Representation theoretic relations between Schur polynomials Текст научной статьи по специальности «Математика»

REPRESENTATION THEORETIC RELATIONS BETWEEN SCHUR POLYNOMIALS

G. F. Helminck, P. C. J. van de Heuvel University of Twente, The Netherlands

Dedicated to Prof. Dr. D.P. Zhelobenko at the occasion of his seventieth birthday

1. Introduction

In this paper we study level one integrable highest weight representations of some Kac-Moody Lie algebras g(k\ and the corresponding vertex operator constructions. We use the technique of fermionic operators, which is developed in e.g. [1] and [2].

A Heisenberg subalgebra (HSA) s plays an important role for the vertex operator representation of a Lie algebra g^k\ A HSA is an algebra on a basis z>0 and the canonical

central element c with commutation relations [p¿,g¿] = $ijC. If V is a representation of such that for all v 6 V there exists an N such that pi1 .. .pi¡v = 0 whenever i\ +... + i/ > N, it is completely reducible with respect to the action of s. The only irreducible ¿-module satisfying this condition is the ring of polynomials C[x] with the assignments p-L —> d/dxi, qi —> X{, c —> 1. Therefore V can be identified with V+ <S> C[ar] where the vacuum space V+ is defined by the set of all vectors which are annihilated by the pt.

In this way one obtains for every HSA s a realization for a given ^W-module V. Such realizations can look very different, which is exemplified by the principal and homogeneous realization of the basic representation of the simplest affine Lie algebra si2.

This paper is organized as follows: In section 2 we recall the prerequisites about the Lie algebras Ax and gln and their representations on the infinite wedge space that will be needed in the sequel. Each choice of a HSA leads to a decomposition V ~ C[a:] <2>F+. In section 3 we look at several of those decompositions. In a number of cases, the main one being the partition n = {n\, «2}, we compute directly the isomorphism of linear spaces 7 : C[a;W] <2> V+ —> G[a.'j, where C[x] corresponds with the principal realization. This gives relations between Schur polynomials. Further we look at the principal degree of the image of a polynomial which culminates in a q-dimension formula.

2. Prerequisites

2.1. The Lie algebra gloo- The associative Lie algebra gl^ is the collection of Z x Z-matrices defined by

(1) gl0o = {(o¿j)ijeZ I aij £ C, all but a finite number of the a¿j are 0}

with the matrix commutator [A, B] — AB — BA as Lie bracket. Denote by £ij the matrix with 1 as the (*, j)**1 entry and all other entries equal to zero. The £ij form a basis for gloo. Let C°° = (BjezCv(j) be an infinite dimensional complex vector space with basis {v(j) | j 6 Z}.

l

The Lie algebra gl& acts on C°° by

(2) Sijv(k') — Sjkv(i)

The group GLqo associated with the Lie algebra gloo is defined as

(3) GLoo = {Id +A | А є gloo, det(Id +A) ф 0}

In order to include the infinitesimal generators of certain flows one passes to the extension gloo of the Lie algebra gloo- It is given by:

(4) gloo = {(aij)ijez І ац = 0 for |і - j\ » 0}

Matrices in gloo have a finite number of nonzero diagonals. The product of two matrices in gloo is well defined, and is again in gloo) so gloo is a Lie algebra containing дІж as a subalgebra. The g/oo-action on C°° extends naturally to an action of gloo-

2.2. The infinite wedge space. Let A^C00 be the vector space with a basis consisting of all semi-infinite exterior products of the basis elements v(k) of C°° of the form:

(5) v(io) A A u(i_2) A ...

such that ¿o > i-\ > 2 > ... and i-i-i = i-i — 1 for I 0. The space A^C00 is called the

infinite wedge space.

One can distinguish the basis elements (5) by their behaviour at large I. An element of the form (5) has charge k if = k — I for all I 0. For instance the vector

Ik) := v(k) A v(k — 1) A v(k — 2) A ...

has charge k. The vector \k) is called the k^1 vacuum. The vector space of all vectors of charge k is denoted by . One has a decomposition of the infinite wedge space in sectors of fixed charge:

лоосоо = 0 F(k) fc€Z

For every k Є Z one defines linear operators ф(к) and ф*(к) on the infinite wedge space by their action on the basis vectors:

(6) ф(к) (v(io) A v(i-1) A v(i-2) A ...) = v(k) A v(io) A v(i-1) A v(i-2) A ...

OO

ф*{к) (и(г0) A v(i-1) A v(i-2) A ...) = ^(-1 )1^к,і_^(іо) A v(i-i) A ... A v(i-i) A ...

1=0

where the notation v(i-i) means that the vector is deleted. These operators satisfy

the anticommutation relations:

(7) {ф(к),ф{1)} = 0 = {Г(к),Ф*(1)} and {ф(к),ф*(1)} = 5kl where the anticommutator {A, B} is defined by {А, В} := AB + BA.

Any element of the infinite wedge space A°°C°° can be written as a finite linear combination of elements of the form

(8) ■ф(к1)---'ф(кг)ф*(11)-.-Г(Ш

where k\ >... > kr > 0 > li >... > ls- This means that one could also have constructed the space A^C00 in a different manner. Namely, let Cl be the Clifford algebra on generators

ip{i), i£Z with relations (7). Define the socalled fermionic Fock space F as the unique irreducible C7-module, which admits a vacuum vector |0) such that

(9) ip(i) |0) =0 V i < 0 and i/>*(i)|0) =0 V i > 0

Then we have F = A^C00. The fermionic Fock space F is also called the spin representation of Cl.

The space A^C00 can be equiped with an inner product ( , ), which is uniquely determined by the requirements

(1°)) |0>) = 1 and = ip*{k),

where denotes the adjoint of a linear operator A on A^C00 w.r.t. the inner product ( , ). Then the elements (8) have length 1. One defines

(10) <0|A|0):=(|0),A|0»

where A is a linear operator on A°°C“. The quantity (10) is called the vacuum expectation value of A. Sometimes one abbreviates it to (^4).

2.3. Representations of gl^ on A^C00. One can define representations p of gl^ and the corresponding one R of GL^ on A^C00 by

p(a)(v(io) A •u(*_i) A v(i-2) A ...) := av(io) A t»(«_i) A v(i-2) A ...

(11) -H>(j'o) A av(i-i) A v(i-2) A ...

+u(io) A v(i_i) A av(i-2) A ... + ...

(12) R(A)(v(io) A v(i-i) A «(¿-2) A ...) := Av(iq) A Av(i-i) A Av(i^2) A ...

The action of the elements £ij can be written as p(£ij) = V,(*)V’*(i)-

The submodule F^ is an irreducible highest weight module for the algebra gl00 ■ One has for j > i

(13) p(£ij)\k) = o

and

(14) p{£u)\k) = 9k(£ii)\k) where the linear mapping 9k : ©,;gzCfri —> C is defined by

(15) 9k(£u) =

0 if * > k 1 if i < k

It is not possible to extend the representation p to the extension gl00 by linearity. For example the identity matrix Id is an element of 3/00 • Its action on the vacuum vector |0) is p(ld)|0) = oo|0), so it is not well defined. A remedy for this problem would be to subtract a term from p and to define

(16) 7r(£ij) := p(£ij) - 6ij6o(£u)I.

For each A = Yhij aii£ij € gloo the operator tt(A) = . otij^{£ij) is well-defined. However,

it is not a representation of gl^ anymore since

(17) br(£»i), *■(£«)] = [p(€ij),p(£ki)}

= &jkp(£il) &liP{£kj)

= $li'K(£kj) “1" (£ii £jj)

— ir(\£ij, £hl\) + £jj)-

This additional term determines a 2-cocycle p : gl00 x gl^ —> C by the bilinear extension of

(18) K£iji£ki) ■= ¿¿fcMo(£i - £jj).

The 2-cocycle p, determines a central extension A^ := gl^ © Cc of gl^ with the Lie bracket on A0o given by

(19) [A + ac, B + /3c] = AB — BA + p(A, B)c A,Be gloo, q,|0 6C.

If one defines ir{A + ac) — tt(A) + aid then 7r is a so-called c = 1 faithful representation of Aoo and one writes gA 7t(Ax>)- 9A-

The representation 7r can be expressed in terms of the fermions ip and ip*. One has ir(£ij) =:

ip{i)ip*{j) where the normal ordering : ip{i)ij}*(j) : is defined by

(20) : ip{i)ip* (j) : := ~ <0|^(*)^*0‘)|0)

_ ii j > 0

1 —ip*(j)'ip(i) otherwise

2.4. The principal degree. On one assigns a degree to a monomial of the form (5) by

00

(21) deg(v(i0) A v(i-1) A v(i-2) A ...) = + s)

s=0

Then the degree is a finite nonnegative integer. This degree is called the principal degree. Let F^ denote the linear span of all vectors in F^ of degree I. Then

and dimqF^ := ^(dimi7^)^ = ,

i> o l

where tp(q) = Ilfc>o(1 -9fc)-

One can express the degree in terms of the ip(i) and the ip*(_j). If one put deg 'ip(i) = i, degip*(j) = —j and deg(|0)) = 0,then the degree of (8) is given by ki + ... + kr —1\ — ... — ls. This degree coincides with the degree above on F. Define

(22) Ho = J2k:^(kW(k):

ke Z

Then [Ho,tl>(k)] = k'ip{k), [Ho,xp*(k)] = —kip*{k) and i/o|0) = 0. The operator Hq is called the Hamiltonian or Energy operator. Its eigenvalues are the degrees of the eigenvectors.

2.5. The oscillator algebra. Define the shift operators Mk : C00 i-> C°° by Mkv(j) = v(j — k). Then the corresponding matrices G gl<*> are

(23) Ah — £jj+k

jez

They form a commutative subalgebra of gl^. It is a straightforward verification that /¿(A*;, Ai) — k6k+ito so the Afc have in A00 the Lie bracket

(24) [Afc, A;] = kSk+iflC

Then (©tezCA/J © Cc is a subalgebra of Aqq, the so-called bosonic oscillator algebra A. Define a(k) := 7r(Afe), then

(25) a(k) = ■ tPtiWij + k) :

jez

They satisfy [a(k),a(l)] = kSk+ifi- The operator a(0) is called the charge operator, its eigenvalues are the charges: a(0)|F(io = k • I. There holds a(fc)|0) = 0 if A; > 0. From [Ho,a(k)] = —ka(k) one sees that a(k) has principal degree —k.

There exists a standard representation of A in the space of polynomials in infinitely many variables xk (k > 1). It is given by:

d

a(k) —> ——,a(—k) —> kxk, for k > 1, a(0) —> ¡j,Id, c —> Id oxk

In this space C[s] = C[sx, x2, 23,...] one has

(26) deg(xA;) = k

2.6. Vertex operators. One defines tp(z) := Y^,ip{k)zk, ip*(z) := Yl^ityz fc> where z is a formal parameter. The ip(z), ip*(z) are generating operators for the ip(k), ip*(k). They are called fermionic fields. These fields can be expressed in terms of the bosons a(k). The fields are eigenvectors for the adjoint action of the a(k):

[a(k),tp(z)] = zkip(z) and [a(k),ip*(z)] = —zkip*(z)

There is a well-known expression for the fermions in terms of the bosons (see e.g. [4]):

(27) = Qza^+1E-{z)E+{z)

*(z) = Q-1z~aWE-(z)-1E+(z)-1,

where

E~(z) = exp [ — \oi{k)z~k J and E+(z) = exp ( — ^-a(k)z~k J

V fcco / V fc>o J

and Q : F^ —> _F(fc+1) is an operator satisfying

(28)

Qif>(z) = z 1ip(z)Q and Q|0) = V’(l)lO)

Qil>*{z) = zx!>*{z)Q Q_1|0) = -0*(O)|O)

The operator Q commutes with all bosons a(k) except for k — 0:

(29) [a(k),Q] = Sk0Q

One can look at the generating operator for the action of on :

)ukv 1

(30) X{u,v) := ^2 'K{£kl)i

k,lez

= ^ : ip(k)ip*(l) : ukv~l =: ip(u)ip*(v) :

k,leZ

Now one uses equations (27) and : tp(u)‘ip*(v) := ip(u)ip*(v) — {0\ip(u)ij)*(v)\0) and obtain

(11 / n\ ^(0) 1

(31) X(u,v) = U-L-—E-{u)E-{VrlE+{u)E+{v)-1--------------------------¡-I

1 — v/u 1 — v/u

(32) = T^T exp (s \ - »>(-*>)

' \k>0 /

exp (- Y.£<“'* - ’'-*)«(*)] - Y^J-UI

\ k>0 / '

Here is a formal power series in u and v: [•••-■- := J2k>o(v/u)k ■ Thus one sees that the

action of the algebra A^ on the infinite wedge space can be completely expressed in terms of the action of the subalgebra A of all oscillators. Combining this with the fact that the charge k sector is an irreducible A^-module, one concludes that F^ must remain irreducible under the action of this oscillator algebra.

2.7. Schur polynomials. The elementary Schur polynomials Sk(x) 6 C[.x] are defined by the generating function

(33) 5>(,)^exp

kez \k>0 J

Then

(34) Sk{x) =0 for k < 0, So{x) = 1

(35) sk{x)= J2 forfc>0

fcl+2fc2 + ...=fc

One denotes the set of partitions by Par. Thus A £ Par is a nonincreasing finite sequence of positive integers Ai > A2 > • • ■ > Ar > 0. In the sequel also a different notation will be used: the integers are labeled by k < 0, and the sequence is extended with zeros; so we have

Par ~ {(Ak) | k < 0, 3N : p > N A_p = 0}.

To each A € Par one associates the Schur polynomial S\(x) defined by the determinant

(36) S\{x) := det{Sxi-i+j(x))

With respect to the principal gradation on A°°C°° the Schur polynomial S\(x) is a homogeneous polynomial of degree |A| := Ai + A2 + — The (S\)\<=par form a basis of C[x].

2.8. Boson-fermion correspondence. One has an isomorphism of .-4-modules

(37) C[x]=C[xux2,...]~{a{-kl)...a(-kr)\0) \ h > 0} =

The isomorphism a : F^ —> C[:r] is called the boson-fermion correspondence.

Every F№ is isomorphic with C[x]. The factor Qk indicates the charge sector F^k\ Then a : /\°°<C°° —> C[x', Q, Q-1] is characterized by (k £ Z, I > 0)

1

cr(lfc)) = Qh,cra(l)<j~1 = ——,cra(—= Ixi, aa(0)a~l = Q—,crldcr_1 = Id.

oxi dQ

Define

(38) #(*):= ]Ta:*a(fc)

fc>0

Then is well defined on A^C00. On the charge zero sector a is given by (see [2] or [1])

(39) cr(A|0)) = (0|eff(xU|0)

In general it is given by

(40) a(A|0)) = Y, = Y,

k€Z kez

One has the following expressions for the transported operators

r_(z) := oE (z)a 1 = exp

and

= exp 1 xb

\k>0

= exp 1 ¡Д>‘

\fe>0

)E+{v)-1a~1

= exp(J2(uk-«*)**!exp(-\(u K~v *ydXkj

/ \ k>0 /

In [3] the following theorem is proved:

Theorem 2.9. For v(io) A v(i-±) A u(i_2) A ... £ there holds

(41) a(v(i0) A v{i-1) A v(i-2) A ...) = S'j0,i_1+i,i_2+2,...(z)-

2.10. The Kac-Moody algebra gln. Let gln denote the Lie algebra of all complex n x n-matrices with the standard basis (^¿j)i<ij<n- One defines the loop algebra gln as

(42) 9ln:=@tk9ln

with commutation relations [Atk,Btl] = [A,B]tk+l. The loop algebra gln acts in a natural way on C[t,t~l]n. Let be the standard basis of Cn. Define unk+j := t~kej. Then

the {uk)kez form a basis of C[t,t 1]" over C. Thus C[t,t 1]n ~ C°°. The Lie algebra gln embeds into gl^ by the Lie algebra homomorphism l : gln —> gl^ determined by

(43) i{t Eij) ^ ' £n(s—k)+i,ns+j

seZ

The image of gln in g/oo consists of

= {(®ij) ^ dlOO | ®i+nj+n = Qij}‘

By restricting the 2-cocycle ¡i to i{gln) one gets a 2-cocycle for gln. It is given by n(i(tkA), i{tlB)) = kSk+lfiTr(AB) =: kSw(A\B)

and determines a central extension gln of gln. This Lie algebra is called the affine Kac Moody algebra associated to gln. It will be viewed as a Lie subalgebra of A^.

Define E G gln by E := Y?k=i Ek,k+1 + Eln. Then

(44) hp := © CEk

l<k<n

is a Cartan subalgebra (CSA) of gln, the principal CSA. This E induces the element E = J2)c=i Ek,k+i + Emt G gln (see e.g. [3]). It is easy to show that i(E ) = A*, k G Z.

We now have a representation tt o i of gln in A^C00. The representation is irreducible under gln since the latter contains the Aj (j G Z) and F^ is irreducible under the action of A. The action of gln in F^ is given by the vertex operator

(45) : ip(uj~ku)'if>*(u)~lu) : 1 < k,l < n

where cj = (see [4]).

3. Isomorphisms of Schur polynomials

3.1. Fermions with various components. It is well-known that the conjugacy classes in W(gln), the Weyl group of gln, are parametrized by partitions of n. Any partition n of the number n in s parts ni, «2,..., ns determines a direct sum decomposition Cn ~ C”1 © • • • © C"s and an associated block decomposition ofanx n-matrix. The diagonal blocks correspond to Lie algebras glni and the principal construction of subsection 1.9 tells us how to make vertex operators describing the action of the affine algebra gln.. So one just takes s copies of the construction above or, which is the same thing, one should work with s-component fermions ?pi(k), ip*{k), 1 < i < s, k G Z. The problem is how to find vertex operators associated to the off diagonal blocks.

A partition n of n leads to s-component fermions:

(46) -ipi(l + mrii) := ip(ni + ... + rii-i + I + mn)

'ip* (I + mrii) := ip* («i + ... + i + I + mn) KKn^mGZ

These fermions satisfy the relations

{#),«l)} = WWil(!)} = 0

(47) {&(*), w} = Mjm

In terms of these fermions the spin module A^C00 can also be defined as the unique Cl-module generated by a vacuum |0) satisfying

ipi(k)\0} = 0, k < 0 Vi and tp*(k)\0) = 0, k > 0 Vi.

Now one defines

(48) <*(*):= £:^(0#(J + *):

lez

Then one obtains in the same way as in the one-component case

(49) ipi{z) = QiZaiW+l exp k ) exp

V k<0 J V k> 0

(50) fâ(z) = Q~lz~ai^ exp f Y2 exP f X/ j:ai(h)z~*

\fc<0 / \fc>0

where the operators Qi satisfy

Qi^iik) — ipi(k + 1 )Qi

Qitiik) = + l )Qi

(51) Qiipj(k) = “ÿjityQi i ^ j

Qitfik) = -4>*(k)Qi i±j

Qi\o) = MW)

{Qi, Qj} = 0 ijij

The basic representation L(Aq) of gloo is isomorphic to F(°). The vacuum space V+ of the

^/„-module Fis spanned by the vectors

(52) r-Cf110), ffljEZ

where Tj := QiQ~+j, 1 < i < n — 1 (see [4]).

3.2. Two types of fermions. Let n be a positive integer. Here we look at partitions of n in two parts: n = ni + n2 where ni,n2 € Z>o- Let {v(k))kez be a basis of C°°. We can relabel this basis with respect to the partition of n mentioned above. We then get:

vi(mni + I) = v(mn + I) m £ Z, 1 < I < n\

V2{mri2 + l) = v(mn + n\ + I) m 6 Z, 1 < I < n2

We can also relabel the fermionic operators ip(k) and ip*(k) in the same way. These relabeled operators then have the anticommutation relations

= = o

{ipi{k),i>j{l)} = Sij8ki i,j = 1,2

These are the anticommutation relations for fermionic operators of two different types. These relabeling leads to the following isomorphism

C[x] ~ F^ ~ F{0) ® F2(0) ® C[T,T-1] ~ C[a;(1\a;(2),T,T-1]

Here T is the operator QiQj1 • It creates a fermion of type 1 and annihilates a fermion of type 2.

It is convenient to define the following functions:

k- 1

fi{k) :=

ni

n2 + k and f2{k) :=

k - 1

n2

ni + ni + k.

Here is k G Z. These functions satisfy the relations Uj(/c) = v(fi(k)) where i — 1,2. Denote by 7 : C[x^^,x(2),T,T_1] —>■ C[x] the above mentioned isomorphism. A basis for C[x^\a;(2\T, T-1] is given by TkS^(x^)S\(x^), where A ,/i 6 Par. We now want to compute the images of the basis elements in C[x]. For k E Z we define kt G Z, 1 < kL < ni by k = kt mod nt. Then we are ready for the following Theorem 3.3. For k > 0, k = mn2 + k2 we have

(53)

where

7 (T^xW)^2))) = (-l)c+5^i/iiA)(x)

c+ = 7p-k2 - i(m + l)fc + ^-k2{n2 - k2)

¿n2 I Zn2

Ar k r — 1 ni

n2 + Xr + k

For —k — mn\ + 1 < r < 0 we have

7 =

= h{K + k + r)-r For p < 0,1 < q < ni,pn + q < —k — mn\ — —mn — k2 we have

A—k+pni+q "t" Q 1

n 1

7 (pn + q,n,\) =

= fl{^-k+pni+q + Q) ~ Q

For p < 0,1 < q < n2,pn + q + n\ < — k — mn\ = —mn — k2 we have

n2

^1 “t" (J>k+pn2+q

y{pn + q + ni,n,\) =

= f2^Pk+pn<2+q q) — q — n 1 In the case that the power ofT is negative we have (k > 0, k = mn\ + k\):

where

7 (T-‘S„(x<l>)SA(*<2>)) = „,»(*)

c_ = ^-k2 + ^(n2 - l)k + ~fci(ni - fci)

Z77-1 Z

For —mn — ki — n2 + 1 < r < 0 we have

Hr + k + r — 1

7(r,//,A) =

= /2(A»r + fc + r) -r For p < 0,1 < q < n\,pn + q < —mn — n2 — k\ we have

n\+ nr + k + ni

7(pn + g,M, A) =

k+pni+q "t" Q 1

ni

n2 + A,

k+pni+q

fl(^k+pni+q + Q)

For p < 0,1 < q < ri2,pn + q + n\ < —mn — k\ — n2 = —(m + 1 )n — k\ + ni we have

j(pn + q + ni,n,X) =

^■1 "I" p—k+pn2+q

n2

= ^2 (/-*—k+pn2+q O) Q ^1

These last formulas are also valid for k — 0. Then we have to take m = — 1 and k\ = ni.

Proof. The formulas have been calculated following the isomorphism <C[a;(1),a;(2),T, T-1] —> Fj0^ ®F2(0) <8>C[T, T_1] —» F^ —> C [:/,'] step by step. We will sketch the proof for a positive power of T, the proof in the other cases is similar.

TkS„(x^)Sx(xW)

—► Tk {... tpi(X-p - pWli-p) ■ ■ • ^i(Ao)^(0)}

{... ip2(n-q ~ q)$z{-q) • • - '02(Mo)'02(°)} |0)

= {■ • ■ i>i(A-p -p + k)ipl(-p + k)...il;i{Xo + k)^\(k)}

{... ip2{li-q -q- k)ip 2 (~q - k)... '02 (/^0 - k)i>2 (-*)} T*|0)

= {... V»i (A-p -p + k)^\ {-p + k)...ip i (A0 + k)ipl (k)}

{... ip2(p-q -q- k)ip2{-q -A:)... Ip2{fi0 - k)ip*2{-k)}

{^lik^i-k + 1)... ipiir^i-r + 1). ..^(1)^(0)} |0>

= {. ..^(A _p_fc -p)V»x(-p)...^i(A_fc)V>i(0)}

{... i>2 (fi-q ~q~ k)i>2(-q - k)... ip2(ii0 - k)^(-k)}

{^i(Ao + k)ip2{—k + 1).. .ipi(X-k+r +r)^l{-r + 1).. .ipi(X_k+i + l)-02(°)} 1°)

= «i(A_/t+i + 1) A ... A Vi(X-k+n2 +n2) Aui(A_fc) A ... A vx (A_fc_„1+i - nx + 1) A ...

Nv\ (A_i+1 + mn2 + 1) A ... A v\ (A0 + A:)

Av2{hq -k) A ... A v2{[i-n2+i+1 - (m + l)n2 + 1)

Aui(A_fc_mni - mni) A ... A vi(A_fc_(m+1)„1+1 - (m + l)rti + 1)

Av2(n-n2+i - (m + l)n2) A ...

Now putting the factors A in decreasing order and using the boson-fermion correspondence a we obtain formula (53). □

To illustrate the result one looks at a concrete example Example 3.4. One considers here the simplest case n = 2 and n\ = n2 = 1. Now we get rid of the entier functions. fi(k) = 2k — 1 and f2(k) = 2k, k\ = k2 = 1. Then we get (k > 0):

c+

k2 — k = 0 mod 2 7k{r) = 2Xr + 2k + r — l (—2k + 2<r<0)

7fc(2p + 1) = 2A_fc+p+i (p < —k)

7fc(2(p + l)) = 2nk+p+1 (p < —k)

c_ = k2 = k mod 2

7-k(r) = 2/j,r + 2k + r (—2k + 1 < r < 0)

T—ki^P + 1) = 2Afc_|_p_)_i (p < — k)

7_fc(2(p + l)) = 2fj,-.k+p+i (p<~k)

We thus have

yk ~ (2Ao + 2k — 1,, 2A_j + 2k j 1,..., 2A_2fc+i, 2/j,q, 2A_2&; ■ • • 12//_;, 2A_2k—h • • •)

7-fc = (2/^o ~t" 2&, • ■., 2¡j,-j + 2k — j,..., 2/i_2fe+i + 1,2/i_2fc, 2Ao, ■.., 2(j,^2k-h 2A_;,...)

In this case we can find a formula for the degree of this polynomial. Remember that on C[x] we have the principal degree deg. Then we easily derive that

(54) deg(-y(Tk SIJi(x^)Sx(x^))) = 2|M| + 2|A| + 2k2 — k

for all k G Z. Here |//| := Ylp<o Ppi ^ie principal degree of . In the general case it is not so easy to find a nice expression for the degree of the image polynomial in terms of the principal degrees of the original Schur polynomials and the integer k. Therefore we first look at 7(Tk). For k > 0 we have

(55) %o,o(r) = fi(k + r) -r = —- n2 + k

for —k — mni + 1 < r < 0. For k < 0 we get a similar formula. A straightforward calculation now gives:

(56) deg{j(Tk)) = nk(nk ~nin2) +n2kx{ni - fci) +nffc2(n2 - k2)

This formula is valid for fceZ.

3.5. The degree of a polynomial. The principal degree of a polynomial in C[x] corresponds in Fwith the eigenvalues of the operator Ho where

(57) H0 = J2k:^kWW:

keZ

This operator has nonnegative integer eigenvalues. It is an element of ft igloo) namely Ho = We have a(k) = Xwez • + k) These a(k) are bosonic oscillators.

The commutator of a(k) with Hq gives: [Hq, a(k)] = —ka(k). This means that we can assign a degree —k to a(k), because a(k) changes the eigenvalue of Ho with —k. The principal degree of xk can then be computed using the isomorphism between C[.x] and F^h This isomorphism gives dcg(i'fc) = k. We can also look at the oscillators oti(k) which are defined by

(58) ai(AO:=£:V’i(0#(i + *):

l€l

Now we can try to compute the commutator of Ho with oti(k). Using

2

Ho = : i’A'py^ip):

j=1pez

we get the following formula

(59) [H0,ai(k)] = Y^Uiip) ~ fiip + k))^iipWiiP + k)

The right hand side is in general not proportional to ai(k), so we cannot assign a principal degree to cti(k). If k = nil then we can assign a degree to it because then [Hq, «¿(«¿i)] = —nlai(nil). So deg(aj(ni/)) = — nl and deg(x^) = nl. In the example we have n\ = n2 = 1, so every k is of the form k = nil. So in this case we have deg(ai(A;)) = —2k and deg(x^) = 2k.

In [4] another degree has been used. That degree corresponds with the eigenvalues of the operator Dq defined by

2 2

(60) Dq := + 0)2 + hHn\21

" \ * / ® V* / 1 rt / ,

n; Z ' Jlj

p>0i=l 1 i=l 1

2 2

(61) = : Mp)i>i(p) ■

/Tit Z I hi ¿i

i=l p£Z i=l

Not all eigenvalues of Do are integers. Therefore we multiply the eigenvalues with a constant N. Call the corresponding degree deg0. Then you can assign a degree deg0 to a,(/c): deg0(a!i(fc)) = —iV~ and deg0(x^) = N^. It is not a surprise that in general one cannot assign a degree deg0 to a(k). One only has deg0(o;(nZ)) = —Nl and deg0(a;ni) = N1. In the next subsections we consider the homogeneous realization.

3.6. The case 3=1+1+1. In this subsection we present some formulas for the isomorphism

7 in the case 3 = 1 + 1 + 1. In this case we have the two operators T\ and T2 which

anticommute.

Theorem 3.7. For k2 > 2ki >0 we have

7 {TklT$2) = )s^(x)

= (—1) 2kl(kl~^+l(k2~kl)(3k2+kl~3) S*j(x)

where

7(3p) — —6p +1 0 > p > —k\

7(3p) = —6p — Ski + 2 — ki >p> —k2

7(9) = 0 otherwise

7 (p) = 3(A;2 - ki) - 1 + 2p 0 >p> -k2 + 2ki 7 (-k2 + 2ki+2p) = k2 + ki - 1 + p 0 >p> ~k2 - kx

7 (~k2 + 2ki + 2p-l) — k2 + ki~l+p 0 >p> ~k2 - kx

7 (q) = 0 otherwise

deg(7{T^Tk2)) = 3kl + 3k22 - 3kik2 -h-k2

Proof. TklTk210) = (—1)C*i-1)(Tir2)fcir*a-fci|0>.

(TiT2)*1T*2~A!l|0)

= {TiT2)kl (tp2(k2 - ki)p3(-k2 + h + l)... ^(1)^(0)) |0)

= (-1)^^ {Mh - ki)tpl(-k2 + 1)... ih(i)P3(-h))

{Mk№(-ki + l)-.-MW№)\0)

= (—l)fci(fc2—fcl)^i(1) A u2(0) A Ui(0) A ... A vi(ki) A v2{—kx + 1) A vi{—ki + 1) A

^2(1) A v2(-ki) A vi(-ki) A ... A v2(k2 — ki) A v2(-k2 + 1) A vi(—k2 + 1) A

V3(~k2) A v2{-k2) A vi(-k2) A ...

(_1)fci(fc2-fci)s-.(x)

Putting the Vi(k) into decreasing order we find the second formula. The degree is found summing the indices of the Schur polynomial Sj. □

These formulas are only valid for k2 > 2k\ > 0. It is easy to calculate the formulas in all other cases, but that is left to the reader. The formulas for the degree of Tkl and can easily be calculated using the isomorphism 7. We then obtain

deg(7(rf1)) = 3k\ - k\ and deg(7(T2fc2)) = 3A:| - k2.

These formulas are valid for k\,k2 6 Z.

3.8. Principal degree in homogeneous realization. In previous sections we generalized the case 1 + 1 = 2 to n\ + n2 = n. In this section we generalize it to 1 + 1 + ... + 1 = n, the homogeneous realization. Here we have the isomorphism

71

(62) C[xM,..., rcW] 0 qrn] ~ ((g) F,f) ® C[fn] ~ F™ ~ C[x]

i=1

Here Tn is a group of operators Tt, 1 < i < n — 1. Tj replaces a fermion of type i + 1 by a

fermion of type i. They satisfy the relations T{Tj — —TjTi when |i — j\ = 1, and T{Tj = TjTi

otherwise, cf. [4].

It is easy to find an expression for the image of a product of Schur-functions:

(63) 7 ^J|5A(i)(xw)^ = 5?(x)

(64) 7 (mn + i) = nA^+1

7 = (nAo"},.. ., nA^J,...)

(65) deg (7(11 ^AtoO^)) J = n^|Aw|

V i=1 / ¿=1

On F^ a(0) acts as 0. Because a(0) = ^«¿(0) we can eliminate an(0). Introduce now

i

(66) Pi := ai(0) 1 < * < n — 1

i=i

We then have ai(0) = /3i, «¿(0) = — /3j_i (1 < i < n) and an(0) = — j8n-i- The eigenvalue

of aj(0) is the charge of the fermions of type i. The eigenvalue of corresponds with the power of Ti. The next proposition and corollary show that in the homogeneous case we can express the principal degree of the image polynomial in terms of the principal degrees of the original Schur polynomials and the powers of the operators Ti.

Proposition 3.9. On FW we have the following equalities:

n n— 1 n—1 n—1

(67)Ff0 = n'^Tyj>2oii(-k)ai(k) +n^«i(0)2 + n ^ «¿(0)^(0) - ^(n - i)ai(0)

i= 1 fc>0 i=l i,j=l,i<j i=1

n n—1 n—1 n—1

(68) = a* (-&)«*(&) + -n^PiPi-i ~ 'Yjh

i=1 k> 0 ¿=1 i—2 t=l

so

We thus have

Proof.

Hq = к : ф(к)ф*(к) :

]Г^(п(А; - 1) + *) : фг{к)ф*(к) :

г=1кеЪ

п п

г(*)

= «Еяо

t=l i=l

Now we use Hq'* = Ylk> о &i(—k)ai(k) + ^«¿(O)2 + ^«¿(0). Then we obtain

Ho = n

fir и lb III

+ -П^(аг(0)2 + аг(0)) + ^(г -п)^(0)

г=1 fc>0 ¿=1 г=1

п , п—1 гг—1 71—1

^ 9Ь 4. ^ IV 1 /4* X

= п^2^2,°ч{-к)оч{к) + -п^аДО)2 + 2n Е °ч(0)°у(0) “

г=1 к>0 г=1 i,j=1 г=1

тг тг—1 гг—1 n—1

= nY2Y2ai{-k)ai(k)+n^2ai(0)2+п «¿(0)^(0) - ^(п - г)аг(О)

г=1 &>0 г=1 i,j=l,i<j г=1

тг .. тг—1 тг—1

= « Е Е oti(—k)a.i(k) + -n(/32 + - Pi-1)2 + Æn-i) - E ^

¿=1 fc>0 г=2 i=l

тг тг—1 тг—1 тг—1

= +nE^? ~nYlPipi-i -

г=1 fc>0 г=1 г=2 г=1

□

Corollary 3.10.

(69) deg ( т ( ■ • • T„V П SA(0 (*«) ) ) = n £ | A« | + n £ A;2 - n £ AiAr^x - À*

V V i=l / / г=1 г=1 г=2 »=1

In corollary 3.10 we see that deg modn only is determined by the ki and not by the partitions A^. We have already seen the formula for n = 2 in equation (54).

Example 3.11. For n = 3 corollary 3.10 gives:

(70) deg (7 (t^T^SxW(x{1))Sx{2)(x^)^ = Sk\ - *1 + 3k\ -k2- Shk2 This formula agrees with the formulas (3.6) and proposition 3.7.

According to corollary 3.10 we have deg(2;^) = nk. In this homogeneous case the degree deg0 is almost equal to the principal degree deg. It is easy to show that deg0 (x^ ) = k.

3.12. q-dimensions. We have the isomorphism

We can now take q-dimensions of equation (71) with respect to the principal degree. Using tp(q) = n^i(1 “ <1%) we obtain the following formula

(72) trzceFm"° =-^ = (-±-Y £

We can rewrite the second part of the equation. We then get

(73) V"* „nYkf-nYkiki-i-JZki _ vil )

ip(q)

Example 3.13. For n = 2 equation (73) becomes

(M> E

*■(«) nr=i(i-92t-')

ifc(fc + l)

The left hand side is equal to Ylk>o? ! so we Se^

(75)

\k{k-1-1)

^(92)

2 s! 2

ft>o *’(■'>

The formulas (74) and (75) can also be derived from the classical Jacobi triple product identity (see [3]):

(76) lit1“ uk~lvk)(l - ukvk~l){ 1 - ukvk) =

k>1 jeZ

Take u = —g3 and u = —r/. Then we get

E^'2+j = na+^a+^d-^)

j€Z fe>l

= + 94fc_3)(i + 94fc_1)(i + q2k)( 1 - <?2fc)

fc>i

2 k

Y[(l+qk)(l-ql

k> 1

(1 — g2fe)2 <p(q2)2

n

^ 1 - <f‘ ¥>(9)

so equation (74) is verified. For equation (75) we can take u = —q and v — — 1 in the Jacobi identity (76).

3.14. Some formulas on F^k\ Because of the isomorphism F^ ~ Fwe can also look at the principal degree on F^k\ We can write this in terms of the rrii, the eigenvalues of the a2(0), or in terms of the ki, the eigenvalues of the /%. We have kt = J2)=i mj- Note that we

know have Y^ai — k. We then obtain for the principal degree on the following formulas:

deg i 7 (ti' ■ ■ ■ n SA(.) (*“’)] J =

(77) = «¿|AW| + + Y2imi - ^nk

1=1 i= 1 ¿=1 n n—1 Tl—1 n—1 -

(78) = n^|Aw| - E — nkkn-1 + -nk(k + 1)

i=l ¿=1 ¿=2 i=l

This leads to the following (/-dimension formulas:

±fc(fc+i)

(79) traceF(fc)g 0 = -

<p{q)

-\nk

(p(qn

mi mi=fc

¥>(<7n)r

^nA;(fc+l) __________

/oi\ _ Q____________

' ' tn(r,n\n q

k\ ,...,kn—\£Z

We can rewrite this formulas to

(82) y(lT = V' afrnrm-Sr+'ZHmi-i)

viq) ^

m,j€mi=k

(83) = q^Zmf+Eirm

mie-^+Z,J2rn.i=0

and

(84) = ^qnUki-i$)2-nUb-i$№-i-(i-i)$)-Uki-i$)

ki€Z

(85) = 9"

kie—i^+Z (1 <i<n)

REFERENCES

1. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa. Transformation groups for soliton equations - Euclidean Lie algebras and reduction of the KP-hierarchy. Publ. RIMS, Kyoto Univ., 1982, vol. 18, 1077-1110.

2. M. Jimbo and T. Miwa. Solitons and infinite dimensional Lie algebras. Publ. RIMS, Kyoto Univ., 1983, vol. 19, 943-1001.

3. V. G. Kac. Infinite Dimensional Lie Algebras. Progress in Mathematics. Birkhauser, vol. 44, 1988. (third edition)

4. A. P. E. ten Kroode and J. W. van de Leur. Bosonic and fermionic realizations of the affine algebra gln. Communications in Mathematical Physics, 1991, vol. 137, 67-107.