p^{ℓ}-TORSION POINTS IN FINITE ABELIAN GROUPS AND COMBINATORIAL IDENTITIES

CHRISTOPHE DELAUNAY AND FRÉDÉRIC JOUHET

Abstract. The main aim of this article is to compute all the moments of the number of p^{ℓ}-torsion elements in some type of finite abelian groups. The averages involved in these moments are those defined for the Cohen-Lenstra heuristics for class groups and their adaptation for Tate-Shafarevich groups. In particular, we prove that the heuristic model for Tate-Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of p-Selmer groups of elliptic curves. For our purpose, we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context.

1. Introduction

In this work, we compute the averages, in some sense, of a large class of functions defined over some families of finite abelian groups. Theses averages come from the well-known Cohen-Lenstra heuristics model for understanding the behavior of class groups of number fields and their adaptation for Tate-Shafarevich^{1} groups of elliptic curves (see [CL84, CM90, Del01, Del07]). In particular, our computations lead to a conjectural formula for all the moments of the number of n-torsion points in Tate-Shafarevich groups. We prove that our prediction is consistent with the recent model of Poonen and Rains that led them to conjecture a formula for all the moments of the number of n-torsion points in the Selmer groups of elliptic curves if n is squarefree (see [PR12]). Furthermore, our approach allows us to conjecture a formula for the moments of the number of n-torsion points in the Selmer groups of elliptic curves for all positive integer n (not only squarefree).

The computation of our averages involves several combinatorial tools. We introduce multivariate polynomials R_{λ}, indexed by integer partitions λ, depending on a fixed complex parameter t and we investigate combinatorial properties satisfied by them. As λ runs through all the set of integer partitions with parts bounded by a fixed integer ℓ, our polynomials form a basis of the polynomial ring ℤ(t)[x_{1}, x_{2}, …, x_{ℓ}]. Hence, these polynomials R_{λ} can be uniquely and explicitly expanded in terms of the monomials, thus defining an infinite ℓ-dimensional matrix, which can be explicitly inverted. The coefficients appearing in this inversion yield nice combinatorial expressions for our averages. Moreover, these coefficients appear to be related to a well-known algebraic combinatorial context, e.g., the problem of counting the number of subgroups of a finite abelian p-group with a given structure. This problem is actually at the heart of the construction of the Hall algebra, and the Hall-Littlewood symmetric polynomials. Furthermore, from the probabilistic model and the averages we computed, we obtain combinatorial identities involving q-series and the theory of Hall-Littlewood functions. A precise combinatorial analysis of the previously mentioned coefficients appearing in the inversion process allows us to prove that the Poonen and Rains model and the heuristic model for Tate-Shafarevich groups are compatible.

Date: April 2, 2013.

1Note that the original heuristic assumption for Tate-Shafarevich groups in [Del01] must be modified a little bit; this correction is explained in section 6.2.

This article is organized as follows: in the next section, we recall the definitions of averages we consider. In Section 3, we fix some combinatorial notations and recall basic facts about integer partitions; we will also study some combinatorial identities related to the theory of Hall-Littlewood symmetric functions. Then, we define in Section 4 the polynomials R_{λ} and prove the above mentioned inversion formula. We also give some properties satisfied by the coefficients appearing in the inversion process, and we show how they are related to the problem of counting subgroups of finite abelian p-groups. In Section 5, we highlight the links between R_{λ} and finite abelian p-groups and deduce some averages. In Section 6, we briefly recall the philosophy of the heuristics on class groups and on Tate-Shafarevich groups and give several consequences of our formulas on heuristics. In particular, concerning the Tate-Shafarevich groups of elliptic curves, we discuss how the heuristics principle and the Poonen and Rains model complement each other. Finally, we end the paper in Section 7 by some combinatorial consequences and related questions.

2. u-averages on finite abelian p-groups and motivations

Throughout the paper, p will always denote a prime number. The letter H (or H_{λ} if H is indexed by an integer partition λ) is used for an isomorphism class of a finite abelian p-group and the letter G (or G_{λ}) is used for an isomorphism class of a (finite abelian) p-group of type S. As in [Del01, Del07], we say that a p-group G is of type S if G is a finite abelian p-group endowed with a non-degenerate, bilinear, alternating pairing:

β : G × G → ℚ/ℤ.

An isomorphism of groups of type S is an isomorphism of groups preserving the pairings. A p-group of type S has the form G = H ×H. Reciprocally, each group G, of the form H ×H has a unique structure of group of type S up to isomorphism. In particular, if G is a group of type S then | Aut^{s}(G)|, the number of automorphisms of G that respect the (underlying) pairing β, does not depend on β. Due to these facts, the pairings associated to groups of type S will never be specified in the sequel.

2.1. Definitions of u-averages. Let h be a complex-valued function defined on isomorphism classes of finite abelian p-groups and let u be a nonnegative real number. In [CL84], Cohen and Lenstra defined an u-average^{2} M_{u}(h), of the function h (in particular, if h is the characteristic function of a property 𝒫, M_{u}(h) is called the u-probability of 𝒫), and we have

M_{u}(h) = \frac{∑_{n⩾1} p^{−nu} ∑_{H(p^{n})} \frac{h(H)}{| Aut(H)|}}{∏_{j⩾1} (1 − p^{−u−j})^{−1}} ,

if the first sum on the numerator converges, and where ∑_{H(p^{n})} means that the sum is over all isomorphism classes of finite abelian groups H of order p^{n}. The number M_{u}(h) is indeed an average since whenever h ≡ 1 we have M_{u}(h) = 1, i.e. (see [Hal38, CL84])

2Actually, the u-averages M_{u}(h), as well as Mu^{S}(h), are defined for all finite abelian groups, not only for p-groups. However, for the functions h we have in mind, all the p-parts behave independently.

∏_{j⩾1} (1 − p^{−u−j})^{−1} = ∑_{n⩾1} p^{−nu} ∑_{H(p^{n})} \frac{1}{| Aut(H)|}.

Cohen and Lenstra introduced such averages as a heuristic model in order to give precise predictions for the behavior of class groups of number fields varying in certain natural families. We will come back to this notion and give some examples later.

The aim of [Del01] was to adapt Cohen-Lenstra’s model, to give predictions for the behavior of Tate-Shafarevich groups of elliptic curves varying in certain natural families. Recall that if they are finite (as it is classically conjectured), Tate-Shafarevich groups are groups of type S. One can define, for a nonnegative real number u, the u-average (in the sense of groups of type S), denoted by Mu^{S}(g), for a function g defined on isomorphism classes of groups of type S. We have

Mu^{S}(g) = \frac{∑_{n⩾1} p^{−nu} ∑_{G(p^{n})} \frac{|G| g(G)}{| Aut^{S}(G)|}}{∏_{j⩾1} (1 − p^{−2u−2j+1})^{−1}} ,

if the first sum on the numerator converges, and where ∑_{G(p^{n})} means that the sum is over all isomorphism classes of groups G of type S and order p^{n} (the sum being empty if n is odd). As above, the number Mu^{S}(g) is an average since if g ≡ 1, then Mu^{S}(g) = 1, i.e. (see [Del01]),

∏_{j⩾1} (1 − p^{−2u−2j+1})^{−1} = ∑_{n⩾1} p^{−nu} ∑_{G(p^{n})} \frac{|G|}{| Aut^{S}(G)|} = ∑_{n⩾1} p^{−2nu} ∑_{G(p^{2n})} \frac{|G|}{| Aut^{S}(G)|}.

As for finite abelian groups, Mu^{S}(g) will be called the u-probability of 𝒫 if g is the characteristic function of a property 𝒫 for groups of type S. It will always be clear in the context whether we are concerned with averages in the sense of finite abelian groups or groups of type S. If h is defined over isomorphism classes of finite abelian p-groups, it is not difficult to deduce the u-average Mu^{S}(h) of h restricted to isomorphism classes of p-groups of type S from M_{u}(h), the u-average of h in the sense of finite abelian groups (see [Del07, Del11]), and reciprocally.

2.2. Motivations. The goal of [Del11] was to study the probability laws of the p^{ℓ}-ranks of abelian p-groups (resp. p-groups of type S) in the sense of the above probability model. In particular, a closed formula was given for the u-averages of the function h(H) = |H[p^{ℓ}]| (resp. g(G) = |G[p^{ℓ}]|) for any integer ℓ ⩾ 0, where H[p^{ℓ}] denotes the p^{ℓ}-torsion elements subgroup of H. Such computations were motivated by a question of M. Bhargava about the heuristics for Tate-Shafarevich groups and their comparison with theoretical results of Bhargava and Shankar ([BS10b, BS10a]) about the distribution of p-Selmer groups (with p = 2 and 3) of all elliptic curves.

From this point of view, our work aims to generalize the study in [Del11], and to give formulas for the u-averages of h(H) = |H[p^{ℓ}]|^{m_{ℓ}} (resp. g(G) = |G[p^{ℓ}]|^{m_{ℓ}}) for all integers ℓ ⩾ 0 and m_{ℓ} ⩾ 0. As we will see later, the computations are much more technical as they involve deeper combinatorial tools. Roughly speaking, in order to evaluate the u-average of h(H) = |H[p^{ℓ}]|^{m_{ℓ}}, we will more generally compute the u-averages of any function h(H) = |H[p]|^{m_{1}}|H[p^{2}]|^{m_{2}} ⋯ |H[p^{ℓ}]|^{m_{ℓ}} for all nonnegative integers m_{1}, m_{2}, …, m_{ℓ}. We are able to find a formula by induction; but actually we directly obtain a closed formula by using a ℓ-dimensional matrix inversion result which takes the place of the recursion principle. The motivation for this computation will be explained in the next section and comes originally from the following question of Poonen: take g(G) = |G[p^{ℓ}]|^{m_{ℓ}}; is it true that M0^{S}(g) = M1^{S}(g) · p^{ℓm_{ℓ}}? We will actually positively answer this question, by proving the following more general result.

Theorem 1. For any positive integer ℓ, let m_{1}, m_{2}, …, m_{ℓ} be ℓ nonnegative integers. Consider the following function g defined on isomorphism classes of p-groups of type S:

g(G) = |G[p]|^{m_{1}}|G[p^{2}]|^{m_{2}} ⋯ |G[p^{ℓ}]|^{m_{ℓ}}.

Then we have

M0^{S}(g) = M1^{S}(g) p^{m_{1}+2m_{2}+···+ℓm_{ℓ}}.

As we will explain, the fact that the answer to Poonen’s question is indeed positive makes the Poonen-Rains model for Selmer groups compatible with the heuristics for Tate-Shafarevich groups.

From a combinatorial point of view, our computations and the tools they require (including the probabilistic model for abelian p-groups), allow us to obtain a family of interesting combinatorial identities. Some of them are related to summation formulas for Hall-Littlewood symmetric functions, and can be proved in a non trivial way through this theory, while others seem to be new, or at least not direct consequences of previous known identities of the same type. At the heart of our computations appears some coefficients, denoted by C_{λ,μ}(q) (where λ and μ are integer partitions), which satisfy nice combinatorial properties, and, setting q = 1/p, are related to the number of subgroups H_{μ} in a finite abelian p-group H_{λ} (a group H_{λ} is often called a group of type λ^{3}). In particular, their symmetry relation given by Theorem 5 yields a proof of Theorem 1, and therefore a positive answer to Poonen’s question. As we will see, Theorem 5 is actually a consequence of a well-known property in the framework of finite abelian p-groups [But94, Del48], and therefore related to Hall-Littlewood symmetric polynomials. We shall however give an alternative proof of this symmetry property by using a general summation formula (due to Lascoux [Las07]) of Hall-Littlewood functions. Finally, to prove Theorem 1, we will also need an evaluation of the number of injective homomorphisms of a group H_{λ} into the finite abelian p-group H, associated to a result of Cohen and Lenstra from [CL84].

3. Integer partitions and Hall-Littlewood functions

3.1. Notations and definitions. Recall that a partition λ := (λ_{1} ⩾ λ_{2} ⩾ ⋯) of a nonnegative integer n is a finite decreasing sequence of nonnegative integers whose sum is equal to n. We often use the notation |λ| := ∑_{i=1}^{ℓ} λ_{i} = n, and the integer ℓ = ℓ(λ) = max{i|λ_{i} > 0} is called the length of the integer partition λ. We will also denote by λ ⊢ n the fact that λ is a partition of the integer n. The integers λ_{i} (for 1 ⩽ i ⩽ ℓ) are the parts of λ. It is also convenient to write an integer partition by its multiplicities: we denote by m_{i} = m_{i}(λ) := #{j|λ_{j} = i} ⩾ 0 the multiplicity of the integer i in λ. Therefore the partition λ can be written λ = 1^{m_{1}}2^{m_{2}} …, and we have n = m_{1} + 2m_{2} + · · ·. Finally, we denote by λ′ the conjugate partition of λ, whose parts are λ′_{i} := #{j|λ_{j} ⩾ i}, for 1 ⩽ i ⩽ λ_{1}. Thus we have |λ′| = |λ|, ℓ(λ′) = λ_{1}, and for any integer i, m_{i}(λ) = λ′_{i} − λ′_{i+1} (and equivalently m_{i}(λ′) = λ_{i} − λ_{i+1}). We will also use the following standard statistic on integer partitions

3If λ := (λ_{1} ⩾ λ_{2} ⋯) is a partition, a p-group H_{λ} is said to be a group of type λ if it is isomorphic to ℤ/p^{λ_{1}}ℤ ⊕ ℤ/p^{λ_{2}}ℤ ⊕ · · ·. We will not use this terminology in order to avoid any confusion with the notion of group of type S we introduced before.

n(λ) := ∑_{i⩾1}(i − 1)λ_{i} = ∑_{i⩾1} \binom{λ′_{i}}{2}.

Next we recall some standard definitions and notations for q-series, which can be found in [GR04]. Let q be a fixed complex parameter (the “base”) with |q| ≠ 1. We define for any real number a and any k ∈ ℤ the q-shifted factorial by

(a)_{k} ≡ (a; q)_{k} := \{ \begin{matrix}1 if k = 0 \\ (1 − a) … (1 − aq^{k−1}) if k > 0 \\ 1/(1 − aq^{−1}) … (1 − aq^{k}) if k < 0, \end{matrix}

and (a)_{∞} := lim_{k→+∞}(a)_{k}. The base q can be omitted when there is no confusion (writing (a)_{k} for (a; q)_{k}, etc). For the sake of simplicity, write for k ∈ ℤ ∪ {∞}

(a_{1}, …, a_{m})_{k} := (a_{1})_{k} × ⋯ × (a_{m})_{k}.

Recall also the definition of the q-binomial coefficient

\left[{n \atop k}\right]_{q} := \frac{(q)_{n}}{(q)_{k}(q)_{n−k}} ,

which is a polynomial in the variable q, with positive integral coefficients.

We will need later a special case of the so-called finite q-binomial theorem (see for instance [GR04]), which for any complex number z and any nonnegative integer n, can be written as follows

(1)  ∑_{k=0}^{n}(−1)^{k}z^{k}q^{k(k−1)/2}\left[{n \atop k}\right]_{q} = (z)_{n}.

Next, we recall some basic facts about the theory of Hall-Littlewood functions, which can be found with more details in [Mac95, Chapter III]. Hall-Littlewood polynomials were introduced by Hall to evaluate the number of subgroups of type μ and cotype ν in a finite abelian p-group of type λ, and expressed explicitly later by Littlewood (see for instance [But94] for detailed explanations of this construction). Let S_{n} be the symmetric group, Λ_{n} = ℤ[x_{1}, …, x_{n}]^{S_{n}} be the ring of symmetric polynomials in n independant variables and Λ be the ring of symmetric functions in countably many independant variables. For a set of n variables x = {x_{1}, …, x_{n}}, the Hall-Littlewood polynomials P_{λ}(x; q) are defined by

P_{λ}(x; q) := ∏_{i⩾1} \frac{(1 − q)^{m_{i}}}{(q)_{m_{i}}} ∑_{w∈S_{n}} w (x_{1}^{λ_{1}} … x_{n}^{λ_{n}} ∏_{i<j} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ) ,

where w acts by permuting the variables. These polynomials are symmetric in x, homogeneous of degree |λ|, with coefficients in ℤ[q], and form a ℤ[q] basis of Λ_{n}[q], which interpolates between monomial (for q = 1) and Schur (for q = 0) polynomials. Moreover, thanks to their classical stability property, they may be extended to the Hall-Littlewood functions in an infinite number of variables in order to form a ℤ[q] basis of Λ[q]. Another useful result is the specialization

(2)  P_{λ}(z, zq, …, zq^{n−1}; q) = \frac{z^{|λ|}q^{n(λ)}(q)_{n}}{(q)_{n−ℓ(λ)}b_{λ}(q)},

where

b_{λ}(q) := ∏_{i⩾1}(q)_{λ′_{i}−λ′_{i+1}}.

Finally, by using the Cauchy identity for Hall-Littlewood polynomials [Mac95, p. 224, (4.4)], the specialization (2), and the Pieri formula [Mac95, p. 224, (3.2)], it is possible to derive the following identity from [Mac95], which can be seen as a q-binomial theorem for Hall-Littlewood functions

(3)  ∑_{λ} q^{n(λ)}(a; q^{−1})_{ℓ(λ)}P_{λ}(x; q) = ∏_{i⩾1} \frac{1 − ax_{i}}{1 − x_{i}} ,

where |q| < 1, a is a complex number, and the summation is taken over all integer partitions.

3.2. Lascoux’s summation of Hall-Littlewood functions. The Rogers-Rama-nujan identities are two of the most famous q-series identities, with deep connections to many branches of mathematics. Lots of proofs of these identities have been given, among which one by Stembridge in [Ste90], who generalized work by Macdonald from [Mac95] to show that the Rogers-Ramanujan identities (among others) may be obtained by appropriate specializations of finite summations for Hall-Littlewood polynomials. In [ASW99], Andrews, Schilling and Warnaar generalized the Rogers-Ramanujan identities to three identities labelled by the Lie algebra A_{2}. Later, seeking a way of proving the latter through the theory of Hall-Littlewood polynomials as Stembridge did for the Rogers-Ramanujan identities, Warnaar showed in [War06] the following A_{2}-type identity for Hall-Littlewood functions

(4) ∑_{λ, μ} q^{n(λ)+n(μ)−(λ′|μ′)}P_{λ}(x; q)P_{μ}(y; q) = ∏_{i⩾1} \frac{1}{(1 − x_{i})(1 − y_{i})} ∏_{i,j⩾1} \frac{1 − x_{i}y_{j}}{1 − x_{i}y_{j}/q} ,

for |q| < 1, where the sum is over all pairs of integers partitions, and (λ|μ) := ∑_{i⩾1} λ_{i}μ_{i}. In [War06], Warnaar explains how (4) yields one of the three A_{2}-type Rogers-Ramanujan identities from [ASW99]. For our purpose, we will actually need a generalization of (4), which was discovered and proved by Lascoux in [Las07]. To this aim, he used the modified Hall-Littlewood polynomials Q′_{μ}, which form the adjoint basis of P_{λ} with respect to the Hall scalar product (see [Las07, Mac95] for details). More precisely, Lascoux’s formula can be written as

(5) ∑_{λ} ∑_{μ⊆λ} P_{λ}(x; q)P_{μ}(y; q)b_{λ}(q)Q′_{λ/μ}(1; q) = ∏_{i⩾1} \frac{1}{1 − x_{i}} ∏_{j⩾1} \frac{1 − qx_{i}y_{j}}{1 − x_{i}y_{j}} , (|q| < 1) ,

where μ ⊆ λ means that μ_{i} ⩽ λ_{i} for all integers i, Q′_{λ/μ} are the skew modified Hall-Littlewood polynomials, and the coefficients Q′_{λ/μ}(1; q) are explicitely given by

(6)  Q′_{λ/μ}(1; q) = q^{|μ|+n(λ)+n(μ)−(λ′|μ′)} ∏_{i⩾1} \left[{λ′_{i} − μ′_{i+1} \atop λ′_{i} − μ′_{i}}\right]_{q}.

It is easy to see that Q′_{λ/μ}(1; q) are polynomials in the variable q, with positive integral coefficients, for which Lascoux gave a combinatorial interpretation in [Las07]. As remarked in [WZ12], setting q = 1/p with p a prime number, Q′_{λ/μ}(1; 1/p) are related to the number of subgroups of type μ in a finite abelian p-group of type λ. We will come back to this link in the next section. Finally, although it is not trivial at first sight, Lascoux showed in [Las07] how Warnaar’s identity (4) is a consequence of (5).

4. The polynomials R_{λ}(x, t)

For our purpose, we fix a positive integer ℓ, and we consider a partition λ = (λ_{1} ⩾ λ_{2} ⩾ ⋯) such that λ_{1} ⩽ ℓ (i.e., the length of the conjugate partition λ′ satisfies ℓ(λ′) ⩽ ℓ). For a set of ℓ variables x = {x_{1}, …, x_{ℓ}} (adding by convention x_{0} := 1) and a complex parameter t, we define the polynomial^{4}

(7)  R_{λ}(x; t) := ∏_{i=1}^{ℓ} ∏_{j=λ′_{i+1}}^{λ′_{i}−1}(x_{i} − t^{j}x_{i−1}),

which can be rewritten through the multiplicities m_{1}, …, m_{ℓ} of the partition λ as follows

R_{λ}(x; t) = ∏_{i=1}^{ℓ} x_{i}^{λ′_{i}−λ′_{i+1}} ∏_{j=0}^{λ′_{i}−λ′_{i+1}−1} (1 − t^{j+λ′_{i+1}} \frac{x_{i−1}}{x_{i}} )

(8)  = ∏_{i=1}^{ℓ} x_{i}^{m_{i}}(t^{m_{i+1}+···+m_{ℓ}}x_{i−1}/x_{i}; t)_{m_{i}}.

It is clear that whenever λ runs through all partitions with λ_{1} ⩽ ℓ, then the polynomials R_{λ}(x; t) form a basis of ℤ(t)[x_{1}, x_{2}, …, x_{ℓ}]. Our goal is to expand the polynomial R_{λ} in terms of the monomials x_{1}^{k_{1}} … x_{ℓ}^{k_{ℓ}}, and to invert this relation. To this aim, writing λ = 1^{m_{1}}2^{m_{2}} …, we have to express the monomial x_{1}^{m_{1}} … x_{ℓ}^{m_{ℓ}} in terms of the polynomials R_{μ}(x; t), where μ = (μ_{1} ⩾ μ_{2} ⩾ …) are integer partitions included in λ, i.e., satisfying μ_{i} ⩽ λ_{i} for any integer i (note that this is equivalent to the conditions μ′_{i} ⩽ λ′_{i} for any integer i).

It is easy to see that R_{λ} can be expanded as a sum of monomials x^{μ} for μ ⊆ λ, and this is what we explicitly do in the next proposition.

Proposition 2. We have

R_{λ}(x; q) = ∑_{μ⊆λ} x_{1}^{μ′_{1}−μ′_{2}} … x_{ℓ}^{μ′_{ℓ}} (−1)^{|λ|−|μ|} q^{n(λ\μ)}

× q^{λ′_{2}(λ′_{1}−μ′_{1})+···+λ′_{ℓ}(λ′_{ℓ−1}−μ′_{ℓ−1})} ∏_{i⩾1} \left[{λ′_{i} − λ′_{i+1} \atop λ′_{i} − μ′_{i}}\right]_{q}.

Proof. We expand the expression (8) of the polynomials R_{λ}(x; q) by using ℓ times the q-binomial theorem (1), which yields

R_{λ}(x; q) = ∑_{k_{1},...,k_{ℓ}} x_{1}^{m_{1}−k_{1}+k_{2}} … x_{ℓ−1}^{m_{ℓ−1}−k_{ℓ−1}+k_{ℓ}}x_{ℓ}^{m_{ℓ}−k_{ℓ}} (−1)^{k_{1}+···+k_{ℓ}}

× q^{\binom{k_{1}}{2}+···+\binom{k_{ℓ}}{2}}q^{(m_{2}+···+m_{ℓ})k_{1}+···+m_{ℓ}k_{ℓ−1}} \left[{m_{1} \atop k_{1}}\right]_{q} … \left[{m_{ℓ} \atop k_{ℓ}}\right]_{q}.

Definig the integer partition μ by μ = 1^{k_{1}}2^{k_{2}} … ℓ^{k_{ℓ}}, and replacing in this expression k_{i} by λ′_{i} − μ′_{i}, we obtain the formula. □

We are now interested in the reciprocal identity, i.e., the expansion of the monomials x^{λ} in terms of the polynomials R_{μ}. For this, we need the following multidimensional matrix inversion, which was communicated to us by Michael Schlosser.

4Note that we take the parameter t in order to avoid confusions since we will need to specialize t = q and t = 1/q later.

Lemma 3. The infinite lower-triangular ℓ-dimensional matrices F = (F_{a,b})_{a,b ∈ℤ^{ℓ}} and G = (G_{a,b})_{a,b ∈ℤ^{ℓ}} are inverses of each other, where

F_{a,b} = (−1)^{|a|−|b|}q^{∑{ℓ \atop i=1}(\binom{a_{i}+1}{2}+\binom{b_{i}}{2}+a_{i}b_{i+1})} ∏_{i=1}^{ℓ} \frac{(q)_{a_{i}−b_{i+1}}}{(q)_{a_{i}−b_{i}}} ,

and

G_{a,b} = q^{− ∑_{i=1}^{ℓ} b_{i}(a_{i}+a_{i+1})} ∏_{i=1}^{ℓ} \frac{1}{(q)_{a_{i}−b_{i}}(q)_{b_{i}−a_{i+1}}} ,

with a_{ℓ+1} = b_{ℓ+1} = 0 and |a| = a_{1} + ⋯ + a_{ℓ}, |b| = b_{1} + ⋯ + b_{ℓ}.

Proof. We will see that the previous inversion follows from a suitable specialization of a multidimensional matrix inversion due to Schlosser in [Sch07], which is itself a special case of the inversion of the Pieri rule for Macdonald polynomials [LS06, Theorem 2.6]. First recall that the infinite lower-triangular ℓ-dimensional matrices F = (F_{a,b})_{a,b ∈ℤ^{ℓ}} and G = (G_{a,b})_{a,b ∈ℤ^{ℓ}} are inverses of each other means that for any pair of infinite ℓ-dimensional vectors α_{a} and β_{b}, we have

(9)  α_{a} = ∑_{b⩽a} F_{a,b} β_{b} ⇔ β_{a} = ∑_{b⩽a} G_{a,b} α_{b},

where b ⩽ a means that b_{i} ⩽ a_{i} for all integers i ∈ {1, …, ℓ}. Now in [Sch07, (A.9)], replace n by ℓ, let t_{0} → 0 and t_{i} ↦ u_{i}/u_{i+1} for i = 1, …, ℓ. We denote by (f_{a,b})_{a,b ∈ℤ^{ℓ}} and (g_{a,b})_{a,b ∈ℤ^{ℓ}} the resulting infinite lower-triangular ℓ-dimensional matrices, which are inverses of each other. Then we define infinite invertible diagonal ℓ-dimensional matrices D and E by

D_{a} := u_{1}^{a_{1}}u_{2}^{2a_{2}−a_{1}} … u_{ℓ}^{ℓa_{ℓ}−(ℓ−1)a_{ℓ−1}}u_{ℓ+1}^{−ℓa_{ℓ}}(−1)^{a_{ℓ}} ∏_{i=1}^{ℓ−1}(q)_{a_{i}−a_{i+1}}

× q^{a_{l}^{2}/2+a_{ℓ}/2−∑_{i=1}^{ℓ}(a_{i}a_{i+1}−ia_{i})}

and

E_{b} := u_{1}^{b_{1}}u_{2}^{2b_{2}−b_{1}} … u_{ℓ}^{ℓb_{ℓ}−(ℓ−1)b_{ℓ−1}}u_{ℓ+1}^{−ℓb_{ℓ}}(−1)^{b_{ℓ}}(q)_{b_{ℓ}} × q^{b_{ℓ}^{2}/2+b_{ℓ}/2−∑_{i=1}^{ℓ}(b_{i}^{2}−ib_{i})}.

It is clear that through the definition in (9), the matrices (D_{b}f_{a,b}E_{a}^{−1})_{a,b ∈ℤ^{ℓ}} and (E_{b}g_{a,b}D_{a}^{−1})_{a,b ∈ℤ^{ℓ}} are still inverses of each other. Finally, we let u_{1} → ∞, then u_{2} → ∞, ... , u_{ℓ} → ∞ (in this order) in the previous matrices, which yields after some complicated but elementary manipulations

lim_{u_{1},...,u_{ℓ}→∞} D_{b}f_{a,b}E_{a}^{−1} = F_{a,b},

and

lim_{u_{1},...,u_{ℓ}→∞} E_{b}g_{a,b}D_{a}^{−1} = G_{a,b}.

This completes the proof. □

Now we are able to prove the following result, which is our desired inversion.

Theorem 4. For any positive integer ℓ, and an integer partition λ = 1^{m_{1}}2^{m_{2}} … ℓ^{m_{ℓ}}, we have

x_{1}^{m_{1}} … x_{ℓ}^{m_{ℓ}} = ∑_{μ⊆λ} C_{λ, μ}(q) R_{μ}(x; q),

where

(10)  C_{λ, μ}(q) := q^{∑_{i=1}^{ℓ} μ′_{i+1}(λ′_{i}−μ′_{i})} ∏_{i⩾1} \left[{λ′_{i} − μ′_{i+1} \atop λ′_{i} − μ′_{i}}\right]_{q}.

Proof. Recall that we have

R_{λ}(x; q) = ∑_{μ⊆λ} x_{1}^{μ′_{1}−μ′_{2}} … x_{ℓ}^{μ′_{ℓ}} (−1)^{|λ|−|μ|} q^{\binom{λ′_{1}−μ′_{1}}{2}+···+\binom{λ′_{ℓ}−μ′_{ℓ}}{2}}

× q^{λ′_{2}(λ′_{1}−μ′_{1})+···+λ′_{ℓ}(λ′_{ℓ−1}−μ′_{ℓ−1})}\left[{λ′_{1} − λ′_{2} \atop λ′_{1} − μ′_{1}}\right]_{q} … \left[{λ′_{ℓ} \atop λ′_{ℓ} − μ′_{ℓ}}\right]_{q}.

In order to use the inversion of Lemma 3, we observe that this relation is equivalent to the following:

(11)  β_{a} = ∑_{b⩽a} G_{a,b} α_{b},

where a_{i} = λ′_{i}, b_{i} = μ′_{i},

β_{a} := (−1)^{|a|} \frac{q^{− ∑_{i=1}^{ℓ}(\binom{a_{i}}{2}+a_{i}a_{i+1})}}{(q)_{a_{1}−a_{2}} … (q)_{a_{l}}} R_{λ}(x; q),

and

α_{b} := (−1)^{|b|}q∑{ℓ \atop i=1} ^{\binom{b_{i}+1}{2}}x_{1}^{b_{1}−b_{2}} … x_{ℓ}^{b_{ℓ}}.

The multidimensional matrix inversion in Lemma 3 applied to (11) yields

α_{a} = ∑_{b⩽a} F_{a,b} β_{b},

which is equivalent to

x_{1}^{a_{1}−a_{2}} … x_{ℓ}^{a_{ℓ}} = ∑_{b⩽a} q^{∑_{i=1}^{ℓ} b_{i+1}(a_{i}−b_{i})} ∏_{i=1}^{ℓ} \frac{(q)_{a_{i}−b_{i+1}}}{(q)_{b_{i}−b_{i+1}}(q)_{a_{i}−b_{i}}} R_{μ}(x; q),

and this completes the proof after replacing the variables a_{i} by λ′_{i} and b_{i} by μ′_{i}, and then recalling the definition of the q-binomial coefficient. □

Remark. It is easy to see from its definition that C_{λ, μ}(q) is a polynomial in the variable q, with positive integral coefficients. Moreover, thanks to the classical transformation \left[{n \atop k}\right]_{1/q} = q^{k(k−n)}\left[{n \atop k}\right]_{q}, it is related to the coefficient Q′_{λ/μ}(1; q) (defined in (6)) by

(12)  C_{λ, μ}(1/q) = q^{n(μ)−n(λ)}Q′_{λ/μ}(1; q),

which, by setting q = 1/p with p a prime number, counts exactly the number of subgroups of type μ in a finite abelian p-group of type λ (see [Del48, But94]).

Thanks to this remark, the following symmetry property satisfied by the coefficients C_{λ, μ}(q) becomes a direct consequence of the Pontryagin duality for finite abelian groups. However, we give below a very natural proof which uses Lascoux’s formula (5), therefore highlighting and exploiting the link, given by (12), between these coefficients and Hall-Littlewood functions.

Theorem 5. Let m and k be two nonnegative integers, and λ be a partition of m. Then we have

∑_{{|μ|=k \atop μ⊆λ}} C_{λ, μ}(q) = ∑_{{|μ|=m−k \atop μ⊆λ}} C_{λ, μ}(q).

Equivalently, the polynomial ∑_{μ⊆λ} C_{λ,μ}(q)T^{|μ|} ∈ ℕ[q][T] is self-reciprocal.

Proof. We will actually show the following equivalent form of the theorem

(13)  ∑_{{|μ|=k \atop μ⊆λ}} C_{λ, μ}(1/q) = ∑_{{|μ|=m−k \atop μ⊆λ}} C_{λ, μ}(1/q).

In Lascoux’s formula (5), consider the specialization y = {z, zq, zq^{2}, … }, and use the n → ∞ case of (2), together with (12) and the homogeneity of the Hall-Littlewood polynomials to derive

(14)  ∑_{λ} ∑_{μ⊆λ} q^{n(λ)}P_{λ}(x; q)z^{|μ|}C_{λ,μ}(1/q) = ∏_{i⩾1} \frac{1}{(1 − x_{i})(1 − zx_{i})},

which is the generating function for the left-hand side of (13). We then compute the same generating function for the right-hand side of (13)

∑_{k⩾0} ∑_{λ} q^{n(λ)}P_{λ}(x; q)z^{|λ|−k} ∑_{{|μ|=|λ|−k \atop μ⊆λ}} C_{λ,μ}(1/q)

= ∑_{λ,μ | μ⊆λ} q^{n(λ)}P_{λ}(x; q)z^{|λ|−|μ|}C_{λ,μ}(1/q)

= ∑_{λ,μ | μ⊆λ} q^{n(λ)}P_{λ}(zx; q)(1/z)^{|μ|}C_{λ,μ}(1/q),

where we have used the homogeneity of the Hall-Littlewood polynomials and the notation zx for the set of variables {zx_{i}, i ⩾ 1}. Therefore we can use (14) with x replaced by zx and z by 1/z, to get

∏_{i⩾1} \frac{1}{(1 − zx_{i})(1 − zx_{i}/z)},

which is obviously equal to the right-hand side of (14). We thus have the identity

∑_{λ,ν | ν⊆λ} q^{n(λ)}P_{λ}(x; q)z^{|ν|}C_{λ,ν}(1/q) = ∑_{λ,ν | ν⊆λ} q^{n(λ)}P_{λ}(x; q)z^{|λ|−|ν|}C_{λ,ν}(1/q),

which yields (13), by using the fact that P_{λ}(x; q) is a ℤ[q]-basis of Λ[q] and extracting the coefficient of z^{k}q^{n(λ)}P_{λ}(x; q) on both sides. □

Examples. 1- First, we have x_{1} = R_{1^{1}} + 1, and more generally

x_{n} = R_{n^{1}} + R_{(n−1)^{1}} + ⋯ + R_{1^{1}} + 1.

That is exactly the relation obtained in [Del11] in order to compute the u-average of the function H ↦ |H[p^{n}]|. 2- One can also see that x_{1}^{2} = R_{1^{2}} + (t + 1)R_{1^{1}} + 1 and that

x_{1}x_{2} = R_{1^{1}2^{1}} + tR_{2^{1}} + R_{1^{2}} + (t + 1)R_{1^{1}} + 1.

Therefore

x_{2}^{2} = R_{2^{2}} + (t + 1)R_{1^{1}2^{1}} + t(t + 1)R_{2^{1}} + R_{1^{2}} + (t + 1)R_{1^{1}} + 1.

3- As we saw the polynomial

∑_{μ⊆λ} C_{λ,μ}(t)T^{|μ|} ∈ ℤ(t)[T]

is in fact a self-reciprocal polynomial. Of course, this can be directly checked in the above two examples.

4- Let us consider the case x_{1}^{m}. Then the partitions μ ⊆ 1^{m} are exactly the partitions 1^{k} for k = 0, 1, …, m. Hence,

(15)  x_{1}^{m} = ∑_{k=0}^{m} \left[{m \atop k}\right]_{t}R_{1^{k}}.

We will come back later to this example.

Remark. As Ole Warnaar communicated to us, there is an interesting context containing generalizations of Theorems 4 and 5. Indeed, studying properties for Macdonald interpolation polynomials (which are symmetric and whose top homogeneous layer are the classical symmetric Macdonald polynomials, themselves being two parameters generalizations of the Hall-Littlewood polynomials), Okounkov defined in [Oko97] the (q, t)-binomial coefficients \left[{λ \atop μ}\right]_{q,t}, where λ and μ are integer partitions and q, t are complex parameters. These (q, t)-binomial coefficients are defined as quotients of special values of the Macdonald interpolation polynomials, and as remarked by Warnaar, it turns out by using (12) that we have

C_{λ,μ}(1/q) = \left[{λ \atop μ}\right]_{0,q}.

Many properties for these (q, t)-binomial coefficients are known, some of which should therefore have interesting algebraic number theoritic applications in our context. As we only need the Hall-Littlewood case for our purpose here, we plan to study further this connection with Macdonald polynomials in a later work. We only mention the following generalization of Theorem 5 (the latter being obtained by setting ν = 0, q = 0 and then replacing t by q), which is proved by Lascoux, Rains and Warnaar in [LRW09] (actually in the more general context of nonsymmetric Macdonald interpolation polynomials):

∑_{|μ|=k} \left[{λ \atop μ}\right]_{q,t}\left[{μ \atop ν}\right]_{q,t} = ∑_{|μ|=|λ|+|ν|−k} \left[{λ \atop μ}\right]_{q,t}\left[{μ \atop ν}\right]_{q,t},

where λ and ν are fixed integer partitions. Moreover, still remarked by Warnaar, the inversion in Theorem 4 can also be proved using a (q → 0) limit case of the inversion for (q, t)-binomial coefficients, which can be found in [Oko97], and can be stated as follows:

∑_{μ}(−1)^{|μ|−|λ|}q^{n(μ′)−n(λ′)}t^{n(λ)−n(μ)}\left[{λ \atop μ}\right]_{q,t}\left[{μ \atop ν}\right]_{1/q,1/t} = δ_{λ,ν},

where δ_{λ,ν} stands for the Kronecker symbol.

5. Links with finite abelian groups

5.1. Finite abelian p-groups and the polynomial R_{λ}. We have already seen in the previous section that the inversion we needed for our computations of uaverages is related to finite abelian p-groups, through the coefficients C_{λ,μ}. We highlight here this link. There is a classical bijection between the set of partitions of a fixed nonnegative integer m and the isomorphism classes of finite abelian pgroups of order p^{m}. Indeed, to any partition λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} of m, we associate the finite abelian p-group

H_{λ} = (ℤ/pℤ)^{m_{1}} ⊕ (ℤ/p^{2}ℤ)^{m_{2}} ⊕ ⋯ ⊕ (ℤ/p^{ℓ}ℤ)^{m_{ℓ}} .

Recall that λ′_{1}, λ′_{2}, …, λ′_{ℓ} denote the parts of the conjugate of λ (therefore λ′_{i} = ∑_{j⩾i} m_{j}), and we have (see [CL84])

(16)  | Aut(H_{λ})| = p^{λ_{1}^{′2}+λ_{2}^{′2}+···+λ_{ℓ}^{′2}} ∏_{j=1}^{ℓ}(1/p; 1/p)_{m_{j}}.

Furthermore if G_{λ} ≃ H_{λ} × H_{λ} is a group of type S, we have (see [Del01]):

| Aut^{S}(G_{λ})| = p^{2(λ_{1}^{′2}+λ_{2}^{′2}+···+λ_{ℓ}^{′2})+m} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{m_{j}}.

Remark also that we have

H_{λ}[p^{k}] ≃ (ℤ/pℤ)^{m_{1}} ⊕ (ℤ/p^{2}ℤ)^{m_{2}} ⊕ ⋯ ⊕ (ℤ/p^{k−1}ℤ)^{m_{k−1}} ⊕ (ℤ/p^{k}ℤ)^{λ′_{k}},

hence

(17)  |H_{λ}[p^{k}]| = p^{λ′_{1}+···+λ′_{k}}.

The following definitions will be useful in the sequel.

Definition 6. For a set of ℓ variables x = {x_{1}, …, x_{ℓ}}, let T (x) = T (x_{1}, …, x_{ℓ}) be a polynomial and H be a finite abelian p-group. We define

T (H) = T (|H[p]|, |H[p^{2}]|, …, |H[p^{ℓ}]|).

Definition 7. If μ = 1^{n_{1}}2^{n_{2}} ⋯ ℓ^{n_{ℓ}} is an integer partition, we denote by x^{μ} the following monomial

x^{μ} := x_{1}^{n_{1}}x_{2}^{n_{2}} ⋯ x_{ℓ}^{n_{ℓ}}.

Hence, with these definitions, the u-average of |H[p^{ℓ}]| can be simply written as M_{u}(x_{ℓ}). More generally, the u-average of |H[p]|^{n_{1}}|H[p^{2}]|^{n_{2}} ⋯ |H[p^{ℓ}]|^{n_{ℓ}} is simply M_{u}(x^{μ}) where μ denotes the integer partition μ = 1^{n_{1}}2^{n_{2}} ⋯ ℓ^{n_{ℓ}}.

In order to give a closed formula for the latter, we state the following result.

Theorem 8. For a positive integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be an integer partition and let H be a finite abelian p-group. Then the number of injective homomorphisms of H_{λ} into H is given by

∏_{i=1}^{ℓ} ∏_{j=λ′_{i+1}}^{λ′_{i}−1} (|H[p^{i}]| − p^{j}|H[p^{i−1}]|) ,

or in other words,

| {Hom_{inj}(H_{λ}, H)} | = R_{λ}(H; p).

Proof. This is probably a known formula that can be found in the literature, however we give here a sketch of proof. Write

H_{λ} = (ℤ/pℤ)^{m_{1}} ⊕ (ℤ/p^{2}ℤ)^{m_{2}} ⊕ ⋯ ⊕ (ℤ/p^{ℓ}ℤ)^{m_{ℓ}} ,

and take e_{1}, e_{2}, …, e_{λ′_{1}} a “canonical basis” in an obvious sense so that e_{1}, …, e_{m_{1}} are of order p and e_{m_{1}+m_{2}+···+m_{ℓ−1}+1}, …, e_{λ′_{1}} are of order p^{ℓ}. In order to define an injective homomorphism of H_{λ} into H, we need to specify the image f_{i} of each e_{i}. The image f_{λ′_{1}} of e_{λ′_{1}} is any point of order p^{ℓ} and there are |H[p^{ℓ}]| − |H[p^{ℓ−1}]| choices for this image (these correspond to the factor i = ℓ and j = 0 = λ′_{ℓ+1} in the theorem). Assume that k + 1 ⩽ λ′_{1}, and that f_{λ′_{1}}, …, f_{k+1} are already defined. Let p^{t} be the order of e_{k}. Note that the order of each f_{λ′_{1}}, …, f_{k+1} is ⩾ p^{t}. Now the image f_{k} of e_{k} must be an element of order p^{t} such that, for any r = 1, …, p^{t} −1, rf_{k} is not in the subgroup ⟨f_{k+1}, …, f_{λ′_{1}}⟩ generated by the images that are already defined. The latter means that the image of f_{k} in the quotient group H̃ = H/⟨f_{k+1}, …, f_{λ′_{1}}⟩ has order p^{t}. The number of elements of order p^{t} in H̃ is |H̃[p^{t}]| − |H̃[p^{t−1}]|. For each of these elements there are |⟨f_{k+1}, …, f_{λ′_{1}}⟩[p^{t}]| corresponding pre-images of order p^{t} in H. Hence the number of possibilities for f_{k} is

(|H̃[p^{t}]| − |H̃[p^{t−1}]) |⟨f_{k+1}, …, f_{λ′_{1}}⟩[p^{t}]| = |H[p^{t}]| − p^{λ′_{1}−k}|H[p^{t−1}]|.

This is exactly the term occurring with i = t and j = λ′_{1} − k in the formula of the theorem (and we indeed have λ′_{t+1} ⩽ λ′_{1} − k ⩽ λ′_{t} − 1). □

We derive the following consequence.

Corollary 9. Let n and m be nonnegative integers, and λ be a fixed partition of m. Then we have

∑_{μ⊢n} \frac{R_{λ}(H_{μ}; p)}{| Aut H_{μ}|} = \frac{1}{p^{n−m}(1/p; 1/p)_{n−m}} ,

which can equivalently be written as follows in terms of generating functions

∑_{n⩾0} z^{n} ∑_{μ⊢n} \frac{R_{λ}(H_{μ}; p)}{| Aut H_{μ}|} = \frac{z^{m}}{(z/p; 1/p)_{∞}} .

Proof. We clearly have

| Hom_{inj}(H_{λ}, H)| = | Aut(H_{λ})| · |{J subgroup of H, J ≃ H_{λ}}|.

Then Proposition 4.1 of [CL84] gives

| Aut(H_{λ})| ∑_{H(p^{n})} \frac{|\{J subgroup of H, J ≃ H_{λ}\}|}{| Aut(H)|} = \frac{1}{p^{n−m}(1/p; 1/p)_{n−m}} .

We obtain the first equality from Theorem 8 and the correspondence (μ ⊢ n) ↔ (H_{μ} of order p^{n}). The second equality comes from Euler’s identity (see for instance [GR04])

∑_{n⩾m} \frac{z^{n}}{p^{n−m}(1/p; 1; p)_{n−m}} = z^{m} ∑_{n⩾0} \frac{z^{n}}{p^{n}(1/p; 1; p)_{n}}

= z^{m} \frac{1}{(z/p; 1/p)_{∞}} .
□

Using Theorem 4, Definitions 6, 7, and Corollary 9, we obtain the following result.

Theorem 10. Let λ be a fixed integer partition, and z a complex number. We have

∑_{μ} \frac{x^{λ}(H_{μ})}{| Aut H_{μ}|} z^{|μ|} = \frac{1}{(z/p; 1/p)_{∞}} ∑_{ν⊆λ} C_{λ,ν}(p) z^{|ν|},

where the sum on the left-hand side is over all integer partitions μ, and the sum on the right-hand side is over all integer partitions ν satisfying ν_{i} ⩽ λ_{i} for all i.

5.2. A combinatorial formula. As a consequence in combinatorics, we derive the following by setting q = 1/p and switching μ into μ′ in Theorem 10, and finally using (16) and (17).

Corollary 11. For a positive integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be a fixed integer partition, and z be a complex number. We have:

(18)  ∑_{μ} \frac{q^{∑_{i⩾1} μ_{i}^{2}}q^{−m_{1}μ_{1}−···−m_{ℓ}(μ_{1}+···+μ_{ℓ})}}{b_{μ}(q)} z^{|μ|} = \frac{1}{(zq)_{∞}} ∑_{ν⊆λ} C_{λ,ν}(1/q) z^{|ν|}.

This identity generalizes [Del11, corollary 4], and we give here an alternative proof through Lascoux’s formula (5), without using the previous computations for finite abelian groups. First, it is possible to rewrite the left-hand side of (18) with the notations of Hall-Littlewood functions. Indeed, replace μ by its conjugate μ′ on the left-hand side, note that

∑_{i=1}^{ℓ} m_{i}(μ′_{1} + ⋯ + μ′_{i}) = ∑_{i=1}^{ℓ}(λ′_{i} − λ′_{i+1})(μ′_{1} + ⋯ + μ′_{i})

= ∑_{i=1}^{ℓ} λ′_{i}μ′_{i} = (λ′|μ′),

and write

∑_{i⩾1} μ_{i}^{′2} = 2n(μ) + |μ|.

Then, using the limit case n → ∞ of the specialization (2), we can rewrite (18) in the following way:

(19)  ∑_{μ} q^{|μ|+n(μ)−(λ′|μ′)}P_{μ}(z, zq …; q) = \frac{1}{(zq)_{∞}} ∑_{ν⊆λ} C_{λ,ν}(1/q) z^{|ν|}.

Now multiply both sides of (19) by q^{n(λ)}P_{λ}(x; q) and sum over all partitions λ. As Hall-Littlewood functions form a ℤ[q]-basis of Λ[q], it is enough to show that

(20) ∑_{λ,μ} q^{n(λ)+n(μ)−(λ′|μ′)}P_{λ}(x; q)P_{μ}(zq, zq^{2} …; q)

= \frac{1}{(zq)_{∞}} ∑_{ν⊆λ} q^{n(λ)}P_{λ}(x; q)C_{λ,ν}(1/q) z^{|ν|}.

Set y = {zq, zq^{2}, … } in Warnaar’s special case (4) of Lascoux’s identity to see that the left-hand side of (20) is equal to

\frac{1}{(zq)_{∞}} ∏_{i⩾1} \frac{1}{(1 − x_{i})(1 − zx_{i})},

which, through the generating function (14), is easily seen to be equal to the righthand side of (20).

5.3. Consequences on u-averages and proof of Theorem 1. Using Theorem 10 above, replacing z by p^{−u}, and recalling the definitions of u-averages, we obtain the following u-average on finite abelian p-groups and on p-groups of type S.

Theorem 12. For a positive integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be an integer partition. Then, the u-average of the function x^{λ}: H ↦ |H[p]|^{m_{1}}|H[p^{2}]|^{m_{2}} ⋯ |H[p^{ℓ}]|^{m_{ℓ}} is equal to

M_{u}(x^{λ}) = ∑_{μ⊆λ} C_{λ,μ}(p)p^{−|μ|u},

and the u-average of the function x^{λ} in the sense of groups of type S is equal to

M_{u}^{S}(x^{λ}) = ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)}.

Theorem 5 implies directly that if λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} is an integer partition, then we have

M0^{S}(x^{λ}) = M1^{S}(x^{λ}) p^{m_{1}+2m_{2}+···+ℓm_{ℓ}},

which is exactly Theorem 1.

6. Consequence on the heuristics on class groups and on Tate-Shafarevich groups

6.1. Class groups of quadratic number fields. The Cohen-Lenstra heuristics ([CL84]) allow to give precise conjectures for the behavior of the class groups of number fields varying in natural families. Here, we only recall briefly the Cohen heuristics in the case of families of quadratic number fields. For K_{d} = ℚ(\sqrt{d}) a quadratic number field with discriminant d, we denote by Cℓ(d) its class group so that Cℓ(d)_{p} is its p-part. First, take the family of imaginary quadratic number fields: these are the fields K_{d} where d < 0 runs through the whole set of fundamental negative discriminants. Assume that p ⩾ 3 (actually, the 2-part of the class groups of quadratic fields behaves in a specific and well-understood way, due to the genus theory, [Coh93]). If h is a ”reasonable” function defined over finite abelian p-groups, then the Cohen-Lenstra heuristics predict that the average of h over the p-part of the class group Cℓ(d)_{p} is the 0-average of h (0 being the rank of the group of units of K_{d}), that is, we should have

lim_{X→∞} \frac{∑_{|d|⩽X, d<0} h(Cℓ(d)_{p})}{∑_{|d|⩽X, d<0} 1} = M_{0}(h).

For the family of real quadratic number fields, i.e., fields K_{d} where d > 0 runs through the set of fundamental positive discriminants, the Cohen-Lenstra heuristics predict that the average of h over the p-part of the class group Cℓ(d)_{p} is the 1average of h (1 being, in that case, the rank of U(K)), that is, we should have:

lim_{X→∞} \frac{∑_{|d|⩽X, d>0} h(Cℓ(d)_{p})}{∑_{|d|⩽X, d>0} 1} = M_{1}(h).

There are strong evidence and numerical data supporting the heuristic principle, however only few results are proved. The first one is for p = 3 and the function h(H) = |H[3]|, the number of 3-torsion elements in H. Davenport and Heilbronn ([DH71]) proved (before the heuristics were formulated) that the average over the 3-part of the class groups of the function h is equal to 2 (resp. 4/3) in the case of imaginary (resp. real) quadratic fields. In their paper, Cohen and Lenstra proved (it is not a trivial result) that M_{0}(h) = 2 and M_{1}(h) = 4/3. There are other results in the literature for other families of number fields, and for the 4-part of class groups of quadratic number fields ([Bha05], [FK07]).

Of course, the case p = 3 and the function H ↦ |H[3]| is a particular case of our work. Now, from our computations, we can predict the average of all the moments of the number of p^{ℓ}-torsion elements in the class groups.

Conjecture 1. For any positive integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be an integer partition, and assume that p ⩾ 3. As d is varying over the set of fundamental negative discriminants the average of | Cℓ(d)[p]|^{m_{1}}| Cℓ(d)[p^{2}]|^{m_{2}} ⋯ | Cℓ(d)[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p).

As d is varying over the set of fundamental positive discriminants the average of | Cℓ(d)[p]|^{m_{1}}| Cℓ(d)[p^{2}]|^{m_{2}} ⋯ | Cℓ(d)[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p)p^{−|μ|}.

In particular, (still for p ≠ 2) and using equation (15), one obtains that the average of | Cℓ(d)[p]|^{n} of class group of imaginary (resp. real) quadratic fields is equal to

M_{0}(x^{1^{n}}) = ∑_{k=0}^{n} \left[{n \atop k}\right]_{p} (resp. M_{1}(x^{1^{n}}) = \frac{1}{p} ∑_{k=0}^{n} \left[{n \atop k}\right]_{p})

Those are exactly the 𝒩(n, p) (resp. ℳ_{n}(p)) used in [FK07, FK06]. In particular, Fouvry and Klueners used 𝒩(n, p) (resp. ℳ_{n}(p)) in [FK06] in order to prove that conjectures (coming from a special case of the Cohen-Lenstra heuristics) related to the moments of the number of p-torsion points in class groups imply conjectures (coming from another special case of the Cohen-Lenstra heuristics) about the probability laws of the p-ranks of the class groups. Our results could be used to generalize their results and also to adapt them in the case of Tate-Shafarevich groups. This will be explained in a forthcoming paper.

6.2. Tate-Shafarevich groups of elliptic curves. Let u ⩾ 0 be a fixed integer, and consider the family ℱ of elliptic curves E defined over ℚ, with rank u, and ordered by the conductor N_{E}. The Tate-Shafarevich group � (E) is conjecturally a finite abelian group. If so, then � (E) is a group of type S. The heuristic principle ([Del01, Del07]) asserts that if g is a ”reasonable” function defined over isomorphism classes of p-group of type S, we should have

lim_{X→∞} \frac{∑_{E∈ℱ, N_{E}⩽X} g(� (E)_{p})}{∑_{E∈ℱ, N_{E}⩽X} 1} = Mu^{S}(g).

Important remark and correction of the first version of the heuristics on Tate-Shafarevich groups. In this paper, the heuristics on Tate-Shafarevich groups, in particular the equation above and Conjecture 2 take into account the following correction. Due to a bad choice of a parameter, the heuristic assumption for Tate-Shafarevich groups stated in [Del01] is wrong and should be corrected for u > 0 by the following: in the Heuristic Assumption of [Del01, page 195], the equality M_{u}(f) = M_{u/2}^{s}(f) must be replaced by M_{u}(f) = M_{u}^{s}(f) (hence, the examples given should also be modified in consequence). Note that this was done in [Del07] but only explicitly for rank 1 elliptic curves. In fact, this correction was suggested by Kowalski observing that only the corrected version of the heuristics is compatible with results of Heath-Brown ([HB94, HB93]) on the 2-rank of the Selmer groups of the family of elliptic curves E_{d}: y^{2} = x^{3} − d^{2}x, d odd and squarefree (see [Kow07, Remark 2.6]). Note that this corrected version is also compatible with other theoretical results and conjectural works, see the discussion after Conjecture 2.

The heuristic principle has many applications and there are numerical evidence supporting them. With our previous computations on u-averages for groups of type S we are led to conjecture the following.

Conjecture 2. For any positive integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be an integer partition, and let u ⩾ 0 be an integer. As E/ℚ, ordered by conductors, is varying over elliptic curves with rank u, the average of |� (E)[p]|^{m_{1}}|� (E)[p^{2}]|^{m_{2}} ⋯ |� (E)[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)}.

In particular, if we take the function g such that g(G) = |G[p^{ℓ}]| then we have:

Mu^{S}(g) = 1 + \frac{1}{p^{2u−1}} + ⋯ + \frac{1}{p^{(2u−1)ℓ}} ,

which for instance, when ℓ = 1, gives M0^{S}(g) = 1 + p and M1^{S}(g) = 1 + 1/p. The rank conjecture in its strong form (asserting that the rank of E is 0 or 1 with probability 1/2 each and that elliptic curves with rank ⩾ 2 do not contribute in the averages), and the results of Bhargava and Shankar [BS10b, BS10a], imply that the average size over all elliptic curves E/ℚ with rank 0 (resp. rank 1) of � (E)[p] is 1 + p (resp. 1 + 1/p) for p = 2 and p = 3. From their work, Bhargava and Shankar also made a conjecture for all p which, together with the rank conjecture, would imply that the average size |� (E)[p]| is 1 + p (resp. 1 + 1/p) for rank 0 (resp. rank 1) elliptic curves. Swinnerton-Dyer [SD08] and Kane [Kan11] obtained results for p = 2 and for the family of quadratic twists, E_{d}, of any fixed elliptic curve E/ℚ with E[2] ⊂ E(ℚ) and having no rational cyclic subgroup of order 4. Their results and the rank conjecture imply that the average of |� (E_{d})[2]| is 3 (resp. 3/2) for the rank 0 (resp. rank 1) curves. There are also in the literature some other theoretical results for other families of elliptic curves, mainly for p = 2.

Actually, all the previous theoretical results are directly concerned with the behavior of p-Selmer groups of elliptic curves. If n ∈ ℕ, the n-Selmer group Sel_{n}(E) and the Tate-Shafarevich group of an elliptic curve are linked by the exact sequence

0 → E(ℚ)/nE(ℚ) → Sel_{n}(E) → � (E)[n] → 0.

So, if E(ℚ)_{tors} = 0 and if the rank of E is u, then we have | Sel_{n}(E)| = n^{u}|� (E)[n]|. Therefore, assuming the rank conjecture, any result on Sel_{n}(E) gives an information on � (E)[n]. Reciprocally, the distribution of Sel_{n}(E) can be deduced from the distribution of � (E)[n] for rank 0 and rank 1 curves.

Recently, Poonen and Rains [PR12] gave a very deep model for the behavior of the p-Selmer groups of all elliptic curves (they do not separate elliptic curves by their rank). In particular, they obtained the following predictions for elliptic curves defined over ℚ.

Conjecture 3 (Poonen-Rains). For any integer m ⩾ 0, the average of | Sel_{p}(E)|^{m} over all E/ℚ is equal to ∏_{j=1}^{m}(1 + p^{j}).

Assuming that the p-parts of Sel_{n}(E) behave independently for p|n, except for the constraint that their parities are equal, they obtained a conjecture for the average of | Sel_{n}(E)|^{m} for all squarefree positive integer n. The Poonen and Rains model suggests that the p-Selmer groups behave in the same way for rank 0 and rank 1 elliptic curves. If we assume that elliptic curves of rank ⩾ 2 do not contribute to the average value, then this is also suggested by the heuristic principle since E(ℚ)_{tors} = 0 with probability 1 and M0^{S}(g) = M1^{S}(g)p (recall that here, g denotes the function G ↦ |G[p]|). It is then natural to ask wether this is still true for all integers n, without the squarefree restriction. Theorem 1 and the discussions above suggest that this is indeed the case, yielding the following conjecture.

Conjecture 4. Let ℓ ⩾ 1 and m ⩾ 0 be integers. The average of | Sel_{p^{ℓ}}(E)|^{m} over all E/ℚ is equal to

∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{|μ|}

where λ is the integer partition λ = ℓ^{m}.

For ℓ = 1, equation (15) gives ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{|μ|} = ∑_{k=0}^{m} \left[{m \atop k}\right]_{p^{2}}p^{k}. It can be easily shown that the last sum is indeed equal to ∏_{j=1}^{m}(1+p^{j}), as claimed in Conjecture 3.

One can clarify the links between the different conjectures involved in the discussion above. If E is an elliptic curve over ℚ, we denote by rk(E) the rank of its Mordell-Weil group. We have in mind:

• The rank conjecture asserting that rk(E) is 0 or 1 with probability 1/2 each.

• The assumption saying that elliptic curves of rank ⩾ 2 contribute nothing to the average value of p^{m rk(E)}. Note that this depends on m, therefore we denote by e_{p,m} the lim sup, as N → ∞, of the sum of p^{m rk(E)} over curves of rank ⩾ 2 and conductor ⩽ N, divided by the total number of curves of conductor ⩽ N.

• The Poonen-Rains conjecture (Conjecture 1.1 in [PR12]).

• The heuristic model for the probability laws dim_{𝔽_{p}} � (E)[p] ([Del01, Del07]).

• The heuristic model for the moment of � (E), i.e., Conjecture 2 for an integer partition λ.

As discussed above and also explained in [PR12], the rank conjecture and Conjecture 1.1 in [PR12] imply the heuristic model for dim_{𝔽_{p}} � (E)[p], and so Conjecture 2 for |� (E)[p]|^{m} (i.e., λ = 1^{m}) for rank 0 and rank 1 curves. Also, Poonen and Rains proved ([PR12, Theorem 5.2]) that the rank conjecture is the only distribution on rk(E) compatible with the Poonen-Rains conjecture and the heuristic model for dim_{𝔽_{p}} � (E)[p]. Furthermore, Conjecture 1.1 in [PR12] also implies that e_{p,m} = 0 for m = 1, 2, 3 (in fact, Poonen and Rains prove it for m = 1 but their argument can be directly adapted for m = 2 and m = 3). On the other hand, one can see that the equality e_{p,m} = 0, the rank conjecture, and Conjecture 2 for the moments of |� (E)[p^{ℓ}]|^{m} (strengthening with the assumption that the error terms implied in the averages are uniformly bounded independently of the rank u), imply Conjecture 4. Indeed (for simplicity we restrict the following computations to elliptic curves E such that E(ℚ)_{tors} = {0}), we first see that

∑_{\begin{matrix}N_{E}⩽X \\ rk(E)⩾2 \end{matrix}} | Sel_{p^{ℓ}}(E)|^{m} = ∑_{u⩾2} p^{mu} \frac{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=u \end{matrix}} |� (E)[p^{ℓ}]|^{m}}{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=u \end{matrix}} 1} ∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=u \end{matrix}} 1.

Now, the fraction in the sum of the right-hand side tends to Mu^{S}(x^{ℓ^{m}}) as X → ∞. Using the fact that u ↦ Mu^{S}(x^{ℓ^{m}}) is decreasing (because the coefficients C_{λ,μ} are positive) and the above assumption on the error terms, we deduce that this fraction is uniformly bounded (and the bound does not depend on u). Hence,

∑_{\begin{matrix}N_{E}⩽X \\ rk(E)⩾2 \end{matrix}} | Sel_{p^{ℓ}}(E)|^{m} = O ( ∑_{\begin{matrix}N_{E}⩽X \\ rk(E)⩾2 \end{matrix}} p^{m rk(E)}) .

Therefore, since by assumption e_{p,m} = 0, we deduce that the average of | Sel_{p^{ℓ}}(E)|^{m} of curves E of rank ⩾ 2 among all elliptic curves is equal to 0. Now, we can write

\frac{∑_{N_{E}⩽X} | Sel_{p^{ℓ}}(E)|^{m}}{∑_{N_{E}⩽X} 1} = \frac{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=0 \end{matrix}} | Sel_{p^{ℓ}}(E)|^{m}}{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=0 \end{matrix}} 1} · \frac{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=0 \end{matrix}} 1}{∑_{N_{E}⩽X} 1}

+ \frac{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=1 \end{matrix}} | Sel_{p^{ℓ}}(E)|^{m}}{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=1 \end{matrix}} 1} · \frac{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)=1 \end{matrix}} 1}{∑_{N_{E}⩽X} 1} + \frac{∑_{\begin{matrix}N_{E}⩽X \\ rk(E)⩾2 \end{matrix}} | Sel_{p^{ℓ}}(E)|^{m}}{∑_{N_{E}⩽X} 1} .

The last term on the right-hand side tends to 0 as X → ∞. Now, replace | Sel_{p^{ℓ}} | by |� (E)[p^{ℓ}]| in the first term and | Sel_{p^{ℓ}} | by p^{ℓ}|� (E)[p^{ℓ}]| in the second one and use Conjecture 2, Theorem 1 and the rank conjecture to conclude.

Remark. In the above discussions, one can replace the number field ℚ by an arbitrary number field K.

7. u-averages and Hall-Littlewood polynomials

7.1. u-averages and combinatorial consequences. In the paper [Del11], the first author computed the u-probabilities laws for the p^{j}-ranks of finite abelian pgroups (recall that the p^{j}-rank of H is the dimension over 𝔽_{p} of the vector space p^{j−1}H/p^{j}H and is denoted r_{p^{j}}(H)), which we rewrite below.

Proposition 13 (Corollary 11 of [Del11]). Let ℓ ⩾ 1 be an integer and μ be an integer partition with ℓ(μ) ⩽ ℓ (here μ is given by its parts μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ} ⩾ 0). Then the u-probability that a finite abelian p-group has its p^{j}-rank equal to μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{∏_{j⩾μ_{ℓ}+1}(1 − 1/p^{u+j})}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+u(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} ,

and the u-probability that a group of type S has its p^{j}-rank equal to 2μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{∏_{j⩾μ_{ℓ}+1}(1 − 1/p^{2u+2j−1})}{p^{2(μ_{1}^{2}+···+μ_{ℓ}^{2})+(2u−1)(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{μ_{j}−μ_{j+1}}} .

Hence, for a fixed integer partition λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, we can also compute the u-average of x^{λ} over finite abelian p-groups (with the notations of Theorem 12) by its moments. Indeed, we have

x^{λ}(H) = p^{r_{p}(H)m_{1}+(r_{p}(H)+r_{p^{2}} (H))m_{2}+···+(r_{p}(H)+···+r_{p^{ℓ}} (H))m_{ℓ}},

therefore

M_{u}(x^{λ}) = ∑_{0⩽μ_{ℓ}⩽···⩽μ_{1}} u-prob(r_{p}(H) = μ_{1}, …, r_{p^{ℓ}}(H) = μ_{ℓ})p^{μ_{1}m_{1}+···+(μ_{1}+···+μ_{ℓ})m_{ℓ}}.

Replacing u-prob(r_{p}(H) = μ_{1}, …, r_{p^{ℓ}}(H) = μ_{ℓ}) by the value given in Proposition 13 and M_{u}(x^{λ}) by its value from Theorem 12, we obtain

∑_{ℓ(μ)⩽ℓ} \frac{∏_{j⩾μ_{ℓ}+1}(1 − 1/p^{u+j})}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+u(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} p^{(λ′|μ)} = ∑_{ν⊆λ} C_{λ,ν}(p)p^{−|ν|u}.

By setting q = 1/p, replacing μ by its conjugate μ′ on the left-hand side, and setting z = q^{−u}, we see that this formula is equivalent to the following combinatorial identity.

Theorem 14. For any nonnegative integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be a fixed integer partition such that λ_{1} ⩽ ℓ (equivalently ℓ(λ′) ⩽ ℓ), and z be a complex number. Then we have

(21)  ∑_{μ_{1}⩽ℓ} \frac{z^{|μ|}q^{2n(μ)+|μ|−(λ′|μ′)}}{b_{μ}(q)} (zq^{μ′_{ℓ}+1})_{∞} = ∑_{ν⊆λ} C_{λ,ν}(1/q)z^{|ν|}.

Remark. One can also replace finite abelian p-groups by groups of type S in the previous discussion. We obtain as a consequence the same combinatorial identity as (21), in which q is replaced by q^{2} and then z is replaced by z^{2}/q.

Note that thanks to Theorem 5, the sum ∑_{ν⊆λ} C_{λ,ν}(1/q)q^{−|ν|u} must be equal to q^{−|λ|u} ∑_{ν⊆λ} C_{λ,ν}(1/q)q^{|ν|u}. By using (21) with z = q^{−u} and z = q^{u}, this yields an identity between the left-hand sides which seems quite mysterious.

In the case where m_{1} = m_{2} = ⋯ = m_{ℓ−1} = 0 and m_{ℓ} = 1 (therefore λ = (ℓ), a row partition), identity (21) corresponds to [Del11, Corollary 12], which can be written as follows

(22)  ∑_{μ_{1}⩽ℓ} \frac{z^{|μ|}q^{2n(μ)}}{b_{μ}(q)} (zq^{μ′_{ℓ}+1})_{∞} = \frac{1 − z^{ℓ+1}}{1 − z} .

In view of the structure of (21), it would be a challenging problem to prove it through the theory of Hall-Littlewood functions, by using Warnaar’s identity (4) or Lascoux’s generalization (5), as we did for proving the symmetry property (13). However, problems occur as we need to bound the first part of the partition μ on the left-hand side in (21), therefore a finite summation of Hall-Littlewood polynomials seems to be required (i.e., a summation over partitions where both the length and the first part are bounded). In his paper [War06], Warnaar raises the question of finding a finite form of (4), which seems out of reach, and therefore finding a finite form of Lascoux’s identity seems hopeless. Note that in (4), specializing the variables y_{i} to 0, one recovers the special case a = 0 of (3), for which Warnaar proves a finite version in [War06, Theorem 6.1]. Unfortunately, the latter neither contains (21) nor even (22) as special cases. Inspired by Macdonald’s partial fraction method [Mac95] used in [Ste90, JZ05, IJZ06, War06], we can prove the following finite form of (3), which will be shown to contain (22) as a special case.

Theorem 15. Let n be a positive integer, and x = {x_{1}, …, x_{n}} be a set of n variables. Then for any complex number a and any positive integer k, we have

(23) ∑_{λ⊆(k^{n})} q^{n(λ)}(a; q^{−1})_{ℓ(λ)}(a; q^{−1})_{n−m_{k}(λ)}P_{λ}(x; q)

= ∑_{I⊆[n]} q^{k\binom{|I|}{2}}(a; q^{−1})_{|I|}(a; q^{−1})_{n−|I|} ∏_{i∈I} x_{i}^{k}

× ∏_{i∈I} \frac{1 − ax_{i}^{−1}q^{1−n}}{1 − x_{i}^{−1}q^{1−|I|}} ∏_{j∉I} \frac{1 − ax_{j}}{1 − x_{j}q^{|I|}} ∏_{i∈I, j∉I} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ,

where the sum on the left-hand side is over partitions λ satisfying λ_{1} ⩽ k and ℓ(λ) ⩽ n, [n] := {1, …, n}, and |I| is the cardinality of the set I.

We postpone the proof of this result to the next subsection, but we give the following usefull consequence.

Corollary 16. For positive integers n, k, and for any complex numbers a, z, we have

(24) ∑_{λ⊆(k^{n})} z^{|λ|}q^{2n(λ)} \frac{(a; q^{−1})_{ℓ(λ)}(a; q^{−1})_{n−m_{k}(λ)}(q)_{n}}{(q)_{n−ℓ(λ)}b_{λ}(q)} = ∑_{r=0}^{n}(−1)^{r}z^{(k+1)r}q^{(2k+3)\binom{r}{2}}

× (1 − zq^{2r−1})\frac{(q^{n−r+1})_{r}(azq^{r})_{n−r}(a; q^{−1})_{r}(aq^{1−n}/z; q^{−1})_{r}(a; q^{−1})_{n−r}}{(q)_{r}(zq^{r−1})_{n+1}} ·

Proof. We consider the specialization x_{i} = zq^{i−1} in (23). In that case note that for any I ⊆ [n],

∏_{i∈I,j∉I} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} ≠ 0 ⇔ ∃r ∈ \{0, ⋯, n\} | I = \{1, …, r\}.

Thus, by using (2), we get the desired result after a few manipulations on the right-hand side. □

Note that when n → ∞, identity (24) reduces to Stembridge’s Theorem 3.4 (b) with b = 0 in [Ste90]. Moreover, when a = 0, (24) corresponds to Warnaar’s identity (6.10) from [War06], which is itself equivalent to Stembridge’s Theorem 1.3 (b) from [Ste90]. Now we can easily see that (24) contains (22) (but unfortunately not (21)) as a consequence. Indeed, on the left-hand side of (24), replace λ by μ, k by ℓ, and take a = zq^{n} to get

∑_{μ_{1}⩽ℓ} z^{|μ|}q^{2n(μ)} \frac{(zq^{n}; q^{−1})_{l(μ)}(zq^{n}; q^{−1})_{n−μ′_{ℓ}}(q)_{n}}{(q)_{n−l(μ)}b_{μ}(q)} = ∑_{r=0}^{n}(−1)^{r}z^{(ℓ+1)r}q^{(2ℓ+3)\binom{r}{2}}

× (1 − zq^{2r−1})\frac{(q^{n−r+1})_{r}(z^{2}q^{r+n})_{n−r}(zq^{n}; q^{−1})_{r}(q; q^{−1})_{r}(zq^{n}; q^{−1})_{n−r}}{(q)_{r}(zq^{r−1})_{n+1}} ·

Note that on the right-hand side, (q; q^{−1})_{r} = 0 unless r = 0 or 1, therefore letting finally n → ∞ yields (22).

7.2. Proof of the finite form of the q-binomial theorem for Hall-Littlewood polynomials. Consider the generating function

S(u) = ∑_{λ_{0},λ} q^{n(λ)}c_{λ_{0}}(λ, a)P_{λ}(x; q) u^{λ_{0}},

where the sum is over all partitions λ = (λ_{1}, …, λ_{n}) and the integers λ_{0} ⩾ λ_{1}, and

c_{k}(λ, a) = (a; q^{−1})_{l(λ)}(a; q^{−1})_{n−m_{k}(λ)}.

Assume λ = (μ_{1}^{r_{1}} μ_{2}^{r_{2}} … μ_{k}^{r_{k}}), where μ_{1} > μ_{2} > ⋯ > μ_{k} ⩾ 0 and (r_{1}, …, r_{k}) is a composition of n. Let S_{n}^{λ} be the set of permutations of S_{n} which fix λ. Each w ∈ S_{n}/S_{n}^{λ} corresponds to a surjective mapping f : x ⟶ {1, 2, …, k} such that |f^{−1}(i)| = r_{i}. For any subset y of x = {x_{1}, …, x_{n}}, let p(y) denote the product of the elements of y (in particular, p(∅) = 1). We can rewrite Hall-Littlewood functions as follows

P_{λ}(x; q) = ∑_{f} p(f^{−1}(1))^{μ_{1}} ⋯ p(f^{−1}(k))^{μ_{k}} ∏_{f(x_{i})<f(x_{j})} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ,

summed over all surjective mappings f : x ⟶ {1, 2, …, k} such that |f^{−1}(i)| = r_{i}. Furthermore, each such f determines a filtration of x

(25)  ℱ : ∅ = F_{0} ⊊ F_{1} ⊊ ⋯ ⊊ F_{k} = x,

according to the rule x_{i} ∈ F_{l} ⟺ f(x_{i}) ⩽ l for 1 ⩽ l ⩽ k. Conversely, such a filtration ℱ = (F_{0}, F_{1}, …, F_{k}) determines a surjection f : x ⟶ {1, 2, …, k} uniquely. Thus we can write

(26)  P_{λ}(x; q) = ∑_{ℱ}π_{ℱ} ∏_{1⩽i⩽k} p(F_{i} \ F_{i−1})^{μ_{i}},

summed over all the filtrations ℱ such that |F_{i}| = r_{1} + r_{2} + ⋯ + r_{i} for 1 ⩽ i ⩽ k, and

π_{ℱ} := ∏_{f(x_{i})<f(x_{j})} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ,

where f is the function defined by ℱ. Now let ν_{i} = μ_{i} − μ_{i+1} if 1 ⩽ i ⩽ k − 1 and ν_{k} = μ_{k}, thus ν_{i} > 0 if i < k and ν_{k} ⩾ 0. Furthermore, let μ_{0} = λ_{0} and ν_{0} = μ_{0} − μ_{1} in the definition of S(u), so that ν_{0} ⩾ 0 and μ_{0} = ν_{0} + ν_{1} + ⋯ + ν_{k}. Since the lengthes of the columns of λ are |F_{j}| = r_{1} + ⋯ + r_{j} with multiplicities ν_{j} for 1 ⩽ j ⩽ k, we have

n(λ) = ∑_{i=1}^{k} ν_{i}\binom{|F_{i}|}{2},

ℓ(λ) = nχ(ν_{k} ≠ 0) + |F_{k−1}|χ(ν_{k} = 0),
m_{k}(λ) = χ(ν_{0} ≠ 0) + |F_{1}|χ(ν_{0} = 0),

and

c_{λ_{0}}(λ, a) = (a; q^{−1})_{n} (χ(ν_{k} ≠ 0) + \frac{(a; q^{−1})_{|F_{k−1}|}}{(a; q^{−1})_{n}} χ(ν_{k} = 0))

× (χ(ν_{0} ≠ 0) + \frac{(a; q^{−1})_{n−|F_{1}|}}{(a; q^{−1})_{n}} χ(ν_{0} = 0)) .

For any filtration ℱ of x define

𝒜_{ℱ} (x, a, u) := (a; q^{−1})_{n}

× ∏_{j=0}^{k} (\frac{q^{\binom{|F_{j} |}{2}}p(F_{j})u}{1 − q^{\binom{|F_{j} |}{2}}p(F_{j})u} + \frac{(a; q^{−1})_{|F_{k−1}|}}{(a; q^{−1})_{n}} χ(F_{j} = x) + \frac{(a; q^{−1})_{n−|F_{1}|}}{(a; q^{−1})_{n}} χ(F_{j} = ∅)) .

Letting F (x) be the set of filtrations of x, we get S(u) = ∑_{ℱ∈F (x)} π_{ℱ} 𝒜_{ℱ} (x, a, u).

Hence S(u) is a rational function of u with simple poles at 1/p(y)q^{\binom{|y|}{2}}, where y is a subset of x. We are now proceeding to compute the corresponding residue c(y) at each pole u = 1/p(y)q^{\binom{|y|}{2}}. Let us start with c(∅). Writing λ_{0} = λ_{1} + k with k ⩾ 0, we see that

S(u) = ∑_{λ} q^{n(λ)}(a; q^{−1})_{l(λ)}P_{λ}(x; q)u^{λ_{1}} ∑_{k⩾0} u^{k} \frac{(a; q^{−1})_{n−m_{λ_{1}+k}}}{(a; q^{−1})_{n}}

= ∑_{λ} q^{n(λ)}(a; q^{−1})_{l(λ)}P_{λ}(x; q)u^{λ_{1}} (\frac{u}{1 − u} + \frac{(a; q^{−1})_{n−m_{λ_{1}}}}{(a; q^{−1})_{n}} ) .

It follows from (3) that

c(∅) = [S(u)(1 − u)]_{u=1} = ∏_{i⩾1} \frac{1 − ax_{i}}{1 − x_{i}} =: ϕ(x; a),

which gives the identity

(27)  ∑_{ℱ∈F (x)} π_{ℱ} 𝒜_{ℱ} (x, a, u)(1 − u)|_{u=1} = ϕ(x; a).

For the computations of other residues, we need some more notations. For any y ⊆ x and any α ∈ ℂ, let q^{α}y = {q^{α}x_{i}, x_{i} ∈ y}, y′ = x\y, and −y = {x_{i}^{−1} : x_{i} ∈ y}. Let y ⊆ x, then

(28)  c(y) = [∑_{ℱ}π_{ℱ} 𝒜_{ℱ} (x, a, u)(1 − p(y)q^{\binom{|y|}{2}}u)]_{u=p(−y)q^{−\binom{|y|}{2}}} .

If y ∉ ℱ, the corresponding summand is equal to 0. Thus we need only to consider the following filtrations ℱ

∅ = F_{0} ⊊ ⋯ ⊊ F_{p} = y ⊊ ⋯ ⊊ F_{k} = x 1 ⩽ p ⩽ k.

We may then split ℱ into two filtrations ℱ_{1} and ℱ_{2}

ℱ_{1} : ∅ ⊊ −q^{−|y|+1}(y \ F_{p−1}) ⊊ ⋯ ⊊ −q^{−|y|+1}(y \ F_{1}) ⊊ −q^{−|y|+1}y,
ℱ_{2} : ∅ ⊊ q^{|y|}(F_{p+1} \ y) ⊊ ⋯ ⊊ q^{|y|}(F_{k−1} \ y) ⊊ q^{|y|}y′.

As

(|y| − 1)(|y| − |F_{j}|) + \binom{|F_{j}|}{2} − \binom{|y|}{2} = \binom{|y| − |F_{j}|}{2} for |y| ⩾ |F_{j}|,

|y|(|F_{j}| − |y|) + \binom{|F_{j}|}{2} − \binom{|y|}{2} = \binom{|F_{j}| − |y|}{2} for |y| ⩽ |F_{j}|,

and

π_{ℱ} (x) = π_{ℱ_{1}} (−q^{−|y|+1}y)π_{ℱ_{2}} (q^{|y|}y′) ∏_{x_{i}∈y,x_{j}∈y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} ,

we can write, setting v = p(y)q^{\binom{|y|}{2}}u, and using (28),

c(y) = [ ∑_{ℱ∈F (x)} π_{ℱ} 𝒜_{ℱ} (x, a, u) (1 − p(y)q^{\binom{|y|}{2}}u)]_{u=p(−y)q^{−\binom{|y|}{2}}}

= \frac{(a; q^{−1})_{n}}{(aq^{|y|−n}; q^{−1})_{|y|}(aq^{−|y|}; q^{−1})_{n−|y|}} ∏_{x_{i}∈y,x_{j}∈y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}}

× [∑_{ℱ_{1}}π_{ℱ_{1}} 𝒜_{ℱ_{1}} (−q^{−|y|+1}y, aq^{|y|−n}, v) (1 − v)]_{v=1}

× [∑_{ℱ_{2}}π_{ℱ_{2}} 𝒜_{ℱ_{2}} (q^{|y|}y′, aq^{−|y|}, v) (1 − v)]_{v=1} .

Then, using (27), we get

c(y) = ϕ (−q^{1−|y|}y; aq^{|y|−n}) ϕ (q^{|y|}y′; aq^{−|y|})

×\frac{(a; q^{−1})_{|y|}(a; q^{−1})_{n−|y|}}{(a; q^{−1})_{n}} ∏_{x_{i}∈y,x_{j}∈y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} .

Encoding each subset y of x by the corresponding subset I ⊆ [n] and extracting the coefficient u^{k} in

S(u) = ∑_{y⊆x} \frac{c(y)}{1 − q^{\binom{|y|}{2}}p(y)u},

we get the result.

8. Acknowledgements

We thank B. Poonen, M. Schlosser and S. O. Warnaar for helpful discussions and suggestions during the preparation of this manuscript.

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Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, CNRS UMR UMR 6623, Facultés des Sciences et Techniques, 16 route de Gray, 25030 Besançon, France.

   E-mail address: christophe.delaunay@univ-fcomte.fr

Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France.

   E-mail address: jouhet@math.univ-lyon1.fr