May 9, 2003

SHORT ANTICHAINS IN ROOT SYSTEMS, SEMI-CATALAN ARRANGEMENTS, AND B-STABLE SUBSPACES

DMITRI I. PANYUSHEV

Independent University of Moscow, Bol’shoi Vlasevskii per. 11 121002 Moscow, Russia e-mail: panyush@mccme.ru

Let G be a complex simple algebraic group with Lie algebra 𝔤. Fix a Borel subalgebra 𝔟. An ideal of 𝔟 is called ad-nilpotent, if it is contained in [𝔟,𝔟]. The goal of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals in the case, where 𝔤 has roots of different length. Let 𝔄𝔡 denote the set of all ad-nilpotent ideals of 𝔟. Any 𝔠 ∈ 𝔄𝔡 is completely determined by the corresponding set of roots. The minimal roots in this set are called the generators of an ideal. The collection of generators of an ideal form an antichain in the poset of positive roots, and the whole theory can be expressed in the combinatorial language, in terms of antichains. An antichain is called strictly positive, if it contains no simple roots. Enumerative results for all and strictly positive antichains were recently obtained in the work Athanasiadis, Cellini-Papi, Sommers, and this author [1], [2], [3], [4], [10], [14]. There are two different theoretical approaches to describing (enumerating) antichains. The first approach consists of constructing a bijection between antichains and the coroot lattice points lying in a certain simplex. An important intermediate step here is a bijection between antichains and the so-called minimal elements of the affine Weyl group, Ŵ. It turns out that the simplex obtained is ”equivalent” to a dilation of the fundamental alcove of Ŵ, so that the problem of counting the coroot lattice points in it can be resolved. For strictly positive antichains, one constructs another bijection and another simplex, and the respective elements of Ŵare called maximal; yet, everything is quite similar. The second approach uses the Shi bijection between the ad-nilpotent ideals (or antichains) and the dominant regions of the Catalan arrangement. Under this bijection, the strictly positive antichains correspond to the bounded regions. There is a powerful result of Zaslavsky allowing one to compute the number of all and bounded regions, if the characteristic polynomial of the arrangement is known. Since the characteristic polynomial of the Catalan arrangement is recently computed in [1], the result follows. If 𝔤 has roots of different length, one can distinguish the length of elements occurring in antichains. We say that an antichain is short, if it consists of only short roots. This notion has a natural representation-theoretic incarnation: the short antichains are in a one-to-one correspondence with the 𝔟-stable subspaces, without nonzero semisimple elements, in the little adjoint G-module. A short analogue of strictly positive antichains, strictly s-positive antichains, is also defined. We are able to carry the above two approaches over to the short antichains. First, we introduce and characterize suitable elements of Ŵ (s-minimal and s-maximal ones), establish bijections between these two sets of elements and the coroot lattice points of certain simplices, and eventually obtain formulae for the number of short and strictly s-positive antichains. Second, we introduce and study the semi-Catalan arrangement, which has the same relation to short and strictly s-positive antichains as the usual Catalan arrangement has to all and strictly positive antichains. The difference between the Catalan and semi-Catalan arrangements is that we ”deform” only the hyperplanes orthogonal to short roots in the latter. We prove various results connecting the dominant regions of the semi-Catalan arrangement and the elements of Ŵ attached to short antichains. Adapting Athanasiadis’ argument from [1], we compute the characteristic polynomial for the extended semi-Catalan arrangements, or in other words, for m-semi-Catalan arrangements, Cat_{s}^{m}(Δ), with m = 0,1,2,.... For m = 0, one obtains the Coxeter arrangement of W, and for m = 1, the semi-Catalan arrangement.

This research was supported in part by R.F.B.I. Grants no. 01–01–00756 and 02–01–01041.

Here is a part of our results. Let α_{1},...,α_{p} be the simple roots of 𝔤 and θ the highest root. Let 𝒜 be the fundamental alcove of Ŵand g the sum of coefficients of the short simple roots in the expression of θ = ∑ c_{i}α_{i}. Then the short (resp. strictly s-positive) antichains are in a one-to-one correspondence with the coroot lattice points in (g + 1)𝒜 (resp. (g − 1)𝒜). If the root system is not of type G_{2}, this leads to a closed formula for the number of the

respective antichains. E.g., the number of short antichains is equal to ∏_{i=1}^{p} \frac{g + e_{i}+ 1}{e_{i}+ 1} , where

e_{i}, i = 1,2,...,p, are the exponents of the Weyl group W. Using this, we found a uniform expression, which covers the G_{2}-case as well, see Eq. (5.6), but it awaits for a conceptual explanation. The characteristic polynomial of Cat_{s}^{m}(Δ) is χ(t) = ∏ _{i=1}^{p} (t −mg−e_{i}) (again, if Δ is not of type G_{2}.) For G_{2}, the formula for χ(t) depends on the parity of m. We also define a ‘short’ analogue of the extended Shi arrangement, which we call, of course, the extended semi-Shi arrangement, and propose a conjectural formula for its characteristic polynomial, see (6.8).

A rough description of the contents is as follows. In Sections 2 and 3, we give a review of results concerning ideals (antichains) and Catalan arrangements, including two approaches described above. In particular, we consider minimal and maximal elements of Ŵand their connection with ideals. Some complements to known results are also given. We attempt to present a unified treatment that can be generalized afterwards, without much pains, to the setting of short antichains. Our main results are gathered in Sections 4-7. After a brief description in Section 4 of the relationship between 𝔟-stable subspaces of the little adjoint G-module and short antichains, we turn, in Section 5, to considering sminimal and s-maximal elements of Ŵand related simplices. In Section 6, we compute the characteristic polynomial for the m-semi-Catalan arrangement with arbitrary m ∈ ℕ and study relationship between the semi-Catalan arrangement (which corresponds to m = 1) and short antichains. As a consequence of our theory, we present, in Section 7, several intriguing results whose proof uses case-by-case verification.

To a great extent, this work was inspired by recent preprints of C. Athanasiadis [1] and E. Sommers [14].

Acknowledgements. This paper was written during my stay at the Max-Planck-Institut für Mathematik (Bonn). I would like to thank this institution for hospitality and excellent working conditions.

1. Notation and other preliminaries

(1.1) Main notation. Δ is the root system of (𝔤,𝔱) and W is the usual Weyl group. For α ∈ Δ, 𝔤_{α} is the corresponding root space in 𝔤.

Δ^{+} is the set of positive roots and ρ = \frac{1}{2} ∑_{α∈Δ^{+}} α.

Π = {α_{1},...,α_{p}} is the set of simple roots in Δ^{+} and θ is the highest root in Δ^{+} . We set V := 𝔱_{ℝ} = ⊕_{i=1}^{p} ℝα_{i} and denote by ( , ) a W -invariant inner product on V. As usual, μ^{∨} = 2μ/(μ,μ) is the coroot for μ ∈ Δ.

𝒞 = {x ∈ V | (x,α) > 0 ∀α ∈ Π} is the (open) fundamental Weyl chamber.

𝒜 = {x ∈ V | (x,α) > 0 ∀α ∈ Π & (x,θ) < 1} is the fundamental alcove.

Q^{+} = {∑ _{i=1}^{p} n_{i}α_{i} | n_{i} = 0,1,2,...} and Q^{∨} = ⊕_{i=1}^{p} ℤα_{i}^{∨} ⊂ V is the coroot lattice. Letting V̂= V ⊕ ℝδ ⊕ ℝλ, we extend the inner product ( , ) on V̂ so that (δ,V) = (λ,V) = (δ,δ) = (λ,λ) = 0 and (δ,λ) = 1.

Δ̂ = {Δ + kδ | k ∈ ℤ} is the set of affine real roots and Ŵis the affine Weyl group. Then Δ̂^{+} = Δ^{+} ∪ {Δ + kδ | k ≥ 1} is the set of positive affine roots and Π̂ = Π ∪ {α_{0}} is the corresponding set of affine simple roots, where α_{0} = δ − θ. The inner product ( , ) on V̂is Ŵ-invariant. The notation β > 0 (resp. β < 0) is a shorthand for β ∈ Δ̂^{+} (resp. β ∈ −Δ̂^{+} ). For α_{i} (0 ≤ i ≤ p), we let s_{i} denote the corresponding simple reflection in Ŵ. If the index of α ∈ Π̂ is not specified, then we merely write s_{α}. The length function on Ŵwith respect to s_{0},s_{1},...,s_{p} is denoted by ℓ. For any w ∈ Ŵ, we set

N(w) = {α ∈ Δ̂^{+} | w(α) ∈ −Δ̂^{+} }.

It is standard that #N(w) = ℓ(w) and N(w) is bi-convex. The latter means that both N(w) and Δ̂^{+} \ N(w) are subsets of Δ̂^{+} that are closed under addition. Furthermore, the assingment w ↦ N(w) sets up a bijection between the elements of Ŵand the finite bi-convex subsets of Δ̂^{+} .

(1.2) Ideals and antichains. Throughout the paper, 𝔟 is the Borel subalgebra of 𝔤 corresponding to Δ^{+} and 𝔲 = [𝔟,𝔟]. Let 𝔠 ⊂ 𝔟 be an ad-nilpotent ideal. Then 𝔠 = ⊕_{α∈I} 𝔤_{α}

for some I ⊂ Δ^{+} . This I is said to be an ideal (of Δ^{+} ). More precisely, a set I ⊂ Δ^{+} is an ideal, if whenever γ ∈ I,μ ∈ Δ^{+} , and γ + μ ∈ Δ, then γ + μ ∈ I. Our exposition will be mostly combinatorial, i.e., in place of ad-nilpotent ideal of 𝔟 we will deal with the respective ideals of Δ^{+} . For μ,γ ∈ Δ^{+} , write μ ≼ γ, if γ − μ ∈ Q^{+} . The notation μ ≺ γ means that μ ≼ γ and γ ≠ μ. We regard Δ^{+} as poset under ‘≼’. Let I ⊂ Δ^{+} be an ideal. An element γ ∈ I is called a generator, if γ − α ∉ I for any α ∈ Π. In other words, γ is a minimal element of I with respect to ‘≼’. We write Γ(I) for the set of generators of I. It is easily seen that Γ(I) is an antichain of Δ^{+} , i.e., γ_{i} ⋠ γ_{j} for any pair (γ_{i},γ_{j}) in Γ(I). Conversely, if Γ ⊂ Δ^{+} is an antichain, then the ideal

I⟨Γ⟩:= {μ ∈ Δ^{+} | μ ≽ γ_{i} for some γ_{i} ∈ Γ}

has Γ as the set of generators. Let 𝔄𝔫 denote the set of all antichains in Δ^{+} . In view of the above bijection 𝔄𝔡 ⟷^{1:1} 𝔄𝔫, we will freely switch between ideals and antichains. An ideal I is called strictly positive, if I ∩ Π = ∅. The set of strictly positive ideals is denoted by 𝔄𝔡_{0}.

2. Ideals, maximal and minimal elements of Ŵ

In this section we review some recent results by Athanasiadis, Cellini-Papi, Sommers, and this author. A few complements is also given. The idea of describing ideals of Δ^{+} through the use of elements of Ŵgoes back to D. Peterson, who exploited minuscule elements for counting Abelian ideals of 𝔟, see [7]. In general case, given I ⊂ Δ^{+} , we want to have w ∈ Ŵsuch that N(w) ⊂ ∪_{k≥1}(kδ − Δ^{+} ) and N(w) ∩ (δ − Δ^{+} ) = δ − I. It turns out that, for any ideal I, there is a unique element of minimal length satisfying these properties. In contrast, the element of maximal length exists if and only if I is strictly positive, and in this case such an element is unique, too. Implementation of this program yields also explicit formulae for the number of all and strictly positive ideals.

As is well known, Ŵis isomorphic to a semi-direct product of W and Q^{∨}. Given w ∈ Ŵ, there is a unique decomposition

(2.1)  w = v·t_{r} ,

where v ∈ W and t_{r} is the translation corresponding to r ∈ Q^{∨}. The word “translation” means the following. The group Ŵhas two natural actions:

(a) the linear action on V̂= V ⊕ ℝδ ⊕ ℝλ;

(b) the affine-linear action on V. We use ‘∗’ to denote the second action. For r ∈ Q^{∨}, the linear action of t_{r} ∈ Ŵon V ⊕ ℝδ is given by t_{r}(x) = x − (x,r)δ (we do not need the formulas for the whole of V̂), while the affine-linear action on V is given by t_{r}∗y = y + r. So that t_{r} is a true translation for the ∗-action on V. Let us say that w ∈ Ŵis dominant, if w(α) > 0 for all α ∈ Π. Obviously, w is dominant if and only if N(w) ⊂ ∪_{k≥1}(kδ − Δ^{+} ). It also follows from [3, 1.1] that w is dominant if and only if w^{−1} ∗𝒜 ⊂ 𝒞. Write Ŵ_{dom}for the set of dominant elements.

2.2 Proposition.

(i) If w = v·t_{r} ∈ Ŵ_{dom}, then r ∈ −\overline{𝒞};
(ii) The mapping Ŵ_{dom}→ Q^{∨} given by w = v·t_{r} ↦ v(r) is a bijection.

Proof. (i) We have w^{−1} ∗x = v^{−1} (x) − r for any x ∈ V. In particular, w^{−1} ∗0 = −r. Since 0 ∈ \overline{𝒜} and w is dominant, we are done.

(ii) Given κ ∈ Q^{∨}, we want to find w = v·t_{r} such that w^{−1} ∗𝒜 = v^{−1} (𝒜) − r ⊂ 𝒞 and v(r) = κ. In view of the last equality, the previous containment reads v^{−1} (𝒜 − κ) ⊂ 𝒞. Therefore v must be the unique element of W taking the alcove 𝒜 − κ into the dominant Weyl chamber 𝒞. Then r = v^{−1} (κ). This argument proves both the injectivity and surjectivity of the mapping in question. □

Letting δ−I := N(w)∩(δ−Δ^{+} ), we easily deduce that I is an ideal, if w ∈ Ŵ_{dom}. We say δ−I is the first layer of N(w) and I is the first layer ideal of w. However, an ideal I may well arise from different dominant elements. To obtain a bijection, one has to impose further constraints on dominant elements. One may attempt to consider either maximal or minimal bi-convex subsets with first layer δ − I. This naturally leads to notions of ‘minimal’ and ‘maximal’ elements. This terminology suggested in [14] is also explained by a relationship between these elements and dominant regions of the Shi arrangement, see Section 3. However, the formal definitions do not require invoking arrangements. Furthermore, we want to stress that many results relating the ideals and these two kinds of dominant elements can be obtained without ever mentioning the Shi (or Catalan) arrangement.

2.3 Definition. w ∈ Ŵis called minimal, if

(i) w is dominant;

(ii) if α ∈ Π̂ and w^{−1} (α) = kδ + μ for some μ ∈ Δ, then k ≥ −1.

Using (i), condition (ii) can be made more precise. If k ∈ {−1,0}, then μ ∈ Δ^{+} . The set of minimal elements is denoted by Ŵ_{min}.

2.4 Proposition [3, Prop. 2.12]. There is a bijection between Ŵ_{min}and 𝔄𝔡. Namely,

• given w ∈ Ŵ_{min}, the corresponding ideal is {μ ∈ Δ^{+} | δ − μ ∈ N(w)};

• given I ∈ 𝔄𝔡, the corresponding minimal element is determined by the finite bi-

convex set

⋃_{k≥1}(kδ − I^{k}) ⊂ Δ̂^{+} .

Here I^{k} is defined inductively by I^{k} = (I^{k−1} + I) ∩ Δ^{+} .

If N ⊂ Δ̂^{+} is a finite convex subset, containing δ−I, then it must also contain ∪_{i≥1} (kδ−I^{k}). So, the latter is the minimal bi-convex subset containing δ − I. The first layer ideal of w ∈ Ŵ_{min}is denoted by I_{w}.

2.5 Proposition [10, Theorem 2.2] [14, 6.3(1)].

If w ∈ Ŵ_{min}, then Γ(I_{w}) = {γ ∈ Δ^{+} | w(δ − γ) ∈ −Π̂ }.

Following [4], we give a ”geometric” description of the minimal elements. Set

D_{min}= {x ∈ V | (x,α) ≥ −1 ∀α ∈ Π & (x,θ) ≤ 2} .

It is a certain simplex in V.

2.6 Proposition [4, Prop. 2 & 3].

1. w = v·t_{r} ∈ Ŵ_{min}⟺ \{ \begin{matrix}w is dominant, \\ v(r) ∈ D_{min}∩ Q^{∨} . \end{matrix}

2. The mapping Ŵ_{min}→ D_{min}∩ Q^{∨}, w = v·t_{r} ↦ v(r), is a bijection.

Proof. 1. ‘⇒’ The first condition is satisfied by the definition. Next, we have w^{−1} (x) = v^{−1} (x) + (x,v(r))δ for any x ∈ V ⊕ ℝδ. In particular,

(2.7)  \begin{matrix}w^{−1} (α_{i}) = v^{−1} (α_{i}) + (α_{i},v(r))δ, i ≥ 1, \\ w^{−1} (α_{0}) = −v^{−1} (θ) + (1 − (θ,v(r)))δ . \end{matrix}

Comparing this with Definition 2.3(ii), one concludes that v(r) ∈ D_{min}.

‘⇐’ The previous argument can be reversed.

2. This follows from part 1 and Proposition 2.2. □

Remark. The above proof applies equally well to Propositions 2.14, 5.3 and 5.10 below. It is a simplified version of the proof of Propositions 2 & 3 in [4]. It follows that #(𝔄𝔡) equals the number of integral points in D_{min}. (Unless otherwise stated, an ’integral point’ is a point lying in Q^{∨}.) A pleasant feature of this situation is that there is an element of Ŵthat takes D_{min}to a dilated closed fundamental alcove. Namely, w(D_{min}) = (h + 1)\overline{𝒜} for some w ∈ Ŵ, see [4, Thm. 1]. Write θ as a linear combination of simple roots: θ = ∑_{i}c_{i}α_{i}. The integers c_{i} are said to be the coordinates of θ. By a result of M. Haiman [6, 7.4], the number of integral points in t\overline{𝒜} is equal to

(2.8)  ∏_{i=1}^{p} \frac{t + e_{i}}{1 + e_{i}}

whenever t is relatively prime with all the coordinates of θ. Since this condition is satisfied for t = h + 1, one obtains

(2.9)  #(𝔄𝔡) = ∏_{i=1}^{p} \frac{h + e_{i}+ 1}{e_{i}+ 1} .

It is the main result of [4]. Combining Propositions 2.5 and Eq. (2.7) yields the assertion that #Γ(I_{w}) = k if and only if v(r) lies on a face of D_{min}of codimension k [10, Thm. 2.9].

Now, we turn to considering maximal (dominant) elements of Ŵ. Most of the results on these elements are due to E. Sommers. However our presentation follows [14] rather freely, since we want to have a uniform treatment for both minimal and maximal elements. Because some assertions have no exact counterparts in [14], we also give some proofs.

2.10 Definition. w ∈ Ŵis called maximal, if

(i) w is dominant; (ii) if α ∈ Π̂ and w^{−1} (α) = kδ + μ for some μ ∈ Δ, then k ≤ 1.

Using (i), condition (ii) can be made more precise. If k = 1, then μ ∈ −Δ^{+} ; if k = 0, then μ ∈ Δ^{+} The set of maximal elements is denoted by Ŵ_{max}.

If I ∈ 𝔄𝔡_{0}, then for any μ ∈ Δ^{+} we define k(μ,I) as the minimal possible number of summands in the expression μ = ∑_{i}ν_{i}, where ν_{i} ∈ Δ^{+} \ I. Notice that this definition only makes sense for strictly positive ideals.

2.11 Proposition [14, Sect. 5]. There is a bijection between Ŵ_{max}and 𝔄𝔡_{0}. Namely,

• given w ∈ Ŵ_{min}, the corresponding strictly positive ideal is {μ ∈ Δ^{+} | δ − μ ∈ N(w)};

• given I ∈ 𝔄𝔡_{0}, the corresponding maximal element is determined by the finite

bi-convex set (◇) {mδ − γ | γ ∈ I & 1 ≤ m ≤ k(γ,I)− 1}.

Proof. 1. Suppose w ∈ Ŵis dominant, and let I be the first layer ideal of w. Assuming that I ∩ Π ∋ α, we show that w cannot be maximal. For any γ ∈ I, let k_{γ} be the maximal integer such that k_{γ}δ − γ ∈ N(w), i.e.,

N(w) = {lδ − γ | γ ∈ I & 1 ≤ l ≤ k_{γ}} .

Let I(α) be the ideal generated by α. Clearly, I(α) ⊂ I. Set

N(w)^{⟨2⟩}= {lδ − γ | γ ∈ I(α) & 1 ≤ l ≤ 2k_{γ}} ∪ {lδ − γ | γ ∈ I \ I(α) & 1 ≤ l ≤ k_{γ}} .

Obviously, N(w)^{⟨2⟩}is finite and has the same first layer as N(w). It is also easy to verify that N^{⟨2⟩}is again bi-convex. Hence N(w)^{⟨2⟩}= N(w′) for some w′∈ Ŵ. Since N(w′) ⊃ N(w), there is a presentation w′= uw, where ℓ(w′) = ℓ(u) + ℓ(w). If s_{ν} (ν ∈ Π̂ ) is the rightmost reflection in a reduced decomposition for u, then w^{−1} (ν) = kδ − μ with k ≥ 2, as the first layers of N(w′) and N(w) are the same. Thus, w is not maximal.

2. Suppose I ∈ 𝔄𝔡_{0}, and let w ∈ Ŵ be any dominant element with first layer ideal I. Since Δ̂^{+} \ N(w) is convex and contains δ − (Δ^{+} \ I), it follows from the very definition of numbers k(γ,I) that lδ − γ ∈ Δ̂^{+} \ N(w) for all l ≥ k(γ,I). Hence N(w) is contained in the finite set given by Eq. 2.11(◇). It only remains to prove that the latter is bi-convex. For this crucial fact, we refer to [14, Lemma 5.2]. □

The strictly positive ideal corresponding to w ∈ Ŵ_{max}(the first layer ideal of w) is denoted by I^{w}. For an ideal I ⊂ Δ^{+} , we write Ξ(I) for the set of maximal elements of Δ^{+} \ I. It is immediate that Ξ(I) is an antichain.

2.12 Proposition [14, 6.3(2)]. If w ∈ Ŵ_{max}, then Ξ(I^{w}) = {γ ∈ Δ^{+} | w(δ − γ) ∈ Π̂ }.

2.13 Remark. Note that antichains of the form Ξ(I^{w}) are not arbitrary. From the definition of a strictly positive ideal it readily follows that, given Ξ ∈ 𝔄𝔫, we have Ξ = Ξ(I) for some I ∈ 𝔄𝔡_{0} if and only if for any α ∈ Π there is a γ ∈ Ξ such that γ ≽ α. We shall say that such an antichain covers the simple roots.

Now, we proceed to a ”geometric” characterization of the maximal elements. Set

D_{max}= {x ∈ V | (x,α) ≤ 1 ∀α ∈ Π & (x,θ) ≥ 0} .

It is a certain simplex in V.

2.14 Proposition (cf. [14, Prop. 5.6]).

1. w = v·t_{r} ∈ Ŵ_{max}⟺ \{ \begin{matrix}w is dominant, \\ v(r) ∈ D_{max}∩ Q^{∨} . \end{matrix}

2. The mapping Ŵ_{max}→ D_{max}∩ Q^{∨}, w = v·t_{r} ↦ v(r), is a bijection.

Proof. 1. The argument is the same as in Proposition 2.6, taking into account that the constraints for D_{max}are different.

2. This follows from part 1 and Proposition 2.2. □

2.15 Proposition. Suppose w = v·t_{r} ∈ Ŵ_{max}. Then #Ξ(I^{w}) = k if and only v(r) lies on a face of codimension k of D_{max}.

Proof. Combine Propositions 2.12 and Eq. (2.7). □

Since Π is the only antichain of cardinality p [10, 2.10(ii)] and it is certainly of the form Ξ(I^{w}), we see that D_{max}has a unique integral vertex. In order to compute #(D_{max}∩ Q^{∨}), we replace D_{max} with another simplex. Let {ϖ_{i}^{∨}}_{i=1}^{p} denote the dual basis of V for {α_{i}}_{i=1}^{p}. Set ρ^{∨} = ∑ _{i=1}^{p} ϖ_{i}^{∨}. Since the sum of the coordinates of θ equals h−1, the translation x ↦ t_{−ρ^{∨}} ∗x = x−∑ _{i=1}^{p} ϖ_{i}^{∨} takes D_{max}to the negative dilated fundamental alcove

−(h − 1)\overline{𝒜} = \{x ∈ V | (x,α) ≤ 0 ∀α ∈ Π; (x,θ) ≥ 1 − h\}.

It may happen that ρ^{∨} does not belong to Q^{∨}, so that this translation, which is in the extended affine Weyl group, does not belong to Ŵ, while we wish to have a transformation from Ŵ. Nevertheless, since h − 1 is relatively prime with the index of connection of Δ, it follows from [4, Lemma 1] that there is an element of Ŵthat takes D_{max}to (1 − h)\overline{𝒜}.

Again, using the above-mentioned result of Haiman, see Eq. (2.8), one obtains

2.16 Theorem [1],[10],[14].

#(𝔄𝔡_{0}) = ∏_{i=1}^{p} \frac{h + e_{i}− 1}{e_{i}+ 1} .

Remark. The proofs in [1] and [10] are based on the fact that the strictly positive ideals correspond to the bounded regions of the Catalan arrangement and that the number of bounded regions of any hyperplane arrangement can be computed via the characteristic polynomial of this arrangement, see Section 3.

3. Ideals and dominant regions of the Catalan arrangement

Recall a bijection between the ideals of Δ^{+} and the dominant regions of the Catalan arrangement. This bijection is due to Shi [13, Theorem 1.4]. For μ ∈ Δ^{+} and k ∈ ℤ, define the hyperplane ℋ_{μ,k} in V as {x ∈ V | (x,μ) = k}. The Catalan arrangement, Cat(Δ), is the collection of hyperplanes ℋ_{μ,k}, where μ ∈ Δ^{+} and k = −1,0,1. The regions of an arrangement are the connected components of the complement in V of the union of all its hyperplanes. Obviously, the dominant regions of Cat(Δ) are the same as those for the Shi arrangement Shi(Δ). The latter is the collection of hyperplanes ℋ_{μ,k}, where μ ∈ Δ^{+} and k = 0,1. But, it will be more convenient for us to deal with the arrangement Cat(Δ), since it is W -invariant. It is clear that 𝒞 is a union of regions of Cat(Δ). Any region lying in 𝒞 is said to be dominant. The Shi bijection takes an ideal I ⊂ Δ^{+} to the dominant region

(3.1)  R_{I} = {x ∈ 𝒞 | (x,γ) > 1, if γ ∈ I & (x,γ) < 1, if γ ∉ I} .

A proof of this result, which is almost self-contained, is given by Athanasiadis in [2, Lemma 3.1]. But for the fact that R_{I} is non-empty he refers to an earlier work Shi [12]. It is not, however, easy to extract the proof from Shi’s paper. What we want to say is that the non-emptiness of R_{I} readily follows from the theory of minimal elements:

If w ∈ Ŵis the minimal element corresponding to I, then w^{−1} ∗𝒜 belong to R_{I}.

Indeed, ℋ_{μ,1} separates 𝒜 and w^{−1} ∗𝒜 if and only if w(δ − μ) ∈ −Δ̂^{+} , see [3, 1.1]. In fact, w^{−1} ∗𝒜 is the nearest to the origin alcove in R_{I}.

A region (of an arrangement) is called bounded, if it is contained in a sphere about the origin.

3.2 Proposition [1],[10],[14]. I ∈ 𝔄𝔡(𝔤)_{0} if and only if the region R_{I} is bounded.

If R_{I} is bounded, then it obviously contains an alcove that is most distant from the origin. It was shown in [14] that if w is the maximal element corresponding to I ∈ 𝔄𝔡_{0}, then w^{−1} ∗𝒜 is the most distant from the origin alcove in R_{I}. The number of regions and bounded regions of any hyperplane arrangement can be counted through the use of a striking result of T. Zaslavsky. Let χ(A,t) denote the characteristic polynomial of a hyperplane arrangement A in V (see e.g. [1, Sect. 2] for precise definitions).

3.3 Theorem (Zaslavsky).

1. The number of regions into which A dissects V equals (−1)^{p}χ(A,−1).

2. The number of bounded regions into which A dissects V equals |χ(A,1)|.

In [1], Athanasiadis gives a nice case-free proof of the following formula for the characteristic polynomial of the Catalan arrangement:

(3.4)  χ(Cat(Δ),t) = ∏_{i=1}^{p} (t − h − e_{i}) .

Since Cat(Δ) is W -invariant, the values \frac{|χ(Cat(Δ),±1)|}{#(W)} give the number of bounded and

all regions in 𝒞, respectively. In this way, one obtains explicit formulae for the cardinality of 𝔄𝔡_{0} and 𝔄𝔡 written already down in Section 2. Thus, the characteristic polynomial of the Catalan arrangement provides an alternative approach to counting ideals and strictly positive ideals.

4. Short antichains and 𝔟-stable subspaces in the little adjoint G-module

At the rest of the paper, we stick to the case in which Δ has roots of different length. Then we naturally have long and short roots, long and short reflections, etc. Our goal is to show that the theory presented in the previous sections can be extended to the setting, where one pays attention to the length of roots involved. A piece of such theory already appeared in [11], where we studied Abelian ideals of Δ^{+} consisting of only long roots. Now, we consider the general case. Our treatment will again be combinatorial. We wish, however, to stress that it has a related representation-theoretic picture. While the ideals (antichains) in Δ^{+} correspond bijectively to the 𝔟-stable subspaces in 𝔤 having no nonzero semisimple elements, our short antichains in Δ^{+} correspond bijectively to the 𝔟-stable subspaces, without nonzero semisimple elements, in the little adjoint 𝔤-module. To distinguish various objects associated with long and short roots, we use the subscripts ‘l’ and ‘s’, respectively. For instance, Π_{l} is the set of long simple roots and Δ_{s}^{+} is the set of short positive roots. Accordingly, each simple reflection s_{i} is either short or long. Since θ is long, the simple root α_{0} and the reflection s_{0} are regarded as long. Therefore, Π̂_{l} = Π_{l} ∪ {α_{0}}. Write θ_{s} for the unique short dominant root in Δ^{+} . A simple 𝔤-module with highest weight θ_{s}, 𝕍(θ_{s}), is said to be little adjoint. The set of nonzero weights of 𝕍(θ_{s}) is Δ_{s}, all nonzero weights are simple, and the multiplicity of the zero weight is #(Π_{s}) [8, 2.8].

4.1 Definition. An antichain Γ ⊂ Δ^{+} is called short, if it consists of short roots, i.e., Γ ⊂ Δ_{s}^{+} . Similarly, one defines a long antichain.

If Γ is a short antichain, then Γ^{∨} is a long antichain in the dual root system Δ^{∨}. Therefore, it suffices, in principle, to consider only short antichains. We write 𝔄𝔫_{s} for the set of all short antichains of Δ^{+} . The respective set of ideals is denoted by 𝔄𝔡_{s}. Recall that, for any finite-dimensional rational G-module 𝕍, there are notions of semisimple and nilpotent elements, generalizing those in 𝔤, see [15]. An element v ∈ 𝕍 is called semisimple, if the orbit Gv is closed; it is called nilpotent, if the closure of Gv contains the origin. We shall say that a subspace of 𝕍 is nilpotent, if it consists of nilpotent elements.

4.2 Proposition. There is a one-to-one correspondence between 𝔄𝔫_{s} and the nilpotent 𝔟-stable subspaces of 𝕍(θ_{s}).

Proof. If Γ is a short antichain, then the corresponding subspace of 𝕍(θ_{s}) is the sum of weight spaces 𝕍(θ_{s})_{μ}, μ ∈ Δ_{s}^{+} , such that γ_{i}≼μ for some γ_{i} ∈ Γ. Conversely, if 𝕌 is a 𝔟-stable subspace of 𝕍(θ_{s}), then it is a sum of weight spaces. If 𝕌 contains no semisimple elements, then all weights of 𝕌 must lie in an open halfspace of V (by the Hilbert-Mumford criterion). From the above description of weights of 𝕍(θ_{s}), it follows that the weights of 𝕌 must form a subset of Δ_{s}^{+} . The minimal elements of this set of weights give us the required short antichain. □

If Γ is a short antichain, then the set of weights of the corresponding nilpotent 𝔟-stable subspace of 𝕍(θ_{s}) is I⟨Γ⟩ ∩ Δ_{s}^{+} .

5. Short antichains, s-minimal and s-maximal elements of Ŵ

Our goal is to show that the theory described in Section 2 extends well to short antichains.

5.1 Definition. w ∈ Ŵis called s-minimal, if

(i) w is dominant; (ii) if α ∈ Π_{s} and w^{−1} (α) = kδ + μ with μ ∈ Δ, then k ≥ −1; (iii)if α ∈ Π̂_{l} and w^{−1} (α) = kδ + μ with μ ∈ Δ, then k ≥ 0;

Using (i), conditions (ii), (iii) can be made more precise. If k = 0 or k = −1 in (ii), then μ ∈ Δ^{+} . We write Ŵ_{min}^{(s)}for the set of all s-minimal elements. Notice that Ŵ_{min}^{(s)}⊂ Ŵ_{min}.

5.2 Proposition. The bijection between 𝔄𝔡 and Ŵ_{min}described in Proposition 2.4 gives rise to a bijection between 𝔄𝔫_{s} (or 𝔄𝔡_{s}) and Ŵ_{min}^{(s)}.

Proof. 1. Suppose w ∈ Ŵ_{min}^{(s)}, and let I_{w} be the corresponding ideal. It follows from Definition 5.1(ii),(iii) and Proposition 2.5 that Γ(I_{w}) ⊂ Δ_{s}^{+} . Thus, we obtain a short antichain.

2. The use of Proposition 2.5 gives also the converse. □

Now, we give a geometric description of s-minimal elements in the spirit of Section 2. Set

D_{min}^{(s)}= {x ∈ V | (x,α) ≥ −1 (α ∈ Π_{s}); (x,α) ≥ 0 (α ∈ Π_{l}); (x,θ) ≤ 1} ,

and recall that w = v·t_{r}, where v ∈ W and r ∈ Q^{∨}.

5.3 Proposition.

1. w = v·t_{r} ∈ Ŵ_{min}^{(s)}⟺ \{ \begin{matrix}w is dominant, \\ v(r) ∈ D_{min}^{(s)}∩ Q^{∨} . \end{matrix}

2. The mapping Ŵ_{min}^{(s)}→ D_{min}^{(s)}∩ Q^{∨}, w = v·t_{r} ↦ v(r), is a bijection.

Proof. The argument is the same as in Proposition 2.6, taking into account that the constraints for D_{min}^{(s)}are different. □

In order to compute the number #(D_{min}^{(s)}∩ Q^{∨}), we perform the following transformation. Set ρ_{s}^{∨} = ∑_{α_{i}∈Π_{s}} ϖ_{i}^{∨}. It is easily seen that the translation t_{ρ_{s}^{∨}} takes D_{min}^{(s)}to the dilated closed fundamental alcove

(g + 1)\overline{𝒜} = \{x ∈ V | (x,α) ≥ 0 ∀α ∈ Π; (x,θ) ≤ g + 1\}.

Here g = (θ,∑_{α_{i}∈Π_{s}} ϖ_{i}^{∨}), i.e., it is the sum of the short coordinates of θ (i.e., those corresponding to the short simple roots). It is easy to obtain other formulae for g. E.g., g = (#Δ_{s})/p = (ρ_{s},θ^{∨}), where ρ_{s} is the half-sum of all positive short roots. If we want to explicitly indicate that g depends on Δ, we write g_{Δ}. Although the above translation may not belong to Ŵ, the very existence of such a transformation and Lemma 1 in [4] show that the following is true

5.4 Lemma. If g + 1 and the index of connection of Δ are relatively prime, then there is an element of Ŵthat takes D_{min}^{(s)}to (g + 1)\overline{𝒜}.

The numbers g for all root systems with roots of different lengths are as follows:

\begin{matrix}Δ & C_{p} & B_{p} & F_{4} & G_{2} \\ g & 2p − 2 & 2 & 6 & 3 \end{matrix}

It follows that Lemma 5.4 always applies and hence #(𝔄𝔫_{s}) = #(Ŵ_{min}^{(s)}) = #((g + 1)\overline{𝒜} ∩ Q^{∨}). In turn, if g+1 is relatively prime with the coordinates of θ, then this number is computed by Eq. (2.8). One sees that the condition of relative primeness does not hold only for G_{2}. (However, this case can be studied by hand.) Thus, we obtain

5.5 Theorem. Suppose |θ|^{2}/|θ_{s}|^{2} = 2. Then

#(𝔄𝔫_{s}) = ∏_{i=1}^{p} \frac{g + e_{i}+ 1}{e_{i}+ 1} .

It is easily seen that #(𝔄𝔫_{s}) = 4 for G_{2}. Looking at the factors occurring in the formula of Theorem 5.5, one may notice that there is a nice formula for 𝔄𝔫_{s}, which resembles Eq. (2.9) and also covers the case of G_{2}. Here is it. Suppose e_{1} < e_{2} < ... < e_{p} and set n = #(Π_{s}). Then for any Δ we have

(5.6)  #(𝔄𝔫_{s}) = ∏_{i=1}^{n} \frac{h + e_{i}+ 1}{e_{i}+ 1} .

But it is not clear how to prove this a priori. Changing the role of long and short roots in Definition 5.1, one may define l-minimal elements, which are in a one-to-one correspondence with the long antichains. Since the proofs here are similar, we state only results. The simplex associated with the l-minimal elements is

D_{min}^{(l)}= {x ∈ V | (x,α) ≥ 0 (α ∈ Π_{s}); (x,α) ≥ 1 (α ∈ Π_{l}); (x,θ) ≤ 2} ,

and the l-minimal elements bijectively correspond to the integral points of D_{min}^{(l)}. The shift in the direction of ρ_{l}^{∨} = ∑_{α_{i}∈Π_{l}} ϖ_{i}^{∨} takes D_{min}^{(l)}to (h + 1 − g)𝒜. Therefore, if Δ is not of type G_{2}, then

(5.7)  #(𝔄𝔫_{l}) = ∏_{i=1}^{p} \frac{h − g + e_{i}+ 1}{e_{i}+ 1} .

Since this number can also be computed as #(𝔄𝔫_{s}) for the dual root system Δ^{∨}, a relation between g for Δ and Δ^{∨} emerges. Namely, g_{Δ} + g_{Δ^{∨}} = h. Now we turn to considering a ‘short’ analogue of maximal elements.

5.8 Definition. w ∈ Ŵis called s-maximal, if

(i) w is dominant; (ii) if α ∈ Π_{s} and w^{−1} (α) = kδ + μ with μ ∈ Δ, then k ≤ 1; (iii)if α ∈ Π̂_{l} and w^{−1} (α) = kδ + μ with μ ∈ Δ, then k ≤ 0.

Using (i), conditions (ii), (iii) can be made precise. If k = 0, then μ ∈ Δ^{+} ; if k = 1 in (ii), then μ ∈ −Δ^{+} . We write Ŵ_{max}^{(s)}for the set of all s-maximal elements. Notice that Ŵ_{max}^{(s)}⊂ Ŵ_{max}.

As in case of maximal elements, we wish to set up a one-to-one correspondence between the s-maximal elements and a certain subset of 𝔄𝔫_{s}. In order to distinguish the right subset we need some preparations. Recall that, although Δ_{s} is not a sub-root system of Δ, it is a root system in its own right. Clearly, Δ_{s}^{+} is the set of positive roots for Δ_{s}. Let us write Π(Δ_{s}^{+} ) for the corresponding set of simple roots. As Π itself, it consists of p roots. Obviously, Π_{s} ⊂ Π(Δ_{s}^{+} ). Other roots in Π(Δ_{s}^{+} ) are in bijection with Π_{l}. Each β ∈ Π_{l} is replaced by a short root as follows. Let α be the closest to β (in the sense of the Dynkin diagram) short simple root. The sum of all simple roots in the string connecting α and β is a short root, which is a simple root for Δ_{s}^{+} . Warning. Although we often consider antichains lying in (certain subsets of) Δ_{s}^{+} , it is always meant that the ordering ‘≼’ is inherited from the whole of Δ^{+} .

5.9 Proposition. The bijection between 𝔄𝔡_{0} and Ŵ_{max}described in Proposition 2.11 gives rise to a bijection between Ŵ_{max}^{(s)}and the short antichains lying in Δ_{s}^{+} \ Π(Δ_{s}^{+} ).

Proof. The correspondence described in Proposition 2.11 attaches to a maximal element w its first layer ideal, I^{w}. But even if w is s-maximal, the generators of I^{w} may not be short roots. So, we do not immediately obtain a required short antichain. To correct this, we take I^{w}∩ Δ_{s}^{+} . (It is also the set of weights of a nilpotent 𝔟-stable subspace of 𝕍(θ_{s}).) The set of generators (minimal elements) of I^{w}∩Δ_{s}^{+} is a short antichain of Δ^{+} , which we attach to w ∈ Ŵ_{max}^{(s)}.

Now, we are to prove that the resulting antichain lies in Δ_{s}^{+} \ Π(Δ_{s}^{+} ) and that this correspondence is really a bijection.

Recall from Section 2 that Ξ(I) is the set of maximal elements of Δ^{+} \ I and that in case of maximal elements Ξ(I^{w}) is described in Proposition 2.12. That description implies that, for w ∈ Ŵ_{max}^{(s)}, Ξ(I^{w}) consists of short roots. Since Ξ(I^{w}) covers all simple roots (see Remark 2.13) and consists of short roots, it also covers all roots from Π(Δ_{s}^{+} ). (Use the explicit description of Π(Δ_{s}^{+} given above.) This means that the short antichain Γ(I^{w}∩ Δ_{s}^{+} ) does not contain roots from Π(Δ_{s}^{+} ).

\underline{Injectivity}. If w,w′∈ Ŵ_{max}^{(s)}are different, then Ξ(I^{w}) ≠ Ξ(I^{w′}). Since these two sets consist of short roots, we obviously have I^{w}∩ Δ_{s}^{+} ≠ I^{w′}∩ Δ_{s}^{+} .

\underline{Surjectivity}. If Γ is an antichain of Δ^{+} lying in Δ_{s}^{+} \ Π(Δ_{s}^{+} ), then take all maximal short roots in Δ^{+} \ I⟨Γ⟩. More precisely, let Ξ be the set of short roots μ such that if ν ∈ Δ_{s}^{+} and ν ≻ μ then there is γ ∈ Γ such that ν ≽ γ. Then Ξ is a short antichain that covers all roots in Π(Δ_{s}^{+} ) and hence the whole of Π. In view of Remark 2.13, Ξ is of the form Ξ(I^{w}) for some w ∈ Ŵ_{max}. Finally, since Ξ consists of short roots, this w is s-maximal. □

The antichains of Δ^{+} lying in Δ_{s}^{+} \Π(Δ_{s}^{+} ) are said to be strictly s-positive. The corresponding subset of 𝔄𝔫_{s} is denoted by 𝔄𝔫_{s,0}. Once again, the next part of our program is a geometric description. Set

D_{max}^{(s)}= {x ∈ V | (x,α) ≤ 1 (α ∈ Π_{s}); (x,α) ≤ 0 (α ∈ Π_{l}); (x,θ) ≥ 1},

5.10 Proposition.

1. w = v·t_{r} ∈ Ŵ_{max}^{(s)}⟺ \{ \begin{matrix}w is dominant, \\ v(r) ∈ D_{max}^{(s)}∩ Q^{∨} . \end{matrix}

2. The mapping Ŵ_{max}^{(s)}→ D_{max}^{(s)}∩ Q^{∨}, w = v·t_{r} ↦ v(r), is a bijection.

Proof. The argument is the same as in Proposition 2.6, taking into account that the constraints for D_{max}^{(s)}are different. □

The translation in the direction of −ρ_{s}^{∨}, which belongs to extended affine Weyl group, takes D_{max}^{(s)}to (1−g)𝒜. Since g−1 is always relatively prime with the index of connection, there is also an element of Ŵthat does the same, cf. Lemma 5.4. As in case of s-minimal elements, we have g−1 is relatively prime with the coordinates of θ, if Δ is not of type G_{2}. Therefore, if Δ ∈ {B_{p}, C_{p}, F_{4}}, then

(5.11)  #(𝔄𝔫_{s,0}) = #(Ŵ_{max}^{(s)}) = ∏_{i=1}^{p} \frac{g + e_{i}− 1}{e_{i}+ 1} .

For G_{2}, this set consists of 2 elements.

6. Short antichains and the semi-Catalan arrangement

In this section, we study a hyperplane arrangement in V that has the same connection with short antichains in Δ^{+} as the Catalan arrangement has with all antichains. This provides yet another approach to counting the short and strictly s-positive antichains.

6.1 Definition. 1. The semi-Catalan arrangement in V, Cat_{s}(Δ), consists of the hyperplanes ℋ_{μ,k} (μ ∈ Δ_{s}^{+} , k = −1,0,1) and ℋ_{ν,0} (ν ∈ Δ_{l}^{+} ).

2. The m-semi-Catalan arrangement in V, Cat_{s}^{m}(Δ), consists of the hyperplanes ℋ_{μ,k} (μ ∈ Δ_{s}^{+} , k = −m,...,−1,0,1,...,m) and ℋ_{ν,0} (ν ∈ Δ_{l}^{+} ).

All these arrangements are deformations of the Coxeter arrangement. Notice also that Cat_{s}^{0}(Δ) is the usual Coxeter arrangement, and Cat_{s}(Δ) = Cat_{s}^{1}(Δ). First, we are interested in the dominant regions of Cat_{s}(Δ) and their relation to short antichains. Define a mapping

ψ: 𝔄𝔫_{s} → {the dominant regions of Cat_{s}(Δ) }

as follows. For Γ ∈ 𝔄𝔫_{s}, let

Γ ↦^{ψ} R_{Γ}^{(s)} := {x ∈ 𝒞 | (x,μ) > 1, if μ ∈ I⟨Γ⟩ ∩ Δ_{s}^{+} & (x,μ) < 1, if μ ∈ Δ_{s}^{+} \ I⟨Γ⟩} .

6.2 Theorem.

(i) The mapping ψ is well-defined, and it is a bijection; (ii) R_{Γ}^{(s)} is bounded if and only if Γ ∈ 𝔄𝔫_{s,0}.

Proof. (i) 1. Regarding Γ as ”usual” antichain, we can construct a region R_{I⟨Γ⟩}, as prescribed by Eq. (3.1). Obviously, R_{I⟨Γ⟩}⊂ R_{Γ}^{(s)}. Hence the latter is non-empty.

2. Since the definition of the set R_{Γ}^{(s)} includes a constraint for any hyperplane in Cat_{s}(Δ) meeting 𝒞, R_{Γ}^{(s)} cannot contain more than one region. It is also clear that R_{Γ}^{(s)} ≠ R_{Γ′}^{(s)}, if Γ ≠ Γ′. For, if γ ∈ Γ \ I⟨Γ′⟩, then ℋ_{γ,1} separates R_{Γ}^{(s)} and R_{Γ′}^{(s)}. Hence ψ is injective.

3. The surjectivity of ψ follows from the existence of the inverse map. Given a region R, take the set of walls of R separating R from the origin. Then the corresponding set of roots form a short antichain.

(ii) If Π(Δ_{s}^{+} ) ∩ I⟨Γ⟩ = ∅, then R_{Γ}^{(s)} belong to the bounded domain {x ∈ 𝒞 | (x,μ) < 1, μ ∈ Π(Δ_{s}^{+} )}.

Conversely, assume β ∈ I ∩ Π(Δ_{s}^{+} ). Recall from Section 5 that Π(Δ_{s}^{+} ) is in bijection with Π (β either belong to Π_{s} or is obtained via a simple procedure from a long simple root). Let β′ be the simple root in Π corresponding to β and φ_{β′} the respective fundamental weight of 𝔤. Then we claim that if x ∈ R_{Γ}^{(s)}, then x + tφ_{β′} ∈ R_{Γ}^{(s)} for any t ∈ ℝ_{≥0} . Indeed, β is the minimal short having nonzero β′-coordinate. Therefore all short roots having nonzero β′-coordinate are in I⟨Γ⟩. This means that R_{Γ}^{(s)} has no upper bound in the direction of φ_{β′}. Thus, R_{Γ}^{(s)} unbounded. □

Let us look at the relationship between s-minimal and s-maximal elements on one hand, and dominant regions of Cat_{s}(Δ) on the other hand.

6.3 Proposition.

(i) Suppose w ∈ Ŵ_{min}^{(s)}, and let Γ ∈ 𝔄𝔫_{s} be the corresponding antichain. Then w^{−1} ∗𝒜 ⊂ R_{Γ}^{(s)}, and it is the nearest to the origin alcove in R_{Γ}^{(s)}.

(ii) Suppose w ∈ Ŵ_{max}^{(s)}, and let Γ ∈ 𝔄𝔫_{s,0} be the corresponding antichain. Then w^{−1} ∗𝒜 ⊂ R_{Γ}^{(s)}, and it is the most distant from the origin alcove in R_{Γ}^{(s)}.

Proof. (i) It was already observed before that w^{−1} ∗𝒜 ⊂ R_{I⟨Γ⟩}⊂ R_{Γ}^{(s)}. Suppose we are inside w^{−1} ∗𝒜. To get in an alcove that is closer to the origin, we must cross a wall separating w^{−1} ∗𝒜 from the origin. These walls correspond to the roots α ∈ Π̂ such that w^{−1} (α) < 0. But then w^{−1} (α) = −δ + μ, where μ ∈ Δ_{s}^{+} . So that having crossed this wall, we get in another dominant region of Cat_{s}(Δ).

(ii) Suppose w ∈ Ŵ_{max}^{(s)}and we are inside w^{−1} ∗𝒜. To get in an alcove that is more distant from the origin, we must cross a wall of w^{−1} ∗𝒜 that does not separate w^{−1} ∗𝒜 from the origin. These walls correspond to the roots α ∈ Π̂ such that w^{−1} (α) > 0. In view of the definition of s-maximal elements, there are two possibilities: (a) if w^{−1} (α) = μ ∈ Δ+, then crossing such a wall we leave 𝒞; (b) if w^{−1} (α) = δ − μ, μ ∈ Δ_{s}^{+} , then crossing such a wall we get in another dominant region of Cat_{s}(Δ). Hence w^{−1} ∗𝒜 is the most distant from the origin alcove in a certain region. The hyperplanes of Cat_{s}(Δ) separating w^{−1} ∗𝒜 from the origin (not necessarily walls of w^{−1} ∗𝒜) correspond to the short roots μ such that w(δ − μ) < 0, i.e., these roots are exactly the short roots in the first layer ideal of w. According to the correspondence described in the proof of Proposition 5.9, the minimal elements of this set form the short antichain Γ attached to w. Thus, w^{−1} ∗𝒜 lies in the required alcove. □

Theorem 6.2 implies that the number of short or strictly s-positive antichains can be found through the use of the characteristic polynomial of Cat_{s}(Δ). In fact, we are able to compute the characteristic polynomial for Cat_{s}^{m}(Δ) with any m. One should just follow the scheme of Athanasiadis’ proof for the usual m-Catalan arrangement, see [1, Theorem 3.1]. Let P^{∨} be the coweight lattice and f = [P^{∨} : Q^{∨}]. (Hence f is the index of connection of Δ.)

6.4 Theorem. Suppose t ∈ ℕ, t > mg, and both t, t − mg are relatively prime with all the coordinates of θ. Then

χ(Cat_{s}^{m}(Δ),t) = \frac{#W}{f}#((t − mg)𝒜 ∩ P^{∨}) .

Proof. We give only a sketch of the proof, where we indicate essential distinction from Athanasiadis’ proof for an m-Catalan arrangement, referring to [1] for all details.

Let 𝒫 denote the fundamental parallelepiped \{∑ _{i=1}^{p} y_{i}ϖ_{i}^{∨} | 0 ≤ y_{i} ≤ 1\}. Set 𝒫_{t} = 𝒫 ∩ \frac{1}{t}P^{∨}. Also, let V_{Δ,t}^{m} be the set of hyperplanes

ℋ_{μ,k+ \frac{n}{t}} (n,k ∈ ℤ, |n| ≤ m, μ ∈ Δ_{s}) and ℋ_{γ,k}(k ∈ ℤ, γ ∈ Δ_{l}).

Note that fractional indices are allowed only for hyperplanes orthogonal to short roots, so that our V_{Δ,t}^{m} is different from that of Athanasiadis.

Given an arrangement A in V, according to a general result (Athanasiadis, Björner-Ekedahl), the value χ(A,t) is equal to the number of points in the complement of all hyperplanes, counted after reduction modulo t, i.e., in (ℤ_{t})^{p}. More precisely, this equality holds for infinitely many t (this can be made even more precise, see [1, Sect. 2]). In our situation, this general principle leads to the equality χ(Cat_{s}^{m}(Δ),t) = #{𝒫_{t}\ V_{Δ,t}^{m}}. Using the fact that 𝒫 contains #W /f alcoves, this can be transformed to

χ(Cat_{s}^{m}(Δ),t) = \frac{#W}{f}#((𝒜 ∩ \frac{1}{t}P^{∨}) \ V_{Δ,t}^{m}) = \frac{#W}{f}· #((t𝒜 ∩ P^{∨}) \ tV_{Δ,t}^{m}) .

It easily follows from the definition of V_{Δ,t}^{m} that (t𝒜∩P^{∨})\tV_{Δ,t}^{m} is obtained from t𝒜∩P^{∨} by deleting the coweights lying on the hyperplanes ℋ_{α,i} with α ∈ Π_{s} and 1 ≤ i ≤ m. That is, the set in question is equal to

{x ∈ P^{∨} | (x,α) > m (α ∈ Π_{s}); (x,α) > 0 (α ∈ Π_{l}); (x,θ) < t} .

Finally, the translation by the negative of mρ_{s}^{∨} (which lies in P^{∨}) takes this set to the points of P^{∨} lying in the open simplex (t − gm)𝒜. □

Let us discuss consequences of this result. We use values of g given in Section 5. If Δ ∈ {B_{p},C_{p},F_{4}} and t is relatively prime with the coordinates of θ, then the same holds for t − mg with any m. It follows that

χ(Cat_{s}^{m}(Δ),t) = χ(Cat_{s}^{0}(Δ),t − mg) = ∏_{i=1}^{p} (t − mg − e_{i}) .

(The first equality holds for infinitely many values of t; hence it holds always, as both parts are polynomials in t. The second equality is a statement about Coxeter arrangements.) In particular,

(6.5)  χ(Cat_{s}(Δ),t) = ∏_{i=1}^{p} (t − g − e_{i}) .

Combining Theorems 3.3 and 6.2, we conclude that

#(𝔄𝔫_{s}) = |χ(Cat_{s}(Δ),−1)|/#W and #(𝔄𝔫_{s,0}) = |χ(Cat_{s}(Δ),1)|/#W ,

which coincides, of course, with the formula in Theorem 5.5 and Eq. (5.11).

For G_{2}, we have g = 3 and the assumption of Theorem 6.4 is satisfied only if m is even. Therefore

χ(Cat_{s}^{2m}(G_{2}),t) = χ(Cat_{s}^{0}(G_{2}),t − 6m) = (t − 6m − 1)(t − 6m − 5) .

Using ad hoc arguments, one may derive the “odd” formula

(6.6)  χ(Cat_{s}^{2m+1}(G_{2}),t) = (t − 6m − 5)(t − 6m − 7) .

It is also easy to compute χ(Cat_{s}^{1}(G_{2}),t) directly from the definition of a characteristic polynomial. Again, it is noteworthy that formulae (6.5), (6.6) for χ(Cat_{s}(Δ),t) admit a uniform presentation for all non-simply laced cases, cf. Eq. (5.6).

6.7 Theorem. If n = #(Π_{s}) and the exponents of Δ are increasingly ordered, then

χ(Cat_{s}(Δ),t) = ∏_{i≤n}(t − h − e_{i}) ∏_{i≥n+1} (t − e_{i}) .

Of course, it would be interesting to have a uniform proof (explanation) for this.

6.8 Remarks. 1. One may consider ‘short’ analogues for other arrangements associated with root systems. For instance, the extended semi-Shi arrangement, Shi_{s}^{m}(Δ), is the collection of hyperplanes ℋ_{μ,k} (μ ∈ Δ_{s}^{+} , k = −m + 1,...,−1,0,...,m) and ℋ_{ν,0} (ν ∈ Δ_{l}^{+} ). It is not hard to compute that, for C_{2}, the characteristic polynomial is equal to (t − 2m − 1)^{2}. For G_{2}, it is equal to (t − 3m − 1)(t − 3m − 2), at least if m ≥ 3. I conjecture that the following formula holds in general:

χ(Shi_{s}^{m}(Δ),t) = ∏_{i=1}^{p} (t − mg_{Δ^{∨}} − e_{i}(Δ_{l})) ,

where {e_{i}(Δ_{l})} are the exponents for the root system Δ_{l}. For instance, in case of F_{4} we have g_{Δ} = g_{Δ^{∨}} = 6 and Δ_{l} is of type D_{4}. Therefore the conjectural expression is (t − 6m − 1)(t − 6m − 3)^{2}(t − 6m − 5).

2. The dominant regions of Cat_{s}(Δ) provide a connection, in the spirit of [5], with nilpotent G-orbits in 𝕍(θ_{s}). I hope to discuss this topic in a forthcoming publication.

7. Some numerical complements

In this section, we collect several results that can be proved in a case-by-case fashion.

7.1. We know the number of all and short antichains for all irreducible reduced root systems. Using this, on may observe that #(𝔄𝔫_{s}) divides #(𝔄𝔫) in all cases. Furthermore, the ratio has, a posteriori, an interesting description. Namely, let Δ(Π_{l}) be the root system whose set of simple roots is Π_{l}. Notice that Δ(Π_{l}) is smaller than Δ_{l}, and that the former is irreducible, since Π_{l} is a connected subset of the Dynkin diagram. Write 𝔄𝔫(Π_{l}) for the set of all antichains in Δ(Π_{l})^{+} .

7.1 Theorem. #(𝔄𝔫) = #(𝔄𝔫_{s})·#(𝔄𝔫(Π_{l})).

Proof. Case-by-case verification. For instance, in case of F_{4} we have #(𝔄𝔫) = 105, #(𝔄𝔫_{s}) = 21, and Δ(Π_{l}) is of type A_{2}, where one has five antichains. □

Of course, this proof is not illuminating. One may consider a natural mapping 𝔄𝔫 → 𝔄𝔫_{s} that takes Γ to the set of minimal elements of I⟨Γ⟩ ∩ Δ_{s}^{+} . For C_{p}, all fibres of this mapping has the same cardinality, which is 2. To some extent, this is an explanation in this case. Unfortunately, the ”equicardinality” property does not hold for F_{4} and G_{2}. The statement of Theorem 7.1 can be compared with another equality, which is easy to prove. The reflection s_{γ} ∈ W is called short, if γ ∈ Δ_{s}^{+} . Let W_{s} be the (normal) subgroup of W generated by all short reflections, and let W (Π_{l}) be the Weyl group of Δ(Π_{l}). Then W ≃ W_{s}⋊ W (Π_{l}) (a semidirect product).

7.2. We have shown that the short antichains of Δ^{+} lying in Δ_{s}^{+} \ Π(Δ_{s}^{+} ) are in a one-to-one correspondence with s-maximal elements, and then computed their number. However, it is also natural to enumerate the short antichains lying in Δ_{s}^{+} \ Π_{s}. (Recall that Π_{s} is a proper subset of Π(Δ_{s}^{+} ).) Set 𝔄𝔫_{ss} = {Γ ∈ 𝔄𝔫_{s} | Γ ∩ Π_{s} = ∅}. I did not find a suitable bijection for 𝔄𝔫_{ss}, but the following formula for the cardinality is true:

(7.2)  #(𝔄𝔫_{ss}) = ∏_{i=1}^{n} \frac{h + e_{i}− 1}{e_{i}+ 1} ,

where the notation is the same as in Theorem 6.7. Again, this formula bears a striking resemblance with Theorem 2.16. Direct calculations show that this gives us the correct number for B_{p} (this is easy, because there is only a few short roots), F_{4}, G_{2}. The argument for C_{p} goes as follows. The set of positive roots Δ^{+} (C_{p}) is naturally represented by the shifted Ferrers diagram of shape (2p − 1,2p − 3,...,1), and the ideals are represented by suitable subdiagrams of it, see slightly different versions in [13, Sect. 2], [3, Sect. 3], [10, Sect. 5]. In these interpretations, the long roots are represented by the boxes in an extreme diagonal of this shifted Ferrers diagram, and the simple roots correspond to the boxes of another (”opposite”) diagonal. These two diagonals have a unique common box, corresponding to the long simple root. If we want to obtain an ideal whose generators are short and contain no short simple roots, then we just erase both these diagonals and consider a subdiagram of the smaller shifted diagram. But this smaller shifted Ferrers diagram, which is of shape (2p − 3,2p − 5,...,1), can be thought of as the set of positive roots for C_{p−1} . Thus, the number #(𝔄𝔫_{ss}) for C_{p} equals the number #(𝔄𝔫) for C_{p−1} . The latter is known to equal \binom{2p−2}{p−1}, which is consistent with Eq. (7.2). Actually, we obtain more. Our bijection between 𝔄𝔫_{ss}(C_{p}) and 𝔄𝔫(C_{p−1} ) preserves the number of elements. Therefore, we conclude that the number of k-element antichains in 𝔄𝔫_{ss}(C_{p}) is equal to \binom{p−1}{k}^{2}, k = 0,1,...,p − 1.

7.3. Counting antichains with respect to the number of generators yields an interesting q-analogue of #(𝔄𝔫), see [2], [10]. In case of two root lengths, one may consider, of course, a 2-parameter refinement. Set

𝔄𝔫⟨k,m⟩ = {Γ ∈ 𝔄𝔫 | #(Γ ∩ Δ_{s}^{+} ) = k & #(Γ ∩ Δ_{l}^{+} ) = m}, a_{k,m}= #𝔄𝔫⟨k,m⟩ ,

and consider the generating function F (t,u) = ∑_{k,m≥0} a_{k,m}t^{k}u^{m}. We have

G_{2}: F (t,u) = 1 + 3t + 3u + tu.

F_{4}: F (t,u) = 1 + 12t + 12u + 8t^{2}+ 39tu + 8u^{2}+ 12t^{2}u + 12tu^{2}+ t^{2}u^{2}.

The symmetry of these polynomials stems from the fact the corresponding root systems are self-dual. Since the root systems of type B and C are dual to each other, the corresponding matrices (a_{k,m}) are mutually transposed. So, it suffices to handle the case of C_{p}. Each pair of long roots in Δ^{+} (C_{p}) is comparable, hence any antichain contains at most one long root. So that we are to determine the coefficients a_{k,0},a_{k,1}, (k = 0,1,...,p − 1). In [10, Sect. 5], we constructed an involution on the set 𝔄𝔫(C_{p}), which maps 𝔄𝔫⟨k,0⟩ onto 𝔄𝔫⟨p − 1 − k,1⟩. Hence a_{p−k−1,1} = a_{k,0} and we have to only count the number of short antichains with k elements. Using shifted Ferrers diagrams, it can be shown that a_{k,0} = \binom{p}{k}\binom{p−1}{k}. (In this situation, short simple roots are allowed, so that one has to erase only one diagonal and work with the shifted Ferrers diagram of shape (2p − 2,2p − 4,...,2).)

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