Rational Shi tableaux and the skew length statistic

We define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. These rational Shi tableaux encode dominant $p$-stable elements in the affine symmetric group. We prove that the rational Shi tableau is injective, that is, each dominant $p$-stable affine permutation is determined uniquely by its Shi tableau. Moreover, we provide a uniform generalisation of rational Shi tableaux to Weyl groups, and conjecture injectivity in the general case.

The skew length is a statistic on simultaneous core partitions, which counts the number of certain cells in the Young diagram of the core. It was invented by Armstrong [AHJ14] who used it to give a combinatorial formula for the rational q, t-Catalan numbers (see also [GM13,ALW15]) C n,p (q, t) = κ∈Cn,p q (κ) t (n−1)(p−1)/2−skl(κ) .
Here n and p are relatively prime, C n,p denotes the set of n, p-cores, (κ) is the length of κ and skl(κ) denotes the skew length (see Section 1 for all definitions). The intriguing property of these polynomials is the apparent symmetry C n,p (q, t) = C n,p (t, q), for which there is no proof except in very special cases.
The skew length was further studied in [Xin15] and [CDH15]. For coprime n and p it was proven that skl(κ) is invariant under conjugation of the core and independent of whether κ is viewed as an n, p-core or as a p, n-core. The first main contribution of our paper is a refinement of the skew length statistic in terms of the multiset of hook lengths of certain cells of the core. We obtain a refined version of the two results mentioned above in Theorem 1.3 providing a conceptual explanation for these phenomena. Moreover our † Email: robin.sulzgruber@univie.ac.at. Research supported by the Austrian Science Fund (FWF), grant S50-N15 in the framework of the Special Research Program "Algorithmic and Enumerative Combinatorics" (SFB F50). method of proof relies on induction and does not use rational Dyck paths. This has the advantage that we do not need to assume that n and p are relatively prime. Theorem 1.3 therefore extends to previously untreated cases.
Shi tableaux [FTV11] encode the dominant regions of the (m-extended) Shi arrangement, which are counted by the Fuß-Catalan numbers. The Shi arrangement and its regions are intimately related to the affine symmetric group. In [GMV14] Gorsky, Mazin and Vazirani define the so called p-stable affine permutations, which can be seen as a generalisation of the regions of the Shi arrangement to the rational level of Catalan combinatorics. In Section 2 we define rational Shi tableaux, which encode dominant p-stable affine permutations. Our main result concerning rational Shi tableaux is Theorem 2.8, asserting that each dominant p-stable affine permutation is uniquely determined by its Shi tableau. Theorem 2.8 is especially interesting in view of the fact that the rational Pak-Stanley labelling [GMV14] can be obtained by taking the row-sums of the rational Shi tableau.
The Shi arrangement can be defined for any irreducible crystallographic root system Φ. Furthermore, p-stable affine Weyl group elements have been considered in [Thi15]. We provide a uniform definition of the rational Shi tableaux of dominant p-stable elements of the affine Weyl group of Φ. We conjecture that dominant p-stable Weyl group elements are determined uniquely by their Shi tableaux (Conjecture 2.7).
In Section 3 we tie our previous results together by relating dominant p-stable affine permutations to rational Dyck paths and simultaneous cores. This is achieved using the Anderson bijection of [GMV14]. We show that the rational Shi tableau of a dominant p-stable affine permutation can be computed from the corresponding rational Dyck path as the codinv tableau of the path (Theorem 3.2). The codinv statistic on rational Dyck paths, that is, the sum of the entries of the codinv tableau, corresponds naturally to the skew length statistic on cores (Corollary 3.4). Hence we find that rational Shi tableaux can in fact be regarded as another refinement of the skew length statistic. The contents of Section 3 and in particular the connection between rational Shi tableaux and the zeta map on rational Dyck paths [GM13,ALW15] (Theorem 3.3) are the main tools used in the proof of Theorem 2.8. This is an extended abstract and some proofs have been omitted. A preprint of the full version is available [Sul15].

Skew length and conjugation
Throughout this section n and p are positive integers (not necessarily coprime!). For any positive integer z we set [z] = {1, . . . , z}. A partition λ is a weakly decreasing sequence λ 1 ≥ λ 2 ≥ · · · ≥ λ > 0 of positive integers. The number of parts or summands λ i is called the length of the partition and is denoted by (λ). The sum i λ i is called the size of λ. A partition is often identified with its Young diagram We call the elements of the Young diagram cells of the partition. The conjugate of the partition λ is the partition λ where λ i = #{j : λ j ≥ i}. Equivalently, if we view partitions as Young diagrams then (j, i) ∈ λ if and only if (i, j) ∈ λ. The hook length of a cell of λ is defined as h λ (i, j) = λ i − j + λ j − i + 1. A partition κ is called n-core if no cell of κ has hook length n. A partition is called n, p-core if it is both an n-core and a p-core. We denote the set of n-core partitions by C n and the set of simultaneous n, p-cores by C n,p . Clearly λ ∈ C n if and only if λ ∈ C n . Let λ be a partition. Given the hook length h of a cell x in the top row of λ we denote by H c (h) the set of hook lengths of cells in the same column as x. Given the hook length h of a cell x in the first column of λ we denote by H r (h) the set of hook lengths of cells in the same row as x. If h is not the hook length A subset A ⊆ Z is called abacus if there exist integers a, b ∈ Z such that z ∈ A for all z with z < a, and z / ∈ A for all z with z > b. We call the elements of A beads and the elements of Z − A gaps. An abacus is normalised if zero is a gap and there are no negative gaps. An abacus is balanced if the number of positive beads equals the number of nonpositive gaps. An abacus A is n-flush if z − n ∈ A for all z ∈ A. The theorem below is a version of the classical result that n-cores correspond to abacus diagrams that are n-flush (see for example [JK81,2.7.13]). Let κ ∈ C n be an n-core with maximal hook length m, that is, m is the hook length of the top left corner. Define α(κ) = {κ i + 1 − i : i ≥ 1} and β(κ) = H c (m) ∪ {z ∈ Z : z < 0}, where by convention κ i = 0 when i > (κ). Note that β(κ) = {z + (κ) − 1 : z ∈ α(κ)}.
Theorem 1.1 The map α is a bijection between n-cores and balanced n-flush abaci. The map β is a bijection between n-cores and normalised n-flush abaci.
For our purposes in this section we need the following simple consequence of Theorem 1.1.
Lemma 1.2 Let κ ∈ C n be an n-core and z ≥ 0. Then z + n ∈ H r (h) implies z ∈ H r (h), and In fact, the set of n-cores is characterised by the property that z + n ∈ H c (m) implies z ∈ H c (m) for all z ≥ 0, where m is the maximal hook length of κ.
Let κ ∈ C n,p be a simultaneous core with maximal hook length m. Moreover choose an element h ∈ H c (m). We call the row of κ whose leftmost cell has hook length h an n-row (resp. p-row) if h + n / ∈ H c (m) (resp. h + p / ∈ H c (m)). Similarly, given h ∈ H r (m) we call the column whose top cell has hook length h an n-column (resp. p-column) if h + n / ∈ H r (m) (resp. h + p / ∈ H r (m)). Denote by H n,p (κ) the multiset of hook lengths of cells that are contained both in an n-row and in a p-column. In Figure 1 the leftmost hook lengths of the 7-rows are 31, 15, 13, 12 and 4, and the top hook lengths of the 7-columns are 31, 29, 20, 12, 11 and 9.
This section's main result is a surprising symmetry property of the multiset H n,p (κ).
Proof: We prove the claim by induction on the size of κ. Denote byκ the partition obtained from κ by deleting the first column. Clearlyκ ∈ C n,p and we may assume that H n,p (κ) = H p,n (κ).
Let m denote the maximal hook length in κ. Note that each n-row ofκ is an n-row of κ. The only pcolumn ofκ that is not a p-column of κ has maximal hook length m−p. Thus there exist sets We conclude that z ∈ D and the proof is complete. 2 The skew length skl(κ) of an n, p-core was defined by Armstrong, Hanusa and Jones [AHJ14] as the number of cells that are contained in an n-row of κ and have hook length less than p. The set H n,p (κ) allows for a new equivalent definition.
From Theorem 1.3 we immediately recover two results that were recently proven by Guoce Xin [Xin15], and independently by Ceballos, Denton and Hanusa [CDH15] when n and p are relatively prime.
Corollary 1.5 The skew length of an n, p-core is independent of the order of n and p.
Indeed, with our alternative definition of skew length given in Proposition 1.4 the statements of Corollaries 1.5 and 1.6 are identical. We close this section by stating two equivalent conjectural properties of the multiset H n,p (κ) that need further investigation.
Conjecture 1.7 Let n and p be relatively prime, and κ ∈ C n,p . Then the hook length of each cell of κ appears in H n,p (κ) with multiplicity at least one.

Rational Shi tableaux
In the beginning of this section we recall some facts about root systems and Weyl groups. For further details we refer the reader to [Hum90]. Let Φ be an irreducible crystallographic root system with ambient space V , positive system Φ + and simple system ∆ = {σ 1 , . . . , σ r }. Any root α ∈ Φ can be written as a unique integer linear combination α = Thereby ht(α) > 0 if and only if α ∈ Φ + , and ht(α) = 1 if and only if α ∈ ∆. Moreover there exists a unique highest rootα such that ht(α) ≥ ht(α) for all α ∈ Φ. The Coxeter number of Φ can be defined as h = ht(α) + 1.
Let δ be a formal variable. We define the set of affine roots as Φ = {α + kδ : α ∈ Φ, k ∈ Z}. The height of an affine root is given by ht(α + kδ) = ht(α) + kh. The sets of positive and simple affine roots are defined as The Coxeter arrangement Cox(Φ) consists of all hyperplanes of the form H α = {x ∈ V : x, α = 0} for α ∈ Φ. Its regions, that is, the connected components of V − α∈Φ H α , are called chambers. We define the dominant chamber as The Weyl group W of Φ is the group of linear automorphisms of V generated by all reflections in a hyperplane in Cox(Φ). The Weyl group acts simply transitively on the chambers. Thus identifying the identity e ∈ W with the dominant chamber, each chamber corresponds to a unique Weyl group element.
The affine arrangement Aff(Φ) consists of all hyperplanes of the form H α,k = {x ∈ V : x, α = k}, where α ∈ Φ and k ∈ Z. Its regions are called alcoves. We define the fundamental alcove as The affine Weyl group W of Φ is the group of affine transformations of V that is generated by all reflections in a hyperplane in Aff(Φ). The affine Weyl group acts simply transitively on the set of alcoves. By identifying the identity e ∈ W with the fundamental alcove, every alcove corresponds to a unique element of W . An element ω ∈ W is called dominant if and only if the alcove ω(A • ) is contained in the dominant chamber C.
Given a root α ∈ Φ we define its coroot as α ∨ = 2α/ α, α . The coroot lattice is the integer span of all corootsQ = α∈Φ Zα ∨ ⊆ V . For each q ∈Q the translation t q : V → V defined by t q (x) = x + q for all x ∈ V is an element of the affine Weyl group. IdentifyingQ with its translation group we obtain W = W Q . Note that if ω ∈ W is dominant and ω = t q s, where q ∈Q and s ∈ W , then in particular q lies in the closure of the dominant chamber. Thus α, q ≥ 0 for each positive root α ∈ Φ + . Fix a root system of type A n−1 by declaring the roots, positive roots and simple roots to be We now describe a combinatorial model for the affine Weyl group of type A n−1 . A detailed exposition is found in [BB05,Sec. 8.3]. The affine symmetric group ∼ S n is the group of bijections ω : Z → Z such that ω(i+n) = ω(i)+n for all i ∈ Z and ω(1)+· · ·+ω(n) = n(n+1)/2. Such a bijection ω is called affine permutation. Each affine permutation is uniquely determined by its window [ω(1), ω(2), . . . , ω(n)]. The group ∼ S n has a set of generators called simple transpositions given by s i = [1, . . . , i + 1, i, . . . , n] for i ∈ [n − 1] and s n = [0, 2, . . . , n − 1, n + 1]. The affine symmetric group and the set of simple transpositions form a Coxeter system isomorphic to the the affine Weyl group of type A n−1 . The symmetric group S n can be seen as the subgroup of ∼ S n consisting of all affine permutations whose window is a permutation of [n].
The affine symmetric group possesses an involutive automorphism owing to the symmetry of the Dynkin diagram of type A n−1 . Set s * i = s n−i for i ∈ [n − 1] and s * n = s n . This correspondence extends to an automorphism ω → ω * on ∼ S n , where ω * is obtained by replacing all instances of s i in any expression of ω in terms of the simple transpositions by s * i . The involutive automorphism has a simple explicit description in window notation and fulfils many desirable properties.
(ii) The involutive automorphism preserves S n , translations, and dominant affine permutations.
Let ω ∈ ∼ S n be an affine permutation. An affine inversion of ω is a pair (i, j) ∈ [n] × N such that i < j and ω(i) > ω(j). For i, j ∈ [n] with i < j define The numbers k i,j (ω) where first considered by Shi [Shi86, Lem. 4.2.2]. We arrange them in a staircase tableau as in Figure 3. (i) Then the number of affine inversions of ω is given by i,j k i,j (ω).
(ii) If ω is dominant then k i,j (ω) equals the number of affine inversions (a, b) of ω, such that ω(a) ≡ i and ω(b) ≡ j modulo n.
Motivated by Lemma 2.2 we call the collection of numbers k i,j (ω) for i, j ∈ [n] with i < j the inversion table of ω ∈ ∼ S n . A simple computation reveals the effect of the involutive automorphism on the inversion table.
Proposition 2.3 Let ω ∈ ∼ S n be an affine permutation. Then the inversion table of ω * is the transpose of the inversion table of ω. That is k n+1−j,n+1−i (ω * ) = k i,j (ω) for all i, j ∈ [n] with i < j.
There is a natural way to generalise the inversion table to affine Weyl groups. Let Φ be an irreducible crystallographic root system with affine Weyl group W . Let ω ∈ W have the decomposition ω = t q s, where q ∈Q and s ∈ W . The affine Weyl group acts on the set of affine roots by ω · (α + kδ) = s · α + (k − q, s · α )δ.
Note that two distinct elements of ∼ S n may well have the same inversion table. However, we achieve a bijective correspondence between permutations and tableaux if we restrict ourselves to dominant affine permutations. We shall see that in case of the rational Shi tableau, which also counts certain inversions of affine permutations, the appropriate set is even smaller, in fact finite.
Let p be a positive integer that is relatively prime to n. An affine permutation ω ∈ ∼ S n is called p-stable if ω(i) < ω(i + p) for all i ∈ Z. The set of p-stable affine permutations in ∼ S n , denoted by ∼ S p n , was first considered by Gorsky, Mazin and Vazirani [GMV14]. As a consequence of Lemma 2.1 we see that the involutive automorphism ω → ω * preserves Let ω ∈ ∼ S p n be a dominant p-stable affine permutation. Let ω −1 = t q s, where q ∈Q and s ∈ S n , and p = mn + r with r ∈ [n − 1]. We define the rational Shi tableau of ω as the collection of integers where i, j ∈ [n] such that i < j.
Proposition 2.5 Let ω ∈ ∼ S p n be a dominant p-stable affine permutation. Then the rational Shi tableau of ω * is the transpose of the rational Shi tableau of ω. That is, t p n+1−j,n+1−i (ω * ) = t p i,j (ω) for all i, j ∈ [n] with i < j.
The rational Shi tableau generalises naturally to affine Weyl groups. Let Φ be an irreducible crystallographic root system, and p be relatively prime to the Coxeter number h of Φ. Let Φ p denote the set of affine roots of height p. In [Thi15, Sec. 8] Thiel defines p-stable affine Weyl group elements as the set W p = {ω ∈ W : ω · Φ p ⊆ Φ + }. Let α ∈ Φ + be a positive root and ω ∈ W p a dominant p-stable element of the affine Weyl group. Define where Φ + <p denotes the set of positive affine roots with height less than p. The rational Shi tableau of ω is the collection of numbers t p α (ω) for α ∈ Φ + . Proposition 2.6 Let n, p be positive coprime integers, i, j ∈ [n] with i < j, and ω ∈ ∼ S p n be a dominant p-stable affine permutation. Then t p i,j (ω) = t p ei−ej (ω), which counts the number of affine inversions (a, b) of ω such that b < a + p, and ω(a) ≡ i and ω(b) ≡ j modulo n.
The following is the main conjecture of this paper.
Conjecture 2.7 Let Φ be an irreducible crystallographic root system with affine Weyl group W and Coxeter number h, and let p be a positive integer relatively prime to h. Then each dominant p-stable element of the affine Weyl group is determined uniquely by its rational Shi tableau.
Conjecture 2.7 is known in to be true in the Fuß-Catalan case where p = mn + 1. In the next section we exploit the connections of the affine symmetric group to the combinatorics of rational Dyck paths to prove the conjecture if Φ is of type A n−1 . That is, we prove the following theorem. (The proof is found at the end of Section 3.) Theorem 2.8 Let n, p be two positive coprime integers. Then each dominant p-stable affine permutation in ∼ S n is determined uniquely by its rational Shi tableau.
The sum of the entries of the Shi tableau generalises the height statistic used by Stump in [Stu10,Conj. 3.14]. Thus the q-Fuß-Catalan numbers proposed therein are generalised by the polynomials the sum being taken over all dominant p-stable elements of the affine Weyl group, after a suitable transformation. To be precise we have q hmr/2 Cat (m) (W ; q −1 ) = C Φ,hm+1 (q), where r is the rank of Φ. Pak and Stanley [Sta96] found a bijection between the regions of the (m-extended) Shi arrangement of type A n−1 and the set of (m-)parking functions of length n. Gorsky, Mazin and Vazirani [GMV14, Def. 3.8] generalised this bijection to a map from ∼ S p n to the set of rational parking functions. Their rational Pak-Stanley labelling f (ω) is defined by Another major open problem concerning the Pak-Stanley labelling is to find an analogous labelling for different affine Weyl groups. Notably also this problem can be solved for rational Shi tableaux (recall the definition in (4)).
To conclude this section we want to connect the world of cores and abaci to dominant affine permutations. This connection is achieved by the next theorem, which follows essentially from the work of Lascoux [Las01]. Let ω ∈ ∼ S n be a dominant affine permutation, and define γ(ω) = {z ∈ Z : ω(z) ≤ 0}. S p n is p-stable. Recall the map α from cores to balanced flush abaci that was introduced in Section 1. We obtain a bijection α −1 • γ between the dominant affine permutations in ∼ S n and n-cores. In particular α −1 • γ restricts to a bijection between dominant p-stable affine permutations and C n,p . The following result appears to be new.

The codinv statistic
Let n and p be positive coprime integers. A rational Dyck path is a lattice path x that starts at (0, 0), consists of n North steps N = (0, 1) and p East steps E = (1, 0), and never goes below the diagonal with rational slope n/p. Denote the set of all rational Dyck paths by D n,p .
For (i, j) with 0 ≤ i ≤ p and 0 ≤ j ≤ n place the label i,j = jp − in in the unit square with bottom right corner (i, j). Given a rational Dyck path x ∈ D n,p , we assign to each of its steps the label i,j , where (i, j) is the starting point of the step. Let ( 1 , 2 , . . . , n ) be the vector consisting of the labels of the North steps of x (indicated in red in Figure 2) ordered increasingly. Denote by H(x) the set of positive labels below x (indicated in green in Figure 2). These correspond exactly to those boxes below x but strictly above the diagonal. Hence we have area(x) = #H(x).
We introduce the following definitions. A codinv pair of x is a pair of integers ( , h) such that is the label of a North step of x and h ∈ H(x) and ≤ h ≤ + p. The codinv tableau of x is the collection of numbers where i, j ∈ [n] with i < j. See Figure 3 for an example. Similar constructions have appeared before. First we remark that the codinv tableau is related to, albeit not the same as, the laser fillings of Ceballos, Denton and Hanusa [CDH15, Def. 5.13]. The row-sums and column-sums of the codinv tableau and the laser filling of a Dyck path agree. However, the codinv tableau is always of staircase shape while the laser filling sits inside the boxes below the rational Dyck path. Secondly we note that codinv pairs have been considered by Gorsky and Mazin [GM13] using slightly different notation. However, they only considered the column-sums of the codinv tableau.
Our next aim is to relate rational Shi tableaux to codinv tableaux. Our starting point is an elegant bijection ϕ : D n,p → C n,p between rational Dyck paths and simultaneous cores discovered by Anderson [And02]. For any finite set H = {h 1 , . . . , h k } of positive integers there exists a unique partition λ such that H is the set of hook lengths of the cells in the first column of λ. For any x ∈ D n,p let ϕ(x) be the partition such that the set of hook lengths of the cells in its first column equals H(x). The Dyck path in Figure 2 thereby corresponds to the core in Figure 1. Gorsky, Mazin and Vazirani [GMV14, Sec. 3.1] defined a generalised Anderson map A that bijectively maps the set of p-stable affine permutations to the set of rational parking functions. We are interested in the restriction of the Anderson map to the set of dominant p-stable affine permutations, which can be written as the composition A = ϕ −1 • α −1 • γ of maps we have already discussed. Here the inverse of Anderson's bijection ϕ makes an appearance, which accounts for the name of the Anderson map. We use the Anderson map to relate rational Shi tableaux to the codinv statistic.
Theorem 3.2 Let n, p be positive coprime integers and ω ∈ ∼ S p n be a dominant p-stable affine permutation. Then the rational Shi tableau of ω equals the codinv tableau of A(ω). That is, t p i,j (ω) = d i,j (A(ω)) for all i, j ∈ [n] with i < j.
The zeta map ζ : D n,p → D n,p is a map on rational Dyck paths that has appeared many times in the literature. Andrews, Krattenthaler, Orsina and Papi [AKOP02] first defined the inverse of the zeta function in the Catalan case D n,n+1 . Haglund [Hag03] studied the Catalan instance of the zeta map in connection with diagonal harmonics. More recently multiple descriptions of the rational zeta map have been found (see [GM13,ALW15]).
The zeta map can be defined as follows. Let x = x 1 x 2 . . . x n+p ∈ D n,p be a rational Dyck path with steps x i ∈ {N, E}. For i ∈ [n+p] let (x i ) be the label of the step x i . Then ζ(x) = x σ(1) x σ(2) . . . x σ(n+p) , where σ ∈ S n+p is the unique permutation such that (x σ(1) ) < (x σ(2) ) < · · · < (x σ(n+p) ). The zeta map is accompanied by a second map on rational Dyck paths that is called the eta map in [CDH15]. It is defined by η(x) = ζ • ϕ −1 ((ϕ(x) ) where λ is the conjugate partition of λ and ϕ is Anderson's bijection.
Using the codinv tableau we reprove a connection between the Anderson map, the zeta map and the Pak-Stanley labelling that was already observed in [GMV14,Thm. 5.3]. Note that each rational Dyck path x ∈ D n,p is the South-West boundary of a partition λ fitting inside the p × n rectangle. We call λ the complement of x. For example, the complement of the Dyck path x in Figure 2 is the partition λ = (11, 6, 6, 4, 3, 2). Theorem 3.3 Let n, p be positive coprime integers and ω ∈ ∼ S p n be a dominant p-stable affine permutation. Then the complement of ζ • A(ω) equals the (reversed (i) ) Pak-Stanley labelling f (ω). Moreover, the complement of η • A(ω) equals the (reversed) dual Pak-Stanley labelling f * (ω).
It is well known that skl(ϕ(x)) = (n − 1)(p − 1)/2 − area(ζ(x)). This follows from Armstrong's definition of the zeta map [ALW15, Thm 4.8], which uses cores. As a first consequence of Theorem 3.3 we obtain a justification for calling the codinv tableau a refinement of the skew length statistic.
Furthermore we may now use a result of Ceballos, Denton and Hanusa [CDH15] to prove that each rational Dyck path is determined uniquely by its codinv tableau, or equivalently, that each dominant pstable affine permutation is determined uniquely by its rational Shi tableau.
Theorem 3.5 Let x, y ∈ D n,p such that d i,j (x) = d i,j (y) for all i, j ∈ [n] with i < j. Then x = y.
Proof of Theorems 2.8 and 3.5: We deduce the claim from [CDH15, Thm. 6.3], which asserts that any rational Dyck path x can be reconstructed from the pair (ζ(x), η(x)). By Theorems 3.2 and 3.3 the codinv tableau of x encodes both ζ(x) and η(x) in terms of column-sums and row-sums. Therefore it contains enough information to determine the path x uniquely. Using the Anderson map again we obtain the analogous statement for rational Shi tableaux and dominant p-stable affine permutations. 2 Note that the exciting recent result due to Williams [Wil15, Cor. 6.4], that the zeta map is bijective, could replace [CDH15, Thm. 6.3] in the proof of Theorem 2.8.