On tensor products of path algebras of type A

Lutz Hille and Jürgen Müller

Abstract

We derive a formula for the Coxeter polynomial of the s-fold tensor product ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] of path algebras of linearly oriented quivers of Dynkin type A_{n_{i}−1}, in terms of the weights n_{1}, …, n_{s} ⩾ 2, and show that conversely the weights can be recovered from the Coxeter polynomial of the tensor product. Our results have applications in singularity theory, in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles.

1 Introduction and results

This paper is motivated by recent work [9, 10] on weighted projective lines and triangle singularities: Let 𝕏 = 𝕏(n_{1}, n_{2}, n_{3}) be the weighted projective line, over an algebraically closed field F, with respect to the weight triple n_{1}, n_{2}, n_{3} ⩾ 2. Associated with 𝕏 there is the stable category \underline{vect}-𝕏 of vector bundles on 𝕏, which by [10, Thm.B, Thm.6.1] turns out to have a tilting object whose endomorphism algebra is isomorphic to the tensor product of algebras

F [\overrightarrow{A}_{n_{1}−1}] ⊗ F [\overrightarrow{A}_{n_{2}−1}] ⊗ F [\overrightarrow{A}_{n_{3}−1}],

where F [\overrightarrow{A}_{n_{i}−1}] denotes the path algebra of the linearly oriented quiver of Dynkin type A_{n_{i}−1}; for the particularly important special case of weight triples [2, 3, n] see also [9, Prop.5.5]. In view of [10, Rem.5.10, Sect.7.3], the Coxeter transformations of these tensor product algebras are of particular interest.

It is tempting to ask whether there is a similar interpretation of the s-fold tensor product ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}], for weights n_{1}, …, n_{s} ⩾ 2 and arbitrary s ∈ ℕ. Indeed, the following interpretation of \underline{vect}-𝕏 as a singularity category generalises: We first consider the hypersurface singularity

S_{[n_{1},n_{2},n_{3}]} := \frac{F [X_{1}, X_{2}, X_{3}]}{⟨X_{1}^{n_{1}} + X_{2}^{n_{2}} + X_{3}^{n_{3}}⟩},

which is graded with respect to the abelian group \frac{ℤ^{4}}{⟨[n_{1},0,0,1],[0,n_{2},0,1],[0,0,n_{3},1]⟩_{ℤ}}. Letting D^{b}(mod_{gr}-S_{[n_{1},n_{2},n_{3}]}) be the bounded derived category of the category of finitely generated graded S_{[n_{1},n_{2},n_{3}]}-modules, and D^{b}(proj_{gr}-S_{[n_{1},n_{2},n_{3}]}) be its full triangulated subcategory of perfect complexes, the associated singularity category is defined as the quotient D_{sg}^{b}(S_{[n_{1},n_{2},n_{3}]}) := \frac{D^{b}(mod_{gr}-S_{[n_{1},n_{2},n_{3}]})}{D^{b}(proj_{gr}-S_{[n_{1},n_{2},n_{3}]})}. Then, by [5] and [16], the category \underline{vect}-𝕏 is equivalent, as a triangulated category, to the singularity category D_{sg}^{b}(S_{[n_{1},n_{2},n_{3}]}); see also [10, App.C].

Now, given weights n_{1}, …, n_{s} ⩾ 2, the Brieskorn-Pham hypersurface singularity

S_{[n_{1},...,n_{s}]} := \frac{F [X_{1}, …, X_{s}]}{⟨X_{1}^{n_{1}} + ⋯ + X_{s}^{n_{s}}⟩},

which again is naturally graded, similarly gives rise to an associated singularity category D_{sg}^{b}(S_{[n_{1},...,n_{s}]}) := \frac{D^{b}(mod_{gr}-S_{[n_{1},...,ns]})}{D^{b}(proj_{gr}-S_{[n_{1},...,ns]})}. Then, by [3, Thm.1.2], we have an equivalence of triangulated categories

D_{sg}^{b}(S_{[n_{1},...,n_{s}]}) ≅ D^{b} (mod- ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}]) .

Finally, we mention that tensor products F [\overrightarrow{A}_{n_{1}−1}] ⊗ F [\overrightarrow{A}_{n_{2}−1}] have been considered, from a representation theoretic perspective, in [11]. Thus, having all of this in mind, in the present paper we are interested in the algebras ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}], where n_{1}, …, n_{s} ⩾ 2 and s ∈ ℕ, in their bounded derived module categories, and in particular in their Coxeter transformations.

Coxeter transformations. Since our investigations will take place in the realm of finite-dimensional algebras no deep knowledge about singularity theory or weighted projective lines will be required in the sequel. But we assume the reader familiar with the notion of Coxeter transformations and their relevance in representation theory; we are content with briefly recalling the setting:

From now on let F be an arbitrary field, and let A be a finite-dimensional (associative unital) F -algebra of finite global dimension. Let mod-A be the category of finite-dimensional (right) A-modules, let D^{b}(mod-A) be the associated bounded derived category. Then their Grothendieck groups K_{0}(mod-A) and K_{0}(D^{b}(mod-A)), respectively, become bilinear ℤ-lattices with respect to the Euler form, and as such can be naturally identified. Let τ_{A} be the Auslander-Reiten translation associated with A, which is a derived auto-equivalence of A, that is an auto-equivalence of D^{b}(mod-A) as a triangulated category. Hence τ_{A} induces an isomorphism Φ_{A} of K_{0}(mod-A), being called the Coxeter transformation of A. Writing Φ_{A} as an element of GL_{l}(ℤ), where l ∈ ℕ is the number of simple A-modules, gives rise to the characteristic polynomial χ_{A} ∈ ℤ[X] of Φ_{A}, being called the Coxeter polynomial of A. Hence in particular the equivalence class of Φ_{A}, as well as χ_{A} are derived invariants of A, that is invariants of D^{b}(mod-A) under equivalences of triangulated categories.

Of particular interest are spectral properties of Φ_{A}: For example, the fixed space of Φ_{A} coincides with the radical of the Euler form; and the trace of Φ_{A} is related to the Hochschild cohomology of A. Moreover, by Kronecker’s Theorem, Φ_{A} is periodic if and only if Φ_{A} is diagonalisable and has spectral radius at most 1. In particular, if A is fractionally Calabi-Yau, then the order of periodicity of Φ_{A} is related to the Calabi-Yau dimension \frac{a}{b} of A as follows: Recalling that the Serre functor of D^{b}(mod-A) induces the isomorphism −Φ_{A} of K_{0}(mod-A), we get (Φ_{A})^{b} = (−id)^{a+b}, implying that the order of periodicity of Φ_{A} divides 2b.

Still, despite their importance, Coxeter transformations and Coxeter polynomials are not too well understood, where for more details we refer the reader, for example, to [6, 12, 13, 14].

Kronecker products. Given s ∈ ℕ and n_{1}, …, n_{s} ⩾ 2, we now consider the tensor product A := ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}]. This of course, up to isomorphism, only depends on the multiset [n_{1}, …, n_{s}], but not on the order of the tensor factors; moreover, it is independent from adding or deleting tensor factors F [\overrightarrow{A}_{1}], hence we will additionally assume that n_{i} ⩾ 3, for all {1, …, s}, whenever appropriate.

Anyway, for n ⩾ 2 let Φ_{[n]} ∈ GL_{n−1}(ℤ) be the Coxeter transformation of the path algebra F [\overrightarrow{A}_{n−1}], and let χ_{[n]} ∈ ℤ[X] be its Coxeter polynomial. Letting

Φ_{[n_{1},...,n_{s}]} := ⨂_{i=1}^{s} Φ_{[n_{i}]} ∈ GL_{l}(ℤ),

where l := ∏_{i=1}^{s}(n_{i} − 1), the Coxeter transformation Φ_{A} associated with the tensor product A = ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] is given as Φ_{A} = (−1)^{s−1} · Φ_{[n_{1},...,n_{s}]} ∈ GL_{l}(ℤ). Thus we consider the map Φ_{[n_{1},...,n_{s}]} first, and subsequently take care of signs in order to come back to the Coxeter transformation Φ_{A}:

Let χ_{[n_{1},...,n_{s}]} ∈ ℤ[X] be the characteristic polynomial of Φ_{[n_{1},...,n_{s}]}. From Φ_{[2]} = [−1] ∈ GL_{1}(ℤ) we get Φ_{[n_{1},...,n_{s}]} = −Φ_{[n_{1},...,n_{s},2]} = Φ_{[n_{1},...,n_{s},2,2]}; as an aside, in view of the interpretation in terms of Brieskorn-Pham hypersurface singularities, we mention that this observation is related to Knörrer periodicity, see [8]. Anyway, the above equalities yield

χ_{[n_{1},...,n_{s},2]} = (−1)^{l} · χ_{[n_{1},...,n_{s}]}(−X) and χ_{[n_{1},...,n_{s},2,2]} = χ_{[n_{1},...,n_{s}]}.

Thus we may additionally assume that n_{i} ⩾ 3, for all {1, …, s − 1}, that is the multiset [n_{1}, …, n_{s}] contains the element 2 at most once, whenever appropriate.

Our main results now are a formula expressing the polynomial χ_{[n_{1},...,n_{s}]} ∈ ℤ[X] as a rational function in ℚ(X), and a recognition result:

(1.1) Theorem. Let [n_{1}, …, n_{s}] be a multiset, and for any subset ∅ ≠ J ⊆ I := {1, …, s} let n_{J} := lcm(n_{i}; i ∈ J) ∈ ℕ, and let n_{∅} := 1. Then we have

χ_{[n_{1},...,n_{s}]} = ∏_{J ⊆I} ((X^{n_{J}} − 1)^{\frac{∏_{i∈J} n_{i}}{n_{J}}} )^{(−1)^{s−|J |}} .

(1.2) Theorem. Let [n_{1}, …, n_{s}] and [ñ_{1}, …, ñs̃] be multisets each containing the element 2 at most once. Then we have χ_{[n_{1},...,n_{s}]} = χ_{[ñ_{1},...,ñs̃]} ∈ ℤ[X] if and only if the multisets [n_{1}, …, n_{s}] and [ñ_{1}, …, ñs̃] coincide.

Actually, in order to prove Theorem (1.2) we prove much more: We give an algorithm to recover the multiset [n_{1}, …, n_{s}], provided it contains the element 2 at most once, from the characteristic polynomial χ_{[n_{1},...,n_{s}]} alone. Both the algorithm, and the formula in Theorem (1.1) to compute χ_{[n_{1},...,n_{s}]} from [n_{1}, …, n_{s}], are easily implemented, for example into the computer algebra system GAP [4]; this has been used to verify the examples given in (2.7), (2.8), (3.8) and (4.8).

Having these general conclusions, and the techniques to prove them, at hand, we are also able to derive a few further properties of the spectrum of Φ_{[n_{1},...,n_{s}]}, which by Theorem (1.1) consists of roots of unity of order dividing n_{I}. Firstly, we deal with the question whether the roots of unity of highest possible order actually are elements of the spectrum:

(1.3) Corollary. Let [n_{1}, …, n_{s}] be a multiset containing the element 2 at most once. Then the order of periodicity of Φ_{[n_{1},...,n_{s}]} equals n_{I}, where moreover all primitive n_{I}-th roots of unity are eigenvalues of Φ_{[n_{1},...,n_{s}]}.

Moreover, we are able to give a general criterion to decide when 1, that is the root of unity of lowest possible order, is an element of the spectrum; due to its technicality it is only stated in Theorem (4.7). Finally, a consideration of the multiplicity of 1 as an element of the spectrum leads to the following result: Recall that for any f ∈ ℤ[X] \ {0} the reciprocal polynomial is defined as f^{∗} := X^{deg(f)} · f(X^{−1}) ∈ ℤ[X] \ {0}, and f is called self-reciprocal if f^{∗} = f.

(1.4) Corollary. Let [n_{1}, …, n_{s}] be a multiset. Then χ_{[n_{1},...,n_{s}]}^{∗} = ±χ_{[n_{1},...,n_{s}]}, where χ_{[n_{1},...,n_{s}]} is self-reciprocal if and only if gcd(s, n_{1}, …, n_{s}) is odd.

Back to tensor products. Returning to the original motivation, we draw a few immediate conclusions about the tensor product A := ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}], where we now assume that n_{i} ⩾ 3 for all i ∈ {1, …, s}:

a) The associated Coxeter polynomial being given as

\{ \begin{matrix}χ_{[n_{1},...,n_{s}]}, & if s is odd, \\ χ_{[n_{1},...,n_{s},2]}, & if s is even, \end{matrix}

by Corollary (1.3) the Coxeter transformation Φ_{A} is periodic of order

\{ \begin{matrix}n_{I} & = & lcm(n_{1}, …, n_{s}), & if s is odd, \\ lcm(n_{I}, 2) & = & lcm(n_{1}, …, n_{s}, 2), & if s is even. \end{matrix}

In particular, for s = 3 we recover [10, Prop.7.6]; note that the proof given there heavily depends on using weighted projective lines.

b) Moreover, since Coxeter polynomials are derived invariants, from Theorem (1.2) we infer that tensor products A = ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] and ⨂s̃ ; i=1 F [\overrightarrow{A}_{ñ_{i}−1}], where s̃ ∈ ℕ, and ñ_{i} ⩾ 3 for all i ∈ {1, …, s̃}, are derived equivalent if and only if the multisets [n_{1}, …, n_{s}] and [ñ_{1}, …, ñs̃] coincide. c) Finally, we note that the algebra A = ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] is fractionally Calabi-Yau, see [11, Sect.1.4], where since F [\overrightarrow{A}_{n_{i}−1}] has Calabi-Yau dimension \frac{n_{i}−2}{n_{i}},

see [15, Thm.4.1], we infer that A = ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] has Calabi-Yau dimension

\frac{∑_{i=1}^{s} \frac{n_{I}}{n_{i}} · (n_{i} − 2)}{n_{I}} = \frac{s · n_{I} − 2 · ∑_{i=1}^{s} \frac{n_{I}}{n_{i}}}{n_{I}} ;

for the case s = 3 see also [10, Prop.7.5].

This yields (Φ_{A})^{n_{I}} = (−id)^{(s−1)·n_{I}}, just implying the obvious fact that the order of periodicity of Φ_{A} divides n_{I} respectively lcm(n_{I}, 2), whenever s is odd respectively even. But comparing with the actual order of periodicity of Φ_{A} as determined above, we observe that we indeed detect the fractional part of the Calabi-Yau dimension of A = ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] in terms of the associated Coxeter transformation.

Outline. This paper is organised as follows: In Section 2 we prove Theorem (1.1); in order to do so, we apply character theory of finite abelian groups, where we recall the necessary facts, but for more details refer the reader, for example, to [7]. In Section 3 we prove Theorem (1.2) and the first half of Corollary (1.3); in order to do so, we are going to apply lattice theoretic Möbius inversion, where again we recall the necessary facts, but for more details refer the reader, for example, to [17]. In Section 4 we finally prove the second half of Corollary (1.3), state and prove Theorem (4.7), and prove Corollary (1.4).

Acknowledgement. We thank Helmut Lenzing for drawing our attention to these questions, for various enlightening discussions, and in particular for sharing his thoughts about the relevance of tensor product algebras in singularity theory with us. We greatfully acknowledge financial support by the German Research Council (DFG) in the framework of the Scientific Priority Program (SPP 1388) ‘Representation Theory’. The first author thanks the Max Planck Institute for Mathematics (MPI) in Bonn for its hospitality.

2 Characteristic polynomials of Coxeter transformations

We proceed towards a proof of Theorem (1.1), where we from now on again allow for n_{i} ⩾ 2 for all i ∈ {1, …, s}.

(2.1) We recall the determination of the Coxeter polynomial of the path algebra F [\overrightarrow{A}_{n−1}], which is of course well-known, see [1, Ch.VIII.5]: Let

C_{[n]} := [\begin{matrix}1 & 1 & 1 & … & 1 \\ & 1 & 1 & … & 1 \\ & & 1 & … & 1 \\ & & & ... & ⋮ \\ & & & & 1 \end{matrix}] ∈ GL_{n−1}(ℤ)

be the Cartan matrix associated with F [\overrightarrow{A}_{n−1}], describing the classes of the projective indecomposable modules in K_{0}(F [\overrightarrow{A}_{n−1}]) ≅ ℤ^{n−1} with respect to the ‘standard’ basis consisting of the classes of the simple modules; the injective indecomposable modules are similarly described by C_{[n]}^{tr}. Hence, by [1, Ch.VIII.2], the Coxeter transformation Φ_{[n]} ∈ ℤ^{(n−1)×(n−1)} is uniquely determined by the equation C_{[n]} · Φ_{[n]} = −C_{[n]}^{tr}, thus

Φ_{[n]} = −C_{[n]}^{−1} · C_{[n]}^{tr} = [\begin{matrix}. & 1 \\ & . & 1 \\ & & ... & ... \\ & & & . & 1 \\ −1 & −1 & … & −1 & −1 \end{matrix}] ∈ GL_{n−1}(ℤ);

in particular, we have Φ_{[2]} = [−1] ∈ GL_{1}(ℤ).

Since Φ_{[n]} is described by a companion matrix, its characteristic polynomial χ_{[n]} ∈ ℤ[X] is given as

χ_{[n]} = ∑_{i=0}^{n−1} X^{i} = \frac{X^{n} − 1}{X − 1} = ∑_{i=1}^{n−1}(X − ζ_{n}^{i}) ∈ ℂ[X],

where ζ_{n} := exp(\frac{2π\sqrt{−1}}{n}) ∈ ℂ is the standard primitive complex n-th root of unity. Hence Φ_{[n]} is diagonalisable over ℂ, is periodic of order n, and its spectrum is given by the pairwise distinct eigenvalues {ζ_{n}^{i} ∈ ℂ; i ∈ (ℤ/n) \ {0}}.

(2.2) Since the Cartan matrix of the tensor product ⨂_{i=1}^{s} F [\overrightarrow{A}_{n_{i}−1}] is given by the Kronecker product ⨂_{i=1}^{s} C_{[n_{i}]}, the associated Coxeter transformation is

− (⨂_{i=1}^{s} C_{[n_{i}]})^{−1} · (⨂_{i=1}^{s} C_{[n_{i}]})^{tr} = − ⨂_{i=1}^{s} (C_{[n_{i}]}^{−1} · C_{[n_{i}]}^{tr}) = (−1)^{s−1} · ⨂_{i=1}^{s} Φ_{[n_{i}]}.

We let Φ_{[n_{1},...,n_{s}]} := ⨂_{i=1}^{s} Φ_{[n_{i}]}. By the above, Φ_{[n_{1},...,n_{s}]} is also diagonalisable over ℂ and periodic, its complex eigenvalues being given as the multiset

M := {∏_{i=1}^{s} ζ_{n_{i}}^{a_{i}} ∈ ℂ; a_{i} ∈ (ℤ/n_{i}) \ {0}} .

In other words, its characteristic polynomial χ_{[n_{1},...,n_{s}]} ∈ ℤ[X] is given as

χ_{[n_{1},...,n_{s}]} = ∏ ; [a_{1},...,a_{s}]∈∏_{i=1}^{s}((ℤ/n_{i})\{0}) (X − ∏_{i=1}^{s} ζ_{n_{i}}^{a_{i}}) ∈ ℂ[X].

Note that these considerations are reminiscent of the approach taken in [6].

Hence in particular Φ_{[n_{1},...,n_{s}]} has spectral radius 1, and since the above description of the Coxeter transformation yields det((−1)^{s−1} · Φ_{[n_{1},...,n_{s}]}) = (−1)^{l}, where l := ∏_{i=1}^{s}(n_{i} − 1), we for later use note that

det(Φ_{[n_{1},...,n_{s}]}) = (−1)^{ls}.

We now proceed towards the asserted explicit formula for χ_{[n_{1},...,n_{s}]}:

(2.3) To this end, let G_{n} := ⟨z⟩ be a (multiplicative) cyclic group of order n ⩾ 2, and let z be a generator. Moreover, let G_{n}^{∗} := {(ζ_{n}^{∗})^{j}; j ∈ ℤ/n} be its (multiplicative) character group, where ζ_{n}^{∗}: G_{n} → ℂ^{∗}: z ↦ ζ_{n}, then the integral group ring ℤ[G_{n}^{∗}], consisting of the formal ℤ-linear combinations of the characters in G_{n}^{∗}, is called the (additive) group of generalised characters of G_{n}.

In particular, let 1_{n} := (ζ_{n}^{∗})^{0} ∈ G_{n}^{∗} be the trivial character, that is the character of the representation G_{n} → GL_{1}(ℂ): z ↦ 1, and let ρ_{n} := ∑_{j=0}^{n−1}(ζ_{n}^{∗})^{j} ∈ ℤ[G_{n}^{∗}] be the regular character, that is the character of the regular action of G_{n} on the complex group algebra ℂ[G_{n}]. Then we have ρ_{n}(1) = n and ρ_{n}(x) = 0 for all x ∈ G_{n} \ {1}, and conversely for any generalised character χ ∈ ℤ[G_{n}^{∗}] such that χ(x) = 0 for all x ∈ G_{n} \ \{1\} we have χ = \frac{χ(1)}{n} · ρ_{n}.

Now, by mapping z ↦ Φ_{[n]} we get a faithful representation of G_{n} → GL_{n−1}(ℂ), whose character is given as φ_{n} := ∑_{j=1}^{n−1}(ζ_{n}^{∗})^{j} = ρ_{n} − 1_{n} ∈ ℤ[G_{n}^{∗}].

(2.4) Now we consider the direct product

G := ∏_{i=1}^{s} G_{n_{i}} = ∏_{i=1}^{s}⟨z_{i}⟩

of cyclic groups G_{n_{i}} = ⟨z_{i}⟩ with chosen generators z_{i}. Let ρ ∈ ℤ[G^{∗}] be the regular character of G, which hence can be written as the outer tensor product ρ = ρ_{n_{1}} ⊗ ⋯ ⊗ ρ_{n_{s}} of the regular characters ρ_{n_{i}} ∈ ℤ[G_{n_{i}}^{∗}]. Moreover, let φ := φ_{n_{1}} ⊗ ⋯ ⊗ φ_{n_{s}} ∈ ℤ[G^{∗}] be the outer tensor product of the characters φ_{n_{i}} ∈ ℤ[G_{n_{i}}^{∗}]. Hence we have φ = (ρ_{n_{1}} − 1_{n_{1}}) ⊗ ⋯ ⊗ (ρ_{n_{s}} − 1_{n_{s}}) ∈ ℤ[G^{∗}].

Considering the representation of G affording φ, since the tensor factors φ_{n_{i}} are afforded by the representations z_{i} ↦ Φ_{[n_{i}]}, we infer that the element z := ∏_{i=1}^{s} z_{i} ∈ G is represented by the matrix Φ_{[n_{1},...,n_{s}]} = ⨂_{i=1}^{s} Φ_{[n_{i}]}. Hence we consider the cyclic subgroup

H := ⟨z⟩ = ⟨∏_{i=1}^{s} z_{i}⟩ ⩽ G,

which in particular has order n_{I} = lcm(n_{1}, …, n_{s}). Thus the restriction φ|_{H} can be viewed as an element of ℤ[H^{∗}], and hence may be written as a ℤ-linear combination of linear characters of H. This will yield the characteristic polynomial χ_{[n_{1},...,n_{s}]} of Φ_{[n_{1},...,n_{s}]} as a rational function in ℂ(X), where the linear characters occurring, evaluated at z ∈ H, describe its zeroes and poles.

(2.5) To this end, for J ⊆ I let

G_{J} := ∏_{i∈J} G_{n_{i}} = ∏_{i∈J}⟨z_{i}⟩ ⩽ G

be the subgroup generated by the direct factors indicated by J. Hence we have a group epimorphism

α_{J} : G → G_{J} : \{ \begin{matrix}z_{i} ↦ z_{i}, & if i ∈ J , \\ z_{i} ↦ 1, & if i ∉ J . \end{matrix}

Let ρ_{J} = ⨂_{i∈J} ρ_{n_{i}} ∈ ℤ[G_{J}^{∗}] be the regular character of G_{J}, and let ρ̂_{J} := ρ_{J} ∘ α_{J} ∈ ℤ[G^{∗}] be its inflation to G via α_{J}. Thus expanding the outer tensor product φ = (ρ_{n_{1}} − 1_{n_{1}}) ⊗ ⋯ ⊗ (ρ_{n_{s}} − 1_{n_{s}}) by distributivity we get

φ = ∑_{J ⊆I}(−1)^{s−|J |} · (ρ_{J} ∘ α_{J}) = ∑_{J ⊆I}(−1)^{s−|J |} · ρ̂_{J} ∈ ℤ[G^{∗}].

(2.6) Finally, let z_{J} := ∏_{i∈J} z_{i} and

H_{J} := ⟨z_{J} ⟩ ⩽ G_{J} ,

which in particular has order n_{J} = lcm(n_{i}; i ∈ J). Then α_{J} restricts to an epimorphism H → H_{J} : z ↦ z_{J}, which we denote by the same symbol. Let ρ_{n_{J}} ∈ ℤ[H_{J}^{∗}] be the regular character of H_{J}, and let ρ̂_{n_{J}} := ρ_{n_{J}} ∘ α_{J} ∈ ℤ[H^{∗}]

be its inflation to H via α_{J}. Then letting ζ_{n_{J}}^{∗} := (ζ_{n_{I}}^{∗})^{\frac{n_{I}}{n_{J}}} ∈ ℤ[H^{∗}] we have

ρ̂_{n_{J}} = ∑_{j=0}^{n_{J} −1}(ζ_{n_{J}}^{∗})^{j} ∈ ℤ[H^{∗}].

Now we have ρ_{J} (1) = |G_{J} | = ∏_{i∈J} n_{i} and ρ_{n_{J}} (1) = |H_{J} | = n_{J}. Since ρ_{J} (x) = 0 for all x ∈ G_{J} \ {1}, we conclude that

ρ_{J} |_{H_{J}} = \frac{∏_{i∈J} n_{i}}{n_{J}} · ρ_{n_{J}} ∈ ℤ[H_{J}^{∗}].

Hence for all x ∈ H we have

ρ̂_{J} (x) = ρ_{J} (α_{J} (x)) = \frac{∏_{i∈J} n_{i}}{n_{J}} · ρ_{n_{J}} (α_{J} (x)) = \frac{∏_{i∈J} n_{i}}{n_{J}} · ρ̂_{n_{J}} (x),

that is

ρ̂_{J} |_{H} = \frac{∏_{i∈J} n_{i}}{n_{J}} · ρ̂_{n_{J}} ∈ ℤ[H^{∗}].

This implies

φ|_{H} = ∑_{J ⊆I}(−1)^{s−|J |} · ρ̂_{J} |_{H} = ∑_{J ⊆I} ((−1)^{s−|J |} · \frac{∏_{i∈J} n_{i}}{n_{J}} · ρ̂_{n_{J}} ) ∈ ℤ[H^{∗}],

where in turn ρ̂_{n_{J}} = ∑_{j=0}^{n_{J} −1}(ζ_{n_{J}}^{∗})^{j}. Hence evaluating at z ∈ H we get

χ_{[n_{1},...,n_{s}]} = ∏_{J ⊆I} (( ∏_{j=0}^{n_{J} −1}(X − ζ_{n_{J}}^{j}))^{(−1)^{s−|J |}· \frac{∏_{i∈J} n_{i}}{n_{J}}} ) ∈ ℂ(X),

which is the desired description, proving Theorem (1.1). ♯

(2.7) Example. We write out the above formula explicitly for small values of s ∈ ℕ, and verify that the polynomials obtained indeed behave as expected:

a) For s = 1, letting n := n_{1}, we of course just recover χ_{[n]} = \frac{X^{n}−1}{X−1}.

b) For s = 2, letting n_{12} := lcm(n_{1}, n_{2}) and g_{12} := gcd(n_{1}, n_{2}) = \frac{n_{1}n_{2}}{n_{12}} , we have

χ_{[n_{1},n_{2}]} = \frac{(X − 1)(X^{n_{12}} − 1)^{g_{12}}}{(X^{n_{1}} − 1)(X^{n_{2}} − 1)} ;

in particular, if n_{1} and n_{2} are coprime this yields

χ_{[n_{1},n_{2}]} = \frac{(X − 1)(X^{n_{1}n_{2}} − 1)}{(X^{n_{1}} − 1)(X^{n_{2}} − 1)}.

Even more specifically, for the multiset [n, 2] we get χ_{[n,2]} = \frac{X^{n}−(−1)^{n}}{X+1} , thus in any case we indeed have χ_{[n,2]} = (−1)^{n−1} · χ_{[n]}(−X) as expected.

c) For s = 3, abbreviating n_{ij} := lcm(n_{i}, n_{j}) and g_{ij} := gcd(n_{i}, n_{j}) = \frac{n_{i}n_{j}}{n_{ij}}, for i < j ∈ {1, 2, 3}, and g_{123} := gcd(n_{1}, n_{2}, n_{3}), and noting that the principle of inclusion-exclusion yields n_{123} := lcm(n_{1}, n_{2}, n_{3}) = \frac{n_{1}n_{2}n_{3}g_{123}}{g_{12}g_{13}g_{23}} , we have

χ_{[n_{1},n_{2},n_{3}]} = \frac{(X^{n_{1}} − 1)(X^{n_{2}} − 1)(X^{n_{3}} − 1)(X^{n_{123}} − 1)^{\frac{g_{12}g_{13}g_{23}}{g_{123}}}}{(X − 1)(X^{n_{12}} − 1)^{g_{12}}(X^{n_{13}} − 1)^{g_{13}}(X^{n_{23}} − 1)^{g_{23}}} ;

in particular, if n_{1}, n_{2} and n_{3} are pairwise coprime this yields

χ_{[n_{1},n_{2},n_{3}]} = \frac{(X^{n_{1}} − 1)(X^{n_{2}} − 1)(X^{n_{3}} − 1)(X^{n_{1}n_{2}n_{3}} − 1)}{(X − 1)(X^{n_{1}n_{2}} − 1)(X^{n_{1}n_{3}} − 1)(X^{n_{2}n_{3}} − 1)}.

Even more specifically, for the multiset [n, 2, 2] we get χ_{[n,2,2]} = \frac{X^{n}−1}{X−1}, independently of whether n is odd or even; thus we indeed have χ_{[n,2,2]} = χ_{[n]} as expected. Moreover, for [n_{1}, n_{2}, 2], by a similar case distinction with respect to the parity of n_{1} and n_{2}, we indeed get, as expected,

χ_{[n_{1},n_{2},2]} = (−1)^{(n_{1}−1)(n_{2}−1)} · χ_{[n_{1},n_{2}]}(−X).

(2.8) Example. We now consider a few examples which play a particular role in the context of weighted projective lines:

a) For the tensor product F [\overrightarrow{A}_{2}] ⊗ F [\overrightarrow{A}_{n−1}], where n ⩾ 2, which is associated with the most prominent series 𝕏(2, 3, n) of weighted projective lines, see [9], we get the Coxeter polynomial

χ_{[2,3,n]} = χ_{[3,n]}(−X) = \{ \begin{matrix}\frac{(X+1)(X^{3n}−(−1)^{n})}{(X^{3}+1)(X^{n}−(−1)^{n})}, & if 3 ∤ n, \\ \frac{(X+1)(X^{n}−(−1)^{n})^{2}}{(X^{3}+1)} , & if 3 | n. \end{matrix}

In particular, for n ∈ {2, …, 5} we get

χ_{[2,3,n]} = \{ \begin{matrix}Φ_{3}, & if n = 2, \\ Φ_{2}^{2}Φ_{6}, & if n = 3, \\ Φ_{3}Φ_{12}, & if n = 4, \\ Φ_{30}, & if n = 5, \end{matrix}

where Φ_{d} ∈ ℤ[X], for d ∈ ℕ, denotes the d-th cyclotomic polynomial. These coincide with the Coxeter polynomials of the path algebras of the quivers of Dynkin type A_{2}, D_{4}, E_{6}, and E_{8}, respectively, see for example [12, Sect.18.5].

This was to be expected, since, apart from 𝕏(2, 2, n), the above cases are precisely those of weighted projective lines having positive Euler characteristic, see [10, Sect.5.1], and the associated tensor products are by the ADE chain phenomenon derived equivalent to the path algebras named, see [9, Prop.5.5, Prop.5.16], or independently [11, Cor.1.2].

b) Moreover, for the tensor products

F [\overrightarrow{A}_{2}] ⊗ F [\overrightarrow{A}_{2}] ⊗ F [\overrightarrow{A}_{2}], F [\overrightarrow{A}_{3}] ⊗ F [\overrightarrow{A}_{3}], F [\overrightarrow{A}_{2}] ⊗ F [\overrightarrow{A}_{5}],

being associated with the weighted projective lines 𝕏(3, 3, 3), 𝕏(2, 4, 4) and 𝕏(2, 3, 6), respectively, we get the Coxeter polynomials

χ_{[3,3,3]} = (X − 1)^{2} · (\frac{X^{3}−1}{X−1})^{3}  = Φ_{1}^{2}Φ_{3}^{3},
χ_{[2,4,4]} = (X − 1)^{2} · (\frac{X^{2}−1}{X−1}) · (\frac{X^{4}−1}{X−1})^{2}  = Φ_{1}^{2}Φ_{2}^{3}Φ_{4}^{2},
χ_{[2,3,6]} = (X − 1)^{2} · (\frac{X^{2}−1}{X−1}) · (\frac{X^{3}−1}{X−1}) · (\frac{X^{6}−1}{X−1}) = Φ_{1}^{2}Φ_{2}^{2}Φ_{3}^{2}Φ_{6},

which are of canonical type, see [12, Prop.18.6].

This was to be expected, since, apart from 𝕏(2, 2, 2, 2), the above cases are precisely those of weighted projective lines having zero Euler characteristic. In other words, these are the tubular weight types, see [10, Sect.5.2], hence the associated tensor products are derived equivalent to the canonical algebras of the respective type, see [10, Thm.5.6], or [12, Sect.18.6.2] for the case [3, 3, 3].

3 Recovering the tensor factors

We now conversely assume only the characteristic polynomial χ_{Φ} ∈ ℤ[X], where Φ := Φ_{[n_{1},...,n_{s}]} is as above, to be known. Thus we hide the multiset from the notation, and we aim to recover the number s ∈ ℕ of tensor factors and the multiset [n_{1}, …, n_{s}] from χ_{Φ} alone. This will in particular entail Theorem (1.2), as soon as we additionally assume that the multiset [n_{1}, …, n_{s}] contains the element 2 at most once.

(3.1) Recall that for any d ∈ ℕ the multiplicity m_{d} ∈ ℕ_{0} of the primitive d-th root of unity ζ_{d} ∈ ℂ as an element of the multiset M of complex eigenvalues of Φ is given as the order of ζ_{d} as a zero of χ_{Φ} ∈ ℂ[X]; note that m_{d} only depends on the order d, but not on the particular choice of a primitive d-th root of unity. Hence, by evaluating χ_{Φ} at the standard primitive d-th root of unity ζ_{d} ∈ ℂ for increasing values of d ∈ ℕ, the multiplicities m_{d} ∈ ℕ_{0}, the number max(d ∈ ℕ; m_{d} > 0) ∈ ℕ, and thus the number

N := lcm(d ∈ ℕ; m_{d} > 0) ∈ ℕ

can be determined from χ_{Φ}; note that N is the order of periodicity of Φ.

To facilitate the subsequent analysis, assume that n ∈ ℕ is a multiple of the currently unknown number n_{I} = lcm(n_{1}, …, n_{s}). Then we may rewrite M as

M = \{∏_{i=1}^{s} ζ_{n}^{\frac{n}{n_{i}} ·a_{i}} ∈ ℂ; a_{i} ∈ (ℤ/n_{i}) \ \{0\}\} .

We have m_{d} > 0 only if d | n_{I}, hence we get N | n_{I} as well, but a priorly we might have m_{n_{I}} = 0 or even N < n_{I}.

Still, a suitable, but crudely large n ∈ ℕ can be found as follows: Since deg(χ_{Φ}) = ∏_{i=1}^{s}(n_{i} − 1) we have n_{i} ⩽ deg(χ_{Φ}) + 1, for all i ∈ {1, …, s}, hence we may choose n ∈ ℕ as a multiple of lcm(1, 2, …, deg(χ_{Φ}), deg(χ_{Φ}) + 1). This a priori choice admittedly in general is much larger than n_{I}, but is sufficient for the moment, and we will come back to that point in (3.6) again.

(3.2) Converting to an additive setting, for any d | n we have the natural embedding ι_{n}^{d}: ℤ/d → ℤ/n: x (mod d) ↦ \frac{n}{d} · x (mod n). Hence we have the natural homomorphism

φ := [ι_{n}^{n_{1}}, …, ι_{n}^{n_{s}}]: ℤ/n_{1} ⊕ ⋯ ⊕ ℤ/n_{s} → ℤ/n: [x_{1}, …, x_{s}] ↦ ∑_{i=1}^{s} \frac{n}{n_{i}} · x_{i}.

Letting

O := ((ℤ/n_{1}) \ {0}) ⊕ ⋯ ⊕ ((ℤ/n_{s}) \ {0}) ⊆ ℤ/n_{1} ⊕ ⋯ ⊕ ℤ/n_{s},

the multiset M is described by the map φ|_{O}, that is, if x ∈ ℤ/n has additive order d | n, then we have |(φ|_{O})^{−1}(x)| = m_{d}. Thus the task is, given this information on the fibres of φ|_{O}, to recover the multiset [n_{1}, …, n_{s}].

(3.3) In order to do so, we are going to apply lattice theoretic Möbius inversion. We recall the necessary facts: Let L be a finite combinatorial lattice, that is a non-empty finite set equipped with a partial order ⩽, such that for any x, y ∈ L there is a smallest upper bound x + y ∈ L and a largest lower bound x ∩ y ∈ L with respect to ⩽.

Given a map f: L → ℂ, the associated sum function is defined as

f^{+} : L → ℂ: x ↦ ∑_{y⩽x} f(y).

To recover f from the knowledge of f^{+}, we may use the Möbius function associated with L, which is recursively defined as

μ_{L}: L × L → ℂ: [x, y] ↦ \{ \begin{matrix}0, & if x ⩽̸ y, \\ 1, & if x = y, \\ − ∑_{x⩽z<y} μ_{L}(x, z), & if x < y. \end{matrix}

Note that μ_{L} is essentially described by the identities ∑_{x⩽z⩽y} μ_{L}(x, z) = 0 whenever x < y ∈ L. Then for all x ∈ L we have the Möbius inversion formula to recover f from f^{+}, see [17, Prop.2],

f(x) = ∑_{y⩽x} f^{+}(y)μ_{L}(y, x).

Moreover, restricting to an interval L′ = {z ∈ L; u ⩽ z ⩽ v}, where u ⩽ v ∈ L, we get the Möbius function μ_{L′} = μ_{L}|_{L′×L′}, see [17, Prop.4]; and letting L^{∗} be the dual lattice associated with L, obtained by reverting the partial order, its Möbius function is μ_{L^{∗}}(x, y) = μ_{L}(y, x), for all x, y ∈ L, see [17, Prop.3].

We are going to apply Möbius inversion in two cases: Firstly, for the partially ordered set of all subsets of a finite set I, where the partial order is given by set theoretic inclusion ⊆, the non-zero values of the associated Möbius function μ_{I} are given as

μ_{I}(K, J) = (−1)^{|J |−|K|} whenever K ⊆ J ⊆ I.

The resulting inversion principle is known as the principle of inclusion-exclusion.

Secondly, given n ∈ ℕ, for the partially ordered set {d ∈ ℕ; d | n} of all divisors of n, where the partial order is given by the divisibility relation, the essential part of the associated (lattice theoretic) Möbius function μ_{n} is given as, see for example [2, Ch.1.4.9],

μ_{n}(x, y) = μ(\frac{y}{x}) whenever x | y | n,

where μ: ℕ → {−1, 0, 1} is the number theoretic Möbius function given as

μ: d ↦ \{ \begin{matrix}0, & if d is not squarefree, \\ (−1)^{t}, & if d is the product of t ∈ ℕ_{0} pairwise distinct primes. \end{matrix} (3.4) We are now prepared for our first application of Möbius inversion: For J ⊆ I := {1, …, s} let

G_{J} := ⨁_{i∈J} ℤ/n_{i} ⩽ ℤ/n_{1} ⊕ ⋯ ⊕ ℤ/n_{s}

be the subgroup generated by the direct summands indicated by J, and let φ_{J} := φ|_{G_{J}} : G_{J} → ℤ/n. Then, given x ∈ ℤ/n, by taking the support of elements of G_{J} into account, the cardinality |φ_{J}^{−1}(x)| ∈ ℕ_{0} is given as

|φ_{J}^{−1}(x)| = ∑_{K⊆J} |(φ|_{G_{K}∩O})^{−1}(x)|.

Hence the principle of inclusion-exclusion, applied to the full set I, yields

|(φ|_{O})^{−1}(x)| = |(φ|_{G_{I}∩O})^{−1}(x)| = ∑_{J ⊆I}(−1)^{s−|J |} · |φ_{J}^{−1}(x)|,

reducing the task of computing the size of fibres of φ|_{O} to determining the size of fibres of the homomorphisms φ_{J}, for all J ⊆ I.

For J ⊆ I let still n_{J} := lcm(n_{i}; i ∈ J), where we again set n_{∅} := 1. Then we have im(φ_{J}) = im(ι_{n}^{n_{J}}) ⩽ ℤ/n, that is |im(φ_{J})| = n_{J}. Moreover, if x ∈ ℤ/n has additive order d | n, then we have x ∈ im(φ_{J}) if and only if d | n_{J}. In this case we have

|φ_{J}^{−1}(x)| = | ker(φ_{J})| = \frac{|G_{J} |}{|im(φ_{J})|} = \frac{∏_{i∈J} n_{i}}{n_{J}} .

Thus the above expression for |(φ|_{O})^{−1}(x)| = m_{d} becomes

(−1)^{s} · m_{d} = ∑_{J ⊆I,d|n_{J}} (\frac{1}{n_{J}} · ∏_{i∈J}(−n_{i})) .

(3.5) This leads us to the second application of Möbius inversion: Letting

k_{c} := ∑_{J ⊆I,n_{J} =c} (\frac{1}{c} · ∏_{i∈J}(−n_{i}))

for all c | n, we for all d | n get

(−1)^{s} · m_{d} = ∑_{c∈{1,...,n},d|c|n} k_{c}.  (∗)

Hence considering this as a sum over the interval {c ∈ {1, …, n}; d | c | n} with respect to the dual divisibility relation, Möbius inversion yields

(−1)^{s} · k_{d} = ∑_{c∈\{1,...,n\},d|c|n} μ(\frac{c}{d}) · m_{c},

implying that (−1)^{s} · k_{d} can be determined from χ_{Φ}.

Moreover, we have

∑_{c∈{1,...,n},c|d} c · k_{c} = ∑_{c∈{1,...,n},c|d} ( ∑_{J ⊆I,n_{J} =c} (∏_{i∈J}(−n_{i}))) ,

yielding

∑_{c∈{1,...,n},c|d} c · k_{c} = ∑_{J ⊆I,n_{J} |d} (∏_{i∈J}(−n_{i})) .

Letting I_{c} := {i ∈ I, n_{i} = c}, for all c | n, the right hand side equals

∑_{J ⊆{i∈I,n_{i}|d}} (∏_{i∈J}(−n_{i})) = ∏_{i∈I,n_{i}|d}(1 − n_{i}) = ∏_{c∈{1,...,n},c|d}(1 − c)^{|I_{c}|}.

Hence in conclusion we get, for all d | n,

∑_{c∈{1,...,n},c|d} c · k_{c} = ∏_{c∈{1,...,n},c|d}(1 − c)^{|I_{c}|}.  (∗∗)

(3.6) We are now in a position to clarify the relationship between the numbers N := lcm(d ∈ ℕ; m_{d} > 0) and n_{I} = lcm(n_{i}; i ∈ I). As was remarked earlier, we have m_{d} > 0 only if d | n_{I}, thus we have N | n_{I}. We consider the converse:

If d ∤ N then, by equation (∗) we conclude that k_{d} = 0. Hence equation (∗∗) applied to n and N yields

∏_{d∈{1,...,n},d|n}(1 − d)^{|I_{d}|} = ∑_{d|n} d · k_{d} = ∑_{d|N} d · k_{d} = ∏_{d∈{1,...,n},d|N}(1 − d)^{|I_{d}|},

showing that |I_{d}| = 0 whenever 2 ≠ d ∤ N, and |I_{2}| is even whenever N is odd.

Thus, if N is even then n_{i} | N for all i ∈ I, hence n_{I} | N, or equivalently N = n_{I}. If N is odd then we have two cases: If n_{I} is odd as well, then we have n_{i} | N for all i ∈ I, thus N = n_{I} again; if n_{I} is even, then the multiset is of the form [n_{1}, …, n_{s′}, 2, …, 2], where n_{1}, …, n_{s′} are odd and s − s′ = |I_{2}| > 0 is even, thus we have 2N = n_{I}.

Hence, without knowing the underlying multiset [n_{1}, …, n_{s}], we are able to specify a small multiple n of n_{I}, using χ_{Φ} alone, as

n = \{ \begin{matrix}N, & if N is even, \\ 2N, & if N is odd. \end{matrix}

Moreover, assuming that [n_{1}, …, n_{s}] contains the element 2 at most once, then the above also shows that the order of periodicity of Φ is given as N = n_{I}, showing the first half of Corollary (1.3).

(3.7) We are now prepared to describe how to recover the multiset [n_{1}, …, n_{s}] from χ_{Φ}: Firstly, for d = 1 we have |I_{1}| = 0, thus by equation (∗∗) we get k_{1} = 0^{0} = 1, hence being able to compute (−1)^{s} · k_{1} = (−1)^{s} implies that the parity of the number s of tensor factors can be determined from χ_{Φ}. Thus we are able to compute the numbers k_{d}, for all d | n.

Next, for d = 2 equation (∗∗) yields 1 + 2k_{2} = (−1)^{|I_{2}|}, thus k_{2} = \frac{(−1)^{|I_{2}|}−1}{2} ∈ {0, −1}, where k_{2} = 0 if and only if |I_{2}| is even. Hence being able to compute k_{2} implies that the parity of the number |I_{2}| of tensor factors equal to 2 can be determined from χ_{Φ} as well. Recall that we assume that n is even, so that the above analysis indeed holds for d = 2, and that since χ_{[n_{1},...,n_{s}]} = χ_{[n_{1},...,n_{s},2,2]} we cannot possibly expect more than finding the parity of |I_{2}|.

Finally, for divisors d | n such that d > 2 we proceed by induction, using equation (∗∗) again, to obtain successively (1 − d)^{|I_{d}|}, which since d − 1 > 1 immediately yields |I_{d}|.

In conclusion, for all d | n, we are able to recover algorithmically the cardinality |I_{d}| of the number of tensor factors equal to d if d > 2, and the parity |I_{2}| of the number of tensor factors equal to 2. This in particular proves Theorem (1.2). ♯

(3.8) Example. For example, for the multisets [3, 4, 5, 6, 7] and [2, 3, 4, 5, 6, 7] we get deg(χ_{Φ}) = 6 · 5 · 4 · 3 · 2 = 720, and the following non-zero multiplicities m_{d} and m′_{d} of zeroes of χ_{Φ}, respectively, entailing n = N = n_{I} = 420:

\begin{matrix}d & 35 & 70 & 105 & 140 & 210 & 420 \\ m_{d} & 2 & 2 & 2 & 4 & 1 & 3 \\ m′_{d} & 2 & 2 & 1 & 4 & 2 & 3 \end{matrix}

This yields the associated numbers k_{d} and k′_{d}, respectively, as follows:

\begin{matrix}d & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 10 & 12 & 14 & 15 & 20 & 21 \\ (−1)^{s} · k_{d} & −1 & 0 & 1 & 1 & 1 & −2 & 1 & 0 & 3 & 0 & −1 & −1 & −1 \\ (−1)^{s} · k′_{d} & 1 & −1 & −1 & 1 & −1 & −1 & −1 & 1 & 3 & 1 & 1 & −1 & 1 \end{matrix}

\begin{matrix}d & 28 & 30 & 35 & 42 & 60 & 70 & 84 & 105 & 140 & 210 & 420 \\ (−1)^{s} · k_{d} & −1 & 2 & −1 & 2 & −3 & 0 & −3 & 1 & 1 & −2 & 3 \\ (−1)^{s} · k′_{d} & −1 & 1 & 1 & 1 & −3 & −1 & −3 & −1 & 1 & −1 & 3 \end{matrix}

These now yield, by considering d = 1, that s is odd respectively even, and by considering d = 2, that |I_{2}| is even respectively odd, and then by induction on d reveal successively |I_{3}| = |I_{4}| = |I_{5}| = |I_{6}| = |I_{7}| = 1, and |I_{d}| = 0 for d | n such that d ⩾ 8.

4 Spectra of Coxeter transformations

We finally comment on two aspects concerning explicit eigenvalues of Coxeter transformations are concerned, thereby proving the second half of Corollary (1.3), and Corollary (1.4). From now on we assume that the multiset [n_{1}, …, n_{s}] contains the element 2 at most once.

(4.1) The root of unity ζ_{n_{I}} as an eigenvalue. In order to prove the second half of Corollary (1.3), we proceed to show that m_{n_{I}} > 0, in particular implying

max(d ∈ ℕ; m_{d} > 0) = lcm(d ∈ ℕ; m_{d} > 0) = N.

Now, asking whether m_{n_{I}} > 0 is equivalent to asking whether the image im(φ|_{O}) ⊆ ℤ/n contains an element of additive order n_{I}. Since the map

φ = [ι_{n}^{n_{1}}, …, ι_{n}^{n_{s}}]: ℤ/n_{1} ⊕ ⋯ ⊕ ℤ/n_{s} → ℤ/n

factors through ι_{n}^{n_{I}}: ℤ/n_{I} → ℤ/n anyway, we may assume that n = n_{I}, and ask whether 1 ∈ im(φ|_{O}) ⊆ ℤ/n.

To this end, let p_{1}, …, p_{t} ∈ ℕ be the rational prime divisors of n, for some t ∈ ℕ, and let q_{j} := p_{j}^{ϵ_{j}}, where ϵ_{j} ∈ ℕ, such that n = ∏_{j=1}^{t} q_{j}. For i ∈ {1, …, s} and j ∈ {1, …, t} let q_{ij} := p_{j}^{e_{ij}}, where e_{ij} ∈ ℕ_{0}, such that n_{i} = ∏_{j=1}^{t} q_{ij}. Thus we have the primary decompositions

ℤ/n_{i} ≅ ℤ/q_{i1} ⊕ ⋯ ⊕ ℤ/q_{it} and ℤ/n ≅ ℤ/q_{1} ⊕ ⋯ ⊕ ℤ/q_{t},

given by the associated natural embeddings. Note that for those we have the following commutative diagram:

ℤ/n_{i} ⟶^{ι_{n}^{n_{i}}} ℤ/n
ι_{n_{i}}^{q_{ij}} ↑  ↑_{ι_{n}^{q_{j}}}
ℤ/q_{ij} ⟶_{ι_{q_{j}}^{q_{ij}}} ℤ/q_{j}

Hence for j ∈ {1, …, t} letting

φ_{j}: ⨁_{i∈{1,...,s};e_{ij}>0} ℤ/q_{ij} → ℤ/q_{j},

we conclude that 1 ∈ im(φ) ⊆ ℤ/n, that is im(φ) contains an element of additive order n, if and only if im(φ_{j}) contains an element of additive order q_{j}, that is 1 ∈ im(φ_{j}) ⊆ ℤ/q_{j}, for all j ∈ {1, …, t}. We consider restrictions to O:

Assume that 1 ∈ im(φ_{j}|_{O}) ⊆ ℤ/q_{j}, for all j ∈ {1, …, t}. Taking a preimage in φ_{j}^{−1}(1) ∩ O, and augmenting by 0 ∈ ℤ/q_{ij} for i ∈ {1, …, s} such that e_{ij} = 0, we obtain an element x_{j} = [x_{1j}, …, x_{sj}] ∈ ℤ/q_{1j} ⊕ ⋯ ⊕ ℤ/q_{sj}. Letting

y_{ij} := ι_{n_{i}}^{q_{ij}}(x_{ij}) ∈ im(ι_{n_{i}}^{q_{ij}}) ⩽ ℤ/n_{i},

for i ∈ {1, …, s}, we have y_{ij} ≠ 0 if and only if e_{ij} > 0, where in this case y_{ij} has additive order q_{ij}, and

φ([y_{1j}, …, y_{sj}]) = ∑_{i=1}^{s} ι_{n}^{n_{i}}(ι_{n_{i}}^{q_{ij}}(x_{ij})) = ∑_{i=1}^{s} ι_{n}^{q_{j}}(ι_{q_{j}}^{q_{ij}}(x_{ij}))

= ι_{n}^{q_{j}} (∑_{i=1}^{s} ι_{q_{j}}^{q_{ij}}(x_{ij})) = ι_{n}^{q_{j}}(φ_{j}(x_{j})) = ι_{n}^{q_{j}}(1) = \frac{n}{q_{j}}

shows that φ([y_{1j}, …, y_{sj}]) ∈ im(ι_{n}^{q_{j}}) ⩽ ℤ/n has additive order q_{j}. Then for

y := ∑_{j=1}^{t} [y_{1j}, …, y_{sj}] = [∑_{j=1}^{t} y_{1j}, …, ∑_{j=1}^{t} y_{sj}] ∈ ℤ/n_{1} ⊕ ⋯ ⊕ ℤ/n_{s}

we have ∑_{j=1}^{t} y_{ij} ≠ 0 ∈ ℤ/n_{i}, that is y ∈ O, and φ(y) = ∑_{j=1}^{t} \frac{n}{q_{j}} ∈ ℤ/n has additive order lcm(q_{1}, …, q_{s}) = n, hence we indeed have 1 ∈ im(φ|_{O}) ⊆ ℤ/n.

(4.2) Thus to show that 1 ∈ im(φ|_{O}) ⊆ ℤ/n we are led to consider the case t = 1: To simplify notation, let p ∈ ℕ be a rational prime, and let n = p^{e} for some e ∈ ℕ. Hence we have n_{i} = p^{e_{i}}, where e_{i} ∈ ℕ, and where we may assume that e_{i} < e for all i ∈ {1, …, s′}, and e_{i} = e for all i ∈ {s′ + 1, …, s}, for some s′ ∈ {0, …, s − 1}. Now we have 1 ∈ im(φ|_{O}) ⊆ ℤ/p^{e} if and only if m_{p^{e}} ≠ 0, which, by equation (∗) applied to d = p^{e}, is equivalent to k_{p^{e}} ≠ 0.

To determine k_{p^{e}}, for J ⊆ I we have n_{J} = p^{e} if and only if J ∩{s′ +1, …, s} ≠ ∅, that is J is of the form J = J′ ∪^{.} J′′ where J′ ⊆ {1, …, s′} and ∅ ≠ J′′ ⊆ {s′ + 1, …, s}. Thus the principle of inclusion-exclusion yields

p^{e} · k_{p^{e}} = ∑_{J ⊆I,J ∩{s′+1,...,s}≠∅} (∏_{i∈J}(−p^{e_{i}}))

= ∑_{J′⊆{1,...,s′}} ∑_{∅≠J′′⊆{s′+1,...,s}} ( ∏_{i∈J′∪^{.}J′′}(−p^{e_{i}}))

= ∑_{J′⊆{1,...,s′}} ( ∏_{i∈J′}(−p^{e_{i}})) · ∑_{∅≠J′′⊆{s′+1,...,s}} ( ∏_{i∈J′′}(−p^{e}))

= ( ∏_{i∈{1,...,s′}}(1 − p^{e_{i}})) · ((1 − p^{e})^{s−s′} − 1) .

Hence we conclude that k_{p^{e}} ≠ 0, except if p^{e} = 2 and s − s′ = s is even.

(4.3) Thus, returning to the general case again, by the above reduction we are done in all cases, except if n is even and ϵ_{1} = 1, where we let p_{1} := 2. In this case, we may assume that there is s′ ∈ {0, …, s − 1} such that n_{1}, …, n_{s′} are odd, and n_{s′+1}, …, n_{s} are even. By the above reduction again we are done if s − s′ > 0 is odd, hence we may additionally assume that s − s′ ⩾ 2 is even.

To simplify notation, let n′_{i} := n_{i} for i ∈ \{1, …, s′\}, and n′_{i} := \frac{n_{i}}{2} ∈ ℕ for i ∈ \{s′ + 1, …, s\}, and let n′ := \frac{n}{2} = lcm(n′_{1}, …, n′_{s}). Hence the n′_{i} are odd, where n′_{i} = 1 if and only if n_{i} = 2. Since the multiset [n_{1}, …, n_{s}] contains the element 2 at most once, we may assume that n′_{i} ⩾ 3 for all i ∈ {1, …, s − 1}, and we have to distinguish the cases n′_{s} = 1 and n′_{s} ⩾ 3:

If n′_{s} ⩾ 3, we consider the map φ′: ℤ/n′_{1} ⊕ ⋯ ⊕ ℤ/n′_{s} → ℤ/n′. Then by the above analysis we have 1 ∈ im(φ′|_{O}) ⊆ ℤ/n′, hence there is

[y_{1}, …, y_{s}] ∈ ι_{n_{1}}^{n′_{1}}((ℤ/n′_{1}) \ {0}) ⊕ ⋯ ⊕ ι_{n_{s}}^{n′_{s}}((ℤ/n′_{s}) \ {0}) ⊆ O

such that φ([y_{1}, …, y_{s}]) ∈ ℤ/n has additive order n′. Since y_{s} ∈ ι_{n_{s}}^{n′_{s}}(ℤ/n′_{s}) ⊆ ℤ/n_{s} has odd order, and n′_{s} = \frac{n_{s}}{2} ∈ ℤ/n_{s} has additive order 2, we conclude that y_{s} +n′_{s} ∈ (ℤ/n_{s})\{0} has even order. Hence we have [y_{1}, …, y_{s−1}, y_{s} +n′_{s}] ∈ O, and φ([y_{1}, …, y_{s−1}, y_{s} + n′_{s}]) ∈ ℤ/n has additive order n = 2n′.

If n′_{s} = 1, we consider the map φ′: ℤ/n′_{1} ⊕ ⋯ ⊕ ℤ/n′_{s−1} → ℤ/n′ instead. Then similarly there is

[y_{1}, …, y_{s−1}] ∈ ι_{n_{1}}^{n′_{1}}((ℤ/n′_{1}) \ {0}) ⊕ ⋯ ⊕ ι_{n_{s−1}}^{n′_{s−1}}((ℤ/n′_{s−1}) \ {0})

such that φ([y_{1}, …, y_{s−1}, 0]) ∈ ℤ/n has additive order n′. Thus, since 1 ∈ (ℤ/n_{s}) \ {0} = {1} has has additive order 2, we again have [y_{1}, …, y_{s−1}, 1] ∈ O, and φ([y_{1}, …, y_{s−1}, 1]) ∈ ℤ/n has additive order n = 2n′. ♯

(4.4) The root of unity 1 as an eigenvalue. Quite to the opposite we now deal with the question whether m_{1} > 0, or equivalently whether 0 ∈ im(φ|_{O}) ⊆ ℤ/n. We proceed similar to (4.1): If 0 ∈ im(φ_{j}|_{O}) ⊆ ℤ/q_{j}, for some j ∈ {1, …, t}, there again are y_{ij} ∈ im(ι_{n_{i}}^{q_{ij}}) ⩽ ℤ/n_{i}, for i ∈ {1, …, s}, such that y_{ij} ≠ 0 if and only if e_{ij} > 0, where in this case y_{ij} has additive order q_{ij}, and φ([y_{1j}, …, y_{sj}]) = 0 ∈ ℤ/n.

Thus to infer 0 ∈ im(φ|_{O}) ⊆ ℤ/n it suffices to assume that 0 ∈ im(φ_{j}|_{O}) ⊆ ℤ/q_{j} where j runs through a subset K ⊆ {1, …, t} such that ∑_{j∈K}[y_{1j}, …, y_{sj}] ∈ O. The latter condition, saying ∑_{j∈K} y_{ij} ≠ 0 ∈ ℤ/n_{i} for all i ∈ {1, …, s}, is equivalent to {j ∈ {1, …, t}; e_{ij} > 0} ∩ K ≠ ∅, for all i ∈ {1, …, s}, in other words {p_{j}; j ∈ K} contains a prime divisor of any of the numbers n_{1}, …, n_{s}.

(4.5) Thus to show that 0 ∈ im(φ|_{O}) ⊆ ℤ/n we again first consider the case t = 1: We keep the notation of (4.2), where we also assume that e_{1} ⩽ ⋯ ⩽ e_{s} ∈ ℕ. Then by equation (∗), applied to d = 1, we have

(−1)^{s} · m_{1} = ∑_{J ⊆I} (\frac{1}{n_{J}} · ∏_{i∈J}(−p^{e_{i}})) = 1 + ∑_{∅≠J ⊆I} (\frac{1}{p^{e_{max(J)}}} · ∏_{i∈J}(−p^{e_{i}})) .

Reordering the sum with respect to the maximum of the subsets ∅ ≠ J ⊆ I considered, the right hand side becomes

1 + ∑_{∅≠J ⊆I} (\frac{1}{p^{e_{max(J)}}} · ∏_{i∈J}(−p^{e_{i}})) = 1 − ∑_{∅≠J ⊆I} ( ∏_{i∈J \max(J)}(−p^{e_{i}}))

= 1 − ∑_{j=1}^{s} ∑_{J ⊆{1,...,j−1}} (∏_{i∈J}(−p^{e_{i}})) = 1 − ∑_{j=1}^{s} (∏_{i=1}^{j−1}(1 − p^{e_{i}})) .

Hence we conclude that

m_{1} = ∑_{j=2}^{s} ((−1)^{s−j} · ∏_{i=1}^{j−1}(p^{e_{i}} − 1)) .

Thus we have m_{1} = 0 whenever s = 1. Hence let s ⩾ 2, then m_{1} is an alternating sum with summands having non-decreasing absolute value, even increasing absolute value for those j such that p^{e_{j−1}} ⩾ 3. Hence we conclude that m_{1} > 0, except if p = 2 and 1 = e_{1} = ⋯ = e_{s−1} ⩽ e_{s} = e, where s is odd.

(4.6) Now, returning to the general case again, to state the criterion to decide whether m_{1} > 0 we need a few preparations, taking the above analysis into account: Let Γ be the gcd graph of the multiset [n_{1}, …, n_{s}], whose vertices are labelled by n_{1}, …, n_{s}, and where vertices n_{i} and n_{i′}, for i ≠ i′ ∈ {1, …, s}, are adjacent if and only if gcd(n_{i}, n_{i′}) > 1.

The graph Γ is the union of the various prime graphs Γ_{j} of [n_{1}, …, n_{s}], for j ∈ {1, …, t}, which also have vertices labelled by n_{1}, …, n_{s}, where vertices n_{i} and n_{i′}, for i ≠ i′ ∈ {1, …, s}, are adjacent in Γ_{j} if and only if p_{j} | gcd(n_{i}, n_{i′}). Hence each of the graphs Γ_{j}, next to isolated vertices, has at most one nontrivial connected component Γ_{j}^{0}. If it exists, Γ_{j}^{0} is a complete graph with at least two vertices, being labelled by the n_{i} having j-th exponent e_{ij} > 0; if it does not exist we for completeness let Γ_{j}^{0} be the empty graph.

In particular, n_{i} is an isolated vertex of Γ if and only if n_{i} does not belong to any of the connected components Γ_{j}^{0}. Moreover, if n_{i} is an isolated vertex of Γ_{j} then we are in the case s = 1 in the analysis in (4.5), thus we have 0 ∉ im(φ_{j}|_{O}), that is all elements of im(φ_{j}|_{O}) ⊆ ℤ/q_{j} have additive order divisible by p_{j}. Thus, in this case, if z = [z_{1}, …, z_{s}] ∈ φ^{−1}(0) ⊆ ℤ/n_{1} ⊕ ⋯ ⊕ ℤ/n_{s}, then writing z_{i} = [z_{i1}, …, z_{it}] ∈ ℤ/q_{i1} ⊕ ⋯ ⊕ ℤ/q_{it} ≅ ℤ/n_{i} we conclude that z_{ij} = 0 ∈ ℤ/q_{ij}.

(4.7) Theorem. Let [n_{1}, …, n_{s}] be a multiset containing the element 2 at most once. Then 1 is not an eigenvalue of Φ_{[n_{1},...,n_{s}]} if and only if

i) Γ has an isolated vertex, or

ii) n_{I} is even, and letting p_{1} := 2 the graph Γ_{1}^{0} has an odd number of vertices, none of which belongs to any of the Γ_{j}^{0}, where j ∈ {2, …, t}, and the associated multiset of exponents is [1, …, 1, ϵ] for some ϵ ⩾ 1.

Proof. We first show that m_{1} = 0 if either of conditions (i) or (ii) holds: Firstly, if Γ has an isolated vertex n_{i}, then it is isolated in Γ_{j} as well, for all j ∈ {1, …, t}. Hence for any z ∈ φ^{−1}(0), keeping the above notation, we have z_{ij} = 0 ∈ ℤ/q_{ij} for all j ∈ {1, …, t}, that is z_{i} = 0 ∈ ℤ/n_{i}, hence z ∉ O.

Secondly, we may assume that n_{1}, …, n_{s′} are odd, and n_{s′+1}, …, n_{s} are even, for some s′ ∈ {0, …, s − 1}. Then the vertices of Γ_{1}^{0} are labelled by [n_{s′+1}, …, n_{s}], hence s − s′ > 0 is odd, and since we may assume that Γ does not have isolated vertices, we may additionally assume that s−s′ ⩾ 3. Letting i ∈ {s′ +1, …, s}, since n_{i} is an isolated vertex of Γ_{j} for all j ∈ {2, …, t}, we may assume that the vertices of Γ_{1}^{0} are labelled by the multiset [2, …, 2, 2^{ϵ}]. Hence for any preimage z = [z_{1}, …, z_{s}] ∈ φ^{−1}(0) ∩ O we infer [z_{s′+1}, …, z_{s}] ∈ φ_{1}^{−1}(0) ∩ O, which is the exceptional case in (4.5), a contradiction.

Now assume that neither of conditions (i) and (ii) hold. Then we aim to show that the set K := {j ∈ {1, …, t}; Γ_{j}^{0} non-empty} fulfils the conditions described in (4.4). Note that, since Γ does not have isolated vertices, the set {p_{j}; j ∈ K} contains a prime divisor of any of the numbers n_{1}, …, n_{s}. We are going to apply (4.5) repeatedly: First of all, we are done if n_{I} is odd. Hence we may assume that n_{I} is even, and let p_{1} := 2, thus q_{1} = 2^{ϵ} for some ϵ ∈ ℕ. Then, if Γ_{1}^{0} is empty, or is non-empty and has an even number of vertices, we are done as well.

Hence we may assume that Γ_{1}^{0} has an odd number of vertices, being labelled by [n_{s′+1}, …, n_{s}], for some s′ ∈ {0, …, s − 1} such that s − s′ ⩾ 3 is odd. Assume next that n_{s}, say, belongs to Γ_{1}^{0} as well as to Γ_{j}^{0} for some j ∈ {2, …, t}. Then we replace the map φ_{1} by

φ′_{1} : ℤ/q_{s′+1,1} ⊕ ⋯ ⊕ ℤ/q_{s−1,1} → ℤ/q_{1} = ℤ/2^{ϵ},

where the left hand side has an even number of summands. Hence we have 0 ∈ im(φ′_{1}|_{O}), and the conditions in (4.4) hold for K and the maps φ′_{1}, φ_{2}, …, φ_{t}.

Finally, assume that none of the vertices of Γ_{1}^{0} belong to any of the Γ_{j}^{0}, where j ∈ {2, …, t}. Thus by assumption the 2-parts [q_{s′+1,1}, …, q_{s1}] are not of the form [2, …, 2, 2^{ϵ}], implying that 0 ∈ im(φ_{1}|_{O}) ⊆ ℤ/2^{ϵ}, and we are done. ♯

(4.8) Example. Since the multisets given (3.8) contain isolated vertices, entailing m_{1} = 0, we here give a few examples without isolated vertices: For the multisets [2, 4, 6] and [2, 3, 4, 6] and [2, 4, 6, 6] the associated characteristic polynomials have degree 15, 30 and 75, respectively, and we get the following associated multiplicities m_{d}, m′_{d} and m′′_{d}, respectively:

\begin{matrix}d & 1 & 2 & 3 & 4 & 6 & 12 \\ m_{d} & 0 & 1 & 1 & 1 & 1 & 2 \\ m′_{d} & 2 & 2 & 1 & 4 & 2 & 3 \\ m′′_{d} & 5 & 4 & 4 & 9 & 4 & 8 \end{matrix}

(4.9) We finally prove Corollary (1.4), where it indeed suffices to assume that the multiset [n_{1}, …, n_{s}] contains the element 2 at most once: The process of taking reciprocal polynomials is an involutory automorphism of the multiplicative semigroup ℤ[X] \ {0}. Moreover, letting Φ_{d} ∈ ℤ[X] be the d-th cyclotomic polynomial, where d ∈ ℕ, then we have Φ_{1}^{∗} = (X − 1)^{∗} = −(X − 1) = −Φ_{1}, while Φ_{d} is self-reciprocal for d ⩾ 2, see for example [14, Sect.1.3]. Since χ_{[n_{1},...,n_{s}]} ∈ ℤ[X] is a product of cyclotomic polynomials, we thus conclude that χ_{[n_{1},...,n_{s}]}^{∗} = ±χ_{[n_{1},...,n_{s}]}.

Hence for the constant term of χ_{[n_{1},...,n_{s}]} we have χ_{[n_{1},...,n_{s}]}(0) = ±1, where χ_{[n_{1},...,n_{s}]} is self-reciprocal if and only if χ_{[n_{1},...,n_{s}]}(0) = 1. Thus from

χ_{[n_{1},...,n_{s}]}(0) = (−1)^{l} · det(Φ_{[n_{1},...,n_{s}]}) = (−1)^{l(s+1)},

where l := ∏_{i=1}^{s}(n_{i} − 1), we infer that χ_{[n_{1},...,n_{s}]} is self-reciprocal if and only if s is odd or at least one of the n_{i} is odd.

(Alternatively, this also follows from recalling that χ_{[n_{1},...,n_{s}]} or χ_{[n_{1},...,n_{s},2]} is a Coxeter polynomial, depending on whether s is odd or even, respectively, and using the fact that Coxeter polynomials always are self-reciprocal.) ♯

Finally, note that by the above analysis χ_{[n_{1},...,n_{s}]} is self-reciprocal if and only if the multiplicity m_{1} ∈ ℕ_{0} of the eigenvalue 1 is even; see also the examples in (4.8). Hence the conditions characterising the case m_{1} = 0 in Theorem (4.7) constitute a special case of the condition in Corollary (1.4): Indeed, the conditions in Theorem (4.7) imply that Γ is not connected, which in turn entails that gcd(n_{1}, …, n_{s}) = 1, truly a special case of gcd(s, n_{1}, …, n_{s}) being odd.

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L.H.: Mathematisches Institut, Universität Münster Einsteinstraße 62, D-48149 Münster, Germany lutz.hille@uni-muenster.de

J.M.: Lehrstuhl D für Mathematik, RWTH Aachen Templergraben 64, D-52062 Aachen, Germany juergen.jueller@math.rwth-aachen.de