November 20, 2003

IDEALS OF HEISENBERG TYPE AND MINIMAX ELEMENTS OF AFFINE WEYL GROUPS

DMITRI I. PANYUSHEV

Let 𝔤 be a simple Lie algebra over ℂ. Fix a Borel subalgebra 𝔟 and a Cartan subalgebra 𝔱 ⊂ 𝔟. The corresponding set of positive (resp. simple) roots is Δ^{+} (resp. Π). Write θ for the highest root in Δ^{+}. An ideal of 𝔟 is called ad-nilpotent, if it is contained in [𝔟, 𝔟]. Consequently, such a subspace is fully determined by the corresponding set of positive roots, and this set is called an ideal (in Δ^{+}). Let 𝔄𝔡 denote the set of all ideals of Δ^{+}. We regard Δ^{+} as poset with respect to the standard root order ‘≼’ (see Section 1). Given I ∈ 𝔄𝔡, the minimal roots in it are said to be the generators. The set of generators of an ideal form an antichain in (Δ^{+}, ≼), and this correspondence sets up a bijection between the ideals and the antichains of Δ^{+}. An ideal or antichain is called strictly positive, if it contains no simple roots. Another interesting class consists of Abelian ideals, i.e., those with the property (I + I) ∩ Δ^{+} = ∅. We write 𝔄𝔡_{0} (resp. 𝔄𝔟) for the set of strictly positive (resp. Abelian) ideals. Various results for 𝔄𝔡, 𝔄𝔡_{0}, and 𝔄𝔟 were recently obtained in [2], [3], [4], [5], [10], [11], [12], [15]. It was shown by Cellini and Papi [4] that there is a bijection between the ideals and certain elements of Ŵ, the affine Weyl group associated with 𝔤. Then Sommers [15] discovered a bijection between the strictly positive ideals (or antichains) and another class of elements of Ŵ. Following [15], the elements in these two classes are said to be minimal and maximal, respectively. Furthermore, the minimal elements of Ŵ are in a bijection with the points of the coroot lattice lying in a certain simplex. The same assertion also holds for the maximal elements (with another simplex!) A different approach to (ad-nilpotent) ideals relates them with the theory of hyperplane arrangements. By a result of Shi [13], there is a bijection between 𝔄𝔡 and the dominant regions of the Shi (or Catalan) arrangement. It was then observed by Athanasiadis [2] and Panyushev [11] that the Shi bijection induces the bijection between 𝔄𝔡_{0} and the bounded dominant regions. We survey these results in Section 2. The goal of this paper is two-fold. First, we study the ideals lying inside ℋ, the set of all positive roots that are not orthogonal to θ. Such ideals are said to be of Heisenberg type. Second, we study the minimax elements of Ŵ, i.e., those that are simultaneously minimal and maximal. The corresponding strictly positive ideals are said to be minimax, too. Both minimal and maximal elements of Ŵ are particular instances of dominant elements. To any dominant element one may attach an ideal, and we consider dominant elements of special sort, which are said to be of Heisenberg type, see Definition 3.2. The corresponding ideals lie in ℋ, hence the term. We obtain a explicit description of dominant elements of Heisenberg type and distinguish minimal and maximal ones among them. Our description says, in particular, that the dominant (resp. minimal) elements of Heisenberg type are parameterized by Δ_{l} (resp. Δ_{l} \ (−Π)), where Δ_{l} is the set of long roots. We also prove that the minimal elements of Heisenberg type are in a bijection with the non-trivial ideals of Heisenberg type and give explicit formulae for these ideals, see Theorem 3.12. It follows that the number of non-trivial ideals inside ℋ is equal to #(Δ_{l} \ Π). In Section 4, combining known results for minimal and maximal elements, we obtain a characterization of minimax elements (ideals), and establish a bijection between the minimax elements of Ŵ and the the coroot lattice points lying in a polytope D_{mm}. If V is the real vector space spanned by Δ in 𝔱^{∗}, then

2000 Mathematics Subject Classification. 17B20, 20F55.

Key words and phrases. Borel subalgebra, ad-nilpotent ideal, affine Weyl group.

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

D_{mm} := {x ∈ V | −1 ≤ (α, x) ≤ 1 ∀α ∈ Π & 0 ≤ (θ, x) ≤ 2}

It is the intersection of two simplices corresponding to the minimal and maximal elements of Ŵ. The geometric meaning of minimax ideals is that, under the Shi bijection, they correspond to the regions consisting of a single alcove. It would be interesting to further investigate the configuration of those dominant regions that consist of a single alcove. We also give an upper bound on the sum #Γ(I) + #Ξ(I), where Γ(I) is the set of generators (minimal elements) of I and Ξ(I) is the set of maximal elements of Δ^{+} \ I. Since Π is the only antichain of cardinality rk 𝔤, one readily sees that the maximal possible value of the above sum is 2rk 𝔤 − 1, which is attained for I = Δ^{+} \ Π. However, for the minimax ideals this sum is at most rk 𝔤 + 1 (at most rk 𝔤, if 𝔤 ≠ 𝔰𝔩_{2n+1}), see Proposition 4.6. Write 𝔄𝔡_{mm} or 𝔄𝔡_{mm}(𝔤) for the set of minimax ideals of 𝔤. In Section 5, we compute #𝔄𝔡_{mm}(𝔤) for all simple Lie algebras. Using the coefficients of θ, one may form a Laurent polynomial (in x). We prove that #𝔄𝔡_{mm} equals the coefficient of x in this polynomial, divided by the index of connection of Δ. It turns out that some famous integer sequences emerge in connection with the minimax ideals in classical Lie algebras. Namely,

#𝔄𝔡_{mm}(𝔰𝔩_{n+1}) = M_{n}, the n-th Motzkin number;

#𝔄𝔡_{mm}(𝔰𝔭_{2n}) = #𝔄𝔡(𝔰𝔬_{2n+1}) = dir_{n}, the number of directed animals of size n. We also show that #𝔄𝔡_{mm}(𝔰𝔬_{2n}) = 2dir_{n−2} + dir_{n−1}. See [1], [6], [7], and [16, Ch. 6] for relevant background and numerous combinatorial interpretations of these numbers. In Section 6, an explicit matrix characterization of the minimax ideals for 𝔰𝔩_{n+1} and 𝔰𝔭_{2n} is obtained. The description for 𝔰𝔩_{n+1} can be stated as follows. Let {α_{i} = ε_{i} − ε_{i+1} | i = 1, …, n} be the standard set of simple roots. Let Γ = {γ_{1}, …, γ_{k}} be the set of generators of an ideal I, where γ_{t} = α_{i_{t}} + α_{i_{t}+1} + … + α_{j_{t}}, t = 1, 2, …, k. Then I is minimax if and only if j_{t} ≠ i_{s} for all pairs (t, s). In case t = s, this means that γ_{t} cannot be a simple root. Using these descriptions, we also compute the generating function for the statistic “the number of generators” on 𝔄𝔡_{mm}(𝔤) for 𝔤 = 𝔰𝔩_{n+1} and 𝔰𝔭_{2n}.

Acknowledgements. A part of 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^{+}.

e_{1}, e_{2}, …, e_{p} are the exponents and h is the Coxeter number of W. 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 Ŵ is 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 V̂ is 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 Ŵ with 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 assignment w ↦ N(w) sets up a bijection between the elements of Ŵ and the finite bi-convex subsets of Δ̂^{+}.

(1.2) Ideals and antichains. Recall that 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. 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}.

Given I ∈ 𝔄𝔡, write Ξ(I) for the set of maximal elements of Δ^{+} \ I.

2. Minimal and maximal elements in affine Weyl groups and dominant regions of the Catalan arrangement

This section is a review of known results. As is well known, Ŵ is isomorphic to a semidirect product of W and Q^{∨} [9]. 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^{∨}. Then w^{−1} = v^{−1}·t_{−v(r)}. The word “translation” means the following. The group Ŵ has 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 the linear action, we have w^{−1}(x) = v^{−1}(x) + (x, v(r))δ for any x ∈ V ⊕ ℝδ. In particular,

(2.2)  \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}

While the affine-linear action is given by w^{−1}∗ x = v^{−1}(x) − r for x ∈ V. In particular, t_{r}∗ y = y + r, so that t_{r} is a true translation for the ∗-action on V. Let us say that w ∈ Ŵ is 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 [4, 1.1] that w is dominant if and only if w^{−1}∗ 𝒜 ⊂ 𝒞. Write Ŵ_{dom} for the set of dominant elements.

2.3 Proposition [12].

(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.

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 = I_{w} 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 [15] is also explained by a relationship between these elements and dominant regions of the Shi arrangement, see Theorem 2.19.

2.4 Definition. w ∈ Ŵ is 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}. For any γ ∈ I, define the number l(γ, I) as follows

l(γ, I) = max{m | γ = ∑_{i=1}^{m} κ_{i}, κ_{i} ∈ I} .

2.5 Proposition [4, 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 biconvex set

⋃_{k≥1}(kδ − I^{k}) = {mδ − γ | γ ∈ I & 1 ≤ m ≤ l(γ, I)} ⊂ Δ̂^{+} .

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

The minimal element corresponding to I ∈ 𝔄𝔡 is denoted by w_{min}(I). Conversely, the first layer ideal of w ∈ Ŵ_{min} (= the ideal corresponding to w) is denoted by I_{w}. If N ⊂ Δ̂^{+} is an arbitrary finite convex subset, containing δ − I, then it must also contain N(w_{min}(I)) = ∪_{i≥1}(kδ−I^{k}). So, the latter is the minimal bi-convex subset containing δ−I. We have also the following

2.6 Corollary. For w ∈ Ŵ_{min}, we have ℓ(w) = ∑_{k≥1} #((I_{w})^{k}).

In terms of minimal elements, one can give an explicit description of the generators of an ideal.

2.7 Proposition [11, Theorem 2.2] [15, Cor. 6.3(1)].

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

A geometric description of the minimal elements relates them to the integral points in a certain simplex.

2.8 Proposition [5, Prop. 2 & 3]. Set D_{min} = {x ∈ V | (x, α) ≥ −1 ∀α ∈ Π & (x, θ) ≤ 2}. Then

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.

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 Ŵ that takes D_{min} to a dilated closed fundamental alcove. Namely, w(D_{min}) = (h + 1)\overline{𝒜} for some w ∈ Ŵ, see [5, 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 [8, 7.4], the number of integral points in t\overline{𝒜} is equal to

(2.9)  ∏_{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, see [5],

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

Maximal elements of Ŵ are introduced by E. Sommers in [15]. He also obtained main results for these elements.

2.11 Definition. w ∈ Ŵ is 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 the number k(μ, I) as follows:

k(μ, I) = min{n | μ = ∑_{i=1}^{n} ν_{i}, ν_{i} ∈ Δ^{+} \ I} .

Notice that this definition makes sense only for strictly positive ideals.

2.12 Proposition [15, 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}.

The maximal element corresponding to I ∈ 𝔄𝔡_{0} is denoted by w_{max}(I). Accordingly, the strictly positive ideal corresponding to w ∈ Ŵ_{max} (the first layer ideal of w) is I_{w}. It follows from the previous discussion that to any I ∈ 𝔄𝔡_{0} one can assign two elements of Ŵ; namely, w_{min}(I) and w_{max}(I). Let w ∈ Ŵ be any dominant element with first layer ideal I ∈ 𝔄𝔡_{0}. Since Δ̂^{+} \ N(w) is convex and contains δ − (Δ^{+} \ I), it follows from the very definition of numbers k(γ, I) that mδ − γ ∈ Δ̂^{+} \ N(w) for all m ≥ k(γ, I). Hence N(w) is contained in the finite set given by Eq. 2.12(◇). This shows that N(w_{max}(I)) is the maximal bi-convex subset with first layer δ −I. Remember also that N(w_{min}(I)) is the minimal bi-convex subset with first layer δ − I. Hence N(w_{min}(I)) ⊂ N(w_{max}(I)) and therefore

(2.13)  l(γ, I) ≤ k(γ, I) − 1 for any γ ∈ I .

Recall that Ξ(I) is the set of maximal elements of Δ^{+} \ I. If I ∈ 𝔄𝔡_{0}, then a description of Ξ(I) can be given in terms of w_{max}(I):

2.14 Proposition [15, 6.3(2)]. If w ∈ Ŵ_{max}, then Ξ(I_{w}) = {γ ∈ Δ^{+} | w(δ − γ) ∈ Π̂}.

Next, we recall a geometric characterization of the maximal elements.

2.15 Proposition (cf. [15, Prop. 5.6]). Set D_{max} = {x ∈ V | (x, α) ≤ 1 ∀α ∈ Π & (x, θ) ≥ 0}. Then

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.

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 [5, Lemma 1] that there is an element of Ŵ that takes D_{max} to (1 − h)\overline{𝒜}.

Using again the above-mentioned result of Haiman, see Eq. (2.9), one obtains, see [2], [11], [15],

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

Recall a bijection between 𝔄𝔡 and the dominant regions of the Catalan arrangement, which is due to Shi [14, 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. Any region lying in 𝒞 is said to be dominant. 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 is sometimes more convenient to deal with the arrangement Cat(Δ), since it is W -invariant. The Shi bijection takes an ideal I ⊂ Δ^{+} to the dominant region

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

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

2.18 Proposition [2],[11]. I ∈ 𝔄𝔡(𝔤)_{0} if and only if the region R_{I} is bounded.

A relationship between the theory of minimal/maximal elements and regions of the Catalan arrangement is as follows.

2.19 Theorem.

(i) [5] Suppose I ∈ 𝔄𝔡 and w = w_{min}(I). Then w^{−1}∗ 𝒜 is the alcove closest to the origin in R_{I}; (ii) [15] Suppose I ∈ 𝔄𝔡_{0} and w = w_{max}(I). Then w^{−1}∗ 𝒜 is the alcove most distant from the origin in R_{I}.

3. Dominant elements and ideals of Heisenberg type

Given w ∈ Ŵ, write w(α_{0}) = −mδ + ν, where m ∈ ℤ and ν ∈ Δ. Since (θ, θ) = (α_{0}, α_{0}) = (ν, ν), the root ν is long. The root ν is called the rootlet of w, denoted rt(w). In what follows, Δ_{l} stands for the set of long roots. In general, the rootlet can be any root in Δ_{l}; but, for various classes of elements of Ŵ, different constraints emerge.

3.1 Lemma.

(i) If 1 ≠ w ∈ Ŵ_{dom}, then m ≥ 1; (ii) If w ∈ Ŵ_{min}, then rt(w) ∉ −Π; (iii) If w ∈ Ŵ_{max}, then rt(w) ∉ Π.

Proof. (i) Recall the “affine-linear decomposition” w = w̄·t_{r}, where w̄ ∈ W and r ∈ −\overline{𝒞}. If w ≠ 1, then r ≠ 0, see Prop. 2.3. Hence (θ, r) < 0. Furthermore, −\overline{𝒞} contains no elements x ∈ Q^{∨} such that (x, θ) = −1. For, Haiman’s formula (2.9) shows that \overline{𝒜} contains a unique element of Q^{∨}, namely, the origin. Hence (r, θ) ≤ −2. It remains to observe that w(α_{0}) = −w̄(θ) + (1 + (θ, r))δ. Thus, −m = 1 + (θ, r) ≤ −1. (ii) If w(α_{0}) = −mδ − α, α ∈ Π, then w^{−1}(α) = −(m + 1)δ + θ and −(m + 1) ≤ −2. Hence w ∉ Ŵ_{min}. (iii) If w(α_{0}) = −mδ + α, α ∈ Π, then w^{−1}(α) = (m + 1)δ − θ and m + 1 ≥ 2. Hence w ∉ Ŵ_{max}. □

Remark. Notice that the formulas for w(α_{0}) shows that rt(w) = −w̄(θ). Because there is a bijection between the minimal elements of Ŵ and the ideals of Δ^{+}, one may define the rootlet for any ideal. That is, given w ∈ Ŵ_{min}, we set rt(I_{w}) := rt(w). It should be noticed that if I ∈ 𝔄𝔡_{0}, then the roots rt(w_{min}(I)) and rt(w_{max}(I)) are usually different. So, if we wish to get an unambiguous notion of the rootlet for any ideal, then the minimal elements have to be exploited. Amongst all ideals of Δ^{+}, we distinguish the ideal consisting of all roots that are not orthogonal to θ. Set

ℋ = {γ ∈ Δ^{+} | (γ, θ) > 0} .

The corresponding subspace of [𝔟, 𝔟] is the standard Heisenberg subalgebra, so that we say that ℋ is the Heisenberg ideal. Notice that ℋ^{2} = {θ} and ℋ^{3} = ∅.

3.2 Definition. We say that w ∈ Ŵ is of Heisenberg type, if w = vs_{0} for some v ∈ W. An ideal is said to be of Heisenberg type, if it is contained in ℋ.

The term for elements of Ŵ is explained by the following

3.3 Lemma. If w ∈ Ŵ_{dom} is of Heisenberg type, then the first layer ideal of w lies in ℋ.

Proof. Recall that the first layer ideal of w consists of all γ ∈ Δ^{+} such that w(δ − γ) < 0. If w is of the form vs_{0} (v ∈ W) and (γ, θ) = 0, then w(δ − γ) = δ − v(γ) > 0. □

The main goal of this section is to give a characterization of the dominant elements of Heisenberg type, and then to describe all ideals in ℋ.

3.4 Proposition. Let w = vs_{0} ∈ Ŵ be of Heisenberg type. Then

(i) w ∈ Ŵ_{dom} ⇔ v(α) > 0 for any α ∈ Π such that (α, θ) = 0; (ii) rt(w) = v(θ); (iii) w ∈ Ŵ_{min} ⇔ w ∈ Ŵ_{dom} and v(θ) ∉ −Π; (iv) w ∈ Ŵ_{max} ⇔ w ∈ Ŵ_{dom}, (v(θ), θ) ≥ 0, and v(θ) ∉ Π.

Proof. (i) The property of being dominant means w(α) > 0 for all α ∈ Π. If (α, θ) > 0, then w(α) = v(δ − (θ − α)) = δ − v(θ − α) > 0, i.e., it is always satisfied. If (α, θ) = 0, then w(α) = v(α). Hence the condition.

(ii) vs_{0}(α_{0}) = v(θ) − δ.

(iii) & (iv). The conditions of minimality and maximality impose constraints on the coefficient of δ in w^{−1}(α), α ∈ Π̂. For α ∈ Π, we have w^{−1}(α) = s_{0}·v^{−1}(α). If v^{−1}(α) = ±θ, then s_{0}(±θ) = ±(2δ − θ), which is bad. More precisely, w is not maximal, if v(θ) = α; and w is not minimal, if v(θ) = −α. These are the only bad possibilities. For α_{0}, we have w^{−1}(α_{0}) = s_{0}(δ − v^{−1}(θ)) = δ − v^{−1}(θ) − (θ, v(θ)^{∨})(δ − θ). It is easily seen that the only bad possibility is (θ, v(θ)) < 0, in which case w fails to be maximal. □

Thus, the conditions of maximality and minimality for the Heisenberg type elements are stated in terms of the rootlet. Recall the following standard fact:

(3.5)  Suppose u, v ∈ W. Then ℓ(u) + ℓ(v) = ℓ(uv) if and only if N(u) ∩ N(v^{−1}) = ∅. If these conditions are satisfied, then N(uv) = N(v) ⊔ v^{−1}(N(u)).

If ℓ(u) + ℓ(v) = ℓ(uv), then we say that uv is a reduced decomposition for this product. We will also repeatedly use the following result, see [10, Theorem 4.1]:

(3.6)  Given ν ∈ Δ_{l}^{+}, there is a unique shortest element in W taking θ to ν, denoted by w_{ν}. We have N(w_{ν}^{−1}) = {γ ∈ Δ^{+} | (γ, ν^{∨}) = −1}.

Given ν ∈ Δ^{+}, write s_{ν} for the corresponding reflection in W.

3.7 Lemma. Suppose ν ∈ Δ_{l}^{+}. Then

(i) ℓ(s_{ν}) = 2(ρ, ν^{∨}) − 1;

(ii) ℓ(s_{ν}w_{ν}) = ℓ(s_{ν}) + ℓ(w_{ν}) = (ρ, θ^{∨} + ν^{∨}) − 1.

(iii) N(s_{ν}) = {ν} ∪ {γ ∈ Δ^{+} | (γ, ν^{∨}) = 1 & γ ≺ ν}.

Proof. (i) We have s_{ν}(−ν) = ν. Recall that (ρ, α^{∨}) = 1 for all α ∈ Π. We will argue by induction on (ρ, ν^{∨}). If ν ∈ Π, then ℓ(s_{ν}) = 1, and the claim is true. A straightforward calculation shows that (ρ, s_{α}(μ)^{∨}) = (ρ, μ^{∨}) − (α, μ^{∨}) for μ ∈ Δ and α ∈ Π. If μ is long, we have (α, μ^{∨}) ≥ −1 unless μ = −α. Hence (ρ, s_{α}(μ)^{∨}) ≤ (ρ, μ^{∨}) + 1, if μ ≠ −α, and (ρ, s_{α}(−α)^{∨}) = (ρ, −α^{∨}) + 2. Let s_{ν} = s_{i_{1}} … s_{i_{k}} be a reduced decomposition. Consider the corresponding sequence of roots:

(−ν) →^{s_{i_{k}}} ∘ →^{s_{i_{k−1}}} ∘ … →^{s_{i_{2}}} ∘ →^{s_{i_{1}}} ν .

The level function μ ↦ (ρ, μ^{∨}) attains 2(ρ, ν^{∨}) values between −(ρ, ν^{∨}) and (ρ, ν^{∨}), because the zero level is missing. Moreover, it follows from the previous inequalities that at each step one can go up at most to the next existing level. Hence, ℓ(s_{ν}) ≥ 2(ρ, ν^{∨}) − 1. On the other hand, s_{ν} = s_{α}s_{γ}s_{α}, where γ = s_{α}(ν). Because ν ∉ Π, one can find an α ∈ Π so that (α, ν^{∨}) = 1. Then we obtain (ρ, γ^{∨}) = (ρ, ν^{∨}) − 1 and by the induction assumption ℓ(s_{ν}) ≤ 2 + ℓ(s_{γ}) = 2(ρ, ν^{∨}) − 1. This completes the inductive step.

(ii) The first equality follows from Eq. (3.5). Indeed, N(w_{ν}^{−1}) = {γ | (γ, ν^{∨}) = −1} and for such γ we have s_{ν}(γ) = γ + ν > 0. Hence N(w_{ν}^{−1}) ∩ N(s_{ν}) = ∅. The second equality follows from (i) and the fact that ℓ(w_{ν}) = (ρ, θ^{∨} − ν^{∨}), see [10, Theorem 4.2].

(iii) Obvious. □

In what follows, we write N(s_{ν})^{0} for {γ ∈ Δ^{+} | (γ, ν^{∨}) = 1 & γ ≺ ν}.

3.8 Proposition. Suppose ν ∈ Δ_{l}^{+}. Then

(i) w_{ν}s_{0} is a minimal element with rootlet ν, and I_{w_{ν}s_{0}} is Abelian. The element w_{ν}s_{0} is also maximal, if ν ∉ Π.

(ii) s_{ν}w_{ν}s_{0} is dominant and rt(s_{ν}w_{ν}s_{0}) = −ν. Next, s_{ν}w_{ν}s_{0} ∈ Ŵ_{min} if and only if ν ∉ Π;

s_{ν}w_{ν}s_{0} ∈ Ŵ_{max} if and only if (θ, ν) = 0. (iii) If ν ∉ Π, then the ideal I_{s_{ν}w_{ν}s_{0}} is not Abelian. (iv) If ν ∈ Π, then I_{s_{ν}w_{ν}s_{0}} = I_{w_{ν}s_{0}} is Abelian.

Proof. (i) Since w_{ν}s_{0}(α_{0}) + δ = ν, we obtain rt(w_{ν}s_{0}) = ν. That w_{ν}s_{0} is minimal (in particular, dominant) and the ideal I_{w_{ν}s_{0}} is Abelian is shown in [10, Theorem 4.2]. The assertion on maximality stems from Proposition 3.4.

(ii) Set w = s_{ν}w_{ν}s_{0}. From the very definition of w_{ν}, it follows that w(α_{0}) + δ = −ν. Hence the rootlet is as required. In order to prove that w is dominant, we apply Proposition 3.4(i). Here v = s_{ν}w_{ν}. If α ∈ Π and (θ, α) = 0, then s_{ν}w_{ν}(α) = w_{ν}(α) − (w_{ν}(α), ν^{∨})ν = w_{ν}(α) > 0. The last inequality follows from the fact that, by part (i), w_{ν}s_{0} is dominant. It is also not hard to give a direct argument. (Indeed, if w_{ν}(α) = −μ < 0, then μ ∈ N(w_{ν}^{−1}). Therefore (μ, ν^{∨}) = −1 by Eq. (3.6). That is, 1 = (w_{ν}(α), ν^{∨}) = (α, θ^{∨}), a contradiction! The assertions on maximality and minimality follows from Proposition 3.4(iii),(iv).

(iii) Notice that s_{ν}w_{ν}s_{0}(2δ − θ) = −ν < 0. Since s_{ν}w_{ν}s_{0} is minimal for ν ∉ Π, it follows from Proposition 2.5 that θ ∈ I^{2}.

(iv) By Eq. (3.5) and Lemma 3.7(ii), we have N(s_{ν}w_{ν}s_{0}) = N(w_{ν}s_{0}) ∪ s_{0}w_{ν}^{−1}(N(s_{ν})). If ν ∈ Π, then N(s_{ν}) = {ν}, and the second set equals {2δ−θ}. Thus, N(s_{ν}w_{ν}s_{0}) and N(w_{ν}s_{0}) have the same first layers. Since I_{w_{ν}s_{0}} is Abelian and N(w_{ν}s_{0}) = δ − I_{w_{ν}s_{0}}, we are done. □

The next assertion yields a converse to Proposition 3.8, and completes our characterization of the dominant elements of Heisenberg type.

3.9 Proposition. Suppose w = vs_{0} ∈ Ŵ_{dom}, where v ∈ W. Set ν= \{ \begin{matrix}v(θ), & if v(θ) > 0 \\ −v(θ), & if v(θ) < 0 \end{matrix} .

Then v = w_{ν}, if v(θ) > 0; and v = s_{ν}w_{ν}, if v(θ) = −ν < 0.

Proof. 1. Suppose ν = v(θ) > 0. Then (γ, ν^{∨}) = (v^{−1}(γ), θ^{∨}). Therefore, if (γ, ν^{∨}) = −1, then v^{−1}(γ) < 0. In view of Eq. (3.6), this means that N(v^{−1}) ⊃ N(w_{ν}^{−1}). It follows that there is a ”reduced decomposition” v^{−1} = κw_{ν}^{−1} (i.e., ℓ(v) = ℓ(κ) + ℓ(w_{ν})). Then v = w_{ν}κ^{−1} and κ^{−1}(θ) = θ. This means that κ^{−1} takes the (possibly reducible) root system Δ ∩ ⟨θ⟩^{⊥} to itself. Therefore, if κ ≠ id, then there is an α ∈ Π ∩ ⟨θ⟩^{⊥} such that κ^{−1}(α) < 0. Since v = w_{ν}κ^{−1} is a reduced decomposition and therefore N(v) ⊃ N(κ^{−1}), we have v(α) < 0 as well. But Proposition 3.4(i) says that v(α) must be positive here. This contradiction shows that κ = id, and we are done.

2. Suppose −ν = v(θ) < 0. We use the same idea as in part 1. If N(v^{−1}) ⊃ N(w_{ν}^{−1}s_{ν}), then we will obtain a reduced decomposition of the form v^{−1} = κw_{ν}^{−1}s_{ν}, with κ^{−1}(θ) = θ, and eventually prove that κ = id. So, it suffices to establish the above containment. By Lemma 3.7(ii) and Eq.(3.5), we have

N(w_{ν}^{−1}s_{ν}) = N(s_{ν}) ∪ s_{ν}·N(w_{ν}^{−1}) = {ν} ∪ {γ ∈ Δ^{+} | (γ, ν^{∨}) = 1 & γ ≺ ν} ∪ s_{ν}·N(w_{ν}^{−1}) .

Let us check that the three sets in the RHS are contained in N(v^{−1}).

• v^{−1}(ν) = −θ < 0;

• If (γ, ν^{∨}) = 1, then 1 = (v^{−1}(γ), −θ^{∨}). Hence v^{−1}(γ) < 0;

• If γ ∈ N(w_{ν}^{−1}), then v^{−1}s_{ν}(γ), θ^{∨}) = (s_{ν}(γ, −ν^{∨}) = (γ, ν^{∨}) = −1. Hence v^{−1}s_{ν}(γ) < 0. □

Combining Propositions 3.8 and 3.9, we obtain

3.10 Theorem. Suppose w = vs_{0}, where v ∈ W. Then w ∈ Ŵ_{dom} if and only if v is either w_{ν} or s_{ν}w_{ν} for some ν ∈ Δ_{l}^{+}. Furthermore, rt(w) is ν (resp. −ν) in the first (resp. second) case.

It follows that the number of dominant elements of Heisenberg type is equal to #(Δ_{l}). Using Proposition 3.8(ii), we also conclude that the number of minimal elements of Heisenberg type is equal to #(Δ_{l} \ Π), and the number of maximal elements of Heisenberg type is equal to #(Δ_{l}^{+} \ Π) + #({γ ∈ Δ_{l}^{+} | (γ, θ) = 0}). However, it is not yet clear that if I is a non-trivial ideal (resp. strictly positive ideal) in ℋ, then the corresponding minimal (resp. maximal) element is necessarily of Heisenberg type. Actually, this appears to be true for minimal elements, but not for maximal.

3.11 Proposition. If w ∈ Ŵ_{min} and I_{w} ⊂ ℋ, then w is of Heisenberg type.

Proof. It is straightforward to verify that s_{θ}s_{0} ∈ Ŵ_{min} and I_{s_{θ}s_{0}} = ℋ (cf. [11, Example 2.6]). It follows that N(w) ⊂ N(s_{θ}s_{0}). Therefore s_{θ}s_{0} = w̃w is a reduced decomposition. Since w = w′s_{0} for some w′ ∈ Ŵ, we conclude that s_{θ} = w̃w′ is a reduced decomposition for s_{θ}. Whence w̃, w′ ∈ W. □

Remark. It happens quite often that w ∈ Ŵ_{max} and I_{w} ⊂ ℋ, but w is not of Heisenberg type. A ”uniform” example can be described as follows. Suppose θ is a fundamental weight and ν is the unique simple root such that (θ, ν) > 0. According to Proposition 3.8(ii), s_{ν}w_{ν}s_{0} is dominant but neither maximal nor minimal. Letting I = I_{s_{ν}w_{ν}s_{0}}, one can show that w_{min}(I) = w_{ν}s_{0} (this follows from Proposition 3.8(iv)) and w_{max}(I) = s_{0}s_{ν}w_{ν}s_{0} (a straightforward computation). That is, w_{max}(I) is not of Heisenberg type. Thus, we have obtained a bijection between

• the roots in Δ_{l}^{+} \ (−Π);

• the minimal elements of Heisenberg type;

• the non-trivial ideals lying in ℋ. Under this bijection, the roots in Δ_{l}^{+} \ (−Π) are obtained as the rootlets of the corresponding ideals. In particular, different ideals of Heisenberg type have different rootlets. An explicit construction of the Heisenberg-type elements is given in Proposition 3.8. It is also possible to explicitly describe the corresponding ideals.

3.12 Theorem. Suppose ν ∈ Δ_{l}^{+}. Then

(i) I_{w_{ν}s_{0}} = {θ} ∪ {θ + w_{ν}^{−1}(N(w_{ν}^{−1})} = {θ} ∪ {θ − N(w_{ν})}; (ii) I_{s_{ν}w_{ν}s_{0}} = {θ} ∪ {θ − N(w_{ν})} ∪ {θ − w_{ν}^{−1}(N(s_{ν})^{0})} = I_{w_{ν}s_{0}} ∪ {θ − w_{ν}^{−1}(N(s_{ν})^{0})}, if

ν ∉ Π.

Proof. (i) Since I_{w_{ν}s_{0}} is Abelian, #(I_{w_{ν}s_{0}}) = ℓ(w_{ν}s_{0}) = (ρ, θ^{∨} − ν^{∨}) + 1. Since the set in the right-hand side has the required cardinality, it suffices to verify that it is contained in I_{w_{ν}s_{0}}. Observe that (γ, θ^{∨}) = (w_{ν}(γ), ν^{∨}) = −1 for any γ ∈ w_{ν}^{−1}(N(w_{ν}^{−1}). Therefore we know how s_{0} acts on w_{ν}^{−1}(N(w_{ν}^{−1})). Then

w_{ν}s_{0}(δ − θ − w_{ν}^{−1}(N(w_{ν}^{−1})) = w_{ν}(−w_{ν}^{−1}(N(w_{ν}^{−1})) = −N(w_{ν}^{−1}) ⊂ −Δ^{+} .

Also, w_{ν}s_{0}(δ − θ) = ν − δ < 0.

(ii) Remember that ℋ^{2} = {θ}. Since I := I_{s_{ν}w_{ν}s_{0}} ⊂ ℋ and is not Abelian by Proposition 3.8(iii), we have I^{2} = {θ} as well. Therefore, using Corollary 2.6, we obtain #(I) = ℓ(s_{ν}w_{ν}s_{0})−1 = ℓ(s_{ν}w_{ν}) = (ρ, θ^{∨}+ν^{∨})−1. Here, again, the set in the right-hand side has the prescribed cardinality, Hence it remains to show that it is contained in I. First, it follows from (3.5) that I_{w_{ν}s_{0}} ⊂ I_{s_{ν}w_{ν}s_{0}}. Next, we have (w_{ν}^{−1}(N(s_{ν})^{0}), θ^{∨}) = (N(s_{ν})^{0}, ν^{∨}) = 1. Therefore, we know how s_{0} acts on w_{ν}^{−1}(N(s_{ν})^{0}), and hence we can compute that all roots in s_{ν}w_{ν}s_{0}(δ − θ + w_{ν}^{−1}(N(s_{ν})^{0})) are negative. □

Remark. In principle, it is harmless to omit the hypothesis in part (ii) that ν ∉ Π. For, N(s_{ν})^{0} = ∅, if ν ∈ Π. Then part (ii) would assert in this case that I_{s_{ν}w_{ν}s_{0}} = I_{w_{ν}s_{0}}, which was already proved in Proposition 3.8(iv).

4. Minimax elements and ideals: general properties

An element w ∈ Ŵ is called minimax, if it is simultaneously minimal and maximal. Combining the corresponding definitions of Section 2, we obtain the following:

4.1 Definition. w ∈ Ŵ is called minimax, if

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

The corresponding (strictly positive) ideal is said to be minimax, too. That is, I ∈ 𝔄𝔡_{0} is minimax, if w_{min}(I) = w_{max}(I). Therefore we merely write w(I), if I is minimax. The set of all minimax elements, which is clearly equal to Ŵ_{min} ∩ Ŵ_{max}, is denoted by Ŵ_{mm}. Accordingly, 𝔄𝔡_{mm} is the set of minimax ideals in Δ^{+}. Combining geometric descriptions of minimal and maximal elements given in Section 2, we obtain the following.

4.2 Proposition. Set D_{mm} = D_{min} ∩ D_{max} = {x ∈ V | −1 ≤ (x, α) ≤ 1 ∀α ∈ Π & 0 ≤ (x, θ) ≤ 2}. Then

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

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

It can be shown that there is an element of Ŵ that takes D_{mm} to

{x ∈ V | 0 ≤ (x, α) ≤ 2 ∀α ∈ Π & h − 1 ≤ (x, θ) ≤ h + 1} .

But, I do not see how it could help us to count the integral points in D_{mm}. Recall that, for any strictly positive ideal I ⊂ Δ^{+} and any γ ∈ I, we have defined the numbers

l(γ, I) = max{m | γ = ∑_{i=1}^{m} κ_{i}, κ_{i} ∈ I} , k(γ, I) = min{n | γ = ∑_{i=1}^{n} ν_{i}, ν_{i} ∈ Δ^{+} \ I} .

Using results of Section 2 we immediately obtain various reformulations of the minimax property.

4.3 Theorem. Suppose I is a strictly positive ideal and w is either w_{min}(I) or w_{max}(I). Then the following conditions are equivalent:

(i) w ∈ Ŵ_{mm} ; (ii) k(γ, I) − 1 = l(γ, I) for all γ ∈ I; (iii) R_{I} = w^{−1}∗ 𝒜; (iv) R_{I} consists of a single alcove.

Proof. Use Eq. (2.13) and Theorem 2.19. □

Thus, the minimax elements (ideals) are in a bijection with the dominant regions of the Catalan arrangement consisting of a single alcove. Now, we give a description of minimax Abelian ideals. Let I ⊂ Δ^{+} be a nontrivial Abelian ideal and w = w_{min}(I) the corresponding minimal element. It was shown in [10, Prop. 2.5] that w(α_{0}) = −δ + ν, where ν ∈ Δ_{l}^{+}. In particular, rt(w) = rt(I) is a positive root.

4.4 Theorem. An Abelian ideal I is minimax if and only if rt(I) is not a simple root.

Proof. 1. Assume w(α_{0}) + δ = α_{i} ∈ Π. Then w^{−1}(α_{i}) = 2δ − θ, i.e., w is not maximal.

2. Since I is Abelian, l(γ, I) = 1 for all γ ∈ I. Therefore, in view of Theorem 4.3(ii), the minimax property is equivalent to that any γ ∈ I is a sum of two elements of Δ^{+} \ I. Assume w(α_{0}) + δ = μ ∉ Π. Then μ = μ_{1} + μ_{2}, where μ_{i} ∈ Δ^{+}. Hence w^{−1}(μ_{i}) ∈ Δ̂^{+} and 2δ − θ = w^{−1}(μ_{1}) + w^{−1}(μ_{2}) is a sum of two positive roots. The only possibility for this is θ = γ_{1} + γ_{2} and w^{−1}(μ_{i}) = δ − γ_{i}. Since w(δ − γ_{i}) = μ_{i} ≻ 0, we have γ_{i} ∉ I. Thus, the required property is satisfied for θ ∈ I. To prove this for any μ ∈ I, we use a descending induction. Indeed, if θ − α ∈ I for some α ∈ Π, then the condition γ_{1} + γ_{2} − α ∈ I ⊂ Δ^{+} implies that α ≠ γ_{i} and at least one of γ_{i} − α is in Δ^{+} (and hence in Δ^{+} \ I), see [11, Lemma 2.3]. Hence θ − α is a sum of two elements from Δ^{+} \ I. This can be continued further. □

It should already be clear that a minimax ideal is not necessarily Abelian. The simplest example is the ideal in 𝔰𝔩_{5} generated by {α_{1}+α_{2}, α_{3}+α_{4}}. More precisely, applying results of Section 3 to minimax elements, we obtain 4.5 Proposition. A Heisenberg type element w = vs_{0} is minimax if and only if either v = w_{ν}, where ν ∈ Δ_{l}^{+} \ Π, or v = s_{ν}w_{ν}, where ν ∈ Δ_{l}^{+} \ Π and (θ, ν) = 0. The minimax ideals of the second kind are not Abelian.

It is natural to look at Δ_{mm}, the set of rootlets of all minimax elements (ideals). Clearly, we have the inclusion {ν ∈ Δ_{l} | (ν, θ) ≥ 0} \ {±Π} ⊆ Δ_{mm}. But, in general, it is a strict containment. This can be expressed in the following way. Consider the mapping ζ : 𝔄𝔡_{mm} → Δ_{mm} that associates the rootlet to a minimax ideal. Then, in general, there exist fibres of ζ that do not contain a Heisenberg type ideal. Here is an example. Let 𝔤 = 𝔰𝔩_{6}. Take the ideal generated by the roots α_{1}+α_{2}, α_{3}+α_{4}. A straightforward computation shows that it is minimax and the rootlet is ν = −(α_{1}+α_{2}+α_{3}). Thus, ν is negative and (ν, θ) ≠ 0. It is intuitively clear that if a (dominant) region of Cat(Δ) consists of a single alcove, then this region must be sufficiently close to the origin. Equivalently, the corresponding ideal must be not too large. We now discuss a precise meaning that can be given to this intuitive feeling. Given I ∈ 𝔄𝔡, consider the quantity #Γ(I) + #Ξ(I). It is not hard to realize that its maximal value is 2p − 1, which is attained for I = Δ^{+} \ Π.

4.6 Proposition. Suppose I ∈ 𝔄𝔡_{mm}; then

(i) #Γ(I) + #Ξ(I) ≤ p + 1; (ii) If #Γ(I) + #Ξ(I) = p + 1, then h is odd.

Proof. (i) Set w = w(I) and write w^{−1}(α) = k_{α}α+ν_{α}. Since w is simultaneously minimal and maximal, k_{α} ∈ {−1, 0, 1} for any α ∈ Π̂. (Notice that ν_{α} ∈ Δ^{+}, if k_{α} = −1; ν_{α} ∈ −Δ^{+}, if k_{α} = 1.) Making use of Propositions 2.7 and 2.14 we obtain

γ ∈ Γ(I) ⟺ there is an α ∈ Π̂ such that w^{−1}(α) = −δ + γ,

μ ∈ Ξ(I) ⟺ there is an α ∈ Π̂ such that w^{−1}(α) = δ − μ. This means that the elements of Γ(I) (resp. Ξ(I)) are in a bijection with the affine simple roots such that k_{α} = −1 (resp. k_{α} = 1). Since #Π̂ = p + 1, we are done.

(ii) It follows from the previous part of proof that if #Γ(I) + #Ξ(I) = p + 1, then k_{α} ≠ 0 for all α. Therefore we obtain the partition Π̂ = Π̂_{+} ⊔ Π̂_{−}, according to whether k_{α} = +1 or −1. Recall that the coefficients c_{α} (α ∈ Π) are defined by the equality θ = ∑_{α∈Π} c_{α}α. We also set c_{α} = 1 for α = α_{0}. Then

δ = ∑_{α∈Π̂} c_{α}α and ∑_{α∈Π̂} c_{α} = h .

Because also δ = ∑ c_{α}w^{−1}(α), we obtain

1 = ∑_{α∈Π̂_{+}} c_{α} − ∑_{α∈Π̂_{−}} c_{α} = h − 2 ∑_{α∈Π̂_{−}} c_{α} .

That is, h is odd. □

As is well-known, h is odd if and only if 𝔤 = 𝔰𝔩_{2n+1}. The following example shows that the above upper bound is exact. Consider the ideal in 𝔰𝔩_{2n+1} such that

Γ(I) = {α_{1}+α_{2}, α_{3}+α_{4}, …, α_{2n−1}+α_{2n}} .

Then

Ξ(I) = {α_{1}, α_{2}+α_{3}, …, α_{2n−2}+α_{2n−1}, α_{2n}} .

It follows from the description of minimax ideals in type A_{p}, see Corollary 6.5, that this ideal is minimax. For all other simple Lie algebras, it is possible to exhibit a minimax ideal such that #Γ(I) + #Ξ(I) = p.

5. Counting minimax elements/ideals in the simple Lie algebras

By Proposition 4.2, the number of minimax elements of Ŵ is equal to the cardinality of D_{mm} ∩ Q^{∨}. Unfortunately, D_{mm} is not a simplex, so that it is not easy to find a uniform expression for #(D_{mm} ∩ Q^{∨}). However, for each simple algebra one has a certain system of inequalities and one may try to solve these systems individually. It turns out that, for practical computations, it is easier to deal with the coweight lattice in V, denoted P^{∨}. The number [P^{∨} : Q^{∨}] is called the index of connection of Δ. Recall from Section 2 that {ϖ_{i}^{∨}}_{i=1}^{p} is the dual basis of V for {α_{i}}_{i=1}^{p}. Then the lattice generated by the ϖ_{i}’s is P^{∨}. If y = ∑_{i} y_{i}ϖ_{i} ∈ P^{∨}, then y ∈ Q^{∨} if and only if a certain congruence condition is satisfied for (y_{1}, …, y_{p}) ∈ ℤ^{p}. Recall that θ = ∑_{i=1}^{p} c_{i}α_{i}. The equations of D_{mm} in Proposition 4.2 show that #(D_{mm}∩P^{∨}) equals the number of solutions of the following system:

(5.1)  \{ \begin{matrix}y_{i} ∈ \{−1, 0, 1\} (i = 1, 2, …, p) \\ 0 ≤ c_{1}y_{1} + … + c_{p}y_{p} ≤ 2 . \end{matrix}

It is convenient to introduce one more variable, y_{0}, which also ranges over {−1, 0, 1}, and add it to the expression in the second inequality, so that the total sum to be equal to 1. (We also set c_{0} = 1.) Then the above system becomes

(5.2)  \{ \begin{matrix}y_{i} ∈ \{−1, 0, 1\} (i = 0, 1, …, p) \\ c_{0}y_{0} + c_{1}y_{1} + … + c_{p}y_{p} = 1 . \end{matrix}

In a sense, this procedure corresponds to taking the extended Dynkin diagram of 𝔤. For this reason, system (5.2) will be referred to as the extended system. It is easily seen that the number of solutions of system (5.2) is nothing but the coefficient of x in the expansion of the Laurent polynomial

(5.3)  ∏_{i=0}^{p}(x^{−c_{i}} + 1 + x^{c_{i}}) .

An important particular case is that where all c_{i} = 1. The coefficients in the expansion of (x^{−1} + 1 + x)^{n} are called trinomials, and we write X_{k}(n) for the coefficient of x^{k}. The coefficient of x^{0} (resp. x) is called the central (resp. next-to-central) trinomial. It is easily seen that

X_{k}(n) := ∑_{l≥0} \frac{n!}{l!(k + l)!(n − 2l − k)!} .

The trinomial coefficients satisfy the following relation

(5.4)  X_{k}(n+1) = X_{k−1}(n) + X_{k}(n) + x_{k+1}(n) .

To get the number #(D_{mm} ∩ Q^{∨}), one should add to both systems the above-mentioned congruence condition. Recall that [P^{∨} : Q^{∨}] is equal to the number of 1’s among the coefficients c_{i}, i = 0, 1, …, p. Our first goal is to show that the congruence condition can be omitted.

5.5 Theorem. #(D_{mm} ∩ P^{∨}) = [P^{∨} : Q^{∨}] · #(D_{mm} ∩ Q^{∨}).

Proof. The only proof I know is case-by case (see also Question 4 in Section 7). In considering particular cases, our numbering of α_{i}’s and hence of c_{i}’s correspond to the numbering adopted in [17, Tables]. The explicit form of the congruence condition can be extracted from Table 3 in loc. cit.

1) 𝔤 = 𝔰𝔩_{n+1}. Here all c_{i} = 1 and [P^{∨} : Q^{∨}] = n + 1. The extended system is

(5.6)  \{ \begin{matrix}y_{0} + y_{1} + … + y_{n} = 1 \\ y_{i} ∈ \{−1, 0, 1\} , \end{matrix}

and the congruence is ny_{1} + (n − 1)y_{2} + … + y_{n} ∈ (n + 1)ℤ. Obviously, the symmetric group 𝕊_{n+1} acts on the set of solutions of system (5.6). Let c̃ ∈ 𝕊_{n+1} be the transformation y_{i} ↦ y_{i−1} (i > 0), y_{0} ↦ y_{n}. (It is a Coxeter element of 𝕊_{n+1}.) It is easily seen that all orbits of c̃ on the set of solutions of Eq. (5.6) have cardinality n + 1. Furthermore, making use of the relation y_{0} + y_{1} + … + y_{n} = 1, one easily verifies that

c̃(ny_{1} + (n − 1)y_{2} + … + y_{n}) − (ny_{1} + (n − 1)y_{2} + … + y_{n}) ≡ 1 (mod n + 1) .

This yields the desired relation.

2) 𝔤 = 𝔰𝔭_{2n}. Here [P^{∨} : Q^{∨}] = 2, the extended system is

(5.7)  \{ \begin{matrix}y_{0} + 2(y_{1} + … + y_{n−1}) + y_{n} = 1 \\ y_{i} ∈ \{−1, 0, 1\} , \end{matrix}

and the congruence is y_{n} ∈ 2ℤ. It follows that y_{0} + y_{n} is always odd for a solution of (5.7) and that the permutation of y_{0} and y_{n} takes the solutions with y_{n} even to those with y_{n} odd. Hence the result.

3) 𝔤 = 𝔰𝔬_{2n+1}. Here [P^{∨} : Q^{∨}] = 2, the extended system is

(5.8)  \{ \begin{matrix}y_{0} + y_{1} + 2(y_{2} + … + y_{n}) = 1 \\ y_{i} ∈ \{−1, 0, 1\} , \end{matrix}

and the congruence is y_{1} + y_{3} + y_{5} + … ∈ 2ℤ. Here one may use the permutation of y_{0} and y_{1}, as in part 2).

4) 𝔤 = 𝔰𝔬_{2n}, n ≥ 4. Now [P^{∨} : Q^{∨}] = 4 and the extended system is

(5.9)  \{ \begin{matrix}y_{0} + y_{1} + 2(y_{2} + … + y_{n−2}) + y_{n−1} + y_{n} = 1 \\ y_{i} ∈ \{−1, 0, 1\} . \end{matrix}

But the congruence condition depends on the parity of n.

(A) n = 2l. Here we actually have two conditions:

(5.10)  \{ \begin{matrix}y_{n−1} + y_{n} ∈ 2ℤ \\ y_{1} + y_{3} + … + y_{n−1} ∈ 2ℤ \end{matrix} .

Here we consider the cyclic permutation c̃ : y_{0} → y_{1} → y_{n−1} → y_{n} → y_{0}. Since the sum y_{0} + y_{1} + y_{n−1} + y_{n} is always odd, all orbits of this permutation on the set of solutions of system (5.9) have cardinality 4. It is not hard to verify that each orbit contains a unique representative satisfying (5.10). Let us give some details. It follows from system (5.9) that −1 ≤ y_{2} + … + y_{n−2} ≤ 2. That is, there are four cases:

(a_{1}) y_{2} + … + y_{n−2} = −1 and hence y_{0} + y_{1} + y_{n−1} + y_{n} = 3; (a_{2}) y_{2} + … + y_{n−2} = 0 and hence y_{0} + y_{1} + y_{n−1} + y_{n} = 1; (a_{3}) y_{2} + … + y_{n−2} = 1 and hence y_{0} + y_{1} + y_{n−1} + y_{n} = −1; (a_{4}) y_{2} + … + y_{n−2} = 2 and hence y_{0} + y_{1} + y_{n−1} + y_{n} = −3;

The number of possibilities for (y_{0}, y_{1}, y_{2}, y_{3}) in these four cases is equal to 4, 16, 16, and 4, respectively. (For instance, in case (a_{2}), this number is X_{1}(4) = 16.) Hence, the total number is 40, and one have to test only 10 orbits of c̃ (actually, only five orbits in view of the symmetry). Each orbit contains two representatives satisfying the first condition in (5.10); for these two representatives, the parity of y_{1} is different, so that only one of them satisfies the second condition. Notice also that we have proved that the total number of the solutions of system (5.9) is equal to

4X_{−1}(n − 3) + 16X_{0}(n − 3) + 16X_{1}(n − 3) + 4X_{2}(n − 3) .

(B) n = 2l + 1. Here we have one congruence condition:

2(y_{1} + y_{3} + … + y_{n−2}) + y_{n−1} − y_{n} ∈ 4ℤ .

However, we can split it in two conditions: y_{n−1} − y_{n} ∈ 2ℤ and y_{1} + y_{3} + … + y_{n−2} + \frac{1}{2}(y_{n−1} − y_{n}) ∈ 2ℤ, so that the previous argument goes through verbatim.

5) 𝔤 = E_{6}. Here [P^{∨} : Q^{∨}] = 3, the extended system is

(5.11)  \{ \begin{matrix}y_{0} + y_{1} + 2y_{2} + 3y_{3} + 2y_{4} + y_{5} + 2y_{6} = 1 \\ y_{i} ∈ \{−1, 0, 1\} , \end{matrix}

and the congruence condition is y_{1} −y_{2} +y_{4} −y_{5} ∈ 3ℤ. Then one may use the permutation

c̃: \{ \begin{matrix}y_{1} → y_{5} → y_{0} → y_{1} \\ y_{2} → y_{4} → y_{6} → y_{2} \end{matrix} ,

since y_{1} − y_{2} + y_{4} − y_{5} − c̃(y_{1} − y_{2} + y_{4} − y_{5}) ≡ 1 (mod 3)

6) 𝔤 = E_{7}. Here [P^{∨} : Q^{∨}] = 2, the extended system is

(5.12)  \{ \begin{matrix}y_{0} + y_{1} + 2y_{2} + 3y_{3} + 4y_{4} + 3y_{5} + 2y_{6} + 2y_{7} = 1 \\ y_{i} ∈ \{−1, 0, 1\} , \end{matrix}

and the congruence condition is y_{1} + y_{3} + y_{7} ∈ 2ℤ. Then one may use the involution

c̃: \{ \begin{matrix}y_{1} → y_{0}, y_{2} → y_{6} \\ y_{3} → y_{5}, y_{4} → y_{4}, y_{7} → y_{7} . \end{matrix}

since y_{1} + y_{3} + y_{7} − c̃(y_{1} + y_{3} + y_{7}) is odd.

7) Finally, we have P^{∨} = Q^{∨} for G_{2}, F_{4}, E_{8}, and the theorem is proved. □

Now, we are prepared to compute the number of minimax elements in all simple Lie algebras. For the classical Lie algebras, we obtain some well-known combinatorial quantities.

5.13 Theorem. The number of minimax elements in Ŵ(𝔰𝔩_{n+1}) is equal to the n-th Motzkin number, M_{n}.

5.14 Theorem. The number of minimax elements in Ŵ(𝔰𝔭_{2n}) or Ŵ(𝔰𝔬_{2n+1}) is equal to the number of directed animals of size n, denoted dir_{n}.

5.15 Theorem. The number of minimax elements in Ŵ(𝔰𝔬_{2n}) is equal to 2dir_{n−2} + dir_{n−1}.

We refer to [1], [6], and [16, Ex. 6.37] for basic facts on Motzkin numbers and to [7], [16, Ex. 6.46] for ”directed animal” results; see also explicit formulae below. The first few terms of these sequences are

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ M_{n} & 1 & 2 & 4 & 9 & 21 & 51 & 127 & 323 \\ dir_{n} & 1 & 2 & 5 & 13 & 35 & 96 & 267 & 750 \end{matrix}

Proof of Theorem 5.13. Consider the extended system (5.6). Here the number of solutions is just

X_{1}(n+1) = ∑_{k≥0} \frac{(n + 1)!}{k!(k + 1)!(n − 2k)!} .

It then follows from Theorem 5.5 that the number of minimax elements is

\frac{X_{1}(n+1)}{n + 1} = ∑_{k≥0} \binom{n}{2k}C_{k} ,

where C_{k} = \frac{1}{k+1} \binom{2k}{k} is the k-th Catalan number. But the right-hand-side is a well-known formula for M_{n}, see [6] and [7, Eq. (21)]. □

Proof of Theorem 5.14. In both cases, the extended systems (5.7), (5.8) are the same, up to renumbering y_{i}. Let us work with (5.7). Here we have the constraint 0 ≤ y_{1} + y_{2} + … + y_{n−1} ≤ 1. That is, there are two cases:

(a_{1}) y_{1} + y_{2} + … + y_{n−1} = 0 and hence y_{0} + y_{n} = 1;

(a_{2}) y_{1} + y_{2} + … + y_{n−1} = 1 and hence y_{0} + y_{n} = −1. This means that system (5.7) has 2X_{0}(n − 1) + 2X_{1}(n − 1) solutions. Thus, the number of minimax elements is the sum of central and next-to-central trinomials

X_{0}(n − 1) + X_{1}(n − 1) = ∑_{k≥0} \frac{(n − 1)!}{k!k!(n − 2k − 1)!} + ∑_{k≥0} \frac{(n − 1)!}{k!(k + 1)!(n − 2k − 2)!} =

= ∑_{k≥0} \binom{2k}{k}\binom{n − 1}{2k} + ∑_{k≥0} \binom{2k + 1}{k}\binom{n − 1}{2k + 1} = ∑_{q≥0} \binom{q}{[q/2]}\binom{n − 1}{q} ,

which is a well-known expression for dir_{n}, see [7, Eq. (27)]. □

Notice that we have also derived the relation

(5.16)  X_{0}(n − 1) + X_{1}(n − 1) = dir_{n} .

Proof of Theorem 5.15. All essential work is already done in the proof of Theorem 5.5, part 4. Namely, the number of minimax elements is one fourth of the total number of solutions of system (5.9), i.e.,

\frac{1}{4}(4X_{−1}(n − 3) + 16X_{0}(n − 3) + 16X_{1}(n − 3) + 4X_{2}(n − 3)) .

Making use of Eq. (5.4) and (5.16), one transforms this sum to 2dir_{n−1} + dir_{n−2}. □

Remarks. 1. Although #𝔄𝔡_{mm}(𝔰𝔭_{2n}) = #𝔄𝔡_{mm}(𝔰𝔬_{2n+1}), the ideals occurring in these two sets have different algebraic properties. For instance, 𝔄𝔡_{mm}(𝔰𝔭_{8}) contains 8 Abelian ideals (of 13), whereas 𝔄𝔡_{mm}(𝔰𝔬_{9}) contains 11.

2. Using the relation dir_{n} = 3 dir_{n−1} − M_{n−2}, we may also write #𝔄𝔡_{mm}(𝔰𝔬_{2n}) = 5 dir_{n−2} − M_{n−3}.

5.17 Theorem. 1. For 𝔤 = E_{p}, p = 6, 7, 8, the number of minimax elements in Ŵ equals 67, 217, and 834, respectively.

2. For 𝔤 = F_{4} and G_{2}, there are 17 and 3 minimax elements, respectively.

Proof. In each case, one can directly compute the coefficient of x in the respective Laurent polynomial (5.3), which gives the number #(D_{mm} ∩ P^{∨}), and then use Theorem 5.5. □

5.18 Example. For 𝔤 = F_{4}, the coweight and coroot lattices are equal and system (5.1) is

\{ \begin{matrix}y_{i} ∈ \{−1, 0, 1\} (i = 1, …, 4) , \\ 0 ≤ 2y_{1} + 4y_{2} + 3y_{3} + 2y_{4} ≤ 2 . \end{matrix}

It has 17 solutions, and a description of the corresponding non-trivial Abelian minimax ideals can be extracted from [10, Table 1], using Theorem 4.4 (there are 12 such ideals). The non-abelian minimax ideals are described in the next table.

\begin{matrix}Γ(I) & #(I) & #(I^{2}) & w(α_{0}) & y \\ [1, 2, 1, 1] & 9 & 1 & −δ − [0, 2, 1, 0] & (1, −1, 0, 1) \\ [1, 1, 1, 1] & 10 & 1 & −δ − [2, 2, 1, 0] & (−1, 0, 0, 1) \\ [0, 2, 2, 1], [2, 2, 1, 0] & 10 & 2 & −2δ + [2, 4, 2, 1] & (1, 1, −1, −1) \\ [0, 2, 1, 1], [2, 2, 1, 0] & 12 & 3 & −2δ + [2, 2, 1, 0] & (0, 1, 0, −1) \end{matrix}

For all cases we have I^{3} = ∅, so that ℓ(w(I)) = #(I) + #(I^{2}). We write [n_{1}, n_{2}, n_{3}, n_{4}] for the root ∑_{i} n_{i}α_{i}. The numbering of simple roots follows [17, Tables]. The last column presents the corresponding solution of the above system. The ideals given in the first two rows are of Heisenberg type, and the explicit description of w(I) for them can be extracted from Proposition 3.8(ii) and Theorem 3.12(ii).

6. A description of the minimax ideals for 𝔰𝔩_{n+1} and 𝔰𝔭_{2n}

In this section, we give an explicit matrix description of the minimax ideals in 𝔰𝔩_{n+1} and 𝔰𝔭_{2n}. The idea is as follows. In both cases, we describe a certain set of ideals containing the minimax ones. Then we count the number of ideals in this, a priori, larger set. It turns out that we again obtain the Motzkin numbers and the number of directed animals of size n, respectively, which solves the problem.

\underline{𝔤 = 𝔰𝔩_{n+1}}. We choose 𝔟 (resp. 𝔱) to be the set of upper-triangular (resp. diagonal) matrices of size n+1. With the usual numbering of rows and columns of n+1 by n+1 matrices, we identify the positive roots with the pairs (i, j), where 1 ≤ i < j ≤ n + 1. Here α_{i} = (i, i + 1) and therefore (a, b) = α_{a} + … + α_{b−1}. Then an antichain in Δ^{+}(𝔰𝔩_{n+1}) is an arbitrary sequence Γ = {(a_{1}, b_{1}), …, (a_{k}, b_{k})}, where a_{i} < b_{i} and the two sequences {a_{i}} and {b_{i}} are strictly increasing.

6.1 Definition. Let us say that the generators of an ideal I ∈ 𝔄𝔡(𝔰𝔩_{n+1}) do not meet or I has non-meeting generators, if the antichain Γ(I) has the property that b_{j} ≠ a_{i} + 1 for all i, j.

The equality b_{j} = a_{i} + 1 means that the last root in the expression for (a_{j}, b_{j}) is equal to the first root in the expression for (a_{i}, b_{i}). This explains the term.

6.2 Proposition. If I is a minimax ideal, then its generators do not meet.

Proof. Suppose I has meeting generators; i.e., Γ(I) ⊃ γ_{1}, γ_{2}, where γ_{1} = α_{i} + α_{i+1} + … + α_{j} and γ_{2} = α_{j} + α_{j+1} + … + α_{k} (i < j < k). Consider γ := γ_{1} + γ_{2} − α_{j} = α_{i} + … + α_{k}. It is easily seen that, for any presentation of γ as a sum of \underline{two} roots, the summand containing α_{j} lies in I, while the other summand is in Δ^{+} \ I. This means that γ ∉ I^{2}, i.e., l(γ, I) = 1, and k(γ, I) ≥ 3. Thus, condition (ii) in Theorem 4.3 is not satisfied, and therefore I is not minimax. □

6.3 Theorem. The number of ideals in Δ^{+}(𝔰𝔩_{n+1}) with k non-meeting generators is equal to \binom{n}{2k}C_{k}. In particular, the total number of ideals with non-meeting generators is M_{n}.

Proof. Let I be an ideal with non-meeting generators and Γ(I) = {(a_{1}, b_{1}), …, (a_{k}, b_{k})}. In order to compute the number of such ideals, it is convenient to replace b_{j} with b̃_{j} = b_{j} − 1. In view of Definition 6.1, the numbers {a_{i}, b̃_{j} | 1 ≤ i, j ≤ n}, which belong to {1, 2, …, , n}, are pairwise different. They also satisfy the conditions

(6.4)  a_{i} < b̃_{i}, a_{1} < … < a_{k}, b̃_{1} < … < b̃_{k} .

To obtain such a collection of numbers, one should first choose arbitrarily 2k numbers from {1, 2, …, n}. Next, given 2k numbers, one should choose k numbers among them and call them a_{1}, …, a_{k} (in the increasing order!). Then the remaining k numbers form the sequence of b̃_{j}’s. But the choice of the a_{i}’s cannot be arbitrary, for Eq. (6.4) must be satisfied. It is easily seen that the number of admissible choices is C_{k}. Indeed, given an ordered sequence of 2k elements v_{1} v_{2} … v_{2k} from {1, 2, …, n}, we assign to v_{l} the value +1, if v_{l} = a_{i}; and −1, if v_{l} = b̃_{j}. Then Eq. (6.4) is satisfied if and only if all partial sums ∑_{i≤m} v_{i} are nonnegative. But the number of such sequences v_{1} v_{2} … v_{2k} is the k-th Catalan number, see [16, Ex. 6.19(r)]. Hence, there are ∑_{k≥0} \binom{n}{2k}C_{k} ideals with non-meeting generators, and it is the same number that occurs in the proof of Theorem 5.13. □

6.5 Corollary. For 𝔰𝔩_{n+1}, the minimax ideals are precisely the ideals with non-meeting generators.

\underline{𝔤 = 𝔰𝔭_{2n}}. We use a standard matrix model of 𝔰𝔭_{2n} corresponding to a Witt basis for alternating bilinear form. For this basis of ℂ^{2n}, the algebra 𝔰𝔭_{2n} has the following block form:

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

where A, B, C, D are n × n matrices and A ↦ Â is the transpose relative to the antidiagonal. If \overline{𝔟} is the standard Borel subalgebra of 𝔰𝔩_{2n}, then 𝔟 := \overline{𝔟} ∩ 𝔰𝔭_{2n} is a Borel subalgebra of 𝔰𝔭_{2n}. (See also [11, 5.1].) We identify the positive roots of 𝔰𝔭_{2n} with the set {(i, j) | i < j, i + j ≤ 2n + 1}. Here the simple roots are α_{i} = (i, i + 1), 1 ≤ i ≤ n, and therefore:

(i, j) = \{ \begin{matrix}α_{i} + … + α_{j−1}, & if j ≤ n + 1 , \\ α_{i} + … + α_{2n−j} + 2(α_{2n−j+1} + … + α_{n−1}) + α_{n}, & if j > n + 1 . \end{matrix}

The ideals for 𝔰𝔭_{2n} can be identified with the ideals for 𝔰𝔩_{2n} that are symmetric with respect to the antidiagonal (= self-conjugate). In other words, there is a natural bijection between the ideals in Δ^{+}(𝔰𝔭_{2n}) and the self-conjugate ideals in Δ^{+}(𝔰𝔩_{2n}). More precisely, suppose Ī ∈ 𝔄𝔡(𝔰𝔩_{2n}) and Γ(Ī) = {(i_{1}, j_{1}), …, (i_{k}, j_{k})} with i_{1} < … < i_{k}, where we use our convention on the roots of 𝔰𝔩_{2n}. Then Ī is self-conjugate if and only if i_{m} + j_{k+1−m} = 2n + 1 for all m. The corresponding ideal I ∈ 𝔄𝔡(𝔰𝔭_{2n}) has the generators Γ(I) = {(i_{m}, j_{m}) | m ≤ [(k + 1)/2]}. We shall say that Ī ∈ 𝔄𝔡(𝔰𝔩_{2n}) is the symmetrization of I ∈ 𝔄𝔡(𝔰𝔭_{2n}).

6.6 Proposition. If I ∈ 𝔄𝔡(𝔰𝔭_{2n}) is a minimax ideal, then the symmetrization Ī has non-meeting generators.

Proof. Suppose the symmetrization Ī has some meeting generators. Then arguing as in Proposition 6.2, we find a root γ̄ ∈ Ī such that l(γ̄, Ī) = 1 and k(γ̄, Ī) ≥ 3. To any root γ̄ ∈ Δ^{+}(𝔰𝔩_{2n}), one naturally associate the root γ ∈ Δ^{+}(𝔰𝔭_{2n}). With our identification for both root systems, this can be formalized as follows. If γ̄ = (i, j) and i + j ≤ 2n + 1, then set γ = γ̄. If i + j ≥ 2n + 2, then set γ = (2n+1−j, 2n+1−i). This yields a surjective mapping Δ^{+}(𝔰𝔩_{2n}) → Δ^{+}(𝔰𝔭_{2n}) and, in particular, Ī → I for any I ∈ 𝔄𝔡(𝔰𝔭_{2n}). The last and easy observation is that (Ī)^{l} = \overline{I^{l}} for any I ∈ 𝔄𝔡(𝔰𝔭_{2n}). So that l(γ, I) = 1 and k(γ, I) ≥ 3 as well. Thus, I is not minimax. □

6.7 Theorem. The number of ideals in Δ^{+}(𝔰𝔭_{2n}) with q generators, whose symmetrization has non-meeting generators, is equal to \binom{2q−1}{q−1}\binom{n−1}{2q−1}+\binom{2q}{q}\binom{n−1}{2q}. In particular, the total number of such ideals is ∑_{q≥0} \binom{q}{[q/2]}\binom{n−1}{q} = dir_{n}.

Proof. Let Ī ∈ 𝔄𝔡(𝔰𝔩_{2n}) be a self-conjugate ideal and Γ(Ī) = {((i_{1}, j_{1}), …, (i_{k}, j_{k})} the sequence of its generators. Then

i_{m} < j_{m}, 1 ≤ i_{1} < i_{2} < … < i_{k}, j_{1} < j_{2} < … < j_{k} ≤ 2n ,

and the symmetry condition i_{m}+j_{k+1−m} = 2n+1 is satisfied for any m. If the generators do not meet, then all the numbers {i_{l}, j̃_{m} = j_{m}−1 | 1 ≤ l, m ≤ k} are different. They form a set E ⊂ {1, 2, …, 2n−1} consisting of 2k elements. Because of the symmetry, E is completely determined by the first k elements, which lie in {1, 2, …, n}. Moreover, the symmetry and “non-meeting” condition readily imply that n ∉ E. Thus, \frac{1}{2}E := E ∩ \{1, …, n\} actually belongs to \{1, 2, …, n − 1\} and #(\frac{1}{2}E) = k. Notice that E (and hence \frac{1}{2}E) arises as a disjoint union of its i-part and j-part. So, the problem is to count the admissible partitions in two parts of all k-element subsets of {1, …, n−1}. To this end, one should first choose arbitrarily k numbers from {1, 2, …, n−1}, and then to choose a partition of this set into i- and j-parts. In order to compute the number of admissible partitions, we restate the problem, as in the proof of Theorem 6.3, in terms of sequences of +1 and −1: Given a kelement subset v_{1} < v_{2} < … < v_{k} of {1, …, n−1}, we assign the value +1 to all elements lying in the i-part, and −1 to all elements lying in the j-part. It is easily seen that such a sequence corresponds to \frac{1}{2}E for a suitable subset E if and only if all partial sums ∑_{s≤m} v_{s} are non-negative. It is not hard to prove (e.g. using a lattice path interpretation and the reflection principle) that the number of such sequences is equal to \binom{k}{[k/2]}. It remains to remember that the ideal I ∈ 𝔄𝔡(𝔰𝔭_{2n}), corresponding to Ī, has [(k+1)/2] generators, so that the ideals with q generators arise if k = 2q−1, 2q. □

6.8 Corollary. For I ∈ 𝔄𝔡(𝔰𝔭_{2n}), the following conditions are equivalent:

(i) I ∈ 𝔄𝔡_{mm}(𝔰𝔭_{2n}); (ii) Ī ∈ 𝔄𝔡_{mm}(𝔰𝔩_{2n}); (iii) Ī has non-meeting generators.

6.9 Example. By Theorem 6.7 and Corollary 6.8, the number of minimax ideals with one generator is equal to (n − 1)^{2}. It is easy to verify that the set of positive roots occurring in this way is Δ^{+}(𝔰𝔭_{2n}) \ (Π ∪ {α_{i} + … + α_{n} | i = 1, 2, …, n − 1}).

In [11], we considered the statistic on 𝔄𝔡 which assigns to an ideal the number of its generators. The corresponding generating functions (polynomials) turn out to be always palindromic. It is also makes sense to compute the respective generating functions for various classes of ideals. Theorems 6.3 and 6.7 give us essentially these generating functions for 𝔄𝔡_{mm}(𝔰𝔩_{n+1}) and 𝔄𝔡_{mm}(𝔰𝔭_{2n}):

F_{mm}(𝔰𝔩_{n+1}; t) = ∑_{k≥0} \binom{n}{2k}C_{k}t^{k} ,

F_{mm}(𝔰𝔭_{2n}; t) = ∑_{k≥0}(\binom{2k − 1}{k − 1}\binom{n − 1}{2k − 1} + \binom{2k}{k}\binom{n − 1}{2k})t^{k} .

It would be interesting to compute the polynomials F_{mm}(𝔤; t) for all simple Lie algebras. It is likely that these polynomials coincide for 𝔰𝔭_{2n} and 𝔰𝔬_{2n+1}, but I have no suggestion for 𝔰𝔬_{2n}.

7. Concluding remarks

Here we state several questions/problems related to minimax elements.

1. Is there a uniform expression for #(Ŵ_{mm}) for all simple Lie algebras?

2. Consider the set ⋃_{w∈Ŵ_{mm}} w^{−1}∗ \overline{𝒜} ⊂ V. It is just the union of the closures of all dominant

regions consisting of a single alcove. Is there a reasonable description of this set? Note that it is not convex in general.

3. Describe combinatorial properties of the polytope D_{mm} defined in Proposition 4.2. Compute the Ehrhart quasi-polynomial for D_{mm}.

4. Find a uniform proof of Theorem 5.5. It is worth noting that the similar statement can be proved a priori for the simplices D_{min} and D_{max}. Unfortunately, this does not immediately imply the validity of Theorem 5.5 for D_{mm} = D_{min} ∩ D_{max}.

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[17] ���� �������� ���� ������. ������� �� ������� �� � �������������� �������. ������� ������� 1988 (Russian). English translation: A.L. Onishchik and E.B. Vinberg. “Lie groups and algebraic groups”, Berlin: Springer, 1990.

Independent University of Moscow, Bol’shoi Vlasevskii per. 11, 121002 Moscow, Russia

E-mail address: panyush@mccme.ru