March 9, 2003

AD-NILPOTENT IDEALS OF A BOREL SUBALGEBRA: GENERATORS AND DUALITY

DMITRI I. PANYUSHEV

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

In this paper, we develop several combinatorial aspects of the theory of ad-nilpotent ideals. Let 𝔟 be a fixed Borel subalgebra of a simple Lie algebra 𝔤. Following [6], we say that an ideal of 𝔟 is ad-nilpotent, if it is contained in [𝔟,𝔟]. Let 𝔄𝔡 of 𝔄𝔡(𝔤) denote the set of all ad-nilpotent ideals of 𝔟. Any 𝔠 ∈ 𝔄𝔡 is completely determined by the the corresponding set of roots. More precisely, let 𝔱 be a Cartan subalgebra of 𝔤 lying in 𝔟 and let Δ be the root system of the pair (𝔤,𝔱). Choose Δ^{+} , the system of positive roots, such that the roots of 𝔟 are positive. Then 𝔠 = ⊕_{γ∈I}𝔤_{γ}, where I is a suitable subset of Δ^{+} and 𝔤_{γ} is the root space for γ ∈ Δ^{+} . In particular, this means that 𝔄𝔡 is finite. Abusing language, we shall say that such I ⊂ Δ^{+} is an ad-nilpotent ideal, too.

In [6], Cellini and Papi proved that there is a bijection between the ad-nilpotent 𝔟-ideals and the elements of the affine Weyl group Ŵsatisfying certain property (see (1.2) below). In our paper, these elements are said to be admissible. Using admissible elements, Cellini and Papi established a bijection between 𝔄𝔡 and the points of the coroot lattice lying in a certain rk𝔤-dimensional simplex D̃ with rational vertices [7]. As a consequence, they obtained a conceptual proof for the explicit formula giving the number of ad-nilpotent ideals in all simple Lie algebras.

In Section 2, we give a characterization of the generators of ad-nilpotent ideals in terms of admissible elements (Theorem 2.2). It is then shown that an ideal I has k generators if and only if the corresponding lattice point lies on the face of D̃ of codimension k (Theorem 2.9). It is curious that D̃ has exactly one integral vertex. We deduce this from the fact that there is only one ad-nilpotent ideal having rk𝔤 generators.

In Section 3, we consider the ‘simple root’ statistic on 𝔄𝔡(𝔤), which assigns to any ideal the number of simple roots in it. Write 𝔄𝔡(𝔤)_{i} for the set of ideals containing exactly i simple roots. We give recurrent formulas for these numbers and then compute them for A_{p} and the exceptional Lie algebras. It is also shown thet the simple root statistic has the same distribution for B_{p} and C_{p}. In case of C_{p} and D_{p}, we give conjectural values for #𝔄𝔡(𝔤)_{i}, which are, no doubt, true. As a consequence of this theory, we observe some similarities between the ad-nilpotent ideals and clusters (see [9] for the latter). It is shown that the simple root statistic on 𝔄𝔡(𝔤) and a certain statistic on the set of clusters have the same distribution (Theorem 3.11).

This research was supported in part by the Alexander von Humboldt-Stiftung and RFBI Grant no. 01– 01–00756.

To obtain a closed formula for #𝔄𝔡(𝔤)_{0} (Proposition 3.10), we exploit a bijection between the ad-nilpotent ideals and the regions of the Catalan arrangement lying in the dominant chamber, see [18]. We show that I ∈ 𝔄𝔡(𝔤)_{0} if and only if the corresponding region is bounded. In turn, the number of bounded regions of any arrangement can be counted using a powerful result of Zaslavsky, once one knows the characteristic polynomial, see Prop. 3.8 for details. After this part was written, I learned that the formula for #𝔄𝔡(𝔤)_{0} was already obtained, in the same way, in a recent preprint of Athanasiadis [4]. The main result of Athanasiadis’ preprint is a case-free simple proof of the formula for the characteristic polynomial of the Catalan arrangement.

In the last three sections, we consider the statistic that assigns to an ideal I ∈ 𝔄𝔡(𝔤) the number of its generators. In case of 𝔤 = 𝔰𝔩_{n}, the ad-nilpotent ideals are identified with Dyck path of semilength n and, therefore, the generating function for this statistic is the famous Narayana polynomial (of degree n − 1). For this reason, we say that the generating function for this statistic for arbitrary 𝔤 is a generalized Narayana polynomial. Motivated by the fact that the Narayana polynomial is palindromic, we were searching for a materialization of this property, i.e., for an involutory mapping (duality) on 𝔄𝔡(𝔰𝔩_{n}) that takes the ideals with k generators to the ideals with n − 1 − k generators. For 𝔰𝔩_{n}, such a materialization does exists, and it has a number of nice properties, see Section 4. The nicety of these properties is that their formulation admits immediate generalization to all simple Lie algebras. We also show that the number of self-dual ideals in 𝔰𝔩_{2m+1} equals C_{m}, the m-th Catalan number. In Section 5, the results concerning duality are extended to series B and C. This clearly implies that the generalized Narayana polynomials for B and C (in fact, they are equal) are palindromic. We conjecture that such a duality exists for any simple Lie algebra. At least, we show that the generalized Narayana polynomials are palindromic for the exceptional Lie algebras. General properties of this conjectural duality are discussed in Section 6.

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

1. Preliminaries on ad-nilpotent ideals

(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^{+} .

C is the fundamental Weyl chamber. We set V := 𝔱_{ℚ} = ⊕_{i=1}^{p} ℚα_{i} and denote by ( , ) a W -invariant inner product on V. As usual, μ^{∨} = 2μ/(μ,μ) is the coroot for μ ∈ Δ.

Q = ⊕_{i=1}^{p} ℤα_{i} ⊂ V is the root lattice and Q^{∨} = ⊕_{i=1}^{p} ℤα_{i}^{∨} is the coroot lattice.

Q^{+} = {∑ _{i=1}^{p} n_{i}α_{i} | n_{i} ∈ ℕ} ⊂ Q. Letting V̂= V ⊕ ℚδ ⊕ ℚλ, 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. Here α_{0} = δ − θ, where θ is the highest root in Δ^{+} . The inner product ( , ) on V̂is Ŵ-invariant. 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

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

Our convention concerning N̂(w) is the same as in [10], [14], but opposite to that in [6], [7], so that our N̂(w) is N̂(w^{−1} ) in the sense of Cellini-Papi.

(1.2) ad-nilpotent ideals. Throughout the paper, 𝔟 is the Borel subalgebra of 𝔤 corresponding to Δ^{+} and 𝔲 = [𝔟,𝔟]. The expression “ad-nilpotent ideal” or just “ideal” always refers to a 𝔟-ideal lying in 𝔲. Let 𝔠 ⊂ 𝔟 be an ad-nilpotent ideal. Then 𝔠 = ⊕_{α∈I} 𝔤_{α}

for a subset I ⊂ Δ^{+} , which is called the set of roots of 𝔠. As our exposition will be mostly combinatorial, an ad-nilpotent ideal will be identified with the respective set of roots. That is, I is said to be an ad-nilpotent ideal, too. Whenever we want to explicitly indicate the context, we say that 𝔠 is a geometric ad-nilpotent ideal, while I is a combinatorial ad-nilpotent ideal. Accordingly, being in combinatorial (resp. geometric) context, we speak about cardinality (resp. dimension) of an ideal. In the combinatorial context, the definition of an ad-nilpotent ideal can be stated as follows. I ⊂ Δ^{+} is an ad-nilpotent ideal, if the following condition is satisfied:

if γ ∈ I, ν ∈ Δ^{+} , and γ + ν ∈ Δ, then γ + ν ∈ I.

We consider Δ^{+} as poset with respect to the standard partial order ‘≼’, i.e., μ≼ν if and only if ν−μ ∈ Q^{+} . Therefore, a combinatorial ad-nilpotent ideal is nothing but a dual order ideal of the poset (Δ,≼). An element w ∈ Ŵis said to be admissible, if it has two properties:

(a) w(α) is positive for any α ∈ Π;

(b) if w^{−1} (α) is negative for an α ∈ Π̂ , then w^{−1} (α) = γ − δ for some γ ∈ Δ^{+} . By [6, Sect. 2], there is a one-to-one correspondence between the admissible elements of Ŵand the ad-nilpotent 𝔟-ideals. This correspondence is obtained as follows:

• Given 𝔠 ∈ 𝔄𝔡, consider the members of the descending central series 𝔠 = 𝔠^{1}, 𝔠^{k}= [𝔠^{k−1},𝔠] (k ≥ 2) and the corresponding sets of roots I_{k}. Clearly, I_{k} ⊃ I_{k+1} and I_{m}= ∅ for m ≫ 0. Set N_{k} = {kδ − γ | γ ∈ I_{k}}. Then Φ := ∪_{k≥1}N_{k} is a closed subset of Δ̂^{+} whose complement is closed as well, and therefore there is a unique w ∈ Ŵsuch that Φ = N̂(w). This w is the required admissible element.

• Conversely, if w ∈ Ŵis admissible, then N̂(w) = ∪_{k≥1}N_{k}, where N_{k} = {kδ − γ | γ ∈ I_{k}} and I_{k} ⊂ Δ^{+} . Then I_{1} is the set of roots of an ad-nilpotent ideal, say 𝔠. Furthermore, the definition of an admissible element also implies that I_{1} ⊃ I_{2} ⊃ ... and I_{k} is the set of roots of 𝔠^{k}. If w ∈ Ŵis admissible, then I_{w} (resp. 𝔠_{w}) stands for the corresponding combinatorial (resp. geometric) Abelian ideal. That is,

I_{w} = {γ ∈ Δ^{+} | δ − γ ∈ N̂(w)} and 𝔠_{w} = ⊕_{α∈I_{w}} 𝔤_{α} .

Conversely, given I ∈ 𝔄𝔡, we write w⟨I⟩ for the respective admissible element. Notice that

dim𝔠_{w} = #(I_{w}) and l(w) = ∑_{k≥1} dim(𝔠_{w})^{k} .

Throughout the paper, I or I_{w} stands for a combinatorial ad-nilpotent ideal. Whenever we wish to stress that 𝔄𝔡 depends on 𝔟 and/or 𝔤, we write 𝔄𝔡(𝔟) or 𝔄𝔡(𝔤) or even 𝔄𝔡(𝔟,𝔤).

2. The generators of ad-nilpotent ideals

Let I be an ad-nilpotent ideal. We say that γ ∈ I is a generator of I, if γ − α ∉ I for all α ∈ Δ^{+} . Obviously, this is equivalent to the fact that I \{γ} is still an ad-nilpotent ideal. Conversely, if κ is a maximal element of Δ^{+} \ I (i.e., (κ + Δ^{+} ) ∩ Δ ⊂ I), then I ∪ {κ} is an ad-nilpotent ideal. These two procedures show that the following is true.

Proposition 2.1. Suppose I ⊂ J are two ad-nilpotent ideals. Then there is a chain of ideals I = I_{0} ⊂ I_{1} ⊂ ... ⊂ I_{m}= J such that #(I_{i+1} ) = #(I_{i}) + 1. In other words, 𝔄𝔡 is a ranked poset, with cardinality (dimension) of an ideal as the rank function.

In the geometric setting, the set of generators has the following description. For an ideal 𝔠 = ⊕_{γ∈I}𝔤_{γ} ⊂ 𝔟, there is a unique 𝔱-stable space 𝔠̃ ⊂ 𝔠 such that 𝔠 = [𝔟,𝔠] ⊕ 𝔠̃. Then γ is a generator of I if and only if it is a root of 𝔠̃. Write Γ(I) for the set of generators of I. It is clear that a subset Γ = {γ_{1},...,γ_{l}} ⊂ Δ^{+} is the set of generators for some ideal if and only if γ_{i}−γ_{j} ∉ Q^{+} for all i,j. This means that Γ ⊂ Δ^{+} is the set of generators for some ad-nilpotent ideal if and only if it is an antichain of (Δ^{+} ,≼). This is a manifestation of a general fact that, for any poset P, there is a canonical bijection between the antichains and the dual order ideals of P, see e.g. [19, 3.1]. In what follows, we also write I(Γ) (resp. 𝔠(Γ)) for the combinatorial (resp. geometric) ad-nilpotent ideal with the set of generators Γ. For instance, the unique maximal element of 𝔄𝔡 has the following presentation:

Geometric:  𝔠(Π) = [𝔟,𝔟] = 𝔲 ;
Combinatorial:  I(Π) = Δ^{+} .

It is helpful to have a description of the generators of I in terms of the respective admissible element. As usual, we write γ > 0 (resp. γ < 0), if γ ∈ Δ̂^{+} (resp. γ ∈ −Δ̂^{+} ).

Theorem 2.2. Suppose γ ∈ I_{w}. Then γ is a generator of I_{w} if and only if w(δ − γ) ∈ −Π̂ .

Proof. “⇐”. Suppose γ is not a generator of I_{w}, i.e., γ = γ̄+ ν, where γ̄∈ I_{w} and ν ∈ Δ^{+} . Then w(δ − γ) = w(δ − γ̄) − w(ν) is the sum of two negative roots.

“⇒”. Set w(δ − γ) = −μ < 0. If μ is not simple, then μ = μ_{1}+ μ_{2}, where both summands are positive. We have w^{−1} (μ_{1}) + w^{−1} (μ_{2}) = −(δ − γ) < 0. Assume for definiteness that w^{−1} (μ_{2}) < 0. Since w^{−1} (−μ_{2}) > 0 and w(w^{−1} (−μ_{2})) < 0, we have w^{−1} (−μ_{2}) ∈ N̂(w), i.e., w^{−1} (μ_{2}) = −kδ + ν, where k ≥ 1 and ν ∈ I_{w} ⊂ Δ^{+} .

(a) k = 1. It follows that w^{−1} (μ_{1}) = γ−ν ∈ Δ and w(ν−γ) = −μ_{1} < 0. Since w is admissible, ν−γ must be negative, i.e., γ − ν ∈ Δ^{+} . This means that γ is not a generator of I_{w}.

(b) k ≥ 2. Let us show that there is another decomposition of μ as a sum of two positive roots such that one has k = 1 for one of the summands. We argue by induction on k. Since w(kδ − ν) < 0, we have ν ∈ (I_{w})_{k}. Therefore there is a decomposition kδ − ν = k′δ − ν′+ k′′δ − ν′′, where k′,k′′> 0 and ν′,ν′′∈ I_{w}. Hence μ_{2} = μ′_{2}+ μ′′_{2}, where w^{−1} (μ′_{2}) = ν′− k′δ and w^{−1} (μ′′_{2}) = ν′′− k′′δ. The following lemma shows that, in this situation, μ′_{2} + μ_{1} ∈ Δ^{+} or μ′′_{2}+ μ_{1} ∈ Δ^{+} . If the latter holds, then μ = μ′_{2} + (μ′′_{2}+ μ_{1}) is a decomposition such that w^{−1} (μ′_{2}) = ν′− k′δ, and k′< k. This completes the induction step. □

Lemma 2.3. Suppose μ_{1},μ_{2},μ_{3} ∈ Δ̂^{+} and μ := μ_{1} + μ_{2} + μ_{3} ∈ Δ̂^{+} . Then μ_{1} + μ_{2} ∈ Δ̂^{+} or μ_{1}+ μ_{3} ∈ Δ̂^{+} .

Proof. If (μ_{2} + μ_{3},μ_{1}) < 0, then (μ_{2},μ_{1}) < 0 or (μ_{3},μ_{1}) < 0, and we are done. If (μ_{2} + μ_{3},μ_{1}) ≥ 0, then (μ_{2}+ μ_{3},μ) > 0. Hence μ − μ_{2} ∈ Δ̂ or μ − μ_{3} ∈ Δ̂. □

Corollary 2.4. The number of generators of I_{w} is equal to the number of roots α ∈ Π̂ such that w^{−1} (α) < 0.

Proof. By the definition of an admissible element, if w^{−1} (α) < 0, then w^{−1} (α) = γ − δ for some γ ∈ Δ^{+} . Hence w(δ − γ) ∈ −Π̂ and γ is a generator of I_{w}. The rest is clear. □

Thus, the set of generators of I_{w} corresponds to a certain subset of Π̂ . More precisely, if w(γ − δ) = α ∈ Π̂ (γ ∈ Δ^{+} ), then we say that γ is the generator of I_{w} corresponding to α. Recall that the class of nilpotence of I ∈ 𝔄𝔡, denoted cl(I), is the maximal k such that I_{k} ≠ ∅. Making use of the admissible element w defining the ad-nilpotent ideal I_{w}, one can readily determine cl(I_{w}).

Proposition 2.5. cl(I_{w}) = k if and only if w(α_{0}) + kδ ∈ Δ^{+} ∪ (δ − Δ^{+} ).

Proof. Since each (I_{w})_{m} is an ad-nilpotent ideal, we have (I_{w})_{m} ≠ ∅ if and only if θ ∈ (I_{w})_{m}. Therefore, the very definition of the admissible element corresponding to an adnilpotent ideal (see (1.2)) implies that cl(I_{w}) = k if and only if w(kδ−θ) < 0 and w((k+1)δ− θ) > 0. In other words, w(α_{0}) + (k − 1)δ < 0 and w(α_{0}) + kδ > 0. Hence the conclusion. □

Remark. If I is a non-trivial Abelian ideal, then cl(I) = 1 and the Proposition asserts that w(α_{0}) + δ ∈ Δ^{+} ∪ (δ − Δ^{+} ). However, Proposition 2.4 in [14] says that only the first possibility actually realizes, i.e., w(α_{0}) + δ ∈ Δ^{+} . But, it can be shown that in case k = cl(I) > 1 we do have both possibilities for w(α_{0}) + kδ.

Example 2.6. Take w = s_{θ}s_{0} ∈ Ŵ, where s_{θ} ∈ W is the reflection with respect to θ. Then

s_{θ}s_{0}(α) = \{ \begin{matrix}α + δ, & (α,θ) ≠ 0 \\ α, & (α,θ) = 0 \end{matrix} for α ∈ Π .

We also have

w^{−1} : \{ \begin{matrix}α_{0} & ↦ & α_{0}+ 2δ, \\ α_{i} & ↦ & α_{i} & if (α_{i},θ) = 0, i ≠ 0, \\ α_{i} & ↦ & α_{i}− δ & if (α_{i},θ) ≠ 0, i ≠ 0 . \end{matrix}

Hence w is admissible. The corresponding combinatorial ad-nilpotent ideal is H = {γ ∈ Δ^{+} | (γ,θ) > 0} and the set of generators is Γ(H) = H ∩ Π. The (geometric) ideal 𝔠 = ⊕_{γ∈H} 𝔤_{γ} is the standard Heisenberg subalgebra of 𝔤. Obviously, cl(H) = 2, and we have s_{θ}s_{0}(α_{0}) + 2δ = δ − θ.

The work of Cellini and Papi [7] establishes a bijection between the ad-nilpotent ideals of 𝔟 and the points of certain simplex in V lying in Q^{∨}, the coroot lattice. This was used for giving a uniform proof of the formula for the number of ad-nilpotent ideals. Below, we describe that bijection in a form adapted to our notation, and show that this can also be used for determining the number of generators of an ideal. As is well known, Ŵis isomorphic to a semi-direct product of W and Q^{∨}. Given w ∈ Ŵ, there is a unique decomposition

(2.7)  w = v_{w}·t_{r_{w}} ,

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

(a) the linear action on V̂= V ⊕ ℚδ ⊕ ℚλ;

(b) the affine-linear action on V. For r ∈ Q^{∨}, the linear action of t_{r} ∈ Ŵon V ⊕ ℚδ 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 this action on V. For instance, the formulae of Example 2.6 show that s_{θ}s_{0} = t_{−θ^{∨}} . There is a simple procedure for obtaining the affine-linear action on V from the linear action on V̂, which is explained in [7], but we do not need this. Using the decomposition (2.7), one can define the mapping Ŵ→ Q^{∨} by w ↦ v_{w}(r_{w}) =: d_{w}. One of the main results of [7] is that the set {d_{w}}, where w ranges over all admissible elements of Ŵ, provides a nice parametrization of ad-nilpotent ideals. Namely, set

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

It is a simplex in V. The following is Proposition 3 in [7].

Theorem 2.8. (Cellini–Papi) The mapping 𝔄𝔡 → Q^{∨}, defined by I ↦ w⟨I⟩ ↦ d_{w⟨I⟩}=: d_{I}, sets up a bijection between 𝔄𝔡 and D̃∩ Q^{∨}.

Remark. Our D̃∩ Q^{∨} is −D in the notation of [7]. Now, we provide a link between the number of generators of I and the position of d_{I} inside of D̃.

Theorem 2.9. The number of generators of I equals the codimension (in V) of the minimal face of D̃ containing d_{I}.

Proof. We have w = w⟨I⟩ is an admissible element of Ŵ. Let us realise how the vector d_{I} = v_{w}(r_{w}) can be determined by the linear action of w. If w = v_{w}·t_{r_{w}}, then w^{−1} = v_{w}^{−1} ·t_{−v_{w}(r_{w})}. In the following computations, we repeatedly use the facts that δ is isotropic and w(δ) = δ for all w ∈ Ŵ. If x ∈ V ⊕ ℚδ, then

w^{−1} (x) = v_{w}^{−1} (t_{−v_{w}(r_{w})}(x)) = v_{w}^{−1} (x + (x,v_{w}(r_{w}))δ) = v_{w}^{−1} (x) + (x,v_{w}(r_{w}))δ = v_{w}^{−1} (x) + (x,d_{I})δ .

In particular, we have

w^{−1} (α_{i}) = v_{w}^{−1} (α_{i}) + (α_{i},d_{I})δ, i ≥ 1,

and

w^{−1} (α_{0}) = v_{w}^{−1} (α_{0}) + (α_{0},d_{I})δ = −v_{w}^{−1} (θ) + (1 − (θ,d_{I}))δ .

Note that v_{w}^{−1} (α_{i}) and −v_{w}^{−1} (θ) are in Δ. Therefore, by the very definition of an admissible element, we have (α_{i},d_{I}) ≥ −1 (i ≥ 1) and 1−(θ,d_{I}) ≥ −1, i.e., (θ,d_{i}) ≤ 2. (In particular, we have recovered the fact that d_{I} ∈ D̃.) Set k_{i} = (α_{i},d_{I}) and k_{0} = 1 − (θ,d_{I}). By Theorem 2.2, we have k_{i} = −1 if and only if v_{w}^{−1} (α_{i}) is a generator of I; that is, I has a generator corresponding to α_{i}. Similarly, k_{0} = −1 if and only if I has a generator corresponding to α_{0}. It remains to observe that k_{i} = −1 (i = 0,1,...,p) are precisely the equations of facets of D̃. □

It follows that an ad-nilpotent ideal has at most n generators, and the ideals having exactly n generators correspond to the integral (i.e., lying in Q^{∨}) vertices of D̃. Next, we give an elementary proof for the first observation and show that D̃ always has a unique integral vertex.

Proposition 2.10. Let Γ ⊂ Δ^{+} be an antichain. Then

(i) The elements of Γ are linearly independent and hence #(Γ) ≤ rk𝔤; (ii) If #(Γ) = rk𝔤, then Γ = Π.

Proof. (i) Suppose Γ = {γ_{1},...,γ_{t}}. Since γ_{i} − γ_{j} ∉ Δ, the angle between any pair of elements of Γ is non-acute. Because all γ_{i}’s lie in open half-space of V, they are linearly independent. (ii) Suppose Γ = {γ_{1},...,γ_{p}}, and let w ∈ Ŵbe the corresponding admissible element.

We argue by induction on p. The case p = 1 being obvious, we assume that p ≥ 2. If Γ ∩ Π ≠ ∅, say γ_{1} ∈ Π, then {γ_{2},...,γ_{p}} is an antichain in a root system whose rank is p − 1. Hence Γ = Π by the induction assumption. So, we have only to prove that the case Γ ∩ Π = ∅ is impossible. Assume not, i.e., ht(γ_{i}) ≥ 2 for all i. By Theorem 2.2,

(2.11)  w(γ_{i}) − δ = α_{l_{i}} ∈ Π̂ .

Since p ≥ 2, we may choose i such that α_{l_{i}} lies in Π. Without loss of generality, we may assume that i = 1. Choose also roots μ,μ̄ ∈ Δ^{+} such that γ_{1} = μ + μ̄. Obviously, then μ,μ̄ ∉ I = I(Γ). By part (i), Γ is a basis for V. Hence,

μ = ∑_{j∈J} d_{j}γ_{j} − ∑_{k∈K}c_{k}γ_{k} ,

where J,K are disjoint subsets of {1,2,...,p} and c_{k},d_{j} > 0. Therefore,

μ̄= − ∑_{j∈J} d_{j}γ_{j} + ∑_{k∈K}c_{k}γ_{k}+ γ_{1} .

Given ν ∈ Δ̂, we say that the level of ν, denoted lev(ν), is m ∈ ℤ, if ν − mδ ∈ Δ. Consider the roots w(μ),w(μ̄) ∈ Δ̂. Since w(δ − μ) > 0 and w(μ) > 0, we have lev(w(μ)) is either 1 or 0, and likewise for μ̄. As w(μ + μ̄) = δ + γ_{1} has level 1, we may assume without loss that lev(w(μ)) = 0 and lev(w(μ̄)) = 1. Using Eq. 2.11 for the w(γ_{i})’s, we obtain

w(μ) = (∑_{j} d_{j} − ∑_{k} c_{k})δ + ∑_{j∈J} d_{j}α_{l_{j}} − ∑_{k∈K}c_{k}α_{l_{k}} ,

and

w(μ̄) = (1 − ∑_{j} d_{j} + ∑_{k} c_{k})δ − ∑_{j∈J} d_{j}α_{l_{j}} + ∑_{k∈K}c_{k}α_{l_{k}}+ α_{l_{1}} .

If one of the roots α_{l_{i}}, i ∈ J ∪ K, is equal to α_{0} = δ − θ, then the equality lev(w(μ̄)) − lev(w(μ)) = 1 cannot be satisfied. Hence all these roots lie in Π and hence ∑_{j} d_{j} − ∑_{k}c_{k} = lev(w(μ)) = 0. But the equality w(μ) = ∑_{j∈J} d_{j}α_{l_{j}} − ∑_{k∈K}c_{k}α_{l_{k}} ∈ Δ contradicts the fact that w(μ) is positive. □

Corollary 2.12. The simplex D̃has a unique integral vertex, corresponding to the unique maximal ad-nilpotent ideal.

The vertices of D̃ can explicitly be described, see [7]. Indeed, let {π_{i}} be the basis for V dual to {α_{i}}, 1 ≤ i ≤ p and h the Coxeter number for Δ. If θ = ∑ _{i=1}^{p} m_{i}α_{i} and ρ^{∨} is the half-sum of all positive coroots, then the vertices of D̃ are −ρ^{∨} and −ρ^{∨} + \frac{h+1}{m_{i}} π_{i}, 1 ≤ i ≤ p. However, it is not immediately clear from this that exactly one vertex lies in Q^{∨}.

3. A combinatorial statistic on 𝔄𝔡(𝔤), Catalan arrangements, and clusters

By [7], the cardinality of 𝔄𝔡(𝔤) is equal to ∏_{i=1}^{p} \frac{h + e_{i}+ 1}{e_{i}+ 1} , where the e_{i}’s are the exponents

and h is the Coxeter number of 𝔤. In this section, we consider the simple root statistic on 𝔄𝔡(𝔤). It is given by

sim(I) = #(I ∩ Π), I ∈ 𝔄𝔡(𝔤) .

Accordingly, we set

𝔄𝔡(𝔤)_{i} = {I ∈ 𝔄𝔡(𝔤) | #(I ∩ Π) = i}, i = 0,1,...,p .

Because we know the number of ad-nilpotent ideals for all simple 𝔤, the number 𝔄𝔡(𝔤)_{0} can be counted via the inclusion-exclusion principle. Indeed, the ideals containing α_{i} ∈ Π can be identified with the ideals of the semisimple subalgebra of 𝔤 whose simple roots are Π \ {α_{i}}. Write 𝔤(J) for the semisimple subalgebra of 𝔤 whose set of simple roots is J ⊂ Π. Then

(3.1)  #𝔄𝔡(𝔤)_{0} = ∑_{J⊂Π} (−1)^{p−#J} #𝔄𝔡(𝔤(J)) .

In turn, the numbers 𝔄𝔡(𝔤)_{i} (i > 0) are easily computed, once one knows 𝔄𝔡(𝔤)_{0}. For instance, the number of ideals containing exactly one simple root, say α_{i}, is equal to the number of all ideals in 𝔤(Π \ {α_{i}}) that do not contain simple roots. Hence

#𝔄𝔡(𝔤)_{1} = ∑_{#J=p−1} #𝔄𝔡(𝔤(J))_{0} .

Similarly, one obtains the general formula:

(3.2)  #𝔄𝔡(𝔤)_{i} = ∑_{#J=p−i} #𝔄𝔡(𝔤(J))_{0} .

Of course, applying Equations (3.1) and (3.2), one should use the relation that if 𝔥 = 𝔥_{1}⊕ 𝔥_{2}, then #𝔄𝔡(𝔥) = #𝔄𝔡(𝔥_{1})·#𝔄𝔡(𝔥_{2}), and likewise for 𝔄𝔡(𝔥)_{0}. The distribution of the simple root statistic over 𝔄𝔡(𝔤) yields the polynomial

S_{𝔤}(q) = ∑_{i=0}^{p} #(𝔄𝔡(𝔤)_{i})q^{i} ,

which is not hard to compute. For instance, the following table contains the relevant data for exceptional Lie algebras.

i  0  1  2 3 4 5 6 7 8
F_{4}  66 24 10 4 1
#𝔄𝔡(𝔤)_{i} E_{6} 418 228 110 50 20 6 1
E_{7} 2431 1001 429 187 77 27 7 1
E_{8} 17342 4784 1771 728 299 112 35 8 1

It immediately follows from Eq. (3.2) that #𝔄𝔡(𝔤)_{p} = 1 and #𝔄𝔡(𝔤)_{p−1} = p. A bit longer analysis yields

Proposition 3.3.

If 𝔤 simply-laced, then #𝔄𝔡(𝔤)_{p−2} = (p − 1)(p + 2)/2; If 𝔤 ∈ {B,C,F }, then #𝔄𝔡(𝔤)_{p−2} = p(p + 1)/2.

Proof. There are p(p−1) subalgebras of the form 𝔤(J) with #J = 2. Of these subalgebras, we have

• p − 1 subalgebra of type A_{2} and (p − 1)(p − 2)/2 subalgebras of type A_{1} × A_{1}, if 𝔤 is simple laced;

• one subalgebra of type C_{2}, p − 2 subalgebras of type A_{2} and (p − 1)(p − 2)/2 subalgebras of type A_{1}× A_{1}, if 𝔤 is doubly laced. □

Our results and conjectures for the classical series are as follows.

Theorem 3.4. #𝔄𝔡(A_{p})_{i} = \frac{i + 1}{p + 1} \binom{2p − i}{p} , i = 0,1,,...,p.

We defer the proof to Section 4. Arguing by induction on p and using Eq. (3.2), one obtains S_{𝔤}(q) = S_{𝔤^{∨}}(q), where 𝔤^{∨} is the Langlands dual Lie algebra. The only practical output of this equality is that the simple root statistic has the same distribution for B_{p} and C_{p}. However, we have only conjectural values for C_{p} and D_{p}, which are verified for p ≤ 8.

Conjecture 3.5.

#𝔄𝔡(C_{p})_{i} = \binom{2p − 1 − i}{p − 1} , i = 0,1,,...,p.

#𝔄𝔡(D_{p})_{i} = \binom{2p − 2 − i}{p − 2} + \binom{2p − 3 − i}{p − 2} , i = 1,2,,...,p.

Notice that the conjecture does not give an expression for #𝔄𝔡(D_{p})_{0}. As we will see below, the right value for #𝔄𝔡(D_{p})_{0} is \binom{2p−2}{p−2} + \binom{2p−3}{p−3}. Using Eq. (3.1), it is easy to compute #𝔄𝔡(𝔤)_{0} for any simple Lie algebra. However, obtaining a closed expression in the classical case requires some work. In order to obtain a more conceptual explanation and the closed formula valid for all 𝔤, we use the theory of arrangements.

Remark 3.6. Having written up Propositions 3.7 and 3.10 below, I found that exactly the same results are obtained in the recent preprint of C. Athanasiadis [4]. In this preprint, he gave a conceptual proof of the formula (3.9) for the characteristic polynomial of the Catalan arrangement. In fact, Eq. (3.9) was known for all simple Lie algebras via case-by-case verification, and this was used in my original argument.

Recall a bijection between the ad-nilpotent ideals and the regions of the Catalan arrangement that are contained in the fundamental Weyl chamber. This bijection is due to Shi [18, Theorem 1.4], see also [7, § 4]. The Catalan arrangement Cat(Δ) is the set of hyperplanes in V having the equations

(x,μ) = 1,  (x,μ) = 0,  (x,μ) = −1  (μ ∈ Δ^{+} ) .

The regions of an arrangement are the connected components of the complement in V of the union of all its hyperplanes. Clearly, C is a union of regions of Cat(Δ). Any region lying in C is said to be dominant. The bijection takes an ideal I ⊂ Δ^{+} to the region

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

Obviously, the dominant regions of Cat(Δ) are the same as those for the Shi arrangement Shi(Δ). Here Shi(Δ) is the set of hyperplanes in V having the equations

(x,μ) = 1,  (x,μ) = 0  (μ ∈ Δ^{+} ) .

It will be more convenient for us to deal with the arrangement Cat(Δ), since it is W - invariant. A region (of an arrangement) is called bounded, if it is contained in a sphere about the origin.

Proposition 3.7. I ∈ 𝔄𝔡(𝔤)_{0} if and only if the region R_{I} is bounded.

Proof. 1. Suppose I ∩ Π = ∅. Then the definition of R_{I} shows that ii is contained in the bounded domain in C given by the inequalities (α,x) < 1, α ∈ Π.

2. Suppose α_{i} ∈ I ∩ Π. Then I also contains all positive roots whose coefficient of α_{i} in the expression through the simple roots is positive. Hence for all roots γ such that (γ,φ_{i}) > 0 we have the constraints (x,γ) > 1. This means that if x ∈ R_{I}, then all constraints are satisfied for x + aφ_{i} with any a ∈ ℝ_{≥0} . Thus, R_{I} is unbounded. □

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. [2], [17] for precise definitions).

Theorem 3.8. (Zaslavsky [22, Sect. 2])

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

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

Recently, Athanasiadis [4] found a rather simple case-free proof of the following formula for the characteristic polynomial of the Catalan arrangement:

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

(For the classical series, it was computed earlier in [2]). Now, combining the preceding results, we arrive at our goal.

Proposition 3.10. #𝔄𝔡(𝔤)_{0} = ∏_{i=1}^{p} \frac{h + e_{i}− 1}{e_{i}+ 1} .

Proof. Since the arrangement Cat(Δ) is W -invariant, the number of its bounded regions lying in C is equal to \frac{1}{#W}|χ(Cat(Δ),1)|. It remains to observe that #W = ∏ _{i}(e_{i}+ 1). □

Similarly, using the value χ(Cat(Δ),−1), as Athanasiadis also did in [4], one obtains the formula for the number of all ad-nilpotent ideals stated at the beginning of this section. This proof is not so elementary as the proof of Cellini-Papi [7], for it requires some deep results from the theory of arrangements.

It is quite interesting that the numbers ∏_{i=1}^{p} \frac{h + e_{i}+ 1}{e_{i}+ 1} and ∏_{i=1}^{p} \frac{h + e_{i}− 1}{e_{i}+ 1} also appear in [9,

Theorem 1.9 & Prop. 3.9] as the numbers of all and positive clusters, respectively. We are not going to discuss the theory of clusters related to the root systems, referring to that paper for all relevant definitions. For our current purposes, it suffices to know that clusters are certain subsets of Δ^{+} ∪(−Π). Each cluster is a linearly independent subset of V having exactly p elements. A cluster is called positive, if all its elements are positive roots. A close relationship between clusters and ad-nilpotent ideals is seen in the following curious fact. Let ℭ𝔩𝔲𝔰(𝔤)_{i} denote the set of clusters having exactly i elements from −Π.

Theorem 3.11. One always has the equality #𝔄𝔡(𝔤)_{i} = #ℭ𝔩𝔲𝔰(𝔤)_{i}.

Proof. From Proposition 3.6 in [9], it follows that the numbers ℭ𝔩𝔲𝔰(𝔤)_{i}, i = 0,1,...,p, also satisfy the recurrent relations Eq. (3.1) and (3.2). □

It is not too brave to suggest that there exists a natural bijection between clusters and ad-nilpotent ideals that takes ℭ𝔩𝔲𝔰(𝔤)_{i} to 𝔄𝔡(𝔤)_{i} for all i.

4. On ad-nilpotent ideals for 𝔤 = 𝔰𝔩_{n}

At the rest of the paper, we are going to study another combinatorial statistic on the set of ad-nilpotent ideals, which is related to the theory developed in Section 2. We first consider the classical series in Sections 4 and 5, and then move to the general case in Section 6. At the rest of this section, 𝔤 = 𝔰𝔩_{n} and hence p = n − 1. We assume that 𝔟 (resp. 𝔱) is standard, i.e., it is the space of upper-triangular (resp. diagonal) matrices. Then the positive roots are identified with the pairs (i,j), where 1 ≤ i < j ≤ n. For instance, α_{i} = (i,i + 1) and θ = (1,n). An ad-nilpotent 𝔟-ideal is represented by a right-justified Ferrers diagram with at most n − 1 rows, where the length of i-th row is at most n − i. If a box of a Ferrers diagram corresponds to a positive root (i,j), then we say that this box has the coordinates (i.j). The unique northeast corner of the diagram corresponds to θ and the southwest corners give rise to the generators of the corresponding ideal, see Figure 1. Such a diagram (ideal) I is completely determined by the coordinates of boxes that contain the southwest corners of the diagram, say (i_{1},j_{1}),...,(i_{k},j_{k}). Then we obviously have Γ(I) = {(i_{1},j_{1}),...,(i_{k},j_{k})} and

1 ≤ i_{1} < i_{2} < ···< i_{k} ≤ n − 1, 2 ≤ j_{1} < j_{2} < ···< j_{k} ≤ n .

Various enumerative results for ad-nilpotent ideals in 𝔰𝔩_{n} are obtained in [1], [6], [13]. In particular, the total number of ad-nilpotent ideals equals C_{n} = \frac{1}{n+1} \binom{2n}{n}, the n-th Catalan number. There is a host of combinatorial objects that are counted by Catalan numbers, see [20, ch. 6, Ex. 6.19] and the “Catalan addendum” at www-math.mit.edu/˜ rstan/ec. We shall use the fact that C_{n} is equal to

(a) the number of all sequences v = v_{1}v_{2}...v_{2n} of n 1’s and n −1’s with all partial sums nonnegative, or

[figure: i_{1}; i_{k}; j_{1}; ···; j_{k}]

Figure 1. An ad-nilpotent ideal in 𝔰𝔩_{n}

(b) the number of lattice paths from (0,0) to (n,n) with steps (1,0) and (0,1), always staying in the domain x ≤ y, i.e., the number of Dyck paths of semilength n.

In our matrix interpretation, we are forced to assume that the x-axis is vertical and directed downstairs, while the y-axis is horizontal. Therefore (0,0) is the upper-left corner and (n,n) is the lower-right corner of the matrix. The Dyck path corresponding to an adnilpotent ideal is the double path in Figure 1. It has 2n steps. The corresponding sequence v is obtained as follows. We start from (0,0) and attach +1 to the horizontal step (i.e., (0,1)) and −1 to the vertical step (i.e., (1,0)).

Remark. Coordinates of boxes of Ferrers diagrams and lattice points considered above are compatible in the sense that the coordinates of a box are equal to the coordinates of its southeast corner.

Proof of Theorem 3.4. Once the relationship between the ad-nilpotent ideals in 𝔰𝔩_{n} and Dyck paths is established, one may appeal to huge combinatorial literature on the latter. It is clear that I ∈ 𝔄𝔡(𝔰𝔩_{n}) contains a simple root if and only if the corresponding Dyck path touches the diagonal somewhere except the points (0,0) and (n,n). In other words, the number of simple roots in I equals the number of (intermediate) returns of the Dyck path. The distribution of this statistic is well-known, see e.g. [8, 6.6]. □

Let 𝔄𝔡_{n} denote the set of all ad-nilpotent ideals for 𝔰𝔩_{n}. From now on, we stick to considering the statistic gen : 𝔄𝔡_{n}→ ℕ, which assigns to an ideal the number of its generators. Let 𝔄𝔡_{n}^{k}, 0 ≤ k ≤ n− 1, be the set of ideals with k generators, i.e., the set of Ferrers diagrams, as above, with exactly k southwest corners.

Proposition 4.1. #(𝔄𝔡_{n}^{k}) = \frac{1}{n}\binom{n}{k} \binom{n}{k + 1} .

Proof. The numbers N(n,k) = \frac{1}{n}\binom{n}{k}\binom{n}{k−1}, k = 1,...,n, are called the Narayana numbers, so that we are to show that #(𝔄𝔡_{n}^{k−1}) = N(n,k). It is known that the Narayana numbers have the following combinatorial interpretation, see [20, ch. 6, Ex. 36(a)]. Let X_{nk} be the set of all sequences v = v_{1}...v_{2n} as in (a) above, such that

k = #{j | v_{j} = 1,v_{j+1} = −1} .

Then #(X_{nk}) = N(n,k). For v ∈ X_{nk}, it is easily seen that

k − 1 = #{j | v_{j} = −1,v_{j+1} = 1} .

The change of sign from 1 to −1 (resp. from −1 to 1) in v corresponds to the turn of the type “horizontal followed by vertical” (resp. “vertical followed by horizontal”) step in the respective lattice path. Geometrically, the steps of second type correspond to the southwest corners of our Ferrers diagram. It follows that the sequences v ∈ X_{nk} are in bijection with the Ferrers diagrams with k − 1 southwest corners, and we are done. □

Since N(n,k) = N(n,n − k + 1), one may suggest that there is a bijective interpretation of this equality. This is really the case.

Theorem 4.2. There is a natural bijection between 𝔄𝔡_{n}^{k} and 𝔄𝔡_{n}^{n−k−1} .

Proof. Let (i_{1},j_{1}),...,(i_{k},j_{k}) be the generators of an ad-nilpotent ideal I ∈ 𝔄𝔡_{n}^{k}. Consider separately the ordered sets of the first and second coordinates for these generators, i.e. put X(I) = {i_{1},...,i_{k}} and Y (I) = {j_{i},...,j_{k}}. We wish to construct two other ordered sets that will form the first and the second coordinates of the generators for the dual ideal. To this end, put

X(I^{∗}) = {1,...,n− 1} \ {j_{1}− 1,...,j_{k}− 1}.

Y (I^{∗}) = {2,...,n} \ {i_{1}+ 1,...,i_{k}+ 1}. For A = {a_{1},...,a_{m}}, it is convenient to introduce notation A[a] = {a_{1}+a,...,a_{m}+a}. Then the previous formulas can be written as

(4.3)  \begin{matrix}X(I^{∗}) = (\{2,...,n\} \ Y (I))[−1], \\ Y (I^{∗}) = (\{1,...,n − 1\} \ X(I))[1]. \end{matrix}

It is then easily seen that the square of this transformation is the identity on 𝔄𝔡_{n}^{k}. Therefore one has only to prove that the ordered sets X(I^{∗}),Y (I^{∗}) determine an ad-nilpotent ideal. The latter means that if X(I^{∗}) = {i_{1}^{∗},...,i_{n−k−1}^{∗} } and Y (I^{∗}) = {j_{1}^{∗},...,j_{n−k−1}^{∗} }, then i_{q}^{∗} < j_{q}^{∗} for all q. (Of course, i_{1}^{∗} < i_{2}^{∗} < ... and likewise for j_{l}^{∗}.)

(a) Given q ∈ {1,...,n − k − 1}, suppose there is m such that i_{m}> m + q − 1. Assume also that m is the minimal number with this property. Then i_{m} ≥ m + q and i_{m−1} < m − 1 + q. Therefore the q-th element of {1,...,n − 1} \ X(I) is m − 1 + q and hence j_{q}^{∗} = m + q. Since j_{m}> i_{m}= m+q, we can find the minimal number l such that j_{l} > l +q. Then l ≤ m and the q-th element of {2,...,n} \ Y (I) is l + q. Thus, i_{q}^{∗} = l + q − 1 < m + q = j_{q}^{∗}.

(b) Suppose i_{m} ≤ m + q − 1 for all m ∈ {1,...,k}, that is, i_{k} ≤ k + q − 1. Then the q-th element of {1,...,n − 1} \ X(I) is k + q and hence j_{q}^{∗} = k + q + 1. On the other hand, the inequalities i_{q}^{∗} < i_{q+1}^{∗} < ...i_{n−k−1}^{∗} ≤ n − 1 show that i_{q}^{∗} ≤ (n − 1) − ((n − k − 1) − q) = q + k.

Thus, X(I^{∗}) and Y (I^{∗}) determine an element of 𝔄𝔡_{n}^{n−k−1} , which we denote by I^{∗}. □

For all k ∈ {0,1,...,n− 1}, we have constructed bijections

𝔄𝔡_{n}^{k} → 𝔄𝔡_{n}^{n−k−1} , I ↦ I^{∗} .

which give rise to an involutory transformation ∗ : 𝔄𝔡_{n} → 𝔄𝔡_{n}. Although this transformation is not order-reversing with respect to the inclusion of ideals, it has interesting properties. The formulation of these properties is “universal”, i.e., it makes sense for any (semi)simple Lie algebra:

Lemma 4.4. Suppose A ⊂ Π is an arbitrary subset, and I = I(A). Then I^{∗} = I(Π \ A).

Proof. Straightforward. Use Formulae (4.3). □

To state one more property, we need some notation. As usual, the height of a root γ ∈ Δ^{+} is denoted by ht(γ). Recall that h = ht(θ)+1 is the Coxeter number of 𝔤. Set Δ^{+} (k) = {γ ∈ Δ^{+} | ht(γ) = k} and Δ_{k}^{+} = {γ ∈ Δ^{+} | ht(γ) ≥ k}. It is clear that Δ_{k}^{+} is a combinatorial ad-nilpotent ideal and Γ(Δ_{k}^{+} ) = Δ^{+} (k). For 𝔰𝔩_{n}, we have ht(i,j) = j − i and the Coxeter number is n.

Lemma 4.5. In case of 𝔰𝔩_{n}, we have (Δ_{k}^{+} )^{∗} = Δ_{h+ 1− k}^{+}= Δ_{n+ 1− k}^{+}.

Proof. Set I = Δ_{k}^{+} . In our notation, the roots in Δ^{+} (k) are (1,k + 1),(2,k + 2),...,(n − k,n). Hence X(I) = {1,2,...,n−k} and Y (I) = {k +1,k +2,...,n}. Therefore X(I^{∗}) = {1,2,...,k − 1} and Y (I^{∗}) = {n − k + 2,...,n}. This means that I^{∗} is generated by the roots (1,n − k + 2),...,(k − 1,n), i.e., all roots of height n − k + 1. □

Examples. In the geometric context, taking k = 1, we obtain 𝔲^{∗} = {0}. For k = 2, we have [𝔲,𝔲]^{∗} = 𝔤_{θ}, because θ is the only root of height h − 1. It is curious that our definition of the dual ad-nilpotent ideal for 𝔰𝔩_{n} leads to another occurrence of Catalan numbers. Namely, let us try to describe and enumerate the selfdual ideals. For I ∈ 𝔄𝔡_{n}^{m}, the necessary condition of self-duality is m = n − m − 1. That is, n = 2m + 1.

Theorem 4.6. There are no self-dual ad-nilpotent ideals for 𝔰𝔩_{2m}. For 𝔰𝔩_{2m+1}, the number

of self-dual ad-nilpotent 𝔟-ideals is equal to \frac{1}{m + 1} \binom{2m}{m} .

Proof. We use the notation introduced in Theorem 4.2. Suppose I ∈ 𝔄𝔡_{2m+1}^{m} and X = X(I) = {i_{1},i_{2},...,i_{m}}, Y = Y (I) = {j_{1},j_{2},...,j_{m}}. The condition I = I^{∗} means that X = X^{∗} = Ȳ[−1] and Y = Y^{∗} = X̄[1]. Clearly, all these equalities are equivalent to the following

{1,2,...,2m} = {i_{1},i_{2},...,i_{m}} ⊔ {j_{1}− 1,j_{2}− 1,...,j_{m}− 1} = X ⊔ Y [−1] .

Therefore Y is determined by X and vice versa. However, X cannot be an arbitrary melement subset of {1,2,...,2m}, since the conditions i_{k} < j_{k}, k = 1,...,m, must also be satisfied. Given X ⊂ {1,2,...,2m} with #(X) = m, define the sequence v = v_{1}v_{2}...v_{2m}by the following rule:

v_{i} = \{ \begin{matrix}1, if i ∈ X, \\ −1, if i ∉ X. \end{matrix}

Then (X,X̄[1]) determine an ad-nilpotent ideal if and only if all partial sums of v are nonnegative. Indeed, ∑ _{i=1}^{2k−1} v_{i} < 0 if and only if i_{k} ≥ j_{k}. As was mentioned above, the number of such sequences is the m-th Catalan number. □

To illustrate Theorem 4.6, we list the generators of all self-dual ideals for 𝔰𝔩_{7}:

Γ_{1} = {(1,5),(2,6),(3,7)}, Γ_{2} = {(1,4),(2,6),(4,7)}, Γ_{3} = {(1,4),(2,5),(5,7)}, Γ_{4} = {(1,3),(3,6),(4,7)}, Γ_{5} = {(1,3),(3,5),(5,7)}.

Remark 4.7. The equality of Theorem 4.6 is (almost) an instance of the so-called “q = −1 phenomenon” studied by J. Stembridge [21]. The distribution of the statistic “number of generators” yields the polynomial

N_{n}(q) = ∑_{k=0}^{n−1} #(𝔄𝔡_{n}^{k})q^{k} = ∑_{k=0}^{n−1} \frac{1}{n}\binom{n}{k} \binom{n}{k + 1} q^{k} ,

which is often called the Narayana polynomial. The q = −1 phenomenon is said to occur if N_{n}(−1) counts the number of fixed points of some natural involution on 𝔄𝔡_{n}. We already have the involution ‘∗’ and know the number of its fixed points. On the other hand, it follows from [5, Prop. 2.2] that

(4.8)  N_{n}(−1) = \{ \begin{matrix}0, & if n is even \\ (−1)^{(n−1)/2} C_{n−1/2} , & if n is odd. \end{matrix}

(Actually, the authors of [5] deal with the polynomial d_{n}(q) = (1 + q)N_{n}(q + 1). However, the sign given there for the value d_{n}(−2) should be opposite.) Thus, we see that the q = −1 phenomenon occurs up to sign. It is interesting that Equality (4.8) appears also in [12, p.276] in connection with a discussion of the Charney-Davis conjecture and properties of the Coxeter zonotope of type A.

The involution on 𝔄𝔡_{n} (and hence on the set of Dyck paths of semilength n) described in Theorem 4.2 seems to be new.

5. ad-nilpotent 𝔟-ideals for orthogonal and symplectic Lie algebras

A possible idea for constructing an involutory mapping ∗ : 𝔄𝔡(𝔤) → 𝔄𝔡(𝔤) for the other classical Lie algebras can be the following:

Consider the standard embedding 𝔤 ⊂ 𝔰𝔩_{N}, and choose a Borel subalgebra \overline{𝔟} ⊂ 𝔰𝔩_{N} such that \overline{𝔟}∩𝔤 = 𝔟 is a Borel subalgebra of 𝔤. Making use of the embedding 𝔟 ⊂ \overline{𝔟}, one can regard 𝔄𝔡(𝔟,𝔤) as a subset of 𝔄𝔡(\overline{𝔟},𝔰𝔩_{N}) consisting of ideals satisfying a symmetry condition. Then we apply to 𝔄𝔡(\overline{𝔟},𝔰𝔩_{N}) the duality procedure described in the previous section. The last step should be to interpret the resulting ideal in 𝔰𝔩_{N} as an element of 𝔄𝔡(𝔟,𝔤).

It turns out that this recipe yields “expected” results for 𝔰𝔭_{2p}, but not immediately for 𝔰𝔬_{p}. The obstacle is that the last step in the above program cannot always be fulfilled in the orthogonal case. Still, one can modify this procedure, so that to get a suitable result for 𝔰𝔬_{2p+1} . However, I do not know how to deal with the case of 𝔰𝔬_{2p}.

(5.1) The symplectic case. Choose a basis for a 2p–dimensional symplectic 𝕜vector space 𝕍 so that the skew-symmetric non-degenerate bilinear form has the matrix ( \begin{matrix}0 & Υ_{p} \\ −Υ_{p} & 0 \end{matrix} ) , where Υ_{p} is the p × p matrix whose only nonzero entries are 1’s along the

antidiagonal.

For any A ∈ Mat_{p}(𝕜), let Âdenote the matrix Υ_{p}(A^{t})Υ_{p}, where A^{t} is the usual transpose of A. The transformation A ↦ Âis the transpose relative to the antidiagonal. In the above basis for 𝕍, the algebra 𝔰𝔭_{2p} has the following block form:

𝔰𝔭_{2p} = \{( \begin{matrix}A & B \\ C & D \end{matrix} ) | B = B̂, C = Ĉ, D = −Â\} ,

where A,B,C,D are p × p matrices. If \overline{𝔟} is the standard Borel subalgebra of 𝔰𝔩_{2p}, then 𝔟:= \overline{𝔟} ∩ 𝔰𝔭_{2p} is a Borel subalgebra of 𝔰𝔭_{2p}. It follows that 𝔄𝔡(𝔰𝔭_{2p}) can be identified with the subset of 𝔄𝔡(𝔰𝔩_{2p}) consisting of all Ferrers diagram that are symmetric relative to the antidiagonal.

Let us say that Ī∈ 𝔄𝔡(𝔰𝔩_{2p}) is self-conjugate, if the corresponding Ferrers diagram is symmetric with respect to the antidiagonal. It is easily seen that if Ī∈ 𝔄𝔡(𝔰𝔩_{2p}) is selfconjugate, then Ī^{∗} is self-conjugate as well, see below. This induces the desired involution on 𝔄𝔡(𝔰𝔭_{2p}), and a straightforward verification shows that this involution satisfy properties (4.4) and (4.5). Since the Ferrers diagram corresponding to an ad-nilpotent 𝔟-ideal has a symmetry property, we may cancel out its part which is below the antidiagonal. What we obtain is a shifted Ferrers diagram.

5.1.1 Example. 𝔤 = 𝔰𝔭_{8}. In our matrix interpretation, the array of positive roots is

1000 1100 1110 1111 1121 1221 2221
0100 0110 0111 0121 0221
0010 0011 0021
0001  ,

where the quadruple c_{1}c_{2}c_{3}c_{4} stands for the root ∑ c_{i}α_{i}. Consider the ad-nilpotent ideal I whose generators are α_{1},α_{2}+ α_{3},2α_{3}+ α_{4}. The corresponding shifted Ferrers diagram is depicted on the left hand side in Figure 2.

The dotted lines demonstrate the positive roots that are not in I, and the whole array

∘ ∘ ∘ I :  ^{∘ I^{∗} :}

Figure 2. An ad-nilpotent ideal in 𝔄𝔡(𝔰𝔭_{8}) and its dual

corresponds to Δ^{+} (or 𝔲). The boxes marked with ‘∘’ represent the generators. The corresponding self-conjugate ideal Ī∈ 𝔄𝔡(𝔰𝔩_{8}) is depicted in Figure 3, where the dotted line is the antidiagonal.

Ī:

Figure 3. The self-conjugate ad-nilpotent ideal Īin 𝔄𝔡(𝔰𝔩_{8})

From the picture representing Ī, we find that X(Ī) = {1,2,3,5,7} and Y (Ī) = {2,4,6,7,8}. Therefore X(Ī^{∗}) = {2,4} and Y (Ī^{∗}) = {5,7}. This leads to the diagram depicted on the right hand side in Figure 2. The sole generator of the ideal I^{∗} is α_{2}+ α_{3}+ α_{4}.

Formally, our recipe for constructing the dual ad-nilpotent ideal in 𝔄𝔡(𝔰𝔭_{2p}) is as follows. We use the same coordinate system as in the 𝔰𝔩_{n}-case. The shifted Ferrers diagram (as in Figure 2) is determined by the coordinates of the boxes that contain its southwest corners, and these boxes give rise to the generators of the respective ad-nilpotent ideal. Suppose Γ = {(i_{1},j_{1}),...,(i_{k},j_{k})} is the set of generators of I ∈ 𝔄𝔡(𝔰𝔭_{2p}), and i_{1} < i_{2} < ... < i_{k}. Then i_{l} < j_{l} for all l, j_{1} < j_{2} < ... < j_{k}, and i_{k}+j_{k}≤ 2p+1. Conversely, if a set Γ satisfies all these inequalities, then it is the set of generators of an ad-nilpotent ideal. Denoting by Īthe corresponding self-conjugate ideal in 𝔄𝔡(𝔰𝔩_{2p}), we obtain

X(Ī) = (i_{1},...,i_{k},2p + 1 − j_{k},...,2p + 1 − j_{1}),

Y (Ī) = (j_{1},...,j_{k},2p + 1 − i_{k},...,2p + 1 − i_{1}).

[If i_{k}+ j_{k} = 2p + 1, then one should cancel out the repetition in the middle.] The coordinates of vectors X(Ī),Y (Ī) can be paired so that the sum in each pair is equal to 2p + 1. Therefore the same property holds for the shifted complements X(Ī^{∗}),Y (Ī^{∗}). That is, Ī^{∗} is again a self-conjugate ideal in 𝔄𝔡(𝔰𝔩_{2p}), and we can define the ideal I^{∗} ∈ 𝔄𝔡(𝔰𝔭_{2p}). Notice that #Γ(I) + #Γ(I^{∗}) = p and the multiset {Γ(I),Γ(I^{∗})} contains a unique long root, i.e., the distribution of long and short roots is always the same as in Π. (A long root corresponds to the generator (i_{k},j_{k}) with i_{k}+ j_{k} = 2p + 1.) In particular, the equality I = I^{∗} is impossible, i.e., there are no self-dual ad-nilpotent ideals.

5.1.2 Example. 𝔤 = 𝔰𝔭_{6}. In Table 1, we list all pairs of dual ad-nilpotent ideals including the ideals with one and two generators. The column with I (resp. I^{∗}) contains all ideals with one (resp.) two generators. The numeration of simple roots is standard: α_{1} = ε_{1}−ε_{2}, α_{2} = ε_{2}−ε_{3}, α_{3} = 2ε_{3}.

It is clearly seen that properties of Lemmas 4.4 and 4.5 are satisfied here.

\begin{matrix}no. & Γ(I) & Γ(I^{∗}) \\ 1–3 & α_{i} & Π \ \{α_{i}\} \\ 4 & α_{1}+ α_{2} & α_{1}+ α_{2},α_{3} \\ 5 & α_{2}+ α_{3} & 2α_{2}+ α_{3},α_{1} \\ 6 & 2α_{2}+ α_{3} & α_{2}+ α_{3},α_{1} \\ 7 & α_{1}+ α_{2}+ α_{3} & α_{1}+ α_{2},2α_{2}+ α_{3} \\ 8 & α_{1}+ 2α_{2}+ α_{3} & α_{1}+ α_{2}+ α_{3},2α_{2}+ α_{3} \\ 9 & 2α_{1}+ 2α_{2}+ α_{3} & α_{1}+ α_{2},α_{2}+ α_{3} \end{matrix}

Table 1. Pairs of dual ad-nilpotent ideals in 𝔰𝔭_{6}

(5.2) The orthogonal case. Choose a basis for an n–dimensional orthogonal 𝕜vector space 𝕍 so that the symmetric non-degenerate bilinear form has the matrix Υ_{n}. In the above basis for 𝕍, we have:

𝔰𝔬_{n} = {A | A = −Â} .

Here we also have 𝔟:= \overline{𝔟} ∩ 𝔰𝔬_{n} is a Borel subalgebra. This means that to any ad-nilpotent 𝔟-ideal in 𝔰𝔬_{n}, one can again attach a self-conjugate ad-nilpotent \overline{𝔟}-ideal in 𝔰𝔩_{n}. But unlike the symplectic case this mapping is not onto. The reason is that the orthogonal matrices have zero antidiagonal entries. Therefore a self-conjugate ad-nilpotent ideal in 𝔰𝔩_{n} having a generator on the antidiagonal cannot correspond to a 𝔟-ideal in 𝔰𝔬_{n}. It may happen that, for I ∈ 𝔄𝔡(𝔰𝔬_{n}), the last element in the sequence I → Ī→ Ī^{∗} cannot be interpreted as an ideal in 𝔰𝔬_{n}. So, a naive attempt to repeat the “symplectic” procedure fails.

In the odd-dimensional case, this difficulty can be circumvented by associating to a 𝔟ideal in 𝔰𝔬_{2p+1} the ideal in 𝔰𝔭_{2p} having the same shape (shifted Ferrers diagram). This is achieved by cancelling out from a symmetric Ferrers diagram both the antidiagonal (which corresponds to zero entries in the matrix) and the part below the antidiagonal. This leads to a satisfactory procedure.

5.2.1 Example. 𝔤 = 𝔰𝔬_{7}. In Table 2, we list all pairs of dual ad-nilpotent ideals including the ideals with one and two generators. The column with I (resp. I^{∗}) contains all ideals with one (resp.) two generators. The numeration of simple roots is standard: α_{1} = ε_{1}−ε_{2}, α_{2} = ε_{2}−ε_{3}, α_{3} = ε_{3}. One can see some small distinctions from Table 1.

Again, the properties of Lemmas 4.4 and 4.5 are satisfied here. In the following section,

\begin{matrix}no. & Γ(I) & Γ(I^{∗}) \\ 1–3 & α_{i} & Π \ \{α_{i}\} \\ 4 & α_{1}+ α_{2} & α_{1}+ α_{2},α_{3} \\ 5 & α_{2}+ α_{3} & α_{2}+ 2α_{3},α_{1} \\ 6 & α_{2}+ 2α_{3} & α_{2}+ α_{3},α_{1} \\ 7 & α_{1}+ α_{2}+ α_{3} & α_{1}+ α_{2},α_{2}+ 2α_{3} \\ 8 & α_{1}+ α_{2}+ 2α_{3} & α_{1}+ α_{2}+ α_{3},α_{2}+ 2α_{3} \\ 9 & α_{1}+ 2α_{2}+ 2α_{3} & α_{1}+ α_{2},α_{2}+ α_{3} \end{matrix}

Table 2. Pairs of dual ad-nilpotent ideals in 𝔰𝔬_{7}

we also summarize some other properties of the duality mapping that are inspired by our computations in classical cases.

6. Towards the general case

In view of Theorem 4.2, it is natural to ask whether there is a natural involutory mapping ∗: 𝔄𝔡(𝔤) → 𝔄𝔡(𝔤) for any simple Lie algebra 𝔤 such that

#(Γ(I)) + #(Γ(I^{∗})) = rk𝔤

and the two properties of Lemmas 4.4 and 4.5 are also satisfied ?

It is plausible that a conjectural definition of duality should exploit somehow admissible elements of Ŵand the simplex D̃. Although my attempts to define such a mapping in a uniform way were unsuccessful, I believe that such a mapping does exist.

Since an ad-nilpotent ideal I ∈ 𝔄𝔡(𝔤) is completely determined by the corresponding antichain Γ = Γ(I) ⊂ Δ^{+} , properties of the conjectural duality on 𝔄𝔡(𝔤) can be restated in terms of antichains in Δ^{+} . Let 𝔄𝔫(Δ^{+} ) denote the set of all antichains in Δ^{+} . For a moment, we assume that Δ is not necessarily irreducible, and Δ = ⊔_{i}Δ_{i}, where each Δ_{i} is an irreducible root system and the rank of Δ_{i} is p_{i}.

Conjecture 6.1. There exists a natural involutory mapping

∗: 𝔄𝔫(Δ^{+} ) → 𝔄𝔫(Δ^{+} )

such that the following holds for Γ ∈ 𝔄𝔫(Δ^{+} ):

(i) Γ^{∗} = ⊔(Γ ∩ Δ_{i})^{∗} and (Γ ∩ Δ_{i})^{∗} depends only on Γ ∩ Δ_{i}; (ii) #(Γ ∩ Δ_{i}) + #(Γ^{∗}∩ Δ_{i}) = p_{i} for all i; (iii)Suppose Γ contains a simple root α. Write Δ(Π \ {α}) for the root subsystem

spanned by the set of simple roots Π \ {α}. Then Γ^{∗} ⊂ Δ(Π \ {α})^{+} and moreover, Γ^{∗} = (Γ \ {α})^{∗}, where Γ \ {α} is regarded as antichain in Δ(Π \ {α})^{+} ; (iv) (≈ a converse to the previous property) If Γ ⊂ Δ(Π \ {α})^{+} , then

Γ^{∗} = {α} ∪ {the dual of Γ taken in Δ(Π \ {α})^{+} };

(v) If Δ is irreducible, then (Δ^{+} (k))^{∗} = Δ^{+} (h + 1 − k), where h is the Coxeter number of Δ (cf. Lemma 4.5).

(vi) the distribution of long and short roots in the multiset {Γ,Γ^{∗}} is the same as in Π. (This condition is vacuous in the simply-laced case)

It is easy to see that the duality defined for the root systems of type A_{p}, B_{p}, C_{p} satisfies all these properties. Also, it is immediate that ‘∗’ can uniquely be defined for G_{2}. Now, we again assume that Δ is irreducible. Clearly, a necessary condition for such a duality to exist is that the number of antichains of cardinality k ought to be equal to the number of antichains of cardinality p−k. This holds in all cases, where the corresponding values are known, see below. If k = 0, then the assertion follows from Proposition 2.10. In case k = 1, one should be able to prove that the number of positive roots is equal to the number of antichains of cardinality p − 1. Unfortunately, the only proof I know amounts to a case-by-case verification,

For each simple Lie algebra 𝔤, we define an analogue of Narayana polynomial as follows. Let d_{k}(𝔤) be the number of all ad-nilpotent ideals with k generators or, equivalently, the number of all k-element antichains in Δ^{+} . Then

(6.2)  N_{𝔤}(q) = ∑_{i=0}^{p} d_{k}(𝔤)q^{k}

is said to be the Narayana polynomial of type 𝔤 (or, a generalized Narayana polynomial). Clearly, d_{0}(𝔤) = d_{p}(𝔤) = 1 and d_{1}(𝔤) = #Δ^{+} . By Theorem 2.9, d_{p−1} (𝔤) equals the number of integral points lying on the edges of the simplex D̃ (except of the unique integral vertex). Below, we list all known to us generalized Narayana polynomials:

N_{A_{p}}(q) = ∑_{k=0}^{p} \frac{1}{p + 1} \binom{p + 1}{k} \binom{p + 1}{k + 1} q^{k};

N_{B_{p}}(q) = N_{C_{p}}(q) = ∑_{k=0}^{p} \binom{p}{k} ^{2}q^{k};

N_{G_{2}}(q) = 1 + 6q + q^{2};

N_{F_{4}}(q) = 1 + 24q + 55q^{2}+ 24q^{3}+ q^{4} ;

N_{E_{6}}(q) = 1 + 36q + 204q^{2}+ 351q^{3}+ 204^{4}+ 36q^{5}+ q^{6} ;

N_{E_{7}}(q) = 1 + 63q + 546q^{2}+ 1470q^{3}+ 1470^{4}+ 546q^{5}+ 63q^{6}+ q^{7} ;

N_{E_{8}}(q) = 1 + 120q + 1540q^{2}+ 6120q^{3}+ 9518^{4}+ 6120q^{5}+ 1540q^{6}+ 120q^{7}+ q^{8} .

In type A, it is the usual Narayana polynomial (cf. 4.7). The result for types B and C follows from [3, Corollary 5.8]. In that place, Athanasiadis computes the number of nonnesting partitions on B_{p} or C_{p} whose ‘type’ has k parts. However, it follows from his previous exposition that a non-nesting partition whose type has k parts is exactly an antichain of cardinality p − k. The case of G_{2} is trivial and that of F_{4} is relatively easy. The case of E_{n} requires more work. The result can be obtained through the counting all integral points in D̃ and use of Theorem 2.9.

I also conjecture that N_{D_{p}}(q) = ∑_{k=0}^{p}( \binom{p}{k} ^{2}− \frac{p}{p − 1} \binom{p − 1}{k} \binom{p − 1}{k − 1} ) q^{k}.

In fact, \binom{p}{k}^{2}− \frac{p}{p−1} \binom{p−1}{k}\binom{p−1}{k−1} is the number of non-crossing partitions on D_{p} whose type has k parts [16, Sect. 4]. And I hope that the similarity between the non-crossing and nonnesting partitions known for A_{p},B_{p} and C_{p} remains true also for D_{p}. Thus, all known generalized Narayana polynomials are palindromic.

By [7], we have N_{𝔤}(1) = #𝔄𝔡(𝔤) = ∏_{i=1}^{p} \frac{h + e_{i}+ 1}{e_{i}+ 1} . It would be interesting to find a uniform

expression for the coefficients of the generalized Narayana polynomials.

Another intriguing feature is that there are nice formulae for the values N_{𝔤}(−1). For A_{p}, we refer again to 4.7. The B_{p}- or C_{p}-case amounts to a well-known combinatorial identity:

∑_{k=0}^{p}(−1)^{k}\binom{p}{k} ^{2} = \{ \begin{matrix}0, & if p is odd, \\ (−1)^{p/2}\binom{p}{p/2}, & if p is even. \end{matrix}

Combining the expressions for A_{p} and B_{p} cases, we conclude that if the conjectural formula for N_{D_{p}}(q) is true, then

N_{D_{p}}(−1) = \{ \begin{matrix}0, & if p is odd, \\ (−1)^{p/2}[\binom{p}{p/2} − 2\binom{p−2}{p/2−1}] = (−1)^{p/2}2\binom{p−2}{p/2}, & if p is even. \end{matrix}

One may also observe that if p is even, then (−1)^{p/2}N_{𝔤}(−1) is positive for all examples.

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