Permutation Weights and Modular Poincare Polynomials for Affine Lie Algebras
aa r X i v : . [ m a t h - ph ] S e p PERMUTATION WEIGHTS AND MODULAR POINCARE POLYNOMIALSFOR AFFINE LIE ALGEBRASMeltem Gungormez
Dept. Physics, Fac. Science, Istanbul Tech. Univ.34469, Maslak, Istanbul, Turkeye-mail: [email protected]
Hasan R. Karadayi
Dept. Physics, Fac. Science, Istanbul Tech. Univ.34469, Maslak, Istanbul, Turkeye-mail: [email protected]. Physics, Fac. Science, Istanbul Kultur University34156, Bakirkoy, Istanbul, Turkey
Abstract
Poincare Polynomial of a Kac-Moody Lie algebra can be obtained by classifying theWeyl orbit W ( ρ ) of its Weyl vector ρ . A remarkable fact for Affine Lie algebras is that thenumber of elements of W ( ρ ) is finite at each and every depth level though totally it hasinfinite number of elements. This allows us to look at W ( ρ ) as a manifold graded by depthsof its elements and hence a new kind of Poincare Polynomial is defined. We give thesepolynomials for all Affine Kac-Moody Lie algebras, non-twisted or twisted. The remarkablefact is however that, on the contrary to the ones which are classically defined,these newkind of Poincare polynomials have modular properties, namely they all are expressed inthe form of eta-quotients. When one recalls Weyl-Kac character formula for irreduciblecharacters, it is natural to think that this modularity properties could be directly relatedwith Kac-Peterson theorem which says affine characters have modular properties.Another point to emphasize is the relation between these modular Poincare Polynomi-als and the Permutation Weights which we previously introduced for Finite and also AffineLie algebras. By the aid of permutation weights, we have shown that Weyl orbits of anAffine Lie algebra are decomposed in the form of direct sum of Weyl orbits of its horizontalLie algebra and this new kind of Poincare Polynomials count exactly these permutationweights at each and every level of weight depths. I. INTRODUCTION
We know that any affine Lie algebra b G N is related with a finite Lie algebra G N whichis called its horizontal Lie algebra. Let λ i ’s and α i ’s be respectively the fundamentalweigths and simple roots of horizontal Lie algebra G N where i = 1 , . . . , N . They aredetermined by 2 κ ( λ i , α j ) κ ( α i , α j ) ≡ δ i,j where κ ( , ) is symmetric scalar product which is known always to be exist via the relation2 κ ( α i , α j ) κ ( α i , α j ) ≡ ( A N ) i,j where A N is the Cartan matrix of G N . We follow the book of Humphreys [1] for finiteand Kac [2] for Kac-Moody Lie algebras.Let b A N be the Cartan matrix and α the extra simple root of b A N . Its dual λ is tobe introduced by hand via the relations κ ( λ , λ ) = 0 κ ( λ , α ) = 1due to the fact that b A N is singular. Note also that κ ( λ , α i ) = 0 , i = 1 , , . . . , N. and hence the name horizontal for G N . Affine Lie algebras are also characterized by theexistence of a unique isotropic root δ defined by δ = N X µ =0 k µ α µ where k µ ’s are known to be Kac labels of b G N . Let W b G N be the weight lattice of b G N .For any element b λ ∈ b G N , we know the following decomposition is always valid: b λ = λ + k λ − M δ ( I. level k is constant for the Weyl orbit W ( b λ ) and the depth M is alwaysdefined to take values M = 0 , , . . . , ∞ . For any fixed value of M, let us now define W M ( b λ ) to be the set of weights with the form(I.1). It is known that the orders of these sets are always finite, that is | W M ( b λ ) | < ∞ ( I. W ( b λ ) is infinite. In view of (I.2), wesuggest that W ( b λ ) can be considered as a manifold graded by weight depths M and hencea Poincare polynomial Q ( b G N ) is attributed by the following definition: Q ( b G N ) ≡ ∞ X M =0 | W M ( b λ ) | t M ( I. P ( b G N ) whichare known to be defined by P ( b G N ) = P ( G N ) N Y i =1 − t d i − . ( I. [3] . In (I.4), P ( G N ) is the Poincare polynomial and d i ’s are exponentsof G N .As we have shown in another work [4] , an explicit calculation of Poincare polynomialsof Hyperbolic Lie algebras can be carried out by classifying the Weyl orbit W ( ρ ) in termsof lengths [5] of Weyl group elements. Such a calculation is extended in a direct way to aclassification in terms of weight depths M. For simply-laced affine Lie algebras, b G N = A (1) N , D (1) N , E (1)6 , E (1)7 , E (1)8 depicted in p.54 of [2], these calculations give the result Q ( b G N ) = | W G N | P N R N ( I. P N = N Y i =1 ∞ Y k =0 (1 − q hk + d i ) R N = ∞ Y k =0 ∞ Y s =0 (1 + q s (2 k +1) h ∨ ) ( s +1) N ( I. h ∨ are coxeter and co-coxeter numbers of G N and | W G N | is the order of Weyl group W G N of finite Lie algebra G N .Although similar expressions could be obtained for a complete list of affine Lie alge-bras, this will be presented in the next section in which we expose modular properties ofQ-Poincare polynomials defined in (I.3). II. POINCARE POLYNOMIALS AS ETA-QUOTIENTS
There is quite vast litterature [6] on eta-quotients which are rational products ofDedekind eta functions with several arguments. Their relation with finite groups is alsostudied [7] . Let ϕ ( q ) = ∞ Y i =1 (1 − q i )be Euler product and η ( τ ) ≡ q / ϕ ( q )Dedekind eta function where q = e πiτ . An eta-quotient is defined [8] to be a function f ( τ ) of the form f ( τ ) ≡ d Y i =1 η ( s i τ ) r i ( II. { s , s , . . . , s d } is a finite set of positive integers and r , r , . . . , r d are arbitraryintegers. Let us denote the collection of integers r , s , r , s , . . . , r d , s d defining f ( τ ) bythe formal product g = s r s r . . . s r d d ( II. η g ( τ ) for the corresponding eta-quotient (II.1).What’s important here is to emphasize eta-products are in general meromorphic mod-ular forms of weight k ≡ P di =1 r i and multiplier system for some congruence subgroupof SL ( Z ) . This study is however outside the scope of this paper so we will only give herethe complete list of Poincare series defined above in the notation of (II.2). To this end, wedefine Q ( b G N ) = | W G N | q φ ( b G N ) η g ( b G N ) . ( II. q φ ( b G N ) which stem from the difference between definitions of Eulerproduct and η -function will also be given. Our results are given in the following Table-1for non-twisted types in Kac’s table Aff 1 ( p.54 of [2] ) and in Table-2 for twisted typesof Table Aff 2 ( p.55 of [2] ):Table-1 g ( A (1) N ) = ( h ∨ ) ( N +1) − , φ ( A (1) N ) = − ( N + 1) h ∨ + 1 g ( B (1) N ) = (2 h ∨ ) ( h ∨ ) ( N − − , φ ( B (1) N ) = − ( N + 1) h ∨ − g ( C (1) N ) = (2 h ∨ ) ( N − ( h ∨ ) − , φ ( C (1) N ) = − ( N + 1) h ∨ − g ( D (1) N ) = ( h ∨ ) ( N +1) ( 12 h ∨ ) − − , φ ( D (1) N ) = − ( N + 12 ) h ∨ − g ( G (1)2 ) = 12 − − , φ ( G (1)2 ) = − (2 + 1) 6 + (6 − g ( F (1)4 ) = (18) − − , φ ( F (1)4 ) = − (4 + 1)12 + (12 − g ( E (1)6 ) = 12 − − − , φ ( E (1)6 ) = − (6 + 1)12 + (12 − g ( E (1)7 ) = 18 − − − , φ ( E (1)7 ) = − (7 + 1)18 + (18 − g ( E (1)8 ) = 30 − − − − , φ ( E (1)8 ) = − (8 + 1)30 + (30 − g ( A (2)2 ) = 12 − − − , φ ( A (2) N ) = − g ( A (2)2 N ) = (4 h ∨ ) (2 h ∨ ) ( N − ( h ∨ ) − − , φ ( A (2)2 N ) = − (2 N + 1) h ∨ + 1 g ( A (2)2 N − ) = (2 h ∨ ) ( h ∨ ) ( N − N − , φ ( A (2)2 N − ) = − ( N + 1) h ∨ − ( N + 1) g ( D (2) N +1 ) = (2 h ∨ ) N − − , φ ( D (2) N +1 ) = − N h ∨ + 1 g ( E (2)6 ) = 12 − − − , φ ( E (2)6 ) = − g ( D (3)4 ) = 18 − − − , φ ( D (3)4 ) = − G (1)2 , F (1)4 , E (1)6 , E (1)7 , E (1)8 and D are interesting due toa theorem [6] concerning modular forms for some congruence subgroups of SL ( Z ) . Weleave however such a study in a subsequent paper. III. POINCARE POLYNOMIALS AND PERMUTATION WEIGHTS FORAFFINE LIE ALGEBRAS
We have defined Permutation Weights previously for finite Lie algebras [9] and alsoAffine Lie algebras [10] . In these works, it is shown that permutation weights can becalculated explicitly by the aid of a contructive corollary ( p.7 of [10] ).Here, it is shown that the polynomials Q ( b G N ) | W G N | ( III. [10] of permutation weights. Let b Λ + and λ + bedominant weights of b G N and G N respectively, W ( b Λ + ) and W ( λ + ) be corresponding Weylorbits. We know that all the elements of W ( b Λ + ) has the form (I.1) and among them thepermutation weights are defined by the following specific form: λ + + k λ − M δ , M = 1 , , . . . . ( III. P M ( b Λ + ) to be the set of permutationweights of b Λ + and | P M ( b Λ + ) | be its order. Let also note that (III.1) can always beexpressed in the following form: Q ( b G N ) | W G N | = ∞ X M =0 c M q M ( III. c M ’s are positive integers, c = 1 and q is an indeterminate. One can show that | P M ( b Λ + ) | = c M , M = 1 , , . . . ( III. Q ( b G N ) states that all the elements λ + k λ − M δ are belong to W ( b Λ + ) where λ ∈ W ( λ + ). In other words, if λ + + k λ − M δ ∈ P M ( b Λ + )is exist for any M, then one finds that W ( λ + ) + k λ − M δ ∈ W ( b Λ + )due to existence of Poincare series given in Table-1 and also Table-2. The existence of thesenew kind of Poincare series is the existence of permutation weights. One can formally saythat this gives us an explicit way to decompose any Weyl orbit of an Affine Lie algebra asa direct sum of Weyl orbits of its horizontal Lie algebra. This reflects our main point ofview to introduce permutation weights.It is now useful to proceed in an example for which all our general framework is to bereflected. Let us consider the simply laced, affine Kac-Moody Lie algebra E (1)6 with thefollowing Dynkin diagram:From Table-1 of Sec.II, one finds that the similar of (III.3) is Q ( E (1)6 ) | W E | = 1 + q + q + q + 2 q + 3 q + 3 q + 4 q + 6 q + 7 q + . . . ( III. c M W ( E (1)6 ) be the Weyl group of E (1)6 . Then, all the elements which give us thepermutation weights numbered in above Table-3 are given explicitly as in the following:Σ , = σ Σ , = σ , Σ , = σ , , Σ , = σ , , , , Σ , = σ , , , Σ , = σ , , , , , Σ , = σ , , , , , Σ , = σ , , , , Σ , = σ , , , , , , Σ , = σ , , , , , , Σ , = σ , , , , , Σ , = σ , , , , , , , Σ , = σ , , , , , , Σ , = σ , , , , , , , Σ , = σ , , , , , , Σ , = σ , , , , , , , , Σ , = σ , , , , , , , Σ , = σ , , , , , , , , Σ , = σ , , , , , , , Σ , = σ , , , , , , , , Σ , = σ , , , , , , , Σ , = σ , , , , , , , , , Σ , = σ , , , , , , , , , Σ , = σ , , , , , , , , , Σ , = σ , , , , , , , , , Σ , = σ , , , , , , , , , Σ , = σ , , , , , , , , , Σ , = σ , , , , , , , , We assume here that Weyl group elements are expressed in the form of σ µ ,µ ,...µ k ≡ σ µ σ µ . . . σ µ k where σ µ ’s are simple Weyl reflections which are defined tobe the Weyl group elements with respect to simple roots α µ of E (1)6 with µ = 0 , , , . . . E (1)6 weight latticeby Σ M,c M ( b Λ + ) ≡ b Λ + + κ ( b Λ + , δ ) λ − M δ.
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