Degenerate Cohomological Hall Algebra and

Quantized Donaldson-Thomas Invariants for

m-Loop Quivers

Markus Reineke

Received: June 7, 2011

Communicated by Peter Schneider

Abstract. We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver.

2010 Mathematics Subject Classification: Primary 16G20, secondary 05E05, 14N35 Keywords and Phrases: Quantized Donaldson-Thomas invariant, Cohomological Hall algebra, Noncommutative Hilbert scheme

1 Introduction

Generalized Donaldson-Thomas invariants of (noncommutative) varieties arise from factorizations of generating series of motivic invariants of Hilbert schemetype varieties into Euler products. For 3-Calabi-Yau manifolds, this principle is developed extensively in [10].

In [13], the author showed that the wall-crossing formulae of [10] can be modelled using Hilbert schemes of path algebras of quivers; explicit calculations for these varieties in [14] allowed to derive relative integrality (that is, preservation of integrality under wall-crossing) of generalized Donaldson-Thomas invariants.

In [9] a general framework for the study of such integrality properties is proposed, the central tools being Cohomological Hall algebras and the geometric concept of factorization systems.

The purpose of the present paper is to develop an explicit, in most parts purely combinatorial, setup for the study of the quantized Donaldson-Thomas invariants of [9] in the very special, but typical, case of the m-loop quiver. The relevant concepts of [9] are discussed in sections 3, 4. Our approach is based on the explicit description of Hilbert schemes attached to this quiver of [12], which is reviewed in Section 2. It allows us to give a combinatorial description of a degenerate version of the Cohomological Hall algebra, whose structure is easily described (see Section 5). Using number-theoretic arguments similar to [14], we obtain explicit formulas for these quantized Donaldson-Thomas invariants (see Theorem 6.8) in terms of cyclic classes of certain integer sequences in Section 6. We also relate this combinatorics to a similar one appearing in the study of Higgs moduli in [7], thereby proving a conjecture of [6] on the Hitchin nullcone; see Section 7.

Roughly speaking, our approach uses (the combinatorics of) noncommutative Hilbert schemes as a transitional tool between the geometric problem of determination of Donaldson-Thomas invariants and the combinatorial object of cyclic configurations. However, the present approach is not strong enough to yield the positivity properties conjectured in [9]. A proof of these is announced in [3]. The quantized Donaldson-Thomas invariants considered here also appear in [16].

Acknowledgments: The author would like to thank S. Mozgovoy for several discussions concerning this work, and in particular for pointing out the potential relation to Higgs moduli. After this work was first submitted to the arXiv, the author learned about the work of A. Efimov [3] and of F. Rodriguez-Villegas [16]. This work was started while the author participated in the workshop “Wall-crossing in Mathematics and Physics” at Urbana-Champaign, and finished during stays at the Issac Newton Institute Cambridge and the Hausdorff Institute Bonn. The author would like to thank the organizers and participants of these programmes for the inspiring atmosphere.

2 Noncommutative Hilbert schemes

In this section, we recall the definition of noncommutative Hilbert schemes and their main properties following [12]. We also relate the relevant combinatorics of trees to a combinatorics of partitions which will play a major role in the following.

Fix an integer m ≥ 1. For n ≥ 0, we call a pair consisting of a tuple (φ_{1}, …, φ_{m}) of linear operators on C^{n} and a vector v ∈ C^{n} stable if v is cyclic for the representation of the free algebra F^{(m)} = C⟨x_{1}, …, x_{m}⟩ on C^{n} defined by the operators φ_{i}, that is, if C⟨φ_{1}, …, φ_{m}⟩v = C^{n}. This defines an open subset of the affine space End(C^{n})^{m} ⊕ C^{n}, for which a geometric quotient by the action of GL_{n}(C) via g(φ_{1}, …, φ_{m}, v) = (gφ_{1}g^{−1}, …, gφ_{m}g^{−1}, gv) exists. This quotient is denoted by Hilb_{n}^{(m)} and is called a noncommutative Hilbert scheme for F^{(m)}: in analogy with the Hilbert scheme of n points of an affine variety X parametrizing codimension n ideals in the coordinate ring of X, the variety Hilb_{n}^{(m)} parametrizes left ideals I in F^{(m)} of codimension n, that is, ideals such that dim_{C} F^{(m)}/I = n. Namely, to a tuple as above we associate the left ideal of polynomials P ∈ F^{(m)} such that P (φ_{1}, …, φ_{m})v = 0. Conversely, given a left ideal I ⊂ F^{(m)}, we choose an isomorphism between F^{(m)}/I and C^{n}. The operators φ_{i} are induced by left multiplication by x_{i} on F^{(m)}/I via this isomorphism, whereas v is induced by the coset of the unit 1 ∈ F^{(m)}. This tuple is stable by definition, and well defined up to the choice of the isomorphism F^{(m)}/I ≃ C^{n}, that is, up to the GL_{n}(C)-action.

Consider the set Ω^{(m)} of words ω = i_{1} … i_{k} in the alphabet {1, ...m}. Composition of words defines a monoid structure on Ω^{(m)}; define ω′ to be a left subword of ω if ω = ω′ω′′ for a word ω′′. The set Ω^{(m)} carries a lexicographic ordering ≤_{lex} induced by the canonical total ordering on the alphabet {1, …, m}. An m-ary tree is a finite subset T ⊂ Ω^{(m)} which is closed under left subwords. This terminology is explained as follows: a subset T is visualized as the tree with nodes ω for ω ∈ T and an edge of colour i from ω to ωi if ω, ωi ∈ T; the empty word corresponds to the root of the tree.

For a tree T, define its corona C(T) as the set of all ω ∈ Ω^{(m)} such that ω ∉ T, but ω′ ∈ T for ω = ω′i for some i. We have |C(T)| = (m − 1)|T | + 1.

Given a word ω = i_{1} … i_{k} and a tuple of operators (φ_{1}, …, φ_{m}) as above, we define φ_{ω} = φ_{i_{k}} ∘ … ∘ φ_{i_{1}}. For a tree T of cardinality n, define Z_{T} ⊂ Hilb_{n}^{(m)} as the set of classes of tuples (φ_{1}, …, φ_{m}, v) such that:

1. the elements φ_{ω}v for ω ∈ T form a basis of C^{n},

2. if ω ∈ C(T), then φ_{ω}v = ∑_{ω′} λ_{ω,ω′}φ_{ω′}v, where the sum ranges over all words ω′ ∈ T such that ω′ <_{lex} ω.

Denote by d(T) the number of pairs (ω, ω′) such that ω ∈ C(T), ω′ ∈ T and ω′ <_{lex} ω.

Theorem 2.1 [12, Theorem 1.3] The following holds:

1. Z_{T} is a locally closed subset of Hilb_{n}^{(m)}, which is isomorphic to an affine space of dimension d(T).

2. The subsets Z_{T}, for T ranging over all trees of cardinality n, define a cell decomposition of Hilb_{n}^{(m)}, that is, there exists a decreasing filtration of Hilb_{n}^{(m)} by closed subvarieties, such that the successive complements are the subsets Z_{T}.

As a corollary to this geometric description, we can derive precise information on the cohomology (singular cohomology with rational coefficients) of Hilb_{n}^{(m)}. The existence of a cell decomposition implies vanishing of odd cohomology (and algebraicity of even cohomology), thus we can consider the following generating series of Poincaré polynomials

F (q, t) = ∑_{n≥0} q^{(m−1)\binom{n}{2}} ∑_{k} dim H^{k}(Hilb_{n}^{(m)})q^{−k/2}t^{n} ∈ Z[q, q^{−1}][[t]],

as well as its specialization

F (t) = F (1, t) = ∑_{n≥0} χ(Hilb_{n}^{(m)})t^{n} ∈ Z[[t]].

We also define

H(q, t) = ∑_{n≥0} \frac{q^{(m−1)\binom{n}{2}}}{(1 − q^{−1}) · . . . · (1 − q^{−n})}t^{n} ∈ Q(q)[[t]],

which is a q-hypergeometric series whose major role for the following will be explained in the next section.

Corollary 2.2 We have the following explicit descriptions of the series F (q, t) and F (t):

1. The series F (q, t) is uniquely determined as the solution in Q(q)[[t]] of the algebraic q-difference equation

F (q, t) = 1 + t ∏_{k=0}^{m−1} F (q, q^{k}t).

2. The series F (t) is uniquely determined as the solution in Q[[t]] of the

algebraic equation

F (t) = 1 + tF (t)^{m}.

3. The Euler characteristic of Hilb_{n}^{(m)} equals the number of m-ary trees with n nodes, which is \frac{1}{(m−1)n+1} \binom{mn}{n}.

4. We have F (q, t) = \frac{H(q,t)}{H(q,q^{−1}t)}.

Proof: In the notation of [12], the series F (q, t) equals the series \overline{ζ}_{1}^{(m)}(q, t) of [12, Section 5] by [12, Corollary 4.4.]. The first statement translates the operation of grafting of trees; see [12, Theorem 5.5.]. Specialization of the functional equation to q = 1 yields the second statement. The third statement follows from an explicit formula for the number of m-ary trees; see [12, Corollary 4.5.]. The fourth statement is a special case of [4, Theorem 5.2.]; in the present case, it is easily derived from the identity

H(q, t) = H(q, q^{−1}t) + tH(q, q^{m−1}t)

(which follows by a direct calculation from the definition of H(q, t)), together with the first statement.

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Remark: These explicit descriptions of the series F (q, t) are also derived in [1, 2]; these references are already used in [12] to derive asymptotic properties of the cohomology of Hilb_{n}^{(m)}.

Denote by T_{n} the set of partitions (see the beginning of Section 5 for a discussion of this non-standard definition of partitions) λ = (0 ≤ λ_{1} ≤ … ≤ λ_{n}) such that λ_{i} ≤ (m − 1)(i − 1) for all i = 1, …, n. Define the weight of λ ∈ T_{n} as wt(λ) = (m − 1)\binom{n}{2} − |λ|. We also define a weight function wt(T) on trees T as above by

wt(T) = (m − 1)\binom{|T |}{2} − |\{(ω′, ω) ∈ C(T) × T : ω′ <_{lex} ω\}|,

thus wt(T) = d(T) − (m − 1)\binom{n+1}{2} − n by definition of d(T).

Given an m-ary tree T ⊂ Ω^{(m)} with n vertices as above, write T = {ω_{1}, …, ω_{n}} with ω_{1} <_{lex} … <_{lex} ω_{n}. We define a partition λ(T) by

λ(T)_{i} = |{ω ∈ C(T) : ω <_{lex} ω_{i}}|.

Proposition 2.3 The map associating λ(T) to T defines a weight-preserving bijection between m-ary trees with n nodes and T_{n}.

Proof: To prove that λ(T) belongs to T_{n}, we observe that an element ω ∈ C(T) such that ω <_{lex} ω_{k} belongs to C(T_{k}) \ {ω_{k}} for the subtree T_{k} = {ω_{1}, …, ω_{k−1}} of T; this is a set of cardinality (m − 1)(k − 1). We reconstruct the tree from the partition λ ∈ T_{n} inductively as follows: we start with the empty tree T_{0}. In the k-th step, we list the elements of the corona of T_{k−1} in ascending lexicographic order as C(T_{k−1}) = {ω_{i}^{k}, …, ω_{(m−1)(k−1)−1}^{k}} and define T_{k} = T_{k−1} ∪ {ω_{λ_{k}+1}^{k}}. We then have {ω ∈ C(T), ω <_{lex} ω_{k}} = {ω_{1}^{k}, …, ω_{λ_{k}}^{k}}, proving that T is reconstructed from λ(T). The equality of the weights of T and λ(T) follows from the definitions.

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3 Donaldson-Thomas type invariants

The following definition of Donaldson-Thomas type invariants for the m-loop quiver (recall that m ≥ 1) is motivated by [10].

Definition 3.1 Define DT_{n}^{(m)} ∈ Q for n ≥ 1 by writing

F ((−1)^{m−1}t) = ∏_{n≥1}(1 − t^{n})^{−(−1)^{(m−1)n}nDT_{n}^{(m)}}.

These numbers are well-defined since F (t) is an integral power series with constant term 1. A priori, we have nDT_{n}^{(m)} ∈ Z.

Theorem 3.2 [14] We have DT_{n}^{(m)} ∈ N; explicitly, these numbers are given by the following formula:

DT_{n}^{(m)} = \frac{1}{n^{2}} ∑_{d|n} μ(\frac{n}{d})(−1)^{(m−1)(n−d)}\binom{mn − 1}{n − 1}.

We make this formula more explicit by giving some examples, in which we observe that DT_{n}^{(m)} is a polynomial in m except if n ≡ 2 mod 4 (this phenomenon will become more transparent in the following sections).

DT_{1}^{(m)} = 1, DT_{2}^{(m)} = ⌊\frac{m}{2}⌋ , DT_{3}^{(m)} = \frac{m(m − 1)}{2},

DT_{4}^{(m)} = \frac{m(m − 1)(2m − 1)}{3}, DT_{5}^{(m)} = \frac{5m(m − 1)(5m^{2} − 5m + 2)}{24},

DT_{6}^{(m)} = \frac{m(m − 1)(36m^{3} − 54m^{2} + 31m − \frac{13+(−1)^{m−1}5}{2})}{20},

DT_{7}^{(m)} = \frac{7m(m − 1)(343m^{4} − 686m^{3} + 539m^{2} − 196m + 36)}{720}.

Remark: In general, DT_{n}^{(m)} has leading term \frac{n^{n−2}}{n!} m^{n−1} considered as a function of m. It would be interesting to give a graph-theoretic explanation of this, in the spirit of the graph-theoretic explanation for the leading term of the polynomial counting isomorphism classes of absolutely indecomposable representations of dimension n of the m-loop quiver in [8].

In [9], a conjecture is formulated which implies the above theorem; we formulate a slight variant of this conjecture.

Conjecture 3.3 [9, Section 2.6] There exists a product expansion

H(q, (−1)^{m−1}t) = ∏_{n≥1} ∏_{k≥0} ∏_{l≥0}(1 − q^{k−l}t^{n})^{−(−1)^{(m−1)n}c_{n,k}}

for nonnegative integers c_{n,k}, such that only finitely many c_{n,k} are nonzero for any fixed n.

Assuming this conjecture, we have

F (q, (−1)^{m−1}t) = ∏_{n≥1} ∏_{k≥0} ∏_{l=0}^{n−1}(1 − q^{k−l}t^{n})^{−(−1)^{(m−1)n}c_{n,k}}

and thus

F ((−1)^{m−1}t) = ∏_{n≥1}(1 − t^{n})^{−(−1)^{(m−1)n}n ∑_{k} c_{n,k}}.

Thus, setting DT_{n}^{(m)}(q) = ∑_{k≥0} c_{n,k}q^{k}, the conjecture implies that DT_{n}^{(m)}(q) is a polynomial with nonnegative coefficients, such that DT_{n}^{(m)}(1) = DT_{n}^{(m)}.

In the following, we will use a simplified notation for product expansions as in the conjecture, using the λ-ring exponential Exp, see Section 8. Using Lemma 8.3, the product of the conjecture can be rewritten as

Exp(\frac{1}{1 − q^{−1}} ∑_{n≥1} DT_{n}^{(m)}(q)((−1)^{m−1}t)).

4 The Cohomological Hall algebra

In this section, we review the definition and the main properties of the Cohomological Hall algebra of [9] for the m-loop quiver. In particular, we formulate the main conjecture of [9] on these algebras and relate it to the conjecture of the previous section.

For a vector space V, we denote by E_{V} = End(V)^{m} the space of m-tuples of endomorphisms of V. The group G_{V} = GL(V) acts on E_{V} by simultaneous conjugation. For complex vector spaces V and W of dimension n_{1} and n_{2}, respectively, we consider the subspace E_{V,W} of E_{V ⊕W} of m-tuples of endomorphisms (φ_{1}, …, φ_{m}) respecting the subspace V of V ⊕W, that is, such that φ_{i}(V) ⊂ V for all i = 1, …, m. We have an obvious projection map p : E_{V,W} → E_{V} × E_{W} mapping (φ_{1}, …, φ_{m}) to ((φ_{1}|_{V} , …, φ_{m}|_{V}), (\overline{φ_{1}}, …, \overline{φ_{m}})), where \overline{φ_{i}} denotes the endomorphism of W induced by φ_{i}. The action of G_{V ⊕W} on E_{V ⊕W} restricts to an action of the parabolic subgroup P_{V,W} of G_{V ⊕W}, consisting of automorphisms respecting the subspace V, on E_{V,W}. The projection p is equivariant, if the action of P_{V,W} on E_{V} × E_{W} is defined through the Levi subgroup G_{V} × G_{W} of P_{V,W}. Moreover, the closed embedding of E_{V,W} into E_{V ⊕W} is P_{V,W} -equivariant.

Using these maps E_{V} × E_{W} ← E_{V,W} → E_{V ⊕W} and their P_{V,W} -equivariance, we can define the following map in equivariant cohomology with rational coefficients:

H_{G_{V}}^{∗} (E_{V}) ⊗ H_{G_{W}}^{∗} (E_{W}) ≃ H_{G_{V} ×G_{W}}^{∗} (E_{V} × E_{W})
≃ H_{P_{V,W}}^{∗} (E_{V,W})
→ H_{P_{V,W}}^{∗+2s_{1}}(E_{V ⊕W})
→ H_{G_{V ⊕W}}^{∗+2s_{1}+2s_{2}}(E_{V ⊕W}),

where the shifts in cohomological degree are s_{1} = dim E_{V ⊕W} − dim E_{V,W} = m dim V dim W and s_{2} = − dim G_{V ⊕W} /P_{V,W} = − dim V dim W (see [9, Section 2.2.] for the details). Then the following holds:

Theorem 4.1 [9, Theorem 1] The above maps induce an associative unital Q-algebra structure on H = ⨁_{n≥0} H_{G_{C^{n}}}^{∗} (E_{C^{n}}), which is N × Z-bigraded if H_{G_{C^{n}}}^{k} (E_{C^{n}}) is placed in bidegree (n, (m − 1)\binom{n}{2} − k/2).

The algebra H is called the Cohomological Hall algebra of the m-loop quiver in [9]. Since all spaces E_{V} are contractible, the vector space underlying H is independent of m, whereas the algebra structure depends on m.

The above bigrading differs slightly from the one in [9]; it is more suited to our purposes of studying the series H(q, t) in relation to the generating series F (q, t) of Poincaré polynomials of Hilb_{n}^{(m)}.

We consider the Poincaré-Hilbert series of H:

P_{H}(q, t) = ∑_{n≥0} ∑_{k∈Z} dim_{Q} H_{n,k}q^{k}t^{n}.

Lemma 4.2 The series P_{H}(q, t) equals H(q, t).

Proof: The homogeneous component of H with respect to the first component of the bidegree equals H_{G_{C^{n}}}^{∗} (E_{C^{n}}) ≃ H_{G_{C^{n}}}^{∗} (pt), which is isomorphic to a polynomial ring in n generators placed in bidegree (n, (m − 1)\binom{n}{2} − i) for i = 1, …, n. Thus, this component has Poincaré-Hilbert series

\frac{q^{(m−1)\binom{n}{2}}t^{n}}{(1 − q^{−1}) · . . . · (1 − q^{−n})}.

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Using torus fixed point localization, one obtains the following algebraic description of H:

Theorem 4.3 [9, Theorem 2] The algebra H is isomorphic to the following shuffle-type algebra structure on ⨁_{n≥0} Q[x_{1}, …, x_{n}]^{S_{n}}, the space of symmetric polynomials in all possible numbers of variables:

(f_{1} ∗ f_{2})(x_{1}, …, x_{n_{1}+n_{2}}) =

∑ f_{1}(x_{i_{1}}, …, x_{i_{n_{1}}} )f_{2}(x_{j_{1}}, …, x_{j_{n_{2}}} )(∏_{k=1}^{n_{1}} ∏_{l=1}^{n_{2}}(x_{j_{l}} − x_{i_{k}}))^{m−1},

the sum ranging over all shuffles {i_{1} < … < i_{n_{1}}} ∪ {j_{1} < … < j_{n_{2}}} = {1, …, n_{1} +n_{2}}. A homogeneous symmetric function of degree k in n variables is placed in bidegree (n, (m − 1)\binom{n}{2} − k).

From this description we see that H is commutative in case m is odd, and supercommutative in case m is even.

Conjecture 4.4 [9, Conjecture 1] The bigraded algebra H is isomorphic to Sym(C ⊗ Q[z]), the (graded) symmetric algebra over a bigraded super vector space. For any fixed n ≥ 1, only finitely many homogeneous components C_{n,k} are nonvanishing and k ≥ 0 in this case, and z is a homogeneous element of bidegree (0, −1).

This conjecture immediately implies Conjecture 3.3 for c_{n,k} = dim_{Q} C_{n,k}, since the Poincaré-Hilbert series of a symmetric algebra has a natural product expansion, namely P_{Sym(V)} = Exp(P_{V}). A proof of Conjecture 4.4 is announced in [3].

5 The degenerate Cohomological Hall Algebra

We introduce a degenerate form of the Cohomological Hall algebra H and show that it is of purely combinatorial nature. We analyze its structure using the combinatorics of partitions in the set T_{n} introduced in Section 2.

In the following, we will adopt a non-standard notation for partitions: a partition of length n is a non-decreasing sequence λ = (0 ≤ λ_{1} ≤ … ≤ λ_{n}) of (not necessarily non-zero) integers. Denote by Λ_{n} the set of partitions of length l(λ) = n, and denote the disjoint union of all Λ_{n} (for n ≥ 0) by Λ. For N ∈ N, define S^{N}λ = (λ_{1}+N, …, λ_{n}+N). Define the union μ∪ν of partitions μ, ν ∈ Λ as the partition with parts μ_{1}, …, μ_{l(μ)}, ν_{1}, …, ν_{l(ν)}, resorted in ascending order.

Generalizing the definition in Section 2, the weight of a partition λ is defined as wt(λ) = (m − 1)\binom{n}{2} − |λ|, where |λ| = λ_{1} + … + λ_{n}. Comparison with the argument in the proof of Lemma 4.2 immediately shows:

Lemma 5.1 The generating function ∑_{λ∈Λ} q^{wt(λ}t^{l(λ)} of Λ by weight and length equals H(q, t).

Definition 5.2 Define an algebra structure ∗ on the vector space A with basis elements λ ∈ Λ by

μ ∗ ν = μ ∪ S^{(m−1)l(μ)}ν

for μ, ν ∈ Λ.

This multiplication is obviously associative, but non-commutative unless m = 1. It is easy to verify that this algebra is bigraded by weight and length of partitions, and thus has H(q, t) as its Poincaré series.

The explicit description of the Cohomological Hall algebra in Theorem 4.3 allows us to define the following (naive) quantization.

Definition 5.3 Define the quantized Cohomological Hall algebra H_{q} as the bigraded Q[q]-module ⨁_{n≥0} Q[q][x_{1}, …, x_{n}]^{S_{n}} with the product

(f_{1} ∗ f_{2})(x_{1}, …, x_{n_{1}+n_{2}}) =

∑ f_{1}(x_{i_{1}}, …, x_{i_{n_{1}}} )f_{2}(x_{j_{1}}, …, x_{j_{n_{2}}} )(∏_{k=1}^{n_{1}} ∏_{l=1}^{n_{2}}(x_{j_{l}} − qx_{i_{k}}))^{m−1}.

Remark: It would be interesting to realize this algebra geometrically, as the convolution algebra in some appropriate cohomology theory on the G_{V} -spaces E_{V} of the previous section.

We can specialize the algebra H_{q} to any q ∈ Q, in particular to q = 0, yielding an algebra H_{0}.

Proposition 5.4 We have an isomorphism of bigraded algebras A ≃ H_{0} by mapping a partition λ to the symmetric polynomial

P_{λ}(x_{1}, …, x_{n}) = ∑_{σ∈S_{n}} x_{σ(1)}^{λ_{1}} · . . . · x_{σ(n)}^{λ_{n}}.

Proof: The polynomial P_{λ} is a suitable multiple of the monomial symmetric polynomial m_{λ}(x_{1}, …, x_{n}). The multiplication in H_{0} reduces to

(f_{1} ∗ f_{2})(x_{1}, …, x_{n_{1}+n_{2}}) =

∑ f_{1}(x_{i_{1}}, …, x_{i_{n_{1}}} )f_{2}(x_{j_{1}}, …, x_{j_{n_{2}}} )(∏_{l=1}^{n_{2}} x_{j_{l}})^{(m−1)n_{1}}.

Identification of shuffles with cosets S_{n_{1}+n_{2}}/(S_{n_{1}} × S_{n_{2}}) immediately shows that P_{λ} ∗ P_{μ} = P_{λ∗μ}.

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Recall from section 2 the subset T_{n} ⊂ Λ_{n} of partitions λ ∈ Λ_{n} such that λ_{i} ≤ (m − 1)(i − 1) for all i = 1, …, n, and define T as the disjoint union of all T_{n}.

Lemma 5.5 The subspace B of A generated by the basis elements indexed by T is stable under the multiplication ∗, thus B is a subalgebra of A. In computing a product λ ∗ μ for λ, μ ∈ T, it suffices to append S^{(m−1)l(λ)}μ to λ (without resorting parts).

Proof: Using the definition of T and of ∗, this is immediately verified.

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Denote by S the linear operator on A induced by the operation S on partitions.

Lemma 5.6 Multiplication induces an isomorphism of bigraded vector spaces B ⊗ SA ≃ A.

Proof: On the level of partitions, this reduces to the statement that multiplication induces a bijection between ⋃_{k+l=n} T_{k} × S(Λ_{l}) and Λ_{n} preserving weights. Suppose λ is given. If λ ∈ T_{n}, we map λ to (λ, ()) ∈ T_{n} × Λ_{0}. Otherwise, let i be maximal such that λ_{i} ≤ (m − 1)(i − 1) (thus i < n). We define μ = (λ_{1}, …, λ_{i}). We have λ_{j} > (m − 1)(j − 1) for all j > i, thus (λ_{i+1}, …, λ_{n}) = S^{(m−1)i+1}ν for the partition ν of length n − i with parts ν_{k} = λ_{i+k} − (m − 1)i − 1 ≥ 0. Then λ is mapped to (μ, ν). By a simple calculation, compatibility of this bijection with the weight is verified.

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We can iterate this lemma to get (the infinite tensor product meaning finite combinations of finite pure tensors):

Corollary 5.7 Multiplication induces an isomorphism

⨂_{i≥0} S^{i}B = B ⊗ SB ⊗ S^{2}B ⊗ … ≃ A.

Proof: Iteration of the previous lemma shows that any λ admits a finite decomposition λ = λ^{1} ∗ … ∗ λ^{s} such that λ^{k} ∈ S^{k}B for degree reasons.

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Next, we analyze the structure of the algebra B. Denote by T_{n}^{0} ⊂ T_{n} the subset of all λ ∈ T_{n} such that λ_{i} < (m − 1)(i − 1) for i = 2, …, n, by T^{0} the disjoint union of all T_{n}^{0}, and by B^{0} the subspace of B linearly generated by T^{0}.

Lemma 5.8 B is isomorphic to the tensor algebra T (B^{0}).

Proof: In a product λ = λ^{1} ∗ … ∗ λ^{k} of partitions λ^{i} ∈ T^{0}, the set of indices l = 2, …, n such that λ_{l} = (m − 1)(l − 1) is precisely the set {l(λ^{1}) + 1, l(λ^{1}) + l(λ^{2}) + 1, …, l(λ^{1}) + … + l(λ^{k−1}) + 1}. This observation shows that any λ ∈ T admits a unique such decomposition.

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We define a total ordering on T^{0} by the lexicographic ordering, viewing partitions as words in the alphabet N. This induces a total ordering, the lexicographic ordering in the alphabet T^{0}, on words in T^{0}. Call a word in the alphabet T^{0} Lyndon if it is strictly bigger than all of its proper cyclic shifts. Denote by T^{L} the set of all λ^{1} ∗ … ∗ λ^{k} for λ^{1} … λ^{k} a Lyndon word in T^{0}, thus T^{L} is the union of all T_{n}^{L} = T^{L} ∩ T_{n}, and by B^{L} the subspace of B generated by T^{L}.

Lemma 5.9 Multiplication induces an isomorphism of bigraded vector spaces Sym(B^{L}) ≃ B.

Proof: By the previous lemma, we have B ≃ T (B^{0}), thus B ≃ Sym(L(B^{0})) as vector spaces by Poincaré-Birkhoff-Witt, where L(B^{0}) is the free Lie algebra in B^{0} (since the free algebra of a vector space is the enveloping algebra of its free Lie algebra). By general results on free Lie algebras [17], the Lyndon words parametrize a basis of the free Lie algebra, since every word can be written uniquely as a product of Lyndon words, weakly increasing with respect to lexicographic ordering on words. This construction provides an isomorphism between B and Sym(B^{L}). The latter inherits its bigrading from B^{L}, and the construction obviously respects the bigradings.

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Combining the above lemmas, we arrive at the following description of the algebra A:

Theorem 5.10 We have an isomorphism of bigraded vector spaces

A ≃ Sym(⨁_{i≥0} S^{i}B^{L}).

Proof: The result follows from the following chain of isomorphisms:

A ≃ ⨂_{i≥0} S^{i}B ≃ ⨂_{i≥0} S^{i}Sym(B^{L}) ≃ ⨂_{i≥0} Sym(S^{i}B^{L}) ≃ Sym(⨁_{i≥0} S^{i}B^{L}).

□

Remark: This result is not a direct analogue of Conjecture 4.4 for the algebra A ≃ H_{0}, since the operator S induces a shift of (0, −n) in bidegree on a homogeneous component B_{(n,k)}^{L} of B^{L}.

Comparing Poincaré-Hilbert series of both sides in the formula of Theorem 5.10, we get:

Corollary 5.11 We have the following product expansion:

H(q, t) = Exp(∑_{n≥1} \frac{1}{1 − q^{−n}} ∑_{λ∈T_{n}^{L}} q^{wt(λ)}t^{n}).

For application to (quantized) Donaldson-Thomas invariants, we have to describe H(q, (−1)^{m−1}t), thus it is necessary to derive a signed analogue of the previous corollary. Define T^{L,+} as T^{L} if m is odd, and as

T^{L,+} = T^{L} ∪ {λ ∗ λ | λ ∈ T^{L}, l(λ) odd}

if m is even. Define T_{n}^{L,+} = T^{L,+} ∩ T_{n}.

Theorem 5.12 We have a product expansion

H(q, (−1)^{m−1}t) = Exp(∑_{n≥1} \frac{1}{1 − q^{−n}} ∑_{λ∈T_{n}^{L,+}} q^{wt(λ)}((−1)^{m−1}t)^{n}).

Proof: If m is odd, there is nothing to prove, so suppose that m is even. From the identity

(1 + q^{a}t^{b})^{−1} = Exp(q^{2a}t^{2b} − q^{a}t^{b})

it follows that H(q, (−1)^{m−1}t) equals

Exp(∑_{n≥1} \frac{1}{1 − q^{−n}} ∑_{λ∈T_{n}^{L}} q^{wt(λ)}((−1)^{m−1}t)^{n} + ∑_{\begin{matrix}n≥1 \\ odd \end{matrix}} \frac{1}{1 − q^{−2n}} ∑_{λ∈T_{n}^{L}} q^{2wt(λ)}t^{2n}).

Now it remains to recall that length and weight double when passing from λ to λ ∗ λ, and the claim follows.

□

Arguing as in Section 3, this implies the following combinatorial interpretation of Donaldson-Thomas invariants.

Corollary 5.13 We have DT_{n}^{(m)} = \frac{1}{n}|T_{n}^{L,+}|.

Define the polynomials \overline{Q}_{n}(q) = ∑_{λ∈T_{n}^{L}} q^{wt(λ)} and Q_{n}(q) = ∑_{λ∈T_{n}^{L,+}} q^{wt(λ)}; we thus have Q_{n}(q) = \overline{Q}_{n}(q) except in case m is even and n = 2\overline{n} for odd \overline{n}, where Q_{n}(q) = \overline{Q}_{n}(q) + \overline{Q}\overline{n}(q^{2}). We can reformulate Theorem 5.12 as

H(q, (−1)^{m−1}t) = Exp(∑_{n≥1} \frac{1}{1 − q^{−n}} Q_{n}(q)((−1)^{m−1}t)^{n}).

Example: To illustrate the classes of partitions T^{0} ⊂ T^{L} ⊂ T ⊂ Λ defined above, we consider the case m = 2, n = 4. The set T_{4} consists of the 14 partitions

\underline{(0000)}, \underline{(0001)}, \underline{(0002)}, (0003), \underline{(0011)}, \underline{(0012)}, (0013),
(0022), (0023), (0111), (0112), (0113), (0122), (0123).

The five underlined partitions belong to T_{4}^{0}; for the other ones, we have the following decompositions:

(0003) = (000) ∗ (0), (0013) = (001) ∗ (0), (0022) = (00) ∗ (00),
(0023) = (00) ∗ (0) ∗ (0), (0111) = (0) ∗ (000), (0112) = (0) ∗ (001),

(0113) = (0) ∗ (00) ∗ (0), (0122) = (0) ∗ (0) ∗ (00), (0123) = (0) ∗ (0) ∗ (0) ∗ (0).

The lexicographic ordering on T^{0} gives (0) <_{lex} (00) <_{lex} (000) <_{lex} (001), thus we have the following eight elements in T_{4}^{L}:

(0000), (0001), (0002), (0003), (0011), (0012), (0013), (0023).

6 Explicit formulas and integrality

Denote by U_{n} the set of all sequences (a_{1}, …, a_{n}) of nonnegative integers which sum up to (m − 1)n. We consider the natural action of the n-element cyclic group C_{n} on U_{n} by cyclic shift; call a sequence primitive if it is different from all its proper cyclic shifts. Every non-primitive sequence can be written as the (n/d)-fold repetition of a primitive sequence in U_{d} for d a proper divisor of n; we denote the corresponding subset of U_{n} by U_{n}^{d−prim}, and in particular by U_{n}^{prim} = U_{n}^{n−prim} the subset of primitive sequences. We relate U_{n}^{prim}/C_{n}, the set of C_{n}-orbits of primitive sequences, to the set T_{n}^{L} of the previous section.

Lemma 6.1 We have an injective map φ from T_{n} to U_{n} given by

(λ_{1}, …, λ_{n}) ↦ (λ_{2} − λ_{1}, λ_{3} − λ_{2}, …, λ_{n} − λ_{n−1}, (m − 1)n − λ_{n}).

Its inverse is given by

(a_{1}, …, a_{n}) ↦ (0, a_{1}, a_{1} + a_{2}, …, a_{1} + … + a_{n−1}).

The image of φ consists of the sequences (a_{1}, …, a_{n}) such that a_{1} + … + a_{i} ≤ (m − 1)i for all i = 1, …, n.

Proof: This is immediately verified using the definitions.

□

Call a sequence (a_{1}, …, a_{n}) as above admissible if the condition of the previous lemma is satisfied, that is, if it belongs to the image of φ.

Lemma 6.2 Every cyclic class in U_{n} contains at least one admissible element.

Proof: Define an auxilliary sequence (b_{1}, …, b_{n}) of integers by b_{i} = a_{i} −(m− 1); then ∑_{i} b_{i} = 0, and the admissibility condition translates into ∑_{j=1}^{i} b_{j} ≤ 0 for all i ≤ n. Choose an index i_{0} such that b_{1} +…+b_{i_{0}} is maximal among these partial sums. Then (a_{i_{0}+1}, …, a_{n}, a_{1}, …, a_{i_{0}}) is admissible: for i_{0} ≤ i ≤ n we have

b_{i_{0}+1} + … + b_{i} = (b_{1} + … + b_{i}) − (b_{1} + … + b_{i_{0}}) ≤ 0.

For i ≤ i_{0}, we have (since the b_{i} sum up to 0):

b_{i_{0}+1} + … + b_{n} + b_{1} + … + b_{i} = (b_{1} + … + b_{i}) − (b_{1} + … + b_{i_{0}}) ≤ 0.

□

Proposition 6.3 The map φ induces a bijection between T_{n}^{L} and U_{n}^{prim}/C_{n}.

Proof: If μ ∈ T_{k} and ν ∈ T_{l} for k + l = n, then φ(μ ∗ ν) is just the concatenation of the sequences φ(μ) and φ(ν). Thus, φ(μ ∗ ν) and φ(ν ∗ μ) are cyclic shifts of each other. Conversely, if a sequence a ∈ U_{n} and a proper cyclic shift a′ = (a_{i+1}, …, a_{n}, a_{1}, …, a_{i}) of a are both admissible, both subsequences (a_{1}, …, a_{i}) and (a_{i+1}, …, a_{n}) are admissible. It follows that a = φ(μ ∗ ν) and a′ = φ(ν ∗ μ) for some μ, ν.

We conclude that the restriction of φ to T_{n}^{L} only maps to primitive classes, and that each such cyclic class is hit precisely once.

□

Define U_{n}^{prim,+} as U_{n}^{prim} ∪ U_{n}^{\overline{n}−prim} if m is even and n = 2\overline{n} ≡ 2 mod 4, and as U_{n}^{prim} otherwise. We have the following variant of the previous proposition:

Corollary 6.4 The map φ induces a bijection between T_{n}^{L,+} and U_{n}^{prim,+}/C_{n}.

Under the above map φ, the weight wt(λ) of a partition translates into the function

wt(a_{1}, …, a_{n}) = ∑_{i=1}^{n}(n − i)(m − 1 − a_{i}).

Lemma 6.5 Considered modulo n, the function wt on U_{n} is invariant under cyclic shift. In each cyclic class, it assumes its maximum at an admissible element. If a ∈ U_{n} is the \frac{n}{d}-fold repetition of a sequence b ∈ U_{d}, then wt(a) = \frac{n}{d}wt(b).

Proof: We have

wt(a_{i+1}, …, a_{n}, a_{1}, …, a_{i}) = wt(a_{1}, …, a_{n}) − n((m − 1)i − a_{1} − … − a_{i}),

proving the first two claims. It follows from a direct calculation that the function wt is additive with respect to concatenation of sequences as above, proving the third claim.

□

Defining wt(C) for a cyclic class C ∈ U_{n}^{prim,+}/C_{n} as the maximal weight of sequences in class C, we can thus rewrite the polynomial Q_{n}(q) of the previous section as Q_{n}(q) = ∑_{C∈U_{n}^{prim,+}/C_{n}} q^{wt(C)}. We also derive the identity

nQ_{n}(q) ≡ ∑_{a∈U_{n}^{prim,+}} q^{wt(a)} mod (q^{n} − 1).

Define P_{n}(q) = ∑_{a∈U_{n}} q^{wt(a)}. Using again the previous lemma, we have

P_{n}(q) = ∑_{d|n} ∑_{a∈U_{n}^{d−prim}} q^{wt(a)} = ∑_{d|n} ∑_{b∈U_{d}^{prim}} q^{\frac{n}{d} wt(b)}

and thus

P_{n}(q) ≡ ∑_{d|n} d\overline{Q}_{d}(q^{\frac{n}{d}}) mod (q^{n} − 1).

By Moebius inversion, this gives

Lemma 6.6 We have

\overline{Q}_{n}(q) ≡ \frac{1}{n} ∑_{d|n} μ(\frac{n}{d})P_{d}(q^{\frac{n}{d}}) mod (q^{n} − 1).

Remark: Arguments like the above also appear in the context of the “cyclic sieving phenomenon” for Gaussian binomial coefficients, see [15].

Theorem 6.7 The polynomial Q_{n}(q) is divisible by [n] = 1 + q + … + q^{n−1}, and the quotient \frac{1}{[n]}Q_{n}(q) is a polynomial in Z[q].

Proof: Unwinding the definitions of the polynomial P_{n}(q), of the set U_{n} and of its weight statistics wt, we see that P_{n}(q) equals the t^{(m−1)n}-term in

∑_{a_{1},...,a_{n}≥0} q^{∑_{i}(n−i)(m−1−a_{i})}t^{∑_{i} a_{i}} = \frac{q^{(m−1)\binom{n}{2}}}{∏_{i=0}^{n−1}(1 − q^{−i}t)}.

Let ζ_{n} be a primitive n-th root of unity. Specializing q at an arbitrary n-th root of unity ζ_{n}^{s} for s = 1, …, n, we see that P_{n}(ζ_{n}^{s}) equals the t^{(m−1)n}-term in

\frac{ζ_{n}^{(m−1)\binom{n}{2}s}}{∏_{i=0}^{n−1}(1 − ζ_{n}^{si}t)} = \frac{ζ_{n}^{(m−1)\binom{n}{2}s}}{(∏_{i=0}^{\frac{n}{g} −1}(1 − ζ_{n}^{si}t))^{g}} = \frac{ζ_{n}^{(m−1)\binom{n}{2}s}}{(1 − t^{\frac{n}{g}})^{g}} =

= ζ_{n}^{(m−1)\binom{n}{2}s} ∑_{k≥0} \binom{k + g − 1}{g − 1}t^{\frac{n}{g} k},

where g = gcd(s, n). The term ζ_{n}^{(m−1)\binom{n}{2}s} is easily seen to equal the sign (−1)^{(m−1)(n−1)s}, thus

P_{n}(ζ_{n}^{s}) = (−1)^{(m−1)(n−1)s}\binom{mgcd(s, n) − 1}{gcd(s, n) − 1}.

Substituting this into the Moebius inversion formula of the previous lemma, we arrive at

\overline{Q}_{n}(ζ_{n}^{s}) = \frac{1}{n} ∑_{d|n} μ(\frac{n}{d})(−1)^{(m−1)(n−1)s}\binom{mgcd(s, n) − 1}{gcd(s, n) − 1}.

In particular, we have \overline{Q}_{n}(1) = ∑_{d|n} μ(\frac{n}{d})\binom{mn−1}{n−1}. Applying Lemma 8.4, we see that Q_{n}(ζ_{n}^{s}) = \overline{Q}_{n}(ζ_{n}^{s}) = 0 except in case m even, n even, s = \overline{n} = \frac{n}{2} odd, where \overline{Q}_{n}(−1) = −\frac{1}{\overline{n}} ∑_{d|\overline{n}} μ(\frac{\overline{n}}{d})\binom{md−1}{d−1}. Using the above formula for \overline{Q}_{n}(1), in this case we thus get Q_{n}(−1) = \overline{Q}_{n}(−1) + \overline{Q}\overline{n}(1) = 0.

We have proved that Q_{n}(ζ_{n}^{s}) = 0 for s = 1, …, n − 1, thus Q_{n}(q) ∈ Z[q] is divisible in Z[q] by all nontrivial cyclotomic polynomials Φ_{d}(q) for 1 ≠ d|n, and thus Q_{n}(q) is divisible in Z[q] by their product, which equals the polynomial [n].

□

We thus arrive at the following explicit formulas:

Theorem 6.8 The following holds for all m ≥ 1 and all m ≥ 1:

1. The quantized Donaldson-Thomas invariant DT_{n}^{(m)}(q) is given by

DT_{n}^{(m)}(q) = q^{1−n} \frac{1}{[n]} ∑_{C∈U_{n}^{prim,+}} q^{wt(C)}

and is a polynomial with integer coefficients.

2. For every i = 0, …, n−1, the (unquantized) Donaldson-Thomas invariant DT_{n}^{(m)} equals the number of classes C ∈ U_{n}^{prim,+} of weight wt(C) ≡ i mod n.

Proof: Using Corollary 5.11 and the definition of DT_{n}^{(m)}(q) of Section 3, the first part follows from Theorem 6.7. The second part follows by comparing coefficients of the polynomials Q_{n}(q) and DT_{n}^{(m)}(q).

□

Remark: There seems to be no natural weight function s on classes C ∈ U_{n}^{prim,+} of weight wt(C) ≡ i mod n such that ∑_{C} q^{s(C)} = DT_{n}^{(m)}(q).

7 Relation to Higgs moduli

For n ∈ N and d ∈ Z, define H_{n,d} as the set of all sequences (l_{1}, …, l_{n}) ∈ Z^{n} with the following properties:

1. l_{k+1} − l_{k} + (m − 1) ≥ 0 for all k = 1, …, n − 1,

2. ∑_{i=1}^{n} l_{i} = d,

3. ^{\frac{∑_{i=1}^{k} l_{i}}{k}} ≥ \frac{d}{n} for all k < n.

These sequences arise as certain fixed points (so-called type (1, …, 1)-fixed points) in the moduli space of SL_{n}-Higgs bundles, for the action of C^{∗} scaling the Higgs field; see [7, Proposition 10.1]. A relation to Conjecture 3.3 is provided by [6, Remark 4.4.6].

Remark: Shifting every entry of such a sequence by 1 defines a bijection H_{n,d} ≃ H_{n,d+n}. We also have a duality H_{n,d} ≃ H_{n,−d} by mapping (l_{1}, …, l_{n}) to (−l_{n}, …, −l_{1}). The elements of H_{n,0} appear in combinatorics as “score sequences of complete tournaments” [11].

The aim of this section is to prove a conjecture of T. Hausel and F. Rodriguez-Villegas originating in [6, Remark 4.4.6] (see also [16]):

Theorem 7.1 If d is coprime to n, the cardinality of H_{n,d} equals DT_{n}^{(m)}.

Proof: We continue to work with the sets U_{n}, U_{n}^{prim}, U_{n}^{prim,+} and U_{n}^{prim(,+)}/C_{n} of the previous section. We define a map Φ : H_{n,d} → U_{n} by associating to l_{∗} = (l_{1}, …, l_{n}) the sequence (a_{1}, …, a_{n}) defined by

a_{k} = l_{k+1} − l_{k} + (m − 1) for k = 1, …, n,

where we formally set l_{n+1} = l_{1} (the second and third of the defining conditions of H_{n,d} have to be used to ensure a_{n} ≥ 0). Obviously this map is injective. It is also compatible with cyclic shifts, from which it follows easily that the image of Φ consists only of primitive sequences, and that each cyclic class in U_{n}^{prim} is hit at most once by the image of Φ (compare the proof of Proposition 6.3 ). In other words, Φ induces an embedding of H_{n,d} into U_{n}^{prim}/C_{n}. The weight of Φ(l_{∗}) is easily computed as nl_{1} − d, thus it is congruent to −d mod n. We want to prove that, conversely, every primitive cyclic class a_{∗} of weight wt(a_{∗}) ≡ −d mod n belongs to the image of Φ. We first choose an arbitrary element a_{∗} in such a cyclic class and associate to it the integers

l_{k} = l_{1} + ∑_{i=1}^{k−1} a_{i} − (m − 1)(k − 1) where l_{1} = \frac{wt(a_{∗}) + d}{n}.

This sequence does not necessarily belong to H_{n,d}; only the first two defining conditions are fulfilled a priori, together with the condition l_{1}−l_{n}+(m−1) ≥ 0. We define k_{0} as the maximal index k ∈ \{0, …, n\} where l_{1}+…+l_{k}− \frac{d}{n}k reaches its minimum. Then the cyclic shift (l_{k_{0}+1}, …, l_{n}, l_{1}, …, l_{k_{0}}) fulfills the third defining condition by definition of k_{0} (compare the proof of Lemma 6.2), and the first two conditions are still valid. From the definition, it also follows that this defines an inverse map to Φ.

We have thus proved that H_{n,d} is in bijection to (U_{n}^{prim}/C_{n})_{−d}, the subset of U_{n}^{prim}/C_{n} of sequences of weight ≡ −d mod n. For parity reasons, sequences of such a weight cannot be twice a shorter sequence, thus we can replace U_{n}^{prim}/C_{n} by U_{n}^{prim,+}/C_{n}. By Theorem 6.8, the cardinality of the latter equals DT_{n}^{(m)}.

□

Remark: As in the case of (U_{n}^{prim}/C_{n})_{−d}, there seems to be no natural weight function on sequences (l_{1}, …, l_{n}) as above which gives DT_{n}^{(m)}(q).

8 Appendix: λ-ring exponential and Moebius inversion

Let R be the ring Z[q, q^{−1}][[t]] of formal power series in t with coefficients being integral Laurent series in q, and denote by R^{+} the ideal of formal series without constant term. Then exp and log define mutually inverse isomorphisms between the additive group of R^{+} and the multiplicative group 1 + R^{+} of formal series with constant term 1.

We can define (see [5]) a λ-ring structure on R with Adams operations ψ_{i} for i ≥ 1 given by ψ_{i}(q) = q^{i}, ψ(t) = t^{i}; thus, in particular, we have ψ_{1} = id and ψ_{i}ψ_{j} = ψ_{ij}. We define Ψ = ∑_{i≥1} \frac{1}{i}ψ_{i}.

Lemma 8.1 The operator Ψ is invertible with inverse Ψ^{−1} = ∑_{i≥1} \frac{μ(i)}{i}ψ_{i}, where μ denotes the number-theoretic Möbius function.

Proof: Composition of Ψ with the operator defined on the right hand side of the claimed formula yields ∑_{n≥1} ∑_{i|n} μ(i)\frac{ψ_{n}}{n}, which equals ψ_{1} = id by properties of the Moebius function.

□

Using this explicit form of the operators Ψ and Ψ^{−1}, we can derive the following q-Moebius inversion formula for polynomials f_{n}(q), g_{n}(q) in q (viewing them as coefficients of formal series):

Lemma 8.2 We have

ng_{n}(q) = ∑_{d|n} f_{d}(q^{n/d)}) ⟺ f_{n}(q) = ∑_{d|n} μ(\frac{n}{d})dg_{d}(q^{n/d}).

We define the λ-ring exponential Exp : R^{+} → 1 + R^{+} by Exp = exp ∘Ψ. Its inverse is the λ-ring logarithm Log = Ψ^{−1} ∘ log. We have the following explicit formula:

Lemma 8.3 For coefficients c_{i,k} ∈ Z such that, for fixed i ∈ N, we have c_{i,k} ≠ 0 for only finitely many k ∈ Z, the following formula holds:

Exp(∑_{i≥1} ∑_{k∈Z} c_{i,k}q^{k}t^{i}) = ∏_{i≥1} ∏_{k∈Z}(1 − q^{k}t^{i})^{−c_{i,k}}.

Proof: It suffices to compute Exp(q^{k}t^{i}), which is

exp(∑_{j≥1} \frac{1}{j}(q^{k}t^{i})^{j}) = exp(− log(1 − q^{k}t^{i})) = (1 − q^{k}t^{i})^{−1}.

The lemma follows.

□

In Section 6, we make use of the following Moebius inversion type result.

Lemma 8.4 Let f : N → Z be function on non-negative integers. For n ≥ 1 and a proper divisor s of n, the sum

\frac{1}{n} ∑_{d|n} μ(\frac{n}{d})(−1)^{(m−1)(d−1)s}f(gcd(d, s))

equals 0, except when m is even, n is even, s = \overline{n} = \frac{n}{2} is odd, where it equals −\frac{1}{\overline{n}} ∑_{d|\overline{n}} μ(\frac{\overline{n}}{d})f(d).

Proof: Suppose first that m is even, n is even, and s = \overline{n} is odd. Every divisor of n is a divisor d of \overline{n} or twice such a d. We can then split the sum in question into

\frac{1}{n}(∑_{d|\overline{n}} μ(\frac{n}{d})(−1)^{d−1}f(d) + ∑_{d|\overline{n}} μ(\frac{\overline{n}}{d})(−1)^{2d−1}f(d)).

Since all divisors d are odd, we have μ(\frac{n}{d}) = −μ(\frac{\overline{d}}{n}), and the sum simplifies to

−\frac{2}{n} ∑_{d|\overline{n}} μ(\frac{\overline{n}}{d})f(d) = −\frac{1}{\overline{n}} ∑_{d|\overline{n}} μ(\frac{\overline{n}}{d})f(d),

as claimed.

Now suppose that s is an arbitrary proper divisor of n, but s ≠ \frac{n}{2} in case m is even and n ≡ 2 mod 4. We write

\frac{1}{n} ∑_{d|n} μ(\frac{n}{d})(−1)^{(m−1)(d−1)s}f(gcd(d, s)) =

\frac{1}{n} ∑_{g|s} ∑_{\begin{matrix}d| \frac{n}{g} \\ gcd(d, \frac{s}{g})=1 \end{matrix}} μ(\frac{n}{gd})(−1)^{(m−1)(gd−1)s}f(g).

We can uniquely decompose \frac{n}{g} as n_{1}n_{2}, where n_{1} collects all prime factors of \frac{n}{g} dividing \frac{s}{g}; we then have gcd(n_{1}, n_{2}) = 1, and the divisors d of \frac{n}{g} such that gcd(d, \frac{s}{g}) = 1 are precisely the divisors of n_{2}. Thus, we can rewrite the above sum as

\frac{1}{n} ∑_{g|s} ∑_{d|n_{2}} μ(n_{1})μ(\frac{n_{2}}{d})(−1)^{(m−1)(gd−1)s}f(g) =

\frac{1}{n}(−1)^{(m−1)s} ∑_{g|s} μ(n_{1})f(g) ∑_{d|n_{2}} μ(\frac{n_{2}}{d})(−1)^{(m−1)gds}.

By Moebius inversion, the inner sum, temporarily called ρ(g), equals zero except in the case n_{2} = 1, or n_{2} = 2 and (m − 1)gs is even.

Now suppose that in the above sum, the summand corresponding to a divisor g of s is non-zero, that is, both μ(n_{1}) and ρ(g) are non-zero. First consider the case n_{2} = 1, thus n_{1} = \frac{n}{g} is squarefree. Since s ≠ n, there exists a prime p dividing \frac{n}{s}, and thus also \frac{n}{g}. Since n_{2} = 1, the prime p also divides \frac{s}{g}, thus p^{2} divides \frac{n}{g}, a contradiction. Now consider the case n_{2} = 2 and (m − 1)gs even, thus n_{1} = \frac{n}{2g} is squarefree. Again, a prime p dividing \frac{n}{2g} also divides \frac{s}{g}. If p is odd, the argument of the first case again yields a contradiction. So suppose that 2 is the only prime divisor of \frac{n}{g}, that n = 2^{k}n′ for odd n′, and g = 2^{l}n′. Then s = 2^{l′}n′ for some l′ ≤ l, and \frac{n}{2g} = 2^{k−l}. Since \frac{n}{2g} is squarefree, we have k = l or k = l + 1. If k = l, then s ≥ g = n, a contradiction. If k = l + 1, then g = \frac{n}{2}, both s and g are odd, and thus m is even. But then n ≡ 2 mod 4, and by assumption s ≠ \frac{n}{2}, a contradiction.

Thus we see that no summand above can be non-zero, proving the claim.

□

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[18] R. P. Stanley: Enumerative Combinatorics, vol. 2. Cambridge University Press, New York/Cambridge, 1999.

Markus Reineke Fachbereich C Mathematik und Naturwissenschaften Bergische Universität Wuppertal Gaußstr. 20 D 42097 Wuppertal Deutschland reineke@math.uni-wuppertal.de