Rend. Sem. Mat. Univ. Poi. Torino Voi. 53, 3 (1995) Number Theory

S. Capparelli

LIE ALGEBRA REPRESENTATIONS, COMBINATORIAL IDENTITIES AND g-TRINOMIAL COEFFICIENTS

Abstract. This paper explains the interaction between the representation theory of affine Lie algebras and combinatorial identities. It shows that new identities of the Rogers-Ramanujan type underlie level 3 representation of A£ . ^ a^so discusses briefly the important connections with conformai fìeld theory and statistical mechanics.

1. Introduction

Arithmetic functions arising from various integrai powers p of the infinite product HnLi(l ~ Qn) a r e °f great importance. For example, p = — 1 gives the classical partition function p(n). Other classical identities of Euler and Jacobi involve the first and third power. In an equivalent form Jacobi's identity is

co

(i) n(i-9n)(i-*«BJ(i-*"i9n_i)=D-i)'«*'('+i)*<

n=l iel,

(Jacobi's triple product identity.) LG. Macdonald gave a generalization of Jacobi's formula for ali those powers that are the dimension of a semisimple Lie algebra. See [33], [20]. Later on, Kac and Moody independently interpreted Macdonald's identities as the analogue for Kac-Moody Lie algebras of Weyl's denominator formula.

An attempt to generalize (1) led Lepowsky and Milne in [28] to discover the dose -relationship between affine Lie algebras and classical partition identities.

Works of Lepowsky and Wilson, and Meurman and Prime, among others, have clarifìed the relationship between the representation theory of these infinite dimensionai Lie algebras and combinatorial identities. For example, Lepowsky and Wilson used the representation theory of the affine Lie algebra s/(2, C)~ to interpret and prove the classical identities of Rogers and Ramanujan. No essentially new combinatorial identity had been discovered along these lines. In a previous work I conjectured that new identities could

232 S. Capparelli

"explain" standard level 3 modules for the affine algebra A22\ A purely Lie theoretic proof

of these identities was given in [15]. G.E. Andrews gave a combinatorial proof of one such identity in [6]. He pointed out the similarity of this identity with Schur's 1926 partition theorem, see also [1] and [14].

In the present paper I will explain the important interaction between the representation theory of certain infinite dimensionai Lie algebras and combinatorial identities. In particular, I will discuss the construction of the level 3 modules for A\ , and how, as a corollary, one gets a proof of the combinatorial identity of [13]. I also discuss various standard modules that are related to classical combinatorial identities. In particular, the famous Rogers-Ramanujan identities and their generali zations. In [16] I study in detail some standard representations of B\ ' and D^ ' and other affine algebras underlying classical combinatorial identities. It is hoped that the extensive bibliography at -the end, though necessarily incomplete, will be useful for a panoramic view of the activity in this line of research.

2. Preliminary defìnitions and notations

Let 0 be the set of 3 by 3 matrices with coeffìcients in C and trace 0. It is well known that 0 is a Lie algebra with respect to the multiplication [x,y] — xy — yx. In fact, it is a simple Lie algebra, that is, essentially, it has no nontrivial ideals. A root space decomposition of 0 is given here in complete detail: 0 — ()©Yl ^xa wne]re 1) = spanja, /?} and

/ l 0 0 \ / 0 0 0 \ a = 0 - 1 0 /? = . 0 1 0

\ 0 . 0 0 / \ 0 0 - 1 / and if eij denotes the 3 by 3 matrix with zero coeffìcients except in the (i, j) position then the xa's vary in the following set {xa = ei)2,xp = e2>3,Xa+p = ei,3,#-a = ^iiV-p — e3,2,x-a-p = e3(i}.

The setting {x,y) = trxy (trace of xy) defìnes on g a nondegenerate, symmetric, bilinear form (a multiple of the Killing form).

Then, if u is the 6-th root of unity, consider the elemerits Hi = 2(a + /?), E\ = (xa-xp)yj^t Fi = (x-a-x-.p)yj2}, E0 = x„a-pt FQ = u)-1xa+l3, H0 = -a~(3. Define an automorphism v on # by vHj — Hj and vEj •= uEj for j = 0,1. Then v has order 6 and 0 decomposes into the sum of the following eigenspaces £(0) — Ci^i, fl(1) = CJE7I 0 CE0, 0(2) = C{x-a + x^p), 0(3) = CO - /?), 0(4) = C(xa + xp),

Q(s) = £Fi(B£F0.

Consider then the vector space 0 <g) C[i1/'6,<~1/'6]. The automorphism v can be thought of as a linear automorphism of this new vector space by setting v{x % tl) =

Lie algebra representations, combinatorial identities and q~trinomial coefficients 233

vx®oj~6itl. So we can consider the subspace of fixed points of v which is L L I G Z ^ O ® ^ 6 -

So far we got a subspace of the space of 3 by 3 matrices with coefficients in Laurent series in the formai variable W6. We now add two extra elements e and d and, on the vector space

. 6M= (IIfl(o®*'/6) e Ce® a/,

we define a Lie algebra structure by [e, x] = 0 \/x E j[i/], [e/, a; <g) f /6] = |o? ® W6, and

[*<g> f/6, y <g> * i /6] = [K, y] ® ^ ; + J ' ) / 6 + ^ j + j ,0( ic , y)c.

Let

/o = F 0 ® r 1 / 6 , / i = Fi(8>r1/6, 1 2

/io = #o + -e , /zi = iJi + -e . 6 3

Then {e,-, .fi, /i7;; « = 0,1} is a set of canonica! generators of fl[z/] which is an affine Lie algebra of type A% , a special example of a Kac-Moody Lie algebra. The representation theory of Kac-Moody algebras is analogous to the representation theory of finite dimensionai semisimple Lie algebras over the complex numbers. Indeed these are a subclass of those. In particular there is a notion of a highest weight module. A highest weight module V with highest weight A is a module generated by a vector VQ / 0 such that CÌ.VQ = 0 for i = 0,1 and h.v0 = \(h)vo for ali h £ () = span{/io, hi,d], and A E ()*. The value A(c) is called the level of the module. It is convenient to consider, for ali \i E rj* , the space

Lfl - {v e L : h.v - fjt(h)v V/?. G t)}

It turns out that these spaces are ali finite dimensionai. We can then define the formai character of L as the series ch(L) — ^(dimXjU)e(^), an element of the ring of formai power series with coefficients in Z. The following character formula is very importanti

THEOREM 1. (WEYL-KAC FORMULA) In the ring of formai power series we

have

e(~A)c,,(i) = n„e.+d-(-)r^) where W is the Weyl group and A+ is the set of positive roots.

In some special cases this formula reduces to the Jacobi triple produci identity and the quintuple product identity.

If we set ali the formai variables in e(—X)ch(L) equal to q we get the principally specialized character of L. It is a formai series whose coefficients are the dimension of the

234 S. Capparelli

d-eigenspaces. For such a specialized character, sometimes called also graded dimension and denoted by dim*L, a numerator formula due to Lepowsky holds.It allows one to express the numerator in the Weyl-Kac formula as an infinite product. This is of great importance for the combinatorial interpretation we shall present in later sections. See [27], [28]. In the case of our algebra we have, if X(h0) ^ X{hi)

dim,I(A) = fJO - «T1 Ilf1 - «T 1

where n runs through ali positive integers congruent to ±1 modulo 6, and m runs through

ali positive integers that are not congruent to 0, X(2ho + hi) + 3, ±X(hi) -f 1, ±X(h0) +

1, \(h0 + hi) modulo X(4h0 + 2hi) + 6 and, if A(/i0) = X(hi) then

dim*L(A) = Y[(l - fT1 Hi1 ~ «m)-1 IK1 " «') where n, m run as above, while s runs through the positive integers congruent to A(/?,o) + 1 . We want to consider the special cases where A is one of the following linear forms

(1) h,Q which is 1 on ho and 0 on hi and d,

(2) 2h* which is 0 on ho and d , and 2 on hi,

Q)h*0 + h*v

(4) 3/iì

The first two cases give level two modules, the other two give level three modules. Correspondingly, the graded dimensions are:

co

dmu£(fcs)= n (i-9nr in( i-(«a)' i B"3)" i( i-(92) , ! B"2)"1 U = l ?7,>1

n = ± l m o d 6

co

dim.£(2ftl) = J [ (! - «n)_1 II*1 J (<r)5"~4r1(l " («2)5"-1)-1

rc = l n > l n = ± l m o d 6

These, up to the factor co

p= n (i-?0)-1. n = l

n = ± l m o d 6

are the product sides of the famous Rogers-Ramanujan identities.

Moreover, dim+L(3/»I) = F J ] ( l - tf12^2)"^ - ^12"+3)"1(1 - ? 12n+9 r l ( 1 _ ?12n+l0)-l

n > l

dim,L(/**0 + h\) = F JJ(1 - g 1 2 "* 1 ) -^ - g 1 2 ^ 1 1 ) - ^ - «12w+3)-1(l,- g 1 2 ^ 9 ) - 1

n > l

(1 - q12^)-1^ ~ q12^7)'1^ ~ dUn+2)(l - <Z12n+1°).

Lie algebra representations, combinatoria! identities and q-trinomial coefficients 235

The factor F is a "hint" to the existence of an infinite dimensionai Heisenberg subalgebra of g. This subalgebra is of fundamental importance. It allows one to employ the well known theory of representation of Heisenberg algebras. In fact, such algebras have an irreducible representation as differential operators on polynomial algebras with infìnitely many variables. This is what physicists cali annihilation and creation operators on a Fock space. The key point is that the action of the Heisenberg subalgebra can be extended to the whole affine Lie algebra by the introduction of the vertex operators. We now proceed to explain this in more detail. Let a be the centralizer of E0 + E\. Consider 5 = rin^o ^(en)®^©^ a n d i t s subalgebras s± = U±n>o fl(6n)®*n and b = s+eCceCd. It is well known that s has an irreducible representation on the symmetric algebra S on 5_, see [21.]. If a G Ci, denote by a(6n) its projection on a(6n) and by o'(6n)(n) the operator on S corresponding to &(en) <8> tn- Then for such a we can defìne

E±(a,z) = exp I ^ a(6n)(™)— ) \±n>0 /

n G |Z , a pair of formai Laurent series with coefficients in End S. Then we set

X(a, z) = r(a)E~ (-a, z)E+(-aì z)

where r(a) is a suitable normalization Constant. The coefficients of this formai series are infinite series of operators on S. They are, however, well defìned operators on S because whenever the infinite series is applied to a vector of S only finitely many terms are nonzero. For this reason the order in which We multiply the series E~ and E+ is essential. Note, indeed, that the coefficients of the series E+(—a, z)E~(—aì z) are not well defìned operators on 5. This is an example of normal order. Although these operators have very complicated expressions we can easily compute their commutator by using a calculus of formai variables. The main point of these computations lies in the properties of the formai <5-function, defìned by 6(z) = 52?:gs z%> a f° rmal Fourier expansion of the Dirac 6-function at z = 1. We then have the following

THEOREM 2. The representation of 5 on S extends uniquely to a Lie algebra representation of Q such that the operator corresponding to (£a)(n) <É> tn is the coefftcient of z~n in the vertex operator X(a, z).

This theorem gives us the level 1 representation L(h\). Before proceeding further we need to recali the concept of uni versai enveloping algebra. For any Lie algebra cj we have an associative algebra £/(#) which contains g, such that for x, y e 9 , [x,y] — xy—yx where on the right hand side we have the associative product in li (gì) and such that for any associative algebra A and a linear map ip : g -+ A for which ip([x, y]) = ip(x)ijj(y) - ip(y)ip(x) there exists a unique homomorphism of algebras # : U(g) -*• A that extends ip. To obtain the

236 5. Cappa re Ili

other standard representations, in particular those of level 2 and 3, we proceed to locate these representations inside the tensor products of various copies of L(hl).

PROPOSITION 4. The tensor product L(h\) ® L(h\) contains a submodule isomorphic to L(hl) and a submodule isomorphic to L(2h\). Moreover, the tensor product L(h\) <gj L(h\) ® L(h\) contains a submodule isomorphic to L{Zh\) and one isomorphic to L(hl+.h\).

Proof By the uniqueness of the standard modules with given weight, it is enough to fìnd weight vectors of the appropriate weight. One then applies U(g) to this vectors to obtain the desired modules. If vo is the highest weight vector of weight h\ then one can easily check that the following are the desired vectors:

(1) vo <g> vo has weight 2h\

(2) /ivo ®v0-v0® fxvo has weight h^

(3) vo ® vo ® ̂ o has weight 3/zJ

(4) fivo <& v0 ®vo-v0<g> fava (g> vo has weight h^ 4 h*

Of great importance is the following theorem.

THEOREM 5. (POINCARE'-BIRKHOFF-WITT) Let (vi,v2, v3ì- • •) be an ordered basis of g then the elements Wi(i)V,-(2) ••• i>i(m) ?(^) ^ z(^) < ••• < i{m) (standard monomials), with m a positive integer, along with 1, form a basis ofU(Q).

It follows from this theorem and the previous proposition that one can get a spanning set for the standard module by applying the standard monomials to the highest weight vector. This turns out to be the appropriate spanning set for our construction. It becomes then evident that appropriate restriction on the standard monomials might actually parameterize a basis of the standard module. This is the source of the combinatorial identities at the heart of this study. In fact, as we have already explained, there is an infinite product that counts the dimension of each graded piece. This infinite product usually has a simple combinatorial interpretation. If we can fìnd some way of parameterizing the basis by means of partitions of integers then we might get a nontrivial partition identity. This is indeed what happens. In the case of level 2 modules we obtain the identities of Rogers and Ramanujan. These identities had already been obtained in an analogous way by Lepowsky and Wilson in their work on the level 3 A\~ modules. The most interesting thing here is that for the first time in this line of research, for level 3 modules, we get a pàir of nontrivial combinatorial identities that were unknown to the specialists. They can be stated as follows:

THEOREM 6.

A. The number P\{n) of partitions (mi , . . . ,m r ) of an integer n into parts different from 1 and such that the difference of two consecutive parts is at least 2 (Le.

Lie algebra representations, combinatorial ìdentities and q-trinomial coefficients 237

rrii — rrii+i > 2), and is exactly 2 or 3 only if their sum is a multiple of 3 (Le. 2 < m?: — ???.?:+i < 3 implies rrn + w^+i = 0 modulo 3) is the same as the number P2(rc) of partitions ofn into parts congruent to ±2, ±3 modulo 12.

B. The number C\(n) of partitions (mi , . . . , rar) of an integer n into parts different from 2 and sudi that the difference of two consecutive parts is at least 2 (Le. mi — mi+i > 2), and is exactly 2 or 3 only if their sum is a multiple of 3 (Le. 2 < rrii — raj+i < 3 implies mi + ra»+i = 0 modulo 3) is the same as the number C2(n) of partitions ofn into distinct parts congruent to 1,3,5,6 modulo 6.

These identities had been conjectured in [13]. Identity A was first proved by Andrews ([6]) using generating functions, ^-trinomial coefficients and Jacobi's identity (1). Both identities were proved by Lie theoretic methods in [15]. B was also proved by combinatorial methods in [14].

3. (/-trinomial coefficients

In their work on a certain generalization of the hard-hexagon model in statistical mechanics, Andrews and Baxter introduced the notion of g-trinomial coefficients. We recali that the ordinary trinomial coefficients are the coefficients that appear in the expansion of (1 +1 + 2 - 1 ) n in integrai powers of t. More precisely

(l + H-*-1)" =£(")** j = -n ^J ' 2

These coefficients have obvious similarities with the binomial coefficients, and they have similar properties, including the existence of a Tartaglia-like triangle for these numbers. There are also several ways to interpret them combinatorially. It is, however, their q-analogue that was found to be essential in the exact solution of a model in statistical mechanics. They were also found to lie at the heart of I.Schur's 1926 partition theorem:

THEOREM 7. (SCHUR) The number of partitions of n into parts congruent to 1 or 5 modulo 6 is the same as the number of partitions ofn into parts that-dijfer by at least 3 with the added condition that multiples of 3 differ by at least 6.

The similarity of this theorem with the one stated above is evident. The reason for this similarity is now explained by the fact that Schur's theorem interprets the level one

(2)

module for the algebra A\ while the previous theorem explains level 3 modules for the same algebra.

We recali the definition of the tf-analogue of the trinomial coefficients, used in [9] in the exact solution of a model in statistical mechanics. Set

n; B] q\ =yvo+B) A , V •

' * 3=0

n- j j +A

238 5. Cappa re Hi

where is the Gaussian polynomial defìned by

(1 - q1l)(l - q"-1) • - • (1 - qn~m+1)

if 0 < m < n, and defìned to be zero if m < 0 or m > n. Also note that

f n; B; q\ T T 1 l i m a c c i ' A J^ni_am

2 1 i~<l 4 771 = 1

the generating function of the classical partition function, see [9].

Let Dn(t,q) be the generating function defìned by

£ ón(KJ)ihqj

h,j>0

where 6n(h,j) equais the number of partitions of j of the type enumerated by C\{j) where,

furthermore, ali parts are less than or equal to ri and h is the number of parts congruent to

1 modulo 3 minus the number of parts congruent to 2 modulo 3. For example,

D7(t, q) = 1 + *« + q3 + tq4 + r V + q6 + ?6

+ ?9 + tq7 + ^ ? + ^1 3 + q12 + V ° + * V

corresponding to the partitions

0 ,1 ,3 ,4 ,5 ,5+ 1,6,6 + 3 , 6 + 1 , 7 , 7 + 5 + 1 , 7 + 5,7 + 3 , 7 + 1.

Now consider

^(*,?)=E^^+3j'2(n; , |tÌ ; "3)9-In [14] we proved

T H E O R E M . For each n > 0,

D3n-2(t,q) = Vn(t,q)-

By comparing the coefficients of V in this equality we get

T H E O R E M . The polynomial

n3f-+2j(n; 2j + l; q3\ 1 V 2i + l ) 2

is the generating functions for partitions in which

(i) no part equais 2;

Lie algebra representations, combinato rial identities and q-trinomial coefficients 239

(ii) the dijference of two consecutive parts is at least 2 and is 2 or 3 only if their sum

is a multiple e>/3;

(Hi) each part is less than or equal to 3?i — 2;

(iv) j equals the number of parts congruent to 2 modulo 3 minus the number of parts congruent to 1 modulo 3.

For example, for j = — 1, n = 3 we have

- 1 ; «") = g ( l + g 8 + 2 g«+ 9» + 9 ia ) = g + g 4 + . g W + 9T + 9 7 + 8 + 9 7 + 5 + l _

In later works Alladi, Andrews and Gordon have found a three parameter generalization of Theorem 6. They now have a general theorem which gives these and other similar results. In particular, for example, they give the following results, which they cali "Capparelli version of the Gòllnitz-Gordon identities":

THEOREM 8. The number of partitions of n info distinct parts congruent to 3,4,5,8 modulo 8 is equal to the number of partitions of n info distinct parts > 1 which are / .2 modulo 4 such that the difference between consecutive parts is > 4 unless they are both multiples of 4 or they add up to a multiple ofS.

THEOREM 9. The number of partitions of n info distinct parts congruent to 1,4, 7,8 modulo 8 is equal to the number of partitions of n info distinct parts / 2 modulo 4 , no part = 3, such that the difference between consecutive parts is > 4 unless they are both multiples of 4 or they add up to a multiple of$.

Identities of this type are obviously important in that they could shed new light on the structure of standard modules for affine algebras just as Theorem 6 did for level 3

( 2)

modules of the A\ ' algebra. In particular there is an infinite series of partition identities underlying standard A£ -modules of level 4 and higher that are waiting to be discovered, just as the generalized Rogers-Ramanujan identities due to Andrews, Bressoud and Gordon underlie the higher level modules for the A{' algebra, the simplest example of affine Kac-Moody Lie algebra. The exciting difference is that in this case the combinatorial identities have been discovered by Lie theoretic means. It is remarkable also that it was found in [36] that the identities of Theorem 6 also appear in a different specialization of the character of a standard module for the algebra of type A\K

4. Standard modules underlying the Rogers-Ramanujan identities

It is easy to compute the graded dimension of some standard modules and verify that one gets the Rogers-Ramanujan identities in the following cases: level 3 modules for the algebra of type A\\ these were the first constructed by Lepowsky and Wilson. Level

240 S. Cappa re Ili

(2)

2 modules for the algebra of type A\ ', these were first constructed in [10]. Later on, another construction of these modules was also given in [44] using the Z-algebra approach of Lepowsky and Wilson. Level 2 modules for the algebra of type Ay and level 1 modules for the algebra of type Q ^ were constructed by Misra. Level 1 modules for the algebra of type G[ \ and for the algebra of type F^\ were constructed by Mandia [34].

For example, the graded dimension of a level 1 standard G2- module can be computed as follows. Let A be a (?2 root system with a being the short simple root and (3 the long simple root. Then the set of short roots is As = {±cv, ±(<x + /?), ±(2a -f /?)} and the set of long roots is AL = {±0, ±(3a '+ /?), ±(3a + 2(3)}. The real roots of D^ are then

$R = {a -f ny : a 6 As] U {a + 3ny : ex E A^} where y is the minimal imaginary root, see [23]. By definition the denominator is

D = ]la€*+(^ "~ e ( — a ) ) m ^ - If w e t a^e ^ = ^o m e n t n e desired specialization is the map £ determined by

e(-a>a)^q

e (—a) H-+ q

e(-p)~q2

where «o = y — fo and /?0 is the highest root of G2. With this specialization we get:

«D)= n .o-f) n (i-^n^-?*) 'n=±2,±5,±7,0modl5 i=±lmod5 fc>0

Now the Weyl-Kac character formula together with Lepowsky's numerator formula allows us to prove:

THEOREM.

dim,L(\) = F n„>o(l ~«"):

Explicitly, dropping the factor F which corresponds to the Heisenberg subalgebra, the generating function we get is

n Q-*nrx= n (i-*3")"1

n>0,n=:fc2modl5 n>0,n = ±lmod5

and this is the product side of one of the classical Rogers-Ramanujan identities in the variable q3.

5. Conclusions

The knowledge of partition identities underlying certain standard modules can give clues to the right path to follow in order to obtain a vertex operator construction of these

Lie algebra representations, combinatorial identities and q-trinomial coefficients 241

representations. One may need to use the kind of relations studied in [11] and [12]. Or

vice versa, given a module constructions one may obtain nontrivial combinatorial identities.

It is cruciai to discern which, among the numerous formai series identities that one could

write down from [12],are the essential relations. For this, we need to embed these identities

in a more conceptual framework. What may be needed is the kind of work done in [18],

[19],[17] and [22]. In these works the Z-algebras of Lepowsky and Wilson are interpreted

in terms of a generalization of the notion of vertex operator algebras. This aspect is the

subject of a joint work in progress with C. Husu. Vertex operator algebras seem to be

the right framework also for the approach of Meurman and Prime, see [36]. We recali

that such algebras are the algebraists' version of the physicists' conformai fìeld theory. In

[21] the Monster simple group was represented as the group of automorphism of a suitable

vertex operator algebra, the so-called "moonshine module". One could say that the main

diffìculty of this line of research consists in constructing a conformai fìeld theory not only

of standard modules but of Verma modules, which is much more subtle.

Another aspect worth emphasizing is the clear relationship between certain statistica!

mechanics models and the Rogers-Ramanujan type identities. These were in fact used by

Andrews and Baxter in the exact solution of the "hard hexagon" model. More general

models corresponding to generalized Rogers-Ramanujan identities exist. One could expect

the relevance of these new combinatorial identities from this point of view as well.

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242 S. Capparelli

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Stefano CAPPARELLI Dipartimento di Matematica Università La Sapienza Pie Aldo Moro, 5. 00185 Rome, Italy