Asymptotics of generalized Galois numbers via affine Kac-Moody algebras

Generalized Galois numbers count the number of flags in vector spaces over finite fields. Asymptotically, as the dimension of the vector space becomes large, we give their exponential growth and determine their initial values. The initial values are expressed analytically in terms of theta functions and Euler's generating function for the partition numbers. Our asymptotic enumeration method is based on a Demazure module limit construction for integrable highest weight representations of affine Kac-Moody algebras. For the classical Galois numbers, that count the number of subspaces in vector spaces over finite fields, the theta functions are Jacobi theta functions. We apply our findings to the asymptotic number of q-ary codes, and conclude with some final remarks about possible future research concerning asymptotic enumerations via limit constructions for affine Kac-Moody algebras.


Introduction
The generalized Galois numbers G (r) N (q) count the number of flags 0 = V 0 ⊆ V 1 ⊆ · · · ⊆ V r = F N q of length r in an N -dimensional vector space over a field with q elements [26]. In particular, when r = 2 these are the classical Galois numbers studied by Goldman and Rota [7] which give the total number of subspaces in F N q .
We show that the generalized Galois numbers grow asymptotically, as r is fixed and N → ∞, exponentially with factor O(N 2 ) in logarithmic "time" scale: G (r) N (q) ∼ I r (q) · e O(N 2 ) log(q) . Here, "time" equals the cardinality of the finite field. Our main result is the explicit description of the initial values I r (q) via theta functions and Euler's generating function for the partition numbers.
This investigation serves three purposes. First, the generalized Galois numbers are of independent interest as they enumerate points in fundamental geometric objects defined over finite fields. For example, by definition the classical Galois numbers |Gr(k, N )(F q )| count the number of F q -rational points in Grassmann varieties. The numbers of solutions of the set of equations for Gr(k, N ) in extension fields F p n of F p are in turn subject to the study of local zeta-functions Z(Gr(k, N ), t) = exp( n≥1 |Gr(k, N )(F p n )| t n n ) in number theory. Let us mention that a generating function for the local zeta-function Z(Gr(k, N ), t) can be given by where the b i = dim H 2i (Gr(k, N )(C), Z) are the even topological Betti numbers of the complex Grassmannian. Consequently, the study of Galois numbers reflects upon many subjects. Second, the Galois numbers enumerate asymptotically the number of equivalence classes of linear q-ary codes in algebraic coding theory as recently shown by Hou and Wild [9,10,11,28,29]. For example, the asymptotic number N S n,q of linear q-ary codes under permutation equivalence is We apply our findings to those asymptotic equivalences, and derive considerable simplifications of the asymptotic enumeration of linear q-ary codes ( §4). Third, our investigation serves the demonstration of the asymptotic enumeration method itself ( §3). We identify the generalized Galois numbers G (r) N (q) as the basic specialization of the Demazure modules V −N ω 1 (Λ 0 ) of the affine Kac-Moody algebra sl r (see (3.14)). Those characters pass via a graded limit construction [6,17,20,21] to the characters of the fundamental representations of our affine Kac-Moody algebra: By a symmetry argument, Kac's [12] character formula for the basic representation then allows us to prove our main result: Consider the generalized Galois number G (r) N (q). For any prime power e δ = p m (in fact for any complex number e δ where δ ∈ −2πiH) and 0 ≤ j < r we have the limit Euler's generating function for the partition numbers, and Θ F j (z) = k∈Z r−1 e 2πizF j (k) are theta functions associated to the quadratic forms F 0 , F 1 , . . . , F r−1 on the lattice Z r−1 given by The exponents u 0 , u 1 , . . . , u r−1 are For the classical Galois numbers our theta functions turn out to be Jacobi theta functions (see Corollary 3.6).
Let us conclude the introduction with the following remark on our asymptotic enumeration method. In the case of generalized Galois numbers we do not make use of the modularity of characters of integrable highest weight modules, since the prime powers p −m < 1 lie in the region of convergence of our modular forms. However, we will discuss, in §5, an important eventual application of our asymptotic enumeration method where modularity has to be exploited.

Notation and Background
The generalized Galois number G  The N -th generalized Rogers-Szegő polynomial H (r) N (z, q) ∈ N[z 1 , . . . , z r , q] [22,23,25] (see [1] for an account) is defined as the generating function of the q-multinomial coefficients: Recall from [26] that the q-multinomial coefficient For general facts about affine Kac-Moody algebras and their representation theory we refer the reader to [3,13], and for Demazure modules to [6]. Let us briefly fix the notation we will use throughout. We consider the affine Kac-Moody algebra sl r . We denote the simple roots by α 0 , α 1 , . . . , α r−1 , the highest root by θ = α 1 + . . . + α r−1 and the imaginary root by δ = α 0 + θ. The affine root lattice is then defined as Q = Zα 0 ⊕ Zα 1 ⊕ . . . ⊕ Zα r−1 and the real span of the simple roots is given by h where h = r is the Coxeter number of sl r . For a dominant integral weight Λ = m 1 Λ 0 + m 2 Λ 1 + . . . + m r−1 Λ r−1 we let V (Λ) be the integrable highest weight representation of weight Λ of sl r and χ(V (Λ)) its character. The Λ 0 , Λ 1 , . . . , Λ r−1 are called fundamental weights, the V (Λ l ) the fundamental representations and V (Λ 0 ) the basic representation. As for the Demazure modules, we will only consider the translations t −kω 1 = (s 1 s 2 . . . s r−1 σ r−1 ) k in the extended affine Weyl group of sl r , where ω 1 = Λ 1 − Λ 0 . Here, σ denotes the automorphism of the Dynkin diagram of sl r which sends 0 to 1, and s 1 , . . . , s r−1 are the simple reflections associated to the simple roots α 1 , . . . , α r−1 . We denote the Demazure module associated to those translations by V −kω 1 (Λ) and its character by χ(V −kω 1 (Λ)). We write the monomials in the characters of our modules as e λ , the coefficient k in the monomial e −kα 0 is referred to as the degree.
H will denote the upper half plane in C. We write ∼ for asymptotic equivalence, that is for f, g :

Asymptotics of generalized Galois numbers
Let us start with a direct consequence of Kac's character formula [12, (3.37)].
This establishes the theorem.
Remark 3.4. One can phrase Theorem 3.2 asymptotically as Note that for fixed r the exponents u j (n, r) lie in O(n 2 ). Remark 3.5. For j = 1, . . . , r − 1 the limits in Theorem 3.2 coincide. In fact, the quadratic forms F 1 , . . . , F r−1 differ only by a cyclic shift of the coordinates. Summation over the complete lattice Z r−1 produces equality.
Let us summarize the implications of Theorem 3.2 for the classical Galois numbers G N (q) = G (2) N (q) that count the number of subspaces in F N q . Corollary 3.6. Consider the classical Galois numbers G N (q). For any prime power q = p m (in fact for any complex number |q| > 1) we have Euler's generating function for the partition numbers, and ϑ 2 , ϑ 3 are the Jacobi theta functions The limits differ by , (3.22) where ϑ 4 is the Jacobi theta function Proof. It remains to prove (3.22). This follows from the identity ϑ 4 (z, q) = ϑ 3 (2z, q 4 ) − ϑ 2 (2z, q 4 ) [27, pp. 464].
For completeness, we take a closer look at the numbers g 2∞+1 (q), g 2∞ (q) and their differences.
where p(k) is the partition function that counts the number of ways to write k as a sum of positive integers. In fact, Kac's character formula [12, (3.37)] reduces to this expression (see [12, (3.39)]), and our proof of Theorem 3.2 reduces to this setting.

Applications to linear q-ary codes
To describe the asymptotic number of non-equivalent binary n-codes in terms of the classical Galois numbers G n (2), Wild [28,29] examines numbers d 1 (q), d 2 (q) (see Lemma 1 in both articles) which, in the notation of Corollary 3.6, are defined as He proves that they are positive constants (depending on q) less than 32, gives a numerical evaluation method by use of the recursion formula of Goldman and Rota [7, (5)], evaluates d 1 (q), d 2 (q) numerically for q = 2, and shows d 1 (q) < d 2 (q) for general q. Now, the detailed analytic behavior of those numbers can be extracted from Corollary 3.6 and Corollary 3.7 (see also Table 1 for examples).
For a general prime power q, Hou [9,11] derives asymptotic equivalences for the numbers of linear q-ary codes under three notions of equivalence. That is, the permutation equivalence (S), the monomial equivalence (M), and semi-linear monomial equivalence (Γ). He proves where a = |Aut(F q )| = log p (q) with p = char(F q ). The asymptotic equivalence N S n,2 ∼ Gn (2) n! concerns binary codes and is previously derived by Wild [28,29]. Based on their results, the transitivity of ∼ and our Corollary 3.6 produce the following list.

Conclusion
The asymptotic enumeration method presented in this article can be summarized as follows. Once a certain specialization of Demazure characters has been identified with an interesting combinatorial function, the limit construction for affine Kac-Moody algebras can be used to carry it along towards the character of the integrable highest weight module, and derive asymptotic identities. There are at least two bottlenecks that one has to pass. First, a suitable character formula (for the limiting integrable highest weight representation) has to be available, that performs well with the chosen specialization. Fortunately, there is a great number of results and literature available, e.g. [4,12,14,15] (see also [2,5,16]). Second, the domain, of the combinatorial function, that enumerates the objects in question must lie in the region of convergence of the limiting expressions. For example, Demazure modules specialize to tensor products of representations of the underlying finite-dimensionsal Lie algebra. Unfortunately, the analytic string functions limiting the tensor product multiplicities cannot be simply evaluated, for reasons of (non-)convergence, at the value 1. A much finer analysis of their asymptotic behavior when q → 1 is needed, that has to exploit the fact that we deal with modular forms [15]. Such an asymptotic analysis must take the maximal weights in the integrable highest weight module into account where those string functions emerge. Possibly borrowing and mimicking terminology from stochastic analysis like the central limit region, moderate and strong deviations region, and region of rare events. An investigation of tensor product multiplicities along those lines is planned in a future publication.
An interesting alternative project could be to re-interpret our asymptotic enumeration method geometrically through the geometric realization of Demazure and integrable highest weight modules via cohomology of Schubert and flag varieties [18].

Acknowledgements
This work was supported by the Swiss National Science Foundation (grant PP00P2-128455), the National Centre of Competence in Research 'Quantum Science and Technology', and the German Science Foundation (SFB/TR12, and grants CH 843/1-1, CH 843/2-1).
I would like to thank Matthias Christandl for his kind hospitality at the ETH Zurich, Peter Littelmann who drew my attention towards the Demazure module limit construction a couple of years ago, Thomas Bliem who pointed me towards Rogers-Szegő polynomials, and Ghislain Fourier for many helpful conversations.