**Fock space representations of U**

**Fock space representations of U**

_{q}## ( sl b

_{n}## )

Bernard Leclerc (Universit´e de Caen) Ecole d’´et´e de Grenoble, juin 2008.

**Contents**

**1** **Introduction** **2**

**2** **Fock space representations of**slb_{n}**3**

2.1 The Lie algebraslb*n*and its wedge space representations . . . . 3

2.2 The level one Fock space representation ofslb* _{n}* . . . . 6

2.3 The higher level Fock space representations ofslb*n* . . . . 11

**3** **Fock space representation of U***q*(bsl*n*)**: level 1** **16**
3.1 *The quantum affine algebra U**q*(bsl*n*) . . . . 16

3.2 The tensor representations . . . . 16

3.3 The affine Hecke algebra*H*b*r* . . . . 17

3.4 Action of*H*b*r**on U*^{⊗r} . . . . 17

3.5 *The q-deformed wedge-spaces . . . .* 19

3.6 The bar involution of∧^{r}_{q}*U* . . . . 20

3.7 Canonical bases of∧^{r}_{q}*U . . . .* 21

3.8 *Action of U**q*(bg)*and Z(**H*b*r*)on∧^{r}_{q}*U* . . . . 22

3.9 The Fock space . . . . 23

*3.10 Action of U**q*(bg)onF . . . . 23

3.11 Action of the bosons onF . . . . 24

3.12 The bar involution ofF . . . . 25

3.13 Canonical bases ofF . . . . 25

3.14 Crystal and global bases . . . . 26

**4** **Decomposition numbers** **27**
4.1 The Lusztig character formula . . . . 27

4.2 Parabolic Kazhdan-Lusztig polynomials . . . . 28

4.3 Categorification of∧^{r}*U . . . .* 30

4.4 Categorification of the Fock space . . . . 31

**5** **Fock space representations of U***q*(bsl*n*)**: higher level** **33**
5.1 *Action on U . . . .* 33

5.2 *The tensor spaces U*^{⊗r} . . . . 34

5.3 *The q-wedge spaces*∧^{r}_{q}*U . . . .* 34

5.4 **The Fock spaces F[m**_{ℓ}] . . . . 34

5.5 **The canonical bases of F[m**_{ℓ}] . . . . 35

5.6 Comparison of bases . . . . 35

5.7 *Cyclotomic v-Schur algebras . . . .* 35

**6** **Notes** **36**

**1** **Introduction**

In the mathematical physics literature, the Fock spaceF is the carrier space of the natural irre-
ducible representation of an infinite-dimensional Heisenberg Lie algebra H. Namely, F is the
polynomial ringC[x*i*|*i*∈N^{∗}], andHis the Lie algebra generated by the derivations∂/∂*x**i*and the
*operators of multiplication by x** _{i}*.

In the 70’s it was realized that the Fock space could also give rise to interesting concrete
realizations of highest weight representations of Kac-Moody affine Lie algebrasbg. Indeedbghas
a natural Heisenberg subalgebrap (the principal subalgebra) and the simplest highest weightbg-
module, called the basic representation ofbg, remains irreducible under restriction top. Therefore,
one can in principle extend the Fock space representation ofpto a Fock space representation F
ofbg. This was first done forbg=slb_{2}**by Lepowsky and Wilson [LW]. The Chevalley generators of**
slb_{2}act onF via some interesting but complicated differential operators of infinite degree closely
related to the vertex operators invented by physicists in the theory of dual resonance models. Soon
after, this construction was generalized to all affine Lie algebrasbg*of A,D,E type [KKLW].*

Independently and for different purposes (the theory of soliton equations) similar results were
**obtained by Date, Jimbo, Kashiwara and Miwa [DJKM] for classical affine Lie algebras. Their**
approach is however different. They first endow F with an action of an infinite rank affine Lie
algebra and then restrict it to various subalgebrasbgto obtain their basic representations. In type
*A for example, they realize in* F the basic representation of gl_{∞} (related to the KP-hierarchy
of soliton equations) and restrict it to natural subalgebras isomorphic toslb*n*(n>2) to get Fock
*space representations of these algebras (related to the KdV-hierarchy for n*=2). In this approach,
the Fock space is rather the carrier space of the natural representation of an infinite-dimensional
Clifford algebra, that is, an infinite dimensional analogue of an exterior algebra. The natural
isomorphism between this “fermionic” construction and the previous “bosonic” construction is
called the boson-fermion correspondence.

The basic representation ofbghas level one. Higher level irreducible representations can also
be constructed as subrepresentations of higher level Fock space representations ofbg**[F1, F2].**

After quantum enveloping algebras of Kac-Moody algebras were invented by Jimbo and Drin-
*feld, it became a natural question to construct the q-analogues of the above Fock space represen-*
**tations. The first results in this direction were obtained by Hayashi [H]. His construction was**
**soon developed by Misra and Miwa [MM], who showed that the Fock space representation of**
*U**q*(slb*n*)has a crystal basis (crystal bases had just been introduced by Kashiwara) and described it
completely in terms of Young diagrams. This was the first example of a crystal basis of an infinite-
dimensional representation. Another construction of the level one Fock space representation of
*U**q*(slb*n*)**was given by Kashiwara, Miwa and Stern [KMS], in terms of semi-infinite q-wedges. This***relied on the polynomial tensor representations of U** _{q}*(slb

*)which give rise to the quantum affine*

_{n}**analogue of the Schur-Weyl duality obtained by Ginzburg, Reshetikhin and Vasserot [GRV], and**

**Chari and Pressley [CP] independently.**

**In [LLT] and [LT1], some conjectures were formulated relating the decomposition matrices**
*of type A Hecke algebras and q-Schur algebras at an nth root of unity on the one hand, and the*
global crystal basis of the Fock space representation F *of U** _{q}*(slb

*)on the other hand. Note that*

_{n}**[LT1] contains in particular the definition of the global basis of**F, which does not follow from the general theory of Kashiwara or Lusztig. The conjecture on Hecke algebras was proved by Ariki

**[A1], and the conjecture on Schur algebras by Varagnolo and Vasserot [VV].**

**Slightly after, Uglov gave a remarkable generalization of the results of [KMS], [LT1], and**
**[VV] to higher levels. Together with Takemura [TU], he introduced a semi-infinite wedge real-**
ization of the levelℓ*Fock space representations of U**q*(slb*n*), and in [U1, U2] he constructed their

canonical bases and expressed their coefficients in terms of Kazhdan-Lusztig polynomials for the affine symmetric groups.

A full understanding of these coefficients as decomposition numbers is still missing. Recently,
**Yvonne [Y2] has formulated a precise conjecture stating that, under certain conditions on the com-**
ponents of the multi-charge of the Fock space, the coefficients of Uglov’s bases should give the de-
**composition numbers of the cyclotomic q-Schur algebras of Dipper, James and Mathas [DiJaMa].**

**Rouquier [R] has generalized this conjecture to all multi-charges. In his version the cyclotomic**
*q-Schur algebras are replaced by some quasi-hereditary algebras arising from the category*O of
*the rational Cherednik algebras attached to complex reflection groups of type G(ℓ,*1,*m).*

In these lectures we first present in Section 2 the Fock space representations of the affine Lie
algebraslb_{n}*. We chose the most suitable construction for our purpose of q-deformation, namely,*
we realizeF as a space of semi-infinite wedges (the fermionic picture). In Section 3 we explain
*the level one Fock space representation of U**q*(slb*n*)and construct its canonical bases. In Section 4
**we explain the conjecture of [LT1] and its proof by Varagnolo and Vasserot. Finally, in Section 5**
we indicate the main lines of Uglov’s construction of higher level Fock space representations of
*U**q*(slb*n*), and of their canonical bases, and we give a short review of Yvonne’s work.

**2** **Fock space representations of** sl b

_{n}**2.1** **The Lie algebra**slb_{n}**and its wedge space representations**

*We fix an integer n*>2.

**2.1.1** **The Lie algebra**sl_{n}

The Lie algebrag=sl*n**of traceless n*×*n complex matrices has Chevalley generators*
*E**i*=*E**i,i+1*, *F**i*=*E**i+1,i*, *H**i*=*E**ii*−*E**i+1,i+1*, (16*i*6*n*−1).

*Its natural action on V*=C* ^{n}*=⊕

^{n}*Cv*

_{i=1}*is*

_{i}*E*_{i}*v** _{j}*=δ

*j,i+1*

*v*

*,*

_{i}*F*

_{i}*v*

*=δ*

_{j}*j,i*

*v*

*,*

_{i+1}*H*

_{i}*v*

*=δ*

_{j}*j,i*

*v*

*−δ*

_{i}*j,i+1*

*v*

*, (16*

_{i+1}*i*6

*n*−1).

We may picture the action ofg*on V as follows*

*v*_{1}−→^{F}^{1} *v*_{2}−→ · · ·^{F}^{2} −→^{F}^{n−1}*v*_{n}**2.1.2** **The Lie algebra L(sl*** _{n}*)

*The loop space L(g) =*g⊗C[z,*z*^{−1}]is a Lie algebra under the Lie bracket
[a⊗*z** ^{k}*,b⊗

*z*

*] = [a,b]⊗*

^{l}*z*

*, (a,*

^{k+l}*b*∈g,

*k,l*∈Z).

*The loop algebra L(g)naturally acts on V*(z) =*V*⊗C[z,z^{−1}]by

(a⊗*z** ^{k}*)·(v⊗

*z*

*) =*

^{l}*av*⊗

*z*

*, (a∈g,*

^{k+l}*v*∈

*V,*

*k,l*∈Z).

**2.1.3** **The Lie algebra**slb*n*

The affine Lie algebrabg=slb_{n}*is the central extension L(g)*⊕Cc with Lie bracket

[a⊗*z** ^{k}*+λ

*c,*

*b*⊗

*z*

*+µ*

^{l}*c] = [a,b]*⊗

*z*

*+*

^{k+l}*k*δ

*k,−l*tr(ab)

*c,*(a,

*b*∈g, λ,µ ∈C,

*k,l*∈Z).

*This is a Kac-Moody algebra of type A*^{(1)}* _{n−1}*with Chevalley generators

*e** _{i}*=

*E*

*⊗1,*

_{i}*f*

*=*

_{i}*F*

*⊗1,*

_{i}*h*

*=*

_{i}*H*

*⊗1, (16*

_{i}*i*6

*n*−1),

*e*0=

*E*

*n1*⊗

*z,*

*f*0=

*E*

*1n*⊗

*z*

^{−1},

*h*0= (E

*nn*−

*E*11)⊗1+

*c.*

We denote byΛ*i*(i=0,1, . . . ,*n*−1)the fundamental weights ofbg. By definition, they satisfy
Λ*i*(h*j*) =δ*i j*, (06*i,j*6*n*−1).

*Let V*(Λ)be the irreducible bg-module with highest weightΛ **[K,**§9.10]. IfΛ=∑*i**a** _{i}*Λ

*i*then the

*central element c*=∑

*i*

*h*

*acts as∑*

_{i}*i*

*a*

_{i}*Id on V*(Λ), and we callℓ=∑

*i*

*a*

_{i}*the level of V(Λ). More*

*generally, a representation V of*bg

*is said to have level*ℓ

*if c acts on V by multiplication by*ℓ.

*The loop representation V*(z) can also be regarded as a representation of bg, in which c acts
trivially. Define

*u** _{i−nk}*=

*v*

*i*⊗

*z*

*, (16*

^{k}*i*6

*n,k*∈Z).

Then(u* _{j}*|

*j*∈Z)is aC-basis of V(z). We may picture the action ofbg

*on V*(z)as follows

· · ·−→^{f}^{n−2}*u*−1
*f*_{n−1}

−→*u*0
*f*0

−→*u*1
*f*1

−→*u*2
*f*2

−→ · · ·−→^{f}^{n−1}*u**n*
*f*0

−→*u**n+1*
*f*1

−→*u**n+2*
*f*2

−→ · · ·

Note that this is not a highest weight representation.

**2.1.4** **The tensor representations**

*For r*∈N^{∗}*, we consider the tensor space V*(z)^{⊗r}. The Lie algebra bgacts by derivations on the
*tensor algebra of V*(z). This induces an action on each tensor power V(z)^{⊗r}, namely,

*x(u*_{i}_{1}⊗ · · · ⊗*u*_{i}* _{r}*) = (xu

_{i}_{1})⊗ · · · ⊗

*u*

_{i}*+· · ·+*

_{r}*u*

_{i}_{1}⊗ · · · ⊗(xu

_{i}*), (x∈bg,*

_{r}*i*

_{1},· · ·,

*i*

*∈Z).*

_{r}*Again c acts trivially on V*(z)^{⊗r}.

*We have a vector space isomorphism V*^{⊗r}⊗C[z^{±}_{1}, . . . ,*z*^{±}* _{r}*]−→

^{∼}

*V*(z)

^{⊗r}given by

(v*i*1⊗ · · · ⊗*v**i**r*)⊗*z*_{1}^{j}^{1}· · ·*z*_{r}^{j}* ^{r}* 7→(v

*i*1⊗

*z*

^{j}^{1})⊗ · · · ⊗(v

*i*

*r*⊗

*z*

^{j}*), (16*

^{r}*i*1, . . . ,

*i*

*r*6

*n,*

*j*1, . . . ,

*j*

*r*∈Z).

**2.1.5** **Action of the affine symmetric group**

The symmetric groupS_{r}*acts on V*^{⊗r}⊗C[z^{±}_{1}, . . . ,*z*^{±}* _{r}*]by

σ(v*i*1⊗ · · · ⊗*v**i**r*)⊗*z*_{1}^{j}^{1}· · ·*z*_{r}^{j}* ^{r}* = (v

*i*

_{σ}

_{−1}

_{(1)}⊗ · · · ⊗

*v*

*i*

_{σ}

_{−1}

_{(r)})⊗

*z*

_{1}

^{j}^{σ}

^{−1}

^{(1)}· · ·

*z*

*r*

^{j}^{σ}

^{−1}

^{(r)}, (σ∈S

*r*).

Moreover the abelian groupZ* ^{r}*acts on this space, namely(k

_{1}, . . . ,

*k*

*)∈Z*

_{r}*acts by multiplication*

^{r}*by z*

^{k}_{1}

^{1}· · ·

*z*

^{k}

_{r}

^{r}*. Hence we get an action on V*(z)

^{⊗r}of the affine symmetric group Sb

*:=S*

_{r}*⋉ Z*

_{r}*. Clearly, this action commutes with the action ofbg.*

^{r}It is convenient to describe this action in terms of the basis
(u** _{i}**=

*u*

_{i}_{1}⊗ · · · ⊗

*u*

_{i}*|*

_{r}**i**= (i

_{1}, . . . ,

*i*

*)∈Z*

_{r}*).*

^{r}*Denote by s**k*= (k,k+1),(k=1, . . . ,*r*−1)the simple transpositions ofS*r*. The affine symmetric
groupSb* _{r}*acts naturally onZ

*via*

^{r}*s**k*·i = (i1, . . . ,*i** _{k+1}*,i

*k*, . . .

*i*

*r*), (16

*k*6

*r*−1),

*z*

*·*

_{j}**i**= (i

_{1}, . . . ,

*i*

*−*

_{j}*n, . . . ,i*

*), (16*

_{r}*j*6

*r),*and we have

*w u***i**=*u** _{w·i}*, (i∈Z

*,*

^{r}*w*∈Sb

*r*).

*Thus the basis vectors u***i***are permuted, and each orbit has a unique representative u***i**with
**i**∈*A**r*:={i∈Z* ^{r}*|16

*i*16

*i*26· · ·6

*i*

*r*6

*n}.*

**Let i**∈*A** _{r}*. The stabilizer S

_{i}*of u*

**inSb**

_{i}*is the subgroup of S*

_{r}

_{r}*generated by the s*

*such that*

_{k}*i*

*k*=

*i*

*k+1*, a parabolic subgroup. Hence we have finitely many orbits, and each of them is of the formSb

*r*/S

**i**for some parabolic subgroupS

**i**ofS

*r*.

**2.1.6** **The wedge representations**

Consider now the wedge product∧^{r}*V*(z). It has a basis consisting of normally ordered wedges

∧u** _{i}**:=

*u*

_{i}_{1}∧

*u*

_{i}_{2}∧ · · · ∧

*u*

_{i}*, (i= (i*

_{r}_{1}>

*i*

_{2}>· · ·>

*i*

*)∈Z*

_{r}*).*

^{r}The Lie algebrabg also acts by derivations on the exterior algebra ∧V(z), and this restricts to an
action on∧^{r}*V*(z), namely,

*x(u*_{i}_{1}∧ · · · ∧*u*_{i}* _{r}*) = (xu

_{i}_{1})∧ · · · ∧

*u*

_{i}*+· · ·+*

_{r}*u*

_{i}_{1}∧ · · · ∧(xu

_{i}*), (x∈bg,*

_{r}*i*

_{1},· · ·,i

*∈Z).*

_{r}*Again c acts trivially on*∧^{r}*V*(z).

We may think of∧^{r}*V*(z)*as the vector space quotient of V*(z)^{⊗r}by the subspace
I* _{r}*:=

*r−1*

### ∑

*k=1*

Im(s* _{k}*+Id)⊂

*V*(z)

^{⊗r}of “partially symmetric tensors”.

**2.1.7** **Action of the center of**CSb_{r}

The spaceI* _{r}*is not stable under the action of Sb

*r*

*. (For example if r*=

*2, we have z*1(u0⊗

*u*0) =

*u*

_{−n}⊗u

_{0}, which is not symmetric.) HenceSb

*does not act on the wedge product∧*

_{r}

^{r}*V*(z). However,

*for every k*∈Z, the element

*b**k*:=

### ∑

*r*

*i=1*

*z*^{k}_{i}

of the group algebraCSb*r*commutes with S*r*⊂ZSb*r*. Therefore it has a well-defined action on

∧^{r}*V*(z), given by

*b**k*(∧u**i**) =*u**i*1−nk∧*u**i*2∧ · · · ∧*u**i**r*+*u**i*1∧*u**i*2−nk∧ · · · ∧*u**i**r*+· · ·+*u**i*1∧*u**i*2∧ · · · ∧*u**i**r*−nk.
This action commutes with the action ofbgon∧^{r}*V*(z). The elements b*k* (k∈Z)generate a subal-
gebra ofCSb*r**isomorphic to the algebra of symmetric Laurent polynomials in r variables.*

**2.2** **The level one Fock space representation of**slb_{n}*We want to pass to the limit r*→∞in the wedge product∧^{r}*V*(z).

**2.2.1** **The Fock space**F

*For s*>*r define a linear map*ϕ*r,s*:∧^{r}*V*(z)−→ ∧^{s}*V*(z)by

ϕ*r,s*(∧u**i**):=∧u**i**∧*u*−r∧*u*−r−1∧ · · · ∧u−s+1.
Then clearlyϕ*s,t*◦ϕ*r,s*=ϕ*r,t* *for any r*6*s*6*t. Let*

∧^{∞}*V*(z):=lim

→ ∧^{r}*V*(z)

be the direct limit of the vector spaces∧^{r}*V*(z)with respect to the mapsϕ*r*,s. Each∧u** _{i}** in∧

^{r}*V*(z) has an imageϕ

*r*(∧u

**i**)∈ ∧

^{∞}

*V*(z), which should be thought of as the “infinite wedge”

ϕ*r*(∧u**i**)≡ ∧u**i**∧*u*_{−r}∧*u*_{−r−1}∧ · · · ∧*u*_{−s}∧ · · ·

The spaceF :=∧^{∞}*V*(z)*is called the fermionic Fock space. It has a basis consisting of all infinite*
wedges

*u**i*1∧*u**i*2∧ · · · ∧*u**i**r*∧ · · ·, (i1>*i*2>· · ·>*i**r*>· · ·),
which coincide except for finitely many indices with the special infinite wedge

|/0i:=*u*0∧*u*_{−1}∧ · · · ∧*u*_{−r}∧*u*_{−r−1}∧ · · ·

*called the vacuum vector. It is convenient to label this basis by partitions. A partition*λ is a finite
weakly decreasing sequence of positive integersλ1>λ2>· · ·>λ*s*>0. We make it into an infinite
sequence by puttingλ*j* =*0 for j*>*s, and we set*

|λi:=*u*_{i}_{1}∧*u*_{i}_{2}∧ · · · ∧*u*_{i}* _{r}*∧ · · ·,

*where i*

*=λ*

_{k}*k*−

*k*+1(k>1).

This leads to a natural N-grading onF given by deg|λi=∑*i*λ*i* for every partition λ. The
*dimension of the degree d component*F^{(d)}ofF is equal to number of partitionsλ*of d. This can*
be encoded in the following generating function

*d>0*

### ∑

dimF^{(d)}*t** ^{d}*=

### ∏

*k>1*

1

1−*t** ^{k}*. (1)

**2.2.2** **Young diagrams**

Let P denote the set of all partitions. Elements of P are represented graphically by Young diagrams. For example the partitionλ = (3,2)is represented by

Ifγ*is the cell in column number i and row number j, we call i*−*j the content of*γ. For example
the contents of the cells ofλ are

−1 0 0 1 2

*Given i*∈ {0, . . . ,*n*−1}, we say thatγ*is an i-cell if its content c(*γ)*is congruent to i modulo n.*

**2.2.3** **The total Fock space**F

*It is sometimes convenient to consider, for m*∈Z, similar Fock spacesF* _{m}*obtained by using the
modified family of embeddings

ϕ*r,s*^{(m)}(∧u**i**):=∧u**i**∧*u**m−r*∧*u**m−r−1*∧ · · · ∧*u**m−s+1*.
The spaceF* _{m}*has a basis consisting of all infinite wedges

*u*_{i}_{1}∧*u*_{i}_{2}∧ · · · ∧*u*_{i}* _{r}*∧ · · ·, (i

_{1}>

*i*

_{2}>· · ·>

*i*

*>· · ·), which coincide, except for finitely many indices, with*

_{r}|/0*m*i:=*u**m*∧*u**m−1*∧ · · · ∧*u**m−r*∧*u**m−r−1*∧ · · ·

The spaceF=⊕* _{m∈Z}*F

_{m}*is called the total Fock space, and*F

_{m}*is the Fock space of charge m.*

Most of the time, we shall only deal withF =F_{0}.
**2.2.4** **Action of f**_{i}

*Consider the action of the Chevalley generators f** _{i}*(i=0,1, . . .

*n*−1)ofbgon the sequence of vector spaces∧

^{r}*V*(z). It is easy to see that, for∧u

**i**∈ ∧

^{r}*V*(z),

*f** _{i}*ϕ

*r,s*(∧u

**) =ϕ**

_{i}*r+1,s*

*f*

*ϕ*

_{i}*r,r+1*(∧u

**)**

_{i}*for all s*>

*r.*

**Exercise 1 Check this.**

*Hence one can define an endomorphism f*_{i}^{∞}ofF by

*f*_{i}^{∞}ϕ*r*(∧u** _{i}**) =ϕ

*r+1*

*f*

*i*ϕ

*r,r+1*(∧u

**). (2)**

_{i}*The action of f*_{i}^{∞}on the basis{|λi |λ∈P}has the following combinatorial translation in terms
of Young diagrams:

*f*_{i}^{∞}|λi=

### ∑

µ |µi, (3)

where the sum is over all partitions µ obtained fromλ *by adding an i-cell.*

* Exercise 2 Check it. Check that, for n*=2, we have

*f*_{0}^{∞}|3,1i=|3,2i+|3,1,1i, *f*_{1}^{∞}|3,1i=|4,1i.

**2.2.5** **Action of e***i*

*Similarly, one can check that putting e*^{∞}* _{i}* ϕ

*r*(∧u

**):=ϕ**

_{i}*r*(e

*∧*

_{i}*u*

**)one gets a well-defined endomor- phism ofF. Its combinatorial description is given by**

_{i}*e*^{∞}* _{i}* |µi=

### ∑

λ

|λi, (4)

where the sum is over all partitionsλ obtained fromµ *by removing an i-cell.*

* Exercise 3 Check it. Check that, for n*=2, we have

*e*^{∞}_{0}|3,1i=|2,1i, *e*^{∞}_{1}|3,1i=|3i.

**Theorem 1 The map**

*e** _{i}*7→

*e*

^{∞}

*,*

_{i}*f*

*7→*

_{i}*f*

_{i}^{∞}, (06

*i*6

*n*−1),

*extends to a level one representation of*bg

*on*F

*.*

*Proof — Let*µ be obtained fromλ *by adding an i-cell*γ. Then we callγ*a removable i-cell of* µ
*or an addable i-cell of*λ*. One first deduces from Eq. (3) (4) that h*^{∞}* _{i}* := [e

^{∞}

*,*

_{i}*f*

_{i}^{∞}]is given by

*h*^{∞}* _{i}* |λi=

*N*

*i*(λ)|λi, (5)

*where N** _{i}*(λ)

*is the number of addable i-cells of*λ

*minus the number of removable i-cells of*λ.

*It is then easy to check from Eq. (3) (4) (5) that the endomorphisms e*

^{∞}

*,*

_{i}*f*

_{i}^{∞},

*h*

^{∞}

*(06*

_{i}*i*6

*n*−1)

*satisfy the Serre relations of the Kac-Moody algebra of type A*

^{(1)}

_{n−1}**(see [K,**§0.3, §9.11]). Now,

*c*=

*h*0+

*h*1+· · ·+

*h*

*n−1*acts by

*c|λ*i=

*n−1*

### ∑

*i=0*

*N**i*(λ)|λi.

Clearly, the difference between the total number of addable cells and the total number of removable
cells of any Young diagram is always equal to 1, hence we have∑^{n−1}_{i=0}*N** _{i}*(λ) =1 for everyλ. 2

Thus the Fock space F is endowed with an action ofbg. Note that, although every ∧^{r}*V*(z) is
a level 0 representation, their limitF is a level 1 representation. This representation is called the
*level 1 Fock space representation of*bg.

Note also that ∧^{r}*V*(z) *has no primitive vector, i.e. no vector killed by every e** _{i}*. ButF has
many primitive vectors.

* Exercise 4 Take n*=

*2. Check that v*

_{0}=|/0i

*and v*

_{1}=|2i − |1,1iare primitive vectors. Can you find an infinite family of primitive vectors ?

**2.2.6** **Action of b**_{k}

*Recall the endomorphisms b** _{k}* (k∈Z) of∧

^{r}*V*(z) (see§2.1.7). It is easy to see that, if k6=0, the vectorϕ

*s*

*b*

*k*ϕ

*r,s*(∧u

**i**)

*is independent of s for s*>

*r large enough. Hence, one can define endomor-*

*phisms b*

^{∞}

*(k∈Z*

_{k}^{∗})ofFby

*b*^{∞}* _{k}*ϕ

*r*(∧u

**i**):=ϕ

*s*

*b*

*k*ϕ

*r*,s(∧u

**i**) (s≫1). (6) In other words

*b*^{∞}* _{k}*(u

_{i}_{1}∧

*u*

_{i}_{2}∧ · · ·) = (u

_{i}_{1}

_{−nk}∧

*u*

_{i}_{2}∧ · · ·) + (u

_{i}_{1}∧

*u*

_{i}_{2}

_{−nk}∧ · · ·) +· · ·

where in the right-hand side, only finitely many terms are nonzero. By construction, these endo- morphisms commute with the action ofbgonF. However they no longer generate a commutative algebra but a Heisenberg algebra, as we shall now see.

**2.2.7** **Bosons**

In fact, we will consider more generally the endomorphismsβ*k*(k∈Z^{∗})ofF defined by

β*k*(u*i*1∧*u**i*2∧ · · ·) = (u*i*1−k∧*u**i*2∧ · · ·) + (u*i*1∧*u**i*2−k∧ · · ·) +· · · (7)
*so that b*^{∞}* _{k}* =β

*nk*.

* Proposition 1 For k,l*∈Z

^{∗}

*, we have*[β

*k*,β

*l*] =δ

*k,−l*

*k Id*F.

*Proof — Recall the total Fock space*F=⊕* _{m∈Z}*F

*introduced in§2.2.3. Clearly, the definition of the endomorphismβ*

_{m}*k*ofF given by Eq. (7) can be extended to anyF

*. So we may consider β*

_{m}*k*as an endomorphism of Fpreserving eachF

_{m}*. For i*∈Z, we also have the endomorphism w

*ofF*

_{i}*defined by w*

*(v) =*

_{i}*u*

*∧*

_{i}*v. It sends*F

*toF*

_{m}

_{m+1}*for every m*∈Z. For any v∈F, we have

β*k**w** _{i}*(v) =β

*k*(u

*∧*

_{i}*v) =u*

*∧*

_{i−k}*v*+

*u*

*∧β*

_{i}*k*(v), hence[β

*k*,w

*i*] =

*w*

*. It now follows from the Jacobi identity that*

_{i−k}[[β*k*,β*l*],w* _{i}*] = [[β

*k*,w

*],β*

_{i}*l*]−[[β

*l*,

*w*

*i*],β

*k*] = [w

*,β*

_{i−k}*l*]−[w

*,β*

_{i−l}*k*] =−w

*+*

_{i−k−l}*w*

*i−l−k*=0.

Let us write for short γ= [β*k*,β*l*]. We first want to show that γ =κId_{F} for some constant κ.
*Consider a basis element v*∈F* _{m}*. By construction we have

*v*=*u**i*1∧ · · · ∧*u**i**N*∧ |/0*m−N*i

*for some large enough N and i*_{1}>*i*_{2}>· · ·>*i** _{N}*>

*m*−

*N. Since*γ

*commutes with every w*

*, we have γ(v) =*

_{i}*u*

*i*1∧ · · · ∧

*u*

*i*

*N*∧γ(|/0

*i). (8) We have*

_{m−N}γ(|/0* _{m−N}*i) =

### ∑

*j*

α*j**u*_{m}_{1,}* _{j}*∧ · · · ∧

*u*

_{m}*∧ |/0*

_{L,j}*i*

_{m−N−L}*for some large enough L and m*_{1,}* _{j}*>· · ·>

*m*

_{L,}*>*

_{j}*m−N*−L. Clearly, m

_{1,}

*6*

_{j}*m−N*+|k|+|l|. Tak-

*ing N large enough, we can assume that*{m−

*N*+1,m−N+2, . . . ,

*m*−N+|k|+|l|} ⊂ {i

_{1}, . . . ,

*i*

*N*}.

*Hence we must have m*_{1,}*j* =*m*−*N for every j, and this forces m*_{2,}*j* =*m*−*N*−1, . . . ,*m*_{L,}*j* =
*m−N*−*L*+1. This shows thatγ(v) =κ*v**v for some scalar*κ*v*. Now Eq. (8) shows thatκ*v*=κ|/0* _{m−N}*i

*for N large enough, so*κ*v*=κ *does not depend on v.*

Finally, we have to calculateκ. For that, we remark thatβ*k* is an endomorphism of degree−k,
for the grading ofF defined in§2.2.1. Hence we see that if k+*l*6=0 then κ =0. So suppose
*k*=−l>0, and let us calculate[β*k*,β*l*]|/0i. For degree reasons this reduces toβ*k*β−k|/0i. It is easy
to see thatβ−k|/0i*is a sum of k terms, namely*

β−k|/0i=*u** _{k}*∧ |/0

_{−1}i+

*u*

_{0}∧

*u*

*∧ |/0*

_{k−1}_{−2}i+· · ·+

*u*

_{0}∧

*u*

_{−1}∧ · · · ∧

*u*

_{−k+2}∧

*u*

_{1}∧ |/0

_{−k}i.

*Moreover one can check that each of these k terms is mapped to*|/0ibyβ*k*. 2
Proposition 1 shows that the endomorphisms β*k* (k∈Z^{∗}) endow F with an action of an
infinite-dimensional Heisenberg Lie algebraH*with generators p**i*,*q**i*(i∈N^{∗}), and K, and defining
relations

[p* _{i}*,q

*] =*

_{j}*iδ*

*i,−*

*j*

*K,*[K,

*p*

*i*] = [K,

*q*

*i*] =0.

*In this representation, K acts by Id*F, and deg(q*i*) =*i*=−deg(p*i*). Its character is given by Eq. (1),
* so by the classical theory of Heisenberg algebras (see e.g. [K,*§9.13]), we obtain thatFis isomor-
phic to the irreducible representation ofHinB=C[x

*i*|

*i*∈N

^{∗}]

*in which p*

*i*=∂/∂

*x*

*i*

*and q*

*i*is the

*multiplication by x*

*i*. Note that we endowB

*with the unusual grading given by deg x*

*i*=

*i. Thus*we have obtained a canonical isomorphism between the fermionic Fock spaceF

*and the bosonic*

*Fock space*B

*. This is called the boson-fermion correspondence.*

**Exercise 5 Identify**B=C[x*i*|*i*∈N^{∗}]*with the ring of symmetric functions by regarding x**i*as the
**degree i power sum (see [Mcd]). Then the Schur function s**_{λ} is given by

*s*_{λ}=

### ∑

µ χλ(µ)x_{µ}/z_{µ},
where forλ andµ = (1^{k}^{1},2^{k}^{2}, . . .)*partitions of m,*

*x*_{µ}=*x*^{k}_{1}^{1}*x*^{k}_{2}^{2}· · ·, *z*_{µ}=1^{k}^{1}*k*1!2^{k}^{2}*k*2!· · ·,

andχ_{λ}(µ)denotes the irreducible character χ_{λ} ofS* _{m}*evaluated on the conjugacy class of cycle-
typeµ. Show that the boson-fermion correspondence F →Bmaps|λi

*to s*

_{λ}

**[MJD,**§9.3], [K,

§14.10].

**2.2.8** **Decomposition of**F

*Let us go back to the endomorphisms b*^{∞}* _{k}* =β

*nk*(k∈Z

^{∗})of§2.2.6. By Proposition 1, they generate a Heisenberg subalgebraH

*n*of EndF with commutation relations

[b^{∞}* _{k}*,

*b*

^{∞}

*] =*

_{l}*nk*δ

*k,−l*IdF, (k,

*l*∈Z

^{∗}). (9) Since the actions ofbgandH

*n*onFcommute with each other, we can regard F as a module over

*the product of enveloping algebras U(*bg)⊗U(H

*). LetC[H*

_{n}^{−}

*]denote the commutative subalgebra*

_{n}*of U(H*

*)*

_{n}*generated by the b*

^{∞}

*(k<0). The same argument as in §2.2.7 shows thatC[H*

_{k}^{−}

*]|/0iis an irreducible representation ofH*

_{n}*n*with character

### ∏

*k>1*

1

1−t* ^{nk}*. (10)

On the other hand, the Chevalley generators ofbgact on|/0iby

*e** _{i}*|/0i=0,

*f*

*|/0i=δ*

_{i}*i,0*|1i,

*h*

*|/0i=δ*

_{i}*i,0*|/0i, (i=0,1, . . . ,

*n*−1).

*It follows from the representation theory of Kac-Moody algebras that U*(bg)|/0iis isomorphic to the
irreduciblebg-module V(Λ0)with level one highest weightΛ0**, whose character is (see e.g. [K, Ex.**

14.3])

*k>0*

### ∏

*n−1*

### ∏

*i=1*

1

1−t* ^{nk+i}*. (11)

By comparing Eq. (1), (10), and (11), we see that we have proved that

* Proposition 2 The U(b*g)⊗U(H

*n*)-modulesF

*and V*(Λ0)⊗C[H

^{−}

*]*

_{n}*are isomorphic.*2 We can now easily solve Exercise 4. Indeed, it follows from Proposition 2 that the subspace of primitive vectors ofF for the action ofbgisC[H

^{−}

*]. Thus, for every k>0,*

_{n}*b*^{∞}_{−k}|/0i=

### ∑

*nk*

*i=1*

(−1)* ^{nk−i}*|i,1

*i*

^{nk−i}is a primitive vector. Here, (i,1* ^{nk−i}*) denotes the partition (i,1,1, . . . ,1)

*with 1 repeated nk*−

*i*times.

**2.3** **The higher level Fock space representations of**slb_{n}

**2.3.1** **The representation**F[ℓ]

Letℓ**be a positive integer. Following Frenkel [F1, F2], we notice that**bgcontains the subalgebra
bg_{[ℓ]}:=g⊗C[z^{ℓ},z^{−ℓ}]⊕C*c,*

and that the linear mapιℓ:bg→bg_{[ℓ]}given by

ιℓ(a⊗*z** ^{k}*) =

*a*⊗

*z*

*, ιℓ(c) =ℓc, (a∈g,*

^{kℓ}*k*∈Z),

is a Lie algebra isomorphism. Hence by restricting the Fock space representation ofbgtobg_{[ℓ]} we
obtain a new Fock space representationF[ℓ]ofbg. In this representation, the central element c acts
viaιℓ(c) =ℓc, therefore by multiplication by ℓ. SoF[ℓ]has levelℓ*and is called the level*ℓ*Fock*
*space representation of*bg. More generally, for every m∈Z we get from the Fock space F* _{m}* of

*charge m a level*ℓrepresentationF

*[ℓ]ofslb*

_{m}*.*

_{n}**2.3.2** **Antisymmetrizer construction**

The representationsF* _{m}*[ℓ]

*can also be constructed step by step from the representation V*(z)as we did in the caseℓ=1. This will help to understand their structure. In particular, it will yield an action ofslb

_{ℓ}

*of level n on*F

*[ℓ], commuting with the levelℓaction ofslb*

_{m}*n*.

**(a)** The new action ofbg*on V(z)*coming from the identificationbg≡bg_{[ℓ]}is as follows:

*e**i**u**k* = δ*k≡i+1**u**k−1*, *f**i**u**k* = δ*k≡i**u**k+1*, (16*i*6*n*−1),

*e*_{0}*u** _{k}* = δ

*k≡1*

*u*

*,*

_{k−1−(ℓ−1)n}*f*

_{0}

*u*

*= δ*

_{k}*k≡0*

*u*

*.*

_{k+1+(ℓ−1)n}Here,δ*k≡i* *is the Kronecker symbol which is equal to 1 if k*≡*i mod n and to 0 otherwise. For*
*example, if n*=2 andℓ=3, this can be pictured as follows:

· · · −→^{f}^{0} *u*_{−5} −→^{f}^{1} *u*_{−4} −→^{f}^{0} *u*_{1} −→^{f}^{1} *u*_{2} −→^{f}^{0} *u*_{7} −→^{f}^{1} *u*_{8} −→ · · ·^{f}^{0}

· · · −→^{f}^{0} *u*_{−3} −→^{f}^{1} *u*_{−2} −→^{f}^{0} *u*_{3} −→^{f}^{1} *u*_{4} −→^{f}^{0} *u*_{9} −→^{f}^{1} *u*_{10} −→ · · ·^{f}^{0}

· · · −→^{f}^{0} *u*_{−1} −→^{f}^{1} *u*_{0} −→^{f}^{0} *u*_{5} −→^{f}^{1} *u*_{6} −→^{f}^{0} *u*_{11} −→^{f}^{1} *u*_{12} −→ · · ·^{f}^{0}
*This suggests a different identification of the vector space V(z)* with a tensor product. Let us
*change our notation and write U :=*⊕* _{i}*Cu

*. We consider again*

_{i}*V* =
M*n*

*i=1*

Cv* _{i}*,

and also

*W*=
Mℓ

*j=1*

C*w**j*.
*Then we can identify W*⊗*V*⊗C[z,z^{−1}]*with U by*

*w**j*⊗*v**i*⊗*z** ^{k}*≡

*u*

*i+(j−1)n−ℓnk*, (16

*i*6

*n,*16

*j*6ℓ,

*k*∈Z). (12)

Now the previous action ofbg=slb*n**on the space U reads as follows:*

*e**i* = Id*W*⊗*E**i*⊗1, *f**i* = Id*W*⊗*F**i*⊗1, *h**i* = Id*W*⊗*H**i*⊗1, (16*i*6*n*−1),
*e*_{0} = Id* _{W}*⊗

*E*

*⊗*

_{n1}*z,*

*f*

_{0}= Id

*⊗*

_{W}*E*

*⊗*

_{1n}*z*

^{−1},

*h*

_{0}= Id

*⊗(E*

_{W}*−*

_{nn}*E*

_{11})⊗1.

But we also have a commuting action ofeg:=slb_{ℓ}given by
e

*e** _{i}* =

*E*e

*⊗Id*

_{i}*⊗1, e*

_{V}*f*

*=*

_{i}*F*e

*⊗Id*

_{i}*⊗1, e*

_{V}*h*

*=*

_{i}*H*e

*⊗Id*

_{i}*⊗1, (16*

_{V}*i*6ℓ−1), e

*e*0 = *E*eℓ1⊗Id*V*⊗*z,* e*f*0 = *E*e1ℓ⊗Id*V*⊗*z*^{−1}, e*h*0 = (*E*eℓℓ−*E*e11)⊗Id*V*⊗1.

Here, we have denoted by*E*e* _{i j}*the matrix units ofgl

_{ℓ}, and we have set

*E*e

*=*

_{i}*E*e

*,*

_{i,i+1}*F*e

*=*

_{i}*E*e

*and*

_{i+1,i}*H*e

*i*=

*E*e

*−*

_{i,i}*E*e

*.*

_{i+1,i+1}**(b)** From this we get as in§2.1.4 some tensor representations. Let r∈N^{∗}. We have
*U*^{⊗r}= (W⊗*V*⊗C[z,z^{−1}])^{⊗r}∼=*W*^{⊗r}⊗V^{⊗r}⊗C[z^{±}_{1}, . . . ,z^{±}* _{r}* ].

*This inherits from U two commuting actions of*bgand ofeg.

*We have endowed V*^{⊗r}⊗C[z^{±}_{1}, . . . ,*z*^{±}* _{r}*]with a left action ofSb

*in§2.1.5. Similarly, we endow*

_{r}*W*

^{⊗r}with a right action ofS

*, given by*

_{r}*w*_{i}_{1}⊗ · · · ⊗*w*_{i}* _{r}*·σ=ε(σ)(w

_{i}_{σ}

_{(1)}⊗ · · · ⊗

*w*

_{i}_{σ}

_{(r)}) (σ∈S

*), whereε(σ)denotes the sign of the permutationσ.*

_{r}**(c)** We can now pass to the wedge product and consider the representation∧^{r}*U , on which*bgand
egact naturally. It has a standard basis consisting of normally ordered wedges

∧u** _{i}**:=

*u*

_{i}_{1}∧ · · · ∧

*u*

_{i}*, (i*

_{r}_{1}>· · ·>

*i*

*∈Z).*

_{r}**Proposition 3 We have the following isomorphism of vector spaces**

∧^{r}*U*∼=*W*^{⊗r}⊗CS*r* *V*^{⊗r}⊗C[z^{±}_{1}, . . . ,*z*^{±}* _{r}*]
.

**Proof — Let 1**^{−}denote the one-dimensional sign representation ofS* _{r}*. We have

∧^{r}*U*∼=**1**^{−}⊗_{CS}_{r}*U*^{⊗r},

whereS_{r}*acts on U*^{⊗r}*by permuting the factors. Let A and B be two vector spaces. Then it is easy*
to see that

**1**^{−}⊗CS*r*(A⊗*B)*^{⊗r}∼=*A*^{⊗r}⊗CS*r**B*^{⊗r},

whereS*r* *acts on B*^{⊗r} *by permuting the factors, and on A*^{⊗r} by permuting the factors and mul-
tiplying by the sign of the permutation. Recall that the left action ofS* _{r}* on(V⊗C[z

^{±}])

^{⊗r}is by permutation of the factors, while the right action ofS

_{r}*on W*

^{⊗r}is by signed permutation of the factors. It follows that

∧^{r}*U*∼=*W*^{⊗r}⊗_{CS}_{r}*V*⊗C[z^{±}]⊗r∼=*W*^{⊗r}⊗_{CS}_{r}*V*^{⊗r}⊗C[z^{±}_{1}, . . . ,*z*^{±}* _{r}*]
.

2

**(d)** *The elements b**k*=∑^{r}*i=1**z*^{k}* _{i}* ∈CSb

*r*commute withS

*r*⊂Sb

*r*, hence their action on

*U*

^{⊗r}∼=

*W*

^{⊗r}⊗

*V*

^{⊗r}⊗C[z

^{±}

_{1}, . . . ,z

^{±}

*]*

_{r}by multiplication on the last factor descends to∧^{r}*U . Using Eq. (12), we see that it is given on the*
basis of normally ordered wedges by

*b**k*(∧u**i**) =*u**i*1−nℓk∧*u**i*2∧ · · · ∧*u**i**r* +*u**i*1∧*u**i*2−nℓk∧ · · · ∧*u**i**r* +· · ·+*u**i*1∧*u**i*2∧ · · · ∧*u**i**r*−nℓk. (13)
This action commutes with the actions ofbgandeg.

**(e)** *Finally, we can pass to the limit r*→∞as in§2.2. For every charge m∈Z, we obtain exactly
in the same way a Fock space with a standard basis consisting of all infinite wedges

∧u**i**:=*u**i*1∧*u**i*2∧ · · · ∧*u**i**r*∧ · · ·, (i= (i1>*i*2>· · ·>*i**r*>· · ·)∈Z^{N}^{∗}),
which coincide, except for finitely many indices, with

|/0* _{m}*i:=

*u*

*∧*

_{m}*u*

*∧ · · · ∧*

_{m−1}*u*

*∧*

_{m−r}*u*

*∧ · · ·*

_{m−r−1}We shall denote that space by F* _{m}*[ℓ], since it is equipped with a level ℓ action of bg, obtained,
as in§2.2.4, §2.2.5, by passing to the limit r→∞

**in the action of (c). It is also equipped with**commuting actions ofeg

*of level n, and of the Heisenberg algebra*H

_{nℓ}*generated by the limits b*

^{∞}

_{k}*of the operators b*

*of Eq. (13), which satisfy*

_{k}[b^{∞}* _{j}*,b

^{∞}

*] =*

_{k}*nℓj*δ

*j,−k*IdF

*[ℓ], (*

_{m}*j,k*∈Z

^{∗}). (14)

**2.3.3**

**Labellings of the standard basis**

It is convenient to label the standard basis ofF* _{m}*[ℓ]by partitions. In fact, we shall use two different
labellings. The first one is as in§2.2.1, namely, for∧u

**∈F**

_{i}*[ℓ]we set*

_{m}λ*k*=*i**k*−*m*−*k*+1, (k∈N^{∗}).

By the definition ofF* _{m}*[ℓ], the weakly decreasing sequence λ = (λ

*k*|

*k*∈N

^{∗})

*is zero for k large*enough, hence is a partition. The first labelling is

∧u** _{i}**=|λ,

*mi.*(15)

*For the second labelling, we use Eq. (12) and write, for every k*∈N^{∗},

*u**i**k*=*w**b**k*⊗*v**a**k*⊗*z*^{c}* ^{k}*, (16

*a*

*k*6

*n,*16

*b*

*k*6ℓ,

*c*

*k*∈Z).

This defines sequences(a*k*), (b*k*), and (c*k*). For each t=1, . . . , ℓconsider the infinite sequence
*k*^{(t}_{1}^{)}<*k*^{(t)}_{2} <· · ·,*consisting of all integers k such that b**k* =*t. Then it is easy to check that the*
sequence

*a** _{k}*(t)
1

−*nc** _{k}*(t)
1

, *a** _{k}*(t)
2

−*nc** _{k}*(t)
2

, . . . (16)

is strictly decreasing and contains all negative integers except possibly a finite number of them.

*Therefore there exists a unique integer m**t* such that the sequence
λ_{1}^{(t)}=*a*

*k*^{(t)}_{1} −*nc*

*k*^{(t)}_{1} −*m** _{t}*, λ

_{2}

^{(t)}=

*a*

*k*^{(t)}_{2} −*nc*

*k*_{2}^{(t)}−*m** _{t}*−1, λ

_{3}

^{(t)}=

*a*

*k*^{(t}_{3}^{)}−*nc*

*k*^{(t)}_{3} −*m** _{t}*−2, . . .
is a partition λ

^{(t}

^{)}. Let λ

_{ℓ}= (λ

^{(1)}, . . . ,λ

^{(ℓ)}) be the ℓ-tuple of partitions thus obtained, and let

**m**

_{ℓ}= (m

_{1}, . . . ,

*m*

_{ℓ}). The second labelling is

∧u** _{i}**=|λ

_{ℓ},

**m**

_{ℓ}i. (17)

* Example 1 Let n*=2 andℓ=3. Consider∧u

**i**given by the following sequence

**i**= (7, 6, 4, 3, 1,−1,−2,−3,−5,−6, . . .)

containing all integers6−5. It differs only in finitely many places from the sequence (3, 2, 1, 0,−1,−2,−3,−4,−5,−6, . . .).

*Hence m*=3 andλ = (4,4,3,3,2,1,1,1).

For the second labelling, we have (use the picture in§2.3.2 (a))

*u*7 = *w*1⊗*v*1⊗*z*^{−1}, *u*6 = *w*3⊗*v*2⊗*z*^{0}, *u*4 = *w*2⊗*v*2⊗*z*^{0},
*u*_{3} = *w*_{2}⊗*v*_{1}⊗*z*^{0}, *u*_{1} = *w*_{1}⊗*v*_{1}⊗*z*^{0}, *u*_{−1} = *w*_{3}⊗*v*_{1}⊗*z*^{1},
*u*−2 = *w*2⊗*v*2⊗*z*^{1}, *u*−3 = *w*2⊗*v*1⊗*z*^{1}, *u*−5 = *w*1⊗*v*1⊗*z*^{1},

. . . .

Hence the sequences in Eq. (16) are

3,1, −1,−2,−3, . . . (t=1), 2,1, 0,−1, −2, . . . (t=2), 2,−1,−2, −3, . . . (t=3).

**It follows that m**3= (1,2,0)andλ_{3}= ((2,1),/0,(2)).

**Exercise 6 Prove that m**_{1}+· · ·+m_{ℓ}=*m. Show that the map*(λ,m)7→(λℓ,m_{ℓ})is a bijection from
P×ZtoP^{ℓ}×Z^{ℓ}.

**2.3.4** **The spaces F[m**_{ℓ}]

DefineZ^{ℓ}(m) ={(m_{1}, . . . ,m_{ℓ})∈Z^{ℓ}|*m*_{1}+· · ·+*m*_{ℓ}=*m}. Given m*_{ℓ}∈Z^{ℓ}(m), let F[m_{ℓ}]be the
subspace of F* _{m}*[ℓ] spanned by all vectors of the standard basis of the form |λ

_{ℓ},m

_{ℓ}i for some ℓ-tuple of partitionsλ

_{ℓ}. Using Exercise 6, we get the decomposition of vector spaces

F* _{m}*[ℓ] =

^{M}

**m**_{ℓ}∈Z^{ℓ}(m)

**F[m**_{ℓ}]. (18)

Moreover, since the action ofbg *on W*⊗*V*⊗C[z^{±}]involves only the last two factors and leaves
**W untouched, it is easy to see that every summand F[m**_{ℓ}]is stable underbg. Similarly, since the
*action of the bosons b**k*involves only the factorC[z^{±}_{1}, . . . ,z^{±}* _{r}*]

*in U*

^{⊗r}=

*W*

^{⊗r}⊗V

^{⊗r}⊗C[z

^{±}

_{1}, . . . ,z

^{±}

*],*

_{r}**each space F[m**

_{ℓ}]is stable under the Heisenberg algebra H

*. Therefore Eq. (18) is in fact a*

_{nℓ}*decomposition of U(*bg)⊗U(H

*)-modules.*

_{nℓ}**The space F[m**_{ℓ}]*is called the level*ℓ* Fock space with multi-charge m*ℓ.

**Remark 1 When**ℓ=

**1, F[m**1] =F

_{m}1 *is irreducible as a U*(bg)⊗*U(H**n*)-module (see Proposi-
tion 2). But forℓ >**1, the spaces F[m**_{ℓ}]*are in general not irreducible as U(*bg)⊗*U(H** _{nℓ}*)-modules.