ Symn�Hom�x� y��, for morphisms a1� � � � � an ∈Hom�x� y�.

Proposition 17. Let be a supercategory and let a1� � � � � an ∈ Mor�x� y� andb1� � � � � bn ∈ Mor�y� z�. In the supercategory Symn�� the compositions of morphismsis given by

�a1a2 · · · an��b1b2 · · · bn� =1n!∑�∈Sn

sgn�a� b� ��a1b�−1�1��a2b�−1�2�� · · · �anb�−1�n��

where sgn�a� b� �� = �−1�e and e = e�a� b� �� =∑i>j aib�−1�j� +

∑��i�>��j� bibj�

4.1. Schur Categories

Let k be a field of characteristic 0, m = �m1� � � � � mk� ∈ �k, �m = �m1× · · · ×

�mkand n ∈ �. We define the Schur supercategory of type �m� n� as follows:

Ob�S�m� n�� = finite dimesional k-supervector spaces�

MorS�m�n��V�W� = �Homk�V⊕�m�W⊕�m�⊗n��n

m�Sn�

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�←→E(V,W )klij

�

6

3

Figure 2 Representation of elementary transformation.

�nm � Sn acts on �Homk�V

⊕�m�W⊕�m�⊗n� as follows:

�nm � Sn × �Homk�V

⊕�m�W⊕�m�⊗n� −→ �Homk�V⊕�m�W⊕�m�⊗n�

��c1� � � � � cn�� (Et1u1

r1s1· · ·Etnun

rnsn

) �−→ (E

t��1��u1+c1�

r��1��s1+c1�· · ·Et��n��un+cn�

r��n��sn+cn�

)where E�V�W�klij are the elementary linear transformation in Homk�V

⊕�m�W⊕�m�,i ∈ dim V, k ∈ dimW and j� l ∈ �k

m. We apply Polya functor to obtain explicitformula for the composition rule

Mor�V�W�⊗Mor�W�Z� −→ Mor�V� Z��

Theorem 18. For any given M = m1 · · ·mk, i� t ∈ dimW, k ∈ dimZ, r ∈ dim V,and j� l� s� u ∈ �n

m, we have

�E�V�W�k1l1i1j1

� � � E�V�W�knlninjn

��E�W�Z�t1u1r1s1� � � E�W�Z�tnunrnsn

�

= 1Mnn!

∑�∈Sn

t��a�=ia

sgn�� i� k� r� t�E�V� Z�k1l1r��1��s1+j1−u1�� E�V� Z�

knlnr��n��sn+jn−un�

where sgn�� i� k� r� t� = �−1�e and

e =∑i>j

�ii + ki��r�−1�i� + t�−1�i��+∑

��i�>��j�

�r�−1�i� + t�−1�i��r�−1�j� + t�−1�j��

Proof. Straightforward using Polya functor and Proposition 17. �

We now develop a graphical notation that make transparent the meaningof Theorem 18. Let us assume that n = 4, k = 4, m1 = 6, m2 = 3, dim�V � = 4,dim�W� = 2 and dim�Z� = 3. We represent an elementary linear transformationE�V�W� as in Figure 2.

Notice that each block corresponds to �6 × �3, �6 acting horizontally,and �3 acting vertically. The number of blocks in the bottom row isdim�V �, and the number of blocks in the top row is dim�W�. Elements of�Hom�V⊕�m�W⊕�m�⊗n��6×�3�

4�S4are depicted by four nonnumbered arrows, and

similarly for elements of �Hom�W⊕�m� Z⊕�m�⊗n��6×�3�4�S4

. Composition is obtainedas follows:

• Fix an arbitrary enumeration of the arrows in �Hom�V⊕�m�W⊕�m�⊗n��6×�3�4�S4

.• Sum over all possible enumerations of the arrows in �Hom�W⊕�m ,Z⊕�m�⊗n��6×�3�

4�S4.

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� ��

�1

��

1

�

2

2

3

3

4

4

=

1�

2

�

4

�

3

�

Figure 3 Example of composition.

• Stacks arrows from �Hom�V⊕�m�W⊕�m�⊗n��6×�3�4�S4

to arrows on �Hom�W⊕�m ,Z⊕�m�⊗n��6×�3�

4�S4taking care of enumeration and using the �2 × �6 symmetry.

Notice that composition is interesting in that no-touching arrows may neverthelessbe composed (due to the �6 × �3 symmetry) as shown in Figure 3.

Definition 19. The Schur superalgebra of type �sdimV�m� is given byHomSn

�V⊗n� V⊗n�, where sdim denotes the superdimension of a supervector space.

See Green (1980) for more on Schur algebras.

Corollary 20. For m = 1 and V = W , MorS�m�n��V� V� is the Schur superalgebraSchur�sdimV�m� of type �sdimV�m�.

Proof. �Hom�V� V�⊗n�Sn � �Hom�V� V�⊗n�Sn � HomSn�V⊗n� V⊗n�. �

5. CLASSICAL SYMMETRIC FUNCTIONS

In this section, we study classical symmetric functions by means of the Polyafunctor. We provide a fairly elementary interpretation of the symmetric functions interms of the symmetric powers of the monoidal algebra associated to the additivemonoid �m. We also consider symmetric odd-functions as well as symmetricBoolean algebras. Symmetric functions have been studied from many points of view;see, for example, Macdonald (1995), Gian-Carlo Rota (1995), and Vaccarino (2004).

5.1. Symmetric Functions of Weyl Type

The classical Weyl groups of type An� Bn� and Dn, are Sn, �n2 � Sn and

�n−12 � Sn, respectively. These groups act on ��m�n as follows:

Sn × ��m�n −→ ��m�n

�� x1� � � � � xn�� −→ �x�−1�1�� x�−1�n��

��n2 � Sn�× ��m�n −→ ��m�n

��t1� � � � � tn�� x1� � � � � xn� �−→ �t1x�−1�1�� tnx�−1�n��

The group Dn is regarded as a subgroup of Bn as follows:

�n−12 � Sn = ��t1� � � � � tn�� ∈ �n

2 � Sn � t1t2 � � � tn = 1��

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Definition 21. Fix m ∈ �. The algebra of symmetric functions of type An, Bn andDn are given by

• SymAn�m� = ��x1� � � � � xn�Sn � ��x1� � � � � xn�

Sn ,• SymBn

�m� = ��x1� � � � � xn��n2�Sn

� ��x1� � � � � xn��n2�Sn ,

• SymDn�m� = ��x1� � � � � xn��n−1

2 �Sn� ��x1� � � � � xn�

�n−12 �Sn ,

where xi = �xi1� � � � � xim�, for i = 1� � � � � n.

The map ��m⊗n −→ �x1� � � � � xn given by a1 ⊗ · · · ⊗ an �−→ xa11 · · · xann

defines an isomorphism of algebras, where ai ∈ �m and xaii = x

ai1i1 � � � x

aimim . We

set �me = �a ∈ �m � �a� is even� and �m

o = �a ∈ �m � �a� is odd�. We denote XA =xa11 · · · xann , for A = �a1� � � � � an� ∈ ��m�n.

Theorem 22 (classical symmetric functions).

a. The following is a commutative diagram

��m⊗n�Sn ⊃ ��me +�m

o ⊗n�Sn ⊃ ��m

e ⊗n�Sn↓ ↓ ↓

SymAn�m� � SymDn

�m� � SymBn�m�

where the vertical arrows are isomorphisms.b. For A�B ∈ ��m�n, the product rule in SymAn

�m� is given by

XA XB = 1n!∑�∈Sn

XA+��B�

and on SymBn�m� and SymDn

�m� by restriction.

Proof. The first row of the diagram above follows from the isomorphism aboveafter taken care of the �n

2 (resp. �n−12 ) symmetries for the groups Bn and Dn,

respectively. It is clear that part b implies the rest of a� We prove b� Given A�B ∈��m�n, using Proposition 17 for the product in SymAn

�m�, we obtain

XA XB = 1n!∑�∈Sn

(xa11 · · · xann )(xb�−1�1�

1 · · · xb�−1�n�n

)= 1

n!∑�∈Sn

XA+��B��

Now consider A�B ∈ ��me �

n, the product in SymBn�m� is given by

XA XB = 12nn!

∑�t��∈�n

2�Sn

xa11 · · · xann t

b11 x

b�−1�1�

1 · · · tbnn xb�−1�n�

n

= 12nn!

∑�∈Sn

( ∑t∈�n

2

�−1�∑

ti=−1 bi

)XA+��B�

= 1n!∑�∈Sn

XA+��B�� since∑t∈�n

2

�−1�∑

ti=−1 bi = 2n�

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Consider A�B ∈ ��ne +�m

o �n, we obtain

XA XB = 12n−1n!

∑�t��∈�n

2�Sn

xa11 · · · xann t

b11 x

b�−1�1�

1 · · · tbnn xb�−1�n�

n

= 12n−1n!

∑�∈Sn

( ∑t∈�n

2∏ti=1

�−1�∑

ti=−1 bi+bn

)XA+��B�

= 1n!∑�∈Sn

XA+��B�� since∑t∈�n

2∏ti=1

�−1�∑

ti=−1 bi+bn = 2n−1�

5.2. Symmetric Odd-Functions

Consider the alternating algebra∧�1� � � � � �m, which we regard as the algebra

of functions on the purely odd super-space �0�m. A basis for∧�1� � � � � �m is

given by ��I = �i1 � � � �ik � I ⊂ m�. The structural coefficients are given by �I�J =c�I� J��I∪J , where c�I� J� = �−1��i∈I�j∈J� i>j��, if I ∩ J = ∅, and 0 otherwise. Thealgebra

∧�1� � � � � �m is �2-graded, with grading �I = I = 0 if �I� is even and �I =

I = 1 if �I� is odd. Now we apply Polya functor to obtain

Proposition 23. The product rule in the algebra of symmetric odd functions�∧�1� � � � � �m

⊗n�Sn is given by

��I1 · · · �In��J1 · · · �Jn� =1n!∑�∈Sn

(sgn�I� J� ��

n∏k=1

c�Ik� J�−1�k��

) n∏i=1

�Ik∪J�−1�k��

where

sgn�I� J� �� = �−1�e and e =∑k>l

IkJ�−1�l� +∑

��k�>��l�

JkJl�

Proof.

��I1 · · · �In��J1 · · · �Jn� =1n!∑�∈Sn

sgn�I� J� ��I1�J�−1�1�� · · · ��In�J�−1�n�

�

= 1n!∑�∈Sn

sgn�I� J� ��( n∏

k=1

c�Ik� J�−1�k��Ik∪J�−1�k�

)

= 1n!∑�∈Sn

(sgn�I� J� ��

n∏k=1

c�Ik� J�−1�k��

) n∏k=1

�Ik∪J�−1�k��

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5.3. Symmetric Boolean algebra

Fix n ∈ � and let P�n� be the free �-vector space generated by the subsets ofn, i.e., P�n� = A � A ⊂ n�. Define a product ∪ on P�n� by

∪ � P�n�⊗ P�n� −→ P�n�

A⊗ B �−→ A ∪ B

�P�n��∪� is a Boolean algebra and dim�P�n�� = 2n. Sn acts naturally on n andthus on P�n�. We call the algebra �P�n��∪�/Sn � �P�1�⊗n�/Sn = Symn�P�1�� thesymmetric Boolean algebra; it has dimension n+ 1, a basis being �0� 1� � � � � n�.We define P�n� k� �= �A ⊂ n � �A� = k�. An application of Polya functor yieldsthe next

Theorem 24.

ab = 1(nb

) m∑k=0

(a

b − k

)(n− ak

)a+ k�

for all a� b ∈ �P�n��∪�Sn and m = min�b� n− a�.

Proof.

ab = 1n!∑�∈Sn

a ∪ �b = 1(nb

) ∑B∈Pn�b

a ∪ B

= 1(nb

) ∑B0⊂P�n−a�k�

B1⊂P�a�b−k�

a ∪ B0 =1(nb

) m∑k=0

(a

b − k

)(n− ak

)a+ k� �

6. QUANTUM SYMMETRIC FUNCTIONS

In this section, we assume the reader is familiar with the notations fromKontsevich (2003). Let us recall the notion of a formal deformation.

Definition 25. Fix a Poisson manifold �M� �� . A formal deformation(deformation quantization) of the algebra of smooth functions on M is anassociative star product

� � C��M��⊗�� C��M�� −→ C��M�� such that

a. f � g =∑�n=0 Bn�f� g��

n, where Bn�−�−� are bi-differential operators.b. f � g = fg + 1

2�f� g�� + O��2�, where O��2� are terms of order �2.

In Kontsevich (2003), a canonical �-product has been constructed for anyPoisson manifold. For manifold ��m� �� with Poisson bivector �, the �-product is

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given by the formula

f � g =�∑n=0

�n

n!∑�∈Gn

��B��f� g��

where Gn is a collection of admissible graphs each of which has n edges, and �� aresome constants (independent of the Poisson manifold). Given a finite group K actingon �C��M�� by algebra automorphisms, we call the algebra �C��M�� K ��C��M�� K the algebra of quantum K-symmetric functions on M .

The next theorem shows how groups of automorphisms of �C��M�� arise in a natural way.

Theorem 26. Assume we are given a Poisson structure �-� -� on �m, and a groupK ⊂ Sm such that �-� -� is K-equivariant. Then K acts on �C��m�� byautomorphisms.

Proof. We assume that �f� g� � = �f �� g ��, for all f� g ∈ C��m�, � ∈ K, orequivalently �ij��x� = ��i��j�, where �ij = �xi� xj� for all i� j ∈ m. Let us show that��f � g� = ��f� � ��g�:

��f � g��x� = �f � g��−1x� =�∑n=0

�n

n!∑�

w�B��f� g��−1x��

On the other hand,

��f� � ��g��x� =�∑n=0

�n

n!∑�

w�B��f� �g��x��

We need to prove that

B��f� g��−1x� = B��f� �g��x�� for all � ∈ Gn�

Using Kontsevich’s formula (see Kontsevich, 2003) we get:

B��f� g��−1x� = ∑

I�E�−→m

[ n∏i=1

( ∏e∈E� �e=�∗�i�

�I�e�

)�I�e

1i �I�e

2i �

]��−1x�

×(( ∏

e∈E� �e=�∗�L��I�e�

)f

)��−1x�×

(( ∏e∈E� �e=�∗�R�

�I�e�

)g

)��−1x�

= ∑I�E�−→m

[ n∏i=1

( ∏e∈E� �e=�∗�i�

��I�e��

)��I�e

1i ��I�e

2i ��

]�x�

×(( ∏

e∈E� �e=�∗�L��I�e��

)f �−1

)�x�

×(( ∏

e∈E� �e=�∗�R��I�e��

)g �−1

)�x� = B��f� �g��x��

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Corollary 27. Under the conditions above, the product rule on �C��m�n�� Kis given by

f � g = ∑�∈K

�∑n=0

�n

n!(∑

�

w�B��f� g �−1�

)(5)

for all f� g ∈ C��m�n�.

Proof. Using Polya functor and Corollary 12, we have

f � g = ∑�∈K

f � �g = ∑�∈K

�∑n=0

�n

n!(∑

�

w�B��f� g �−1�

)�

�

Definition 28. Given a Poisson manifold ��m� �� and a subgroup K ⊂ Sn thealgebra of quantum symmetric functions on ��m�n is set to be �C��m�n�� K ��C��m�n�� K .

Notice that if ��m� �� is a Poisson manifold, then ��m�n is a Poisson manifoldin a natural way. Moreover, the Poisson structure on ��m�n is Sn-equivariant, andthus K-equivariant for all subgroup K of Sn.

6.1. Weyl algebra

The Kontsevich �-product given by formula

f � g =�∑n=0

�n

n!∑�∈Gn

��B��f� g�

is notoriously difficult to compute. Nevertheless, there are two main examples (seeKontsevich, 2003) in which a fairly explicit knowledge of the start product isavailable:

a. If � is a constant nondegenerated Poisson bracket on �2n, then the quantumalgebra of polynomial functions on �2n, i.e., ��x1� � � � � x2n�� is isomorphicto W ⊗�� · · · ⊗�� W , where W is the Weyl algebra (see definition below).

b. If � is linear Poisson bracket in �n, then ��n� �� is isomorphic as a Poissonmanifold to �∗ for some Lie algebra �. In this case the quantum algebra ofpolynomial functions on �∗, i.e., ��∗�� is isomorphic to the universalenveloping algebra Uh�� of �.

Case a will be considered in this section. Case b for � = �2 is considered in Diazand Pariguan (2004). The case of a classical Lie algebra will be treated by ourmeans elsewhere. The algebra W = �x� y��/yx − xy − �� is called the Weylalgebra; it is isomorphic to the canonical deformation quantization of ��2� dx ∧dy� if we consider only polynomial functions on �2. This algebra admits a naturalrepresentation as indicated in the following.

Proposition 29. The map � � W −→ End��x�� given by ��x��f� = xf and��y��f� = �

�f

�x, for any f ∈ �x� defines an irreducible representation of the

Weyl algebra.

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We order the letters of the Weyl algebra as follows: x < y < �. Assume we aregiven Ai = �ai� bi� ∈ �2, for i ∈ n. Set A = �A1� � � � � An� ∈ ��2�n, XAi = xaiybi andlet � � � �n −→ � be the function such that �x� �=∑n

i=1 xi, for all x ∈ �n. Given x ∈�n and i ∈ �, we denote by x<i the vector �x1� � � � � xi−1� ∈ �i−1, by x≤i the vector�x1� � � � � xi� ∈ �i and by x>i the vector �xi+1� � � � � xn� ∈ �n−i. We write a � n ifa ∈ �k for some k and �a� = n. Using this notation we have the following.

Definition 30. The normal coordinates N�A� k� of∏n

i=1 XAi ∈ W are defined

through the identity

n∏i=1

XAi =min∑k=0

N�A� k�x�a�−ky�b�−k�k (6)

for 0 ≤ k ≤ min = min��a�� b��. For k > min, we set N�A� k� equal to 0.

Recall that given finite sets N and M with n and m elements, respectively, thenumber of one-to-one functions f � N −→ M is m�m− 1� � � � �m− n+ 1� = �m�n.The number �m�n is called the nth falling factorial of m. The nth rising factorialm�n� of m is given by m�n� = m�m+ 1� · · · �m+ n− 1�. For any a� b ∈ �n, we define� ab � �= �

a1b1��

a2b2� · · · � an

bn�, and a! = a1!a2! · · · an!

Definition 31. A k-pairing from set E to set F is an injective function from a k-elements subset of E to F . We denote by Pk�E� F� the set of k-pairings from E to F .

Definition 32. Fix variables t = �t1� � � � � tn� and s = �s1� � � � � sn�. The generatingseries N of the normal coordinates in the Weyl algebra is given by:

N = ∑a�b�c

N�A� c�sa

a!tb

b!uc ∈ �s� t� u (7)

where the sum runs over a� b ∈ �n, c ∈ � and A = �A1� � � � � An� ∈ ��2�n.

Theorem 33. Let A� k be as in the Definition 30, the following identity holds:

a. N�A� k� =∑p�k(b

p

)∏n−1i=1 ��a>i� − �p>i��pi , where p ∈ �n−1.

b. Let E1� � � � � En� F1� � � � � Fn be disjoint sets such that ��Ei� = ai, ��Fi� = bi, fori ∈ n, set E = ⋃n

i=1 Ei, and F = ⋃ni=1 Fi, then N�A� k� = ��p ∈ Pk�E� F�� if

�a� p�a�� ∈ Ei × Fj then i > j��c. N = exp

(∑i>j utisj +

∑i ti +

∑j sj)�

Proof. Using induction one shows that the following identity hold in the Weylalgebra:

ybxa =min∑k=0

(bk

)�a�kx

a−kyb−k�k� (8)

where min = min�a� b�. Several applications of identity �8� imply a� Noticethat for given sets E� F such that ��E� = a and ��F� = b, � b

k ��a�k is equal to

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E1 F1 E2 F2 E3 F3

Figure 4 Combinatorial interpretation of N .

��p ∈ Pk�E� F��, showing part b for n = 2. The general formula follows frominduction. This prove b� It follows from standard combinatorial facts (see Stanley,1999 and Wehrhahn, 1990 for more details) that

exp(∑

i>j

utisj +∑i

ti +∑j

sj

)= ∑

a�b�c

ca�b�csa

a!tb

b!uc

c!

where ca�b�c = ��p� �� p ∈ Pc�A� B� and � � c → p, a bijection}) which isequivalent to formula (7). �

Figure 4 illustrates the combinatorial interpretation of the normal coordinateN�A� k� of an element

∏3i=1 X

Ai ∈ W , it shows are of the possible pairing contributingto N�A� k�.

Corollary 34. For any given �t� a� b� ∈ �×�n ×�n, the following identity holds

n∏i=1

�t + �a>i� − �b>i��bi =∑p�k

(bp

) n−1∏i=1

��a>i� − �p>i��pi t�b�−k�

Proof. Consider the identity (6) in the representation of the Weyl algebra definedin Proposition 29. Apply both sides of the identity (6) to xt for t ∈ � and useTheorem 33. �

Theorem 35. Let A� c be as in the Definition 30. The following identity holds

N�A� c� = ∑∑

cij=c

k∏j=2

(aj∑i cij

)( ∑i cij

cij · · · c�j−1�j

) k−1∏i=1

(bi∑j cij

)( ∑j cij

ci�i+1� · · · cik)∏

i>j

cij!

where cij are integers with 1 ≤ i ≤ k− 1, 2 ≤ j ≤ k and i > j.

Proof. Consider maps Fij � �p ∈ Pc�E� F� � if �a� p�a�� ∈ Ei × Fj then i > j� −→� given by Fij�p� = ��a� p�a�� ∈ Ei × Fj��. Notice that

N�A� c� = ∑∑

cij=c

��p ∈ Pc�E� F� � Fij�p� = cij� i > j��

Moreover,

��p ∈ Pc�E� F� � Fij�p� = cij� i > j��

=k∏

j=2

(aj∑i cij

)( ∑i cij

cij · · · c�j−1�j

) k−1∏i=1

(bi∑j cij

)( ∑j cij

ci�i+1� · · · cik

)∏i>j

cij!��

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6.2. Quantum Symmetric Functions of Weyl Type

The following theorem provides explicit formula for the product of m elementsof the nth symmetric power of the Weyl algebra. Let us explain our notation: fixa matrix A � m× n −→ �2, �Aij� = ��aij�� bij��. Given � ∈ �Sn�

m and j ∈ n, A�j

denotes the vector �A1�−11 �j�� Am�−1

m �j�� ∈ ��2�n and set XAij

j = xaijj y

bijj for j ∈ n.

Set �A�j � = ��a�

j �� b�j �� where

�a�j � =

m∑i=1

ai�−1i �j� and �b�j � =

m∑i=1

bi�−1i �j��

We have the following.

Theorem 36. For any A � m× n −→ �2, the following identity

�n!�m−1m∏i=1

( n∏j=1

XAij

j

)= ∑

��k�p

(∏i�j

(b�jpj

)��a�

j �>i� − �pj

>i��pji) n∏

j=1

X�A�

j �−�kj �kj �

j ��k� (9)

where � ∈ �id�× Sm−1n , k ∈ �n, �i� j� ∈ m− 1× n and p = p

ji ∈ ��m−1�n, holds in

Symn�W�.

Proof. We use Theorem 14 and Theorem 33

�n!�m−1m∏i=1

( n∏j=1

XAij

j

)= ∑

�∈�id�×Sm−1n

n∏j=1

( m∏i=1

XAi�−1i �j�

j

)

= ∑�∈�id�×Sm−1

n

n∏j=1

( minj∑k=0

N�A�j � k�X

�A�j �−�k�k�

j �k

)

= ∑��k

( n∏j=1

N�A�j � kj�

) n∏j=1

X�A�

j �−�kj �kj �

j ��k�

= ∑��k�p

(∏i�j

(b�j

pj

)��a�

j �>i� − �pj

>i��pji) n∏

j=1

X�A�

j �−�kj �kj �

j ��k��

where minj = min��a�j �� b�j ��. �

Now we proceed to state and prove the quantum analogue of Theorem 22.We begin by introducing a �-product on ��2m⊗n��Sn , which is motivated bythe proof of Theorem 39 below.

Definition 37. The �-product on ��2m⊗n��Sn for A ∈ ��2m�n and C ∈ ��2m�n

is given by the formula

A � C = 1n!∑I��

(bI

)��c��IA+ ��C�− �I� I��I� (10)

where I � n× m → � and � ∈ Sn.

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Definition 38. Fix m ∈ �. The algebra of quantum symmetric functions of typeAn, Bn and Dn are given by:

• QSymAn�m� = ��x1� y1� � � � � xn� yn�� Sn � ��x1� y1� � � � � xn� yn��

Sn ,

• QSymBn�m� = ��x1� y1� � � � � xn� yn�� n

2�Sn

� ��x1� y1� � � � � xn� yn�� n2�Sn ,

• QSymDn�m� = ��x1� y1� � � � � xn� yn�� n−1

2 �Sn

� ��x1� y1� � � � � xn� yn�� n−12 �Sn ,

where xi = �xi1� � � � � xim� and yi = �yi1� � � � � yim�.

Theorem 39 (quantum symmetric functions).

a. The following is a commutative diagram:

��2m⊗n�� Sn ⊃ ��2me +�2m

o ⊗n�� Sn ⊃ ��2me ⊗n�� Sn↓ ↓ ↓

QSymAn�m� � QSymDn

�m� � QSymBn�m�

where the vertical arrows are isomorphisms.b. For Ai = �ai� bi� ∈ ��2�m and Ci = �ci� di� ∈ ��2�m i ∈ n, we set XAi

i = xaii y

bii and

XCii = x

cii y

dii where xi = �xi1� � � � � xim� and y = �yi1� � � � � yim�, the product rule in

QSymAn�m� is given by

XA11 X

A22 · · ·XAn

n � XC11 X

C22 · · ·XCn

n = 1n!∑I��

(bI

)��c��IX

A+��C�−�I�I��I��

where I � n× m → �, � ∈ Sn. The product on QSymBn�m� and QSymDn

�m� isgiven by restriction.

Proof. We prove b which implies a. Given Ai� Ci ∈ ��2�m, using (10), we obtain:

XA11 X

A22 · · ·XAn

n � XC11 X

C22 · · ·XCn

n = 1n!∑�∈Sn

XA11 X

C�−1�1�

1 XA22 X

C�−1�2�

2 · · ·XAnn X

C�−1�n�

n

= 1n!∑�∈Sn

xa11 y

b11 x

c�−1�1�

1 yd�−1�1�

1 · · · xann ybnn x

c�−1�n�n y

d�−1�n�

n

= 1n!∑�∈Sn

(n∏

j=1

minj∑ij=0

(bjij

)�c�−1�j��ijX

Aj+C�−1�j�−�ij �ij �

j �ij

)

= 1n!∑I��

((bI

)��c��I

)XA+��C�−�I�I��

�I�

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where minj = min�bj� c�−1�j��. Now consider Ai� Ci ∈ �2me

XA11 · · ·XAn

n � XC11 · · ·XCn

n = 12nn!

∑�t��∈�n

2�Sn

�xa11 y

b11 � � � x

ann y

bnn ��t� ��x

c11 y

d11 � � � x

cnn y

dnn �

= 12nn!

∑I��

( ∑t∈�n

2

�−1�∑

ti=−1 ci+di

(bI

)��c��I

)XA+��C�−�I�I��

�I�

= 1n!∑I��

((bI

)��c��I

)XA+��C�−�I�I��

�I�

since∑

t∈�n2�−1�

∑ti=−1 ci+di = 2n. Finally consider Ai� Ci ∈ �2m

e +�2mo

XA11 · · ·XAn

n � XC11 · · ·XCn

n = 12n−1n!

∑�t��∈�n

2�Sn

�xa11 y

b11 � � � x

ann y

bnn ��t� ��x

c11 y

d11 � � � x

cnn y

dnn

= 12n−1n!

∑I��

(k�t� c� d�

(bI

)��c��I

)XA+��C�−�I�I��

�I�

= 1n!∑I��

((bI

)��c��I

)XA+��C�−�I�I��

�I�

since

k�t� c� d� = ∑t∈�n2∏ti=1

�−1�∑

ti=−1 ci+di+cn+dn = 2n−1� �

6.3. Quantum Symmetric Functions on ��n/��nm �� Sn

In this section, we shift to complex analytic notation to study the Poissonorbifolds �n/�n

m � Sn and �n/�nm � Sn. The deformation quantization of �n

provided with the canonical symplectic structure is isomorphic to the Weyl algebraW = �z1� z1� � � � � zn� zn��/zz− zz− 2i��. The group �n

m � Sn acts on �n asfollows:

��nm � Sn�×�n −→ �n

��w1� � � � � wn�� z1� � � � � zn� �−→ �w1z�−1�1�� wnz�−1�n��

where wj = e2�ikjm , j = 1� � � � � n. Thus �n

m � Sn acts on W = �z1� z1� � � � ,zn� zn��/zz− zz− 2i��. We denote Z

Aii = z

aii z

bii and �2

m = ��a� b� � there is k ∈� such that b − a = km�.

Definition 40. The �-product on ��2m

⊗n��nm�Sn

for A ∈ ��2�n and C ∈ ��2�n

is given by the formula

A � C = 1n!∑I��

�−2i��I�(bI

)��c��IA+ ��C�− �I� I��I�

where I � n → � and � ∈ Sn.

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Theorem 41. The map

��2m

⊗n�� nm�Sn

−→ ��z� z⊗n�� nm�Sn

�A1� � � � � An� �−→ ZA11 � � � ZAn

n

is an algebra isomorphism.

Proof. Let Ai = �ai� bi� ∈ �2, and Ci = �ci� di� ∈ �2 i ∈ n, we have

ZA11 · · ·ZAn

n � ZC11 · · ·ZCn

n = za11 z

b11 · · · zann zbnn z

c11 z

d11 · · · zcnn zdnn

= 1mnn!

∑�w��∈�n

m�Sn

za11 z

b11 · · · zann zbnn ��w� ��z

c11 z

d11 · · · zcnn zdnn ��

= 1mnn!

∑�∈Sn

( n∏j=1

m−1∑kj=0

�e2�im �dj−cj��kj

) n∏j=1

zajj z

bjj z

c�−1�j�

j zd�−1�j�

j

= 1n!

∑I�n→��∈Sn

(�−2i��I�

(bI

)��c��I

)ZA+��C�−�I�I��

�I��

Let �m be the dihedral group of order 2n, where �m = �R0� R1� � � � � Rm−1�S0� S1� � � � � Sm−1�, Rk�z� = e

2�ikn z, and Sk�z� = e

2�ikn z� for k = 0� � � � � m− 1. �n

m � Snacts on �n as follows:

��nm � Sn�×�n −→ �n

��d1� d2� � � � � dn�� z1� z2� � � � � zn� �−→ �d1z�−1�1�� dnz�−1�n��

We will apply Polya functor to obtain a product rule on �C��n�� n�Sn

Definition 42. The �-product on ��2m

⊗n�� n�Snfor A ∈ ��2�n and

C ∈ ��2�n is given by the formula

A � C = 12n!

∑I��

(bI

)��c��IA+ ��c�� d��− �I� I��I�

+ 12n!

∑I��

(b + cI

)��d��IA+ ��d�� c��− �I� I��I�

where I � n → � and � ∈ Sn.

Theorem 43. The map ��2m

⊗n�� n�Sn−→ ��z� z⊗n�� n�Sn

given by�A1� � � � � An� �−→ Z

A11 · · ·ZAn

n is an algebra isomorphism.

Proof. For any Ai = �ai� bi� ∈ �2, Ci = �ci� di� ∈ �2, we have

ZA11 · · ·ZAn

n ZC11 · · ·ZCn

n

= 1�2n�n!

∑�d��∈�n�Sn

za11 z

b11 · · · zann z

bnn ��d� ��z

c11 z

d11 · · · zcnn zdnn �

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= 1�2n�n!

∑�∈Sn

( n∏j=1

n−1∑kj=0

e2�ikjn �dj−cj�

)( n∏j=1

zajj zj

bj zc�−1�j�

j zjd�−1�j� + z

ajj zj

bj+c�−1�j� z

d�−1�j�

j

)

= 12n!

∑��I

(bI

)��c��IZ

A+��c��d��−�I�I��I�

+ 12n!

∑��I

(b + cI

)��d��IZ

A+��d��c��−�I�I��I��

where � ∈ Sn and I � n → �. �

6.4. Quantum Super-Functions

We denote by Mat�n� the algebra of �-matrices of order n× n. It is well-known (see Deligne et al., 1999) that the canonical quantization of

∧��1� � � � � �m

is isomorphic to the Clifford algebra C�m�, i.e., the complex free algebra on mgenerators �1� � � � � �m subject to the relations �i�j + �j�i = 2�ij , for i� j ∈ m. It isalso known that

C�m� � Mat�2m2 � if m is even, and

C�m� � Mat�2m−12 �⊕Mat�2

m−12 � if m is odd.

Thus the algebra of quantum symmetric functions on the �0�m is isomorphic to

Symn�Mat�2m�� Schur�n� 2m� for m even,

Symn�Mat�2m−12 �⊕Mat�2

m−12 �� ⊕

a+b=n

Syma�Mat�2m−12 ��⊗ Symb�Mat�2

m−12 ��

� ⊕a+b=n

Schur�a� 2m−12 �⊗ Schur�b� 2

m−12 � for m odd�

Definition 44 and Proposition 45 below are taken from Merkulov (2000).

Definition 44. We denote by �� the algebra of all matrices �aij� such that aij ∈�� if i ≥ j, and aij ∈ ��j−i, if i < j. We set Schur�� n� �= Symn��.

For any a� b ∈ �≥0, we define the matrices

Ea�b�i� j� =�b + k�!

k! �b� if �i� j� = �a� b�+ �k� k�� k = 0� 1� 2� � � �

0� otherwise�

Proposition 45. There is a canonical isomorphism � � W −→ �� defined ongenerators by ��x� = E1�0, ��y� = E0�1 and �� = �I .

Proof. The linear map � is a well defined algebra homomorphism since the Weylalgebra is the quotient of a (formal) free algebra by the ideal generated by therelation yx= xy+ �, and the following identity holds E0�1E1�0 =E1�0E0�1 + �I in ��.

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Similarly, � is a bijection since Ea�b is a basis for �� and ��xayb� = Ea�b asconsequence of the fact that in �� the following identities are satisfied: E0�aE0�1 =E0�a+1, Ea�0E1�0 = Ea+1�0 and Ea�0E0�b = Ea�b, for all a� b ∈ �≥0. �

We denote by W the formalWeyl algebra, i.e. W = �x� y��/yx − xy − ��and QSymAn

�1� = Symn�W �.

Theorem 46.

a. The algebra QSymAn�1� of formal quantum symmetric functions on ��2�n�∑

dxi ∧ dyi� is isomorphic to Schur�� n�.b. The algebra of formal quantum symmetric functions on the superspaces �2�n is

isomorphic to Symn��⊗ C�m��.

Proof.

a. QSymAn�1� � Symn�W � � Schur�� n�.

b. The algebra of formal quantum functions on the superspace�2�n is isomorphic to

W ⊗ C�m� � ��⊗ C�m��

Thus, the algebra of formal quantum symmetric function on the superspaces�2m�n is isomorphic to Symn��⊗ C�m��. �

6.5. M -Weyl Algebra

In this section, we introduce the M-Weyl algebra. Although closely relatedto the Weyl algebra, the M-Weyl algebra does not arises as an instance of theKontsevich star product.

Definition 47. The M-Weyl algebra is the algebra MW = �x� y��/yx − xy −x2��. The letter M stands for meromorphic or mimetic.

We have the following analogue of Proposition 29.

Proposition 48. The map � � MW −→ End��x�� given by ��x��f� = x−1f and��y��f� = −�

�f

�x, for any f ∈ �x� defines a representation of the M-Weyl algebra.

We order the letters of the M-Weyl algebra as follows: x < y < �. Assumewe are given Ai = �ai� bi� ∈ �2, for i ∈ n. Set A = �A1� � � � � An� ∈ ��2�n and XAi =xaiybi , for i ∈ n. Using this notation we have the following.

Definition 49. The normal coordinates NM�A� k� of∏n

i=1 XAi ∈ MW are defined

through the identity

n∏i=1

XAi =min∑k=0

NM�A� k�x�A�+�k�−k�

�k (11)

where 0 ≤ k ≤ min = min��a�� b��. For k > min, we set NM�A� k� = 0.

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Theorem 50. Let A� k be as in the previous definition, the following identity holds:

a. NM�A� k� =∑p�k

(bp

) n−1∏i=1

��a>i� + �p>i��pi�� (12)

b. Let E1� � � � � En� F1� � � � � Fn be disjoint sets such that ��Fi� = ai, ��Ei� = bi, for i ∈n. Set E = ⋃

Ei, F = ⋃Fi and consider the set Mk of all functions f � F −→ P�E�

such that

• f�x� ∩ f�y� = ∅, for all x� y ∈ F ,• If x ∈ Fi, y ∈ Ej , and y ∈ f�x�, then j < i,• ∑

a∈F ��f�a�� = k.

Then NM�A� k� = ��Mk�.

Proof. Using induction, one shows that the following identity hold in the M-Weylalgebra:

ybxa =min∑k=0

(bk

)a�k�xa+kyb−k

�k� (13)

where min = min�a� b�. Several application of the identity (13) imply a. Notice thatfor given sets E� F such that �F � = a and �E� = b, � b

k � a�k� is equal to ��f � F −→

P�E� � f�x� ∩ f�y� = ∅� for all x� y ∈ F� and∑

a∈F ��f�a�� = k��. This shows b, forn = 1. The general formula follows from induction. �

Figure 5 illustrates the combinatorial interpretation of the normal coordinatesNM�A� 6� of

∏3i=1 X

Ai ∈ MW , it shows an example of a function contributing toNM�A� 6�.

Corollary 51. For any given �t� a� b� ∈ �×�n ×�n, the following identity holds:

n∏i=1

�t − �a>i� − �b>i��bi =∑p�k

�−1�k(bp

) n−1∏i=1

��a>i� − �p>i��pi�t�b�−k�

Proof. Consider the identity �11� in the representation of the M-Weyl algebradefined in Proposition 48. Apply both sides of the identity �11� to xt and useTheorem 50 formula (12). �

The following theorem provides explicit formula for the product of m elementsof the M-Weyl algebra. Using the same notation as in the Theorem 36, we havethe following.

E1 F1 E2 F2 E3 F3

Figure 5 Combinatorial interpretation of NM .

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Theorem 52 (symmetric powers of M-Weyl algebra). For any A� m× n−→�2,the following identity:

�n!�m−1m∏i=1

( n∏j=1

XAij

j

)= ∑

��k�p

(∏i�j

(b�jpj

)��a�

j �>i� + �pj

>i��pji �

) n∏j=1

X�A�

j �−�kj �kj �

j ��k� (14)

where � ∈ �id�× Sm−1n , k ∈ �n, �i� j� ∈ m− 1× n and p = p

ji ∈ ��m−1�n, holds in

Symn�MW�.

Proof. By Theorem 14 and Theorem 50, we have

�n!�m−1m∏i=1

( n∏j=1

XAij

j

)= ∑

�∈�id�×Sm−1n

n∏j=1

( m∏i=1

XAi�−1i �j�

j

)

= ∑�∈�id�×Sm−1

n

n∏j=1

( minj∑k=0

NM�A�j � k�X

�A�j �+�k�−k�

j �k

)

= ∑��k

( n∏j=1

NM�A�j � kj�

) n∏j=1

X�A�

j �+�kj �−kj�

j ��k�

= ∑��k�p

(∏i�j

(b�jpj

)��a�

j �>i� + �pj

>i��pji �

) n∏j=1

X�A�

j �−�kj �kj �

j ��k��

where minj = min��a�j �� b�j ��. �

6.6. Cohomological Interpretation of Symmetric Functions

Let G be a finite group acting on a compact differentiable manifold X. G actson H•�X�, the singular homology groups with complex coefficients of X, as follows:

G×H•�X� −→ H•�X��g� �� −→ g∗��

where g∗�� denotes the push-forward of � by the map g. Similarly, G acts on H•�X�,the singular cohomology groups with complex coefficients of X, as follows:

G×H•�X� −→ H•�X��g� �� −→ g∗��

where g∗�� denotes the pull-back of � by the map g. It is well-known thatH•�X/G� = H•�X�G (see Hirzebruch and Höfer, 2002). Identifying H•�X�G withH•�X�G, we obtain H•�X/G� = H•�X�G. Consider Xn/Gn

� K where K ⊂ Sn andXn = X × X × · · · × X. We have that

H•�Xn/Gn� K� � H•�X�⊗n/Gn

� K � PG�K�H•�X��

The next theorem shows how symmetric and supersymmetric functions arise asthe cohomology groups of global orbifolds (quotient of manifolds by finite group

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actions). We denote by �� the inductive limit of the complex projective spaces��n. We also let S1 be the unit circle in �.

Theorem 53.

a. H•��m�n/Sn� � SymAn�m�.

b. H•��S1�m�n/Sn� � Symn�∧�1� � � � � �m�.

c. H•��m × �S1�k�n/Sn� � Symn��x1� � � � � xn⊗∧�1� � � � � �k�.

Proof. Recall that H•��n� = �x/�xn� (see Raoul Bott and Tu, 1982), whichimplies that H•�� = �x. Thus, by the remarks above

H•��m�n/Sn� � ��H•��⊗m�⊗n�Sn � H•��x⊗m�⊗n�Sn

� ��x1� � � � � xm�⊗n/Sn � SymAn

�m��

Since H•�S1� � ��/�2� as graded superalgebras with � of degree 1, then

H•��S1�m�n/Sn�

� ��H•�S1�⊗m�⊗n�Sn �(∧

�1� � � � � �m⊗n)Sn� Symn

(∧�1� � � � � �m

)�

Finally,

H•��m × �S1�k�n/Sn� � �H•��⊗m ⊗H•�S1�⊗k�⊗n/Sn

� Symn��x1� � � � � xm⊗∧�1� � � � � �k��

Theorem 53 together with the quantizations of SymAn�2m� and of

Symn�∧�1� � � � � �m� provided in sections 6.2 and 6.4, respectively, give a quantum

product on the cohomology of ��2m�n/Sn and ��S1�m�n/Sn, respectively. Noticethat the quantum product on H•��m�n/Sn� is noncommutative and thereforeis different from the quantum cohomology product defined, for example, in Duffand Salamon (1991).

ACKNOWLEDGMENTS

We thank Nicolas Andruskiewitsch for his advice and encouragement. We alsothank Delia Flores de Chela. Refael Diaz was partially supported by UCV and EddyPariguan was partially supported by FONACIT.

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Deligne, P., Etingof, P., Freed, D., Jeffrey, L., Kazhdan, D., Morgan, J., Morrison, D.,Witten, E. (1999). Quantum Fields and Strings: A Course for Mathematicians. Vol. 1.American Mathematical Society.

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Quantum Symmetric Functions

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