Séminaire Lotharingien de Combinatoire 59 (2008), Article B59c

NUMBER OF “udu”S OF A DYCK PATH AND ad-NILPOTENT IDEALS OF PARABOLIC SUBALGEBRAS OF sl_{ℓ+1}(ℂ)

CÉLINE RIGHI

Abstract. For an ad-nilpotent ideal 𝔦 of a Borel subalgebra of sl_{ℓ+1}(ℂ), we denote by I_{𝔦} the maximal subset I of the set of simple roots such that 𝔦 is an ad-nilpotent ideal of the standard parabolic subalgebra 𝔭_{I}. We use the bijection of Andrews, Krattenthaler, Orsina and Papi [Trans. Amer. Math. Soc. 354 (2002), 3835– 3853] between the set of ad-nilpotent ideals of a Borel subalgebra in sl_{ℓ+1}(ℂ) and the set of Dyck paths of length 2ℓ + 2, to exhibit a bijection between ad-nilpotent ideals 𝔦 of the Borel subalgebra such that ♯I_{𝔦} = r and the Dyck paths of length 2ℓ + 2 having r occurrences of “udu”. We obtain also a duality between antichains of cardinality p and ℓ − p in the set of positive roots.

1. Introduction

Let M_{ℓ+1}(ℂ) be the set of (ℓ+1)-by-(ℓ+1) matrices with coefficients in ℂ, and 𝔤 be the simple Lie algebra sl_{ℓ+1}(ℂ) consisting of elements of M_{ℓ+1}(ℂ) whose trace is equal to zero. Let 𝔥 be the maximal toral subalgebra of 𝔤 consisting of trace zero diagonal matrices. Let (E_{i,j}) be the canonical basis of M_{ℓ+1}(ℂ) and (E_{i,j}^{∗}) be its dual basis. For 1 ⩽ i ⩽ ℓ + 1, set ϵ_{i} = E_{i,i}^{∗}. Then Δ = {ϵ_{i} − ϵ_{j}; 1 ⩽ i, j ⩽ ℓ + 1, i ≠ j} is the root system associated to (𝔤, 𝔥), and Δ^{+} = {ϵ_{i} − ϵ_{j}; 1 ⩽ i < j ⩽ ℓ + 1} is a system of positive roots. Denote by α_{i} = ϵ_{i} − ϵ_{i+1}, for i = 1, …, ℓ. Then Π = {α_{1}, …, α_{ℓ}} is the corresponding set of simple roots. For each α ∈ Δ, let 𝔤_{α} = {x ∈ 𝔤; [h, x] = α(h)x for all h ∈ 𝔥} be the root space of 𝔤 relative to α.

For I ⊂ Π, set Δ_{I} = ℤI ∩ Δ. We fix the corresponding standard parabolic subalgebra,

𝔭_{I} = 𝔥 ⊕ ( ⨁_{α∈Δ_{I}∪Δ^{+}} 𝔤_{α}) .

Note that 𝔭_{∅} is a Borel subalgebra 𝔟 associated to the choice of Δ^{+}.

An ideal 𝔦 of 𝔭_{I} is ad-nilpotent if and only if for all x ∈ 𝔦, ad_{𝔭_{I}}x is nilpotent. Since any ideal of 𝔭_{I} is 𝔥-stable, we can deduce easily that an ideal is ad-nilpotent if and only if it is nilpotent. Moreover, we have 𝔦 = ⨁_{α∈Φ} 𝔤_{α}, for some subset Φ ⊂ Δ^{+} \ Δ_{I}.

A Dyck path of length 2n can be defined as a word of 2n letters u or d, having the same number of u and d, and such that there is always more u’s than d’s to the left of a letter.

Andrews, Krattenthaler, Orsina and Papi established in [AKOP] a bijection between the set of ad-nilpotent ideals of the Borel subalgebra 𝔭_{∅} and the set of Dyck paths of length 2ℓ+2 which allows them to enumerate ad-nilpotent ideals of a fixed class of nilpotence. The purpose of this paper is to explain some applications of this correspondence for the ad-nilpotent ideals of parabolic subalgebras.

More precisely, let 𝔦 be an ad-nilpotent ideal of the Borel subalgebra 𝔭_{∅}. Denote by I_{𝔦} the maximal subset I ⊂ Π such that 𝔦 is an adnilpotent ideal of 𝔭_{I}. The main result we prove here is the following theorem.

Theorem 1. There is a bijection between the ad-nilpotent ideals 𝔦 of 𝔟 such that ♯I_{𝔦} = r and the Dyck paths of length 2ℓ + 2 having r occurrences of “udu”.

We can deduce a formula for the desired number of ideals since the number of Dyck paths having r occurrences of “udu” have been calculated in [Sun].

This paper is organized as follows: we first recall the natural bijection between ℓ-partitions and Dyck paths of length 2ℓ + 2, as in [Pa]. In Section 3, we recall the iterative construction of the bijection of [AKOP]. Then, in Section 4, we explain how to calculate the number of occurrences of “udu” of a Dyck path obtained by the previous construction. In Section 5, we recall some facts of [R] and [CP] on ad-nilpotent ideals and we prove Theorem 1. Finally, in Section 6, we establish a duality between ad-nilpotent ideals of 𝔭_{∅}. Such a duality has already been constructed by Panyushev in [Pa], however, it is not the same as the one we have here.

Acknowledgment. This work was realized while I was visiting the Istituto Guido Castelnuvo di Matematica (Roma). I would like to thank the European program Liegrits for offering me the possibility to go there and the institute for its hospitality.

2. Partitions and Dyck paths

In this section, we shall see how to generate a Dyck path from a partition.

NUMBER OF “udu”S OF A DYCK PATH AND ad-NILPOTENT IDEALS 3

Recall that a partition is an ℓ-tuple λ = (λ_{1}, λ_{2}, …, λ_{ℓ}) ∈ ℕ^{ℓ} such that λ_{1} ⩾ λ_{2} ⩾ ⋯ ⩾ λ_{ℓ}. A partition will be called an ℓ-partition if λ_{i} ⩽ i for i = 1, …, l.

Partitions are usually represented by their Ferrers diagrams. Let T_{ℓ} be the Ferrers diagram of the ℓ-partition (ℓ, ℓ − 1, …, 1). Then the Ferrers diagram F of any ℓ-partition λ can be viewed as a subdiagram of T_{ℓ}. For example, for ℓ = 5, the Ferrers diagram of λ = (3, 1, 1, 0, 0) is the subdiagram of T_{ℓ}, whose boxes are denoted by some ⋆:

\overbrace{⋆ ⋆ ⋆ ; ⋆ ; ⋆}^{ℓ}

Let λ = (λ_{1}, …, λ_{ℓ}) be an ℓ-partition and let F be its Ferrers diagram. We draw a dotted horizontal line from the top of the line x + y = ℓ + 1 to F and a dotted vertical line from F to the bottom of the line x + y = ℓ + 1. For example, when λ = (5, 3, 1, 1, 1, 0, 0), we have:

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[figure: x + y = ℓ + 1]
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Figure 1

If we rotate the figure clockwise by 45 degrees, we can easily see that we obtain a Dyck path of length 2ℓ + 2 called P (λ) as in [Pa]. This construction defines clearly a bijection P : λ ↦ P (λ) between ℓ-partitions and Dyck paths of length 2ℓ + 2. In the above example, the Dyck path P (λ) is:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 1; 2; 3]

3. AKOP-bijection

Let λ = (λ_{1}, …, λ_{ℓ}) be an ℓ-partition whose Ferrers diagram is F. We shall draw a dotted line associated to λ. We start at the top of the line x + y = ℓ + 1. We go left until we meet F. Then, we continue downwards until we reach x + y = ℓ + 1. Then we iterate the procedure until we reach the bottom. For example, for ℓ = 13 and λ = (10, 10, 9, 6, 5, 4, 4, 3, 1, 1, 1, 1, 0):

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[figure: x + y = ℓ + 1]
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Figure 2

Let n(λ) be the number of points of the dotted line on x + y = ℓ + 1, which are not at the top or bottom. For example, we have n((0, …, 0)) = 0, and for the ℓ-partition λ of Figure 2, we have n(λ) = 3.

We shall describe the construction of this line in a more formal way. Let k = n(λ). Set i_{n} = ℓ + 1 for all n > k, i_{k} = λ_{1}, i_{k−1} = λ_{ℓ−i_{k}+2},

i_{k−2} = λ_{ℓ−i_{k−1}+2}, …, i_{1} = λ_{ℓ−i_{2}+2} and i_{p} = 0 for all p ⩽ 0. We have 0 < i_{1} < ⋯ < i_{k} < ℓ + 1. The dotted line describes the shape of an ℓ-partition

(1)  λ^{M} = (i_{k}^{ℓ−i_{k}+1}, i_{k−1}^{i_{k}−i_{k−1}}, …, i_{1}^{i_{2}−i_{1}}, 0^{i_{1}−1}).

Any ℓ-partition λ whose associated dotted line gives the partition λ^{M} must necessarily contain the cells

(1, i_{k}), (ℓ − i_{k} + 2, i_{k−1}), (ℓ − i_{k−1} + 2, i_{k−2}), …, (ℓ − i_{2} + 2, i_{1}).

The “minimal” ℓ-partition in the sense of inclusion of diagrams that contains these cells is

(2)  λ^{m} = (i_{k}, i_{k−1}^{ℓ−i_{k}+1}, i_{k−2}^{i_{k}−i_{k−1}}, …, i_{1}^{i_{3}−i_{2}}, 0^{i_{2}−2}).

For example, take ℓ = 13 and λ = (10, 10, 9, 6, 5, 4, 4, 3, 1, 1, 1, 1, 0), as above, we have n(λ) = k = 3, i_{3} = 10, i_{2} = 5, i_{1} = 1. The three distinguished cells above are

(1, 10), (5, 5), (10, 1).

So we have

\begin{matrix}λ^{M} = (10, 10, 10, 10, 5, 5, 5, 5, 5, 1, 1, 1, 1), and \\ λ^{m} = (10, 5, 5, 5, 5, 1, 1, 1, 1, 1, 0, 0, 0). \end{matrix}

These partitions are illustrated in the figure below, where the distinguished cells are marked with ×, and λ^{M} is the partition corresponding to the dotted line outside λ, while λ^{m} is the one which corresponds to the dotted line inside λ.

[figure: ×]
[figure: x + y = ℓ + 1]
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[figure: ×]
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[figure: ×]
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Observe that the difference λ^{M} \λ^{m} is a disjoint union of k rectangles, denoted by R_{k}, …, R_{1} from the top to the bottom. More precisely,

R_{j} = {(s, t); ℓ − i_{p+1} + 2 < s < ℓ − i_{p} + 2 and i_{p−1} < t ⩽ i_{p}}.

Inside each rectangle R_{j}, the shape of λ could be described by a word M_{j}, whose letters are d and l, where d indicates a down step and l indicates a left step.

Let h_{j} be the number of d’s in M_{j}, which is at most the height of R_{j} and let l_{j} be the number of l’s in M_{j}, which is the length of R_{j}. Then we have

\begin{matrix}h_{j} = i_{j+1} − i_{j} − 1 if j ≠ 1, and h_{j} ⩽ i_{j+1} − i_{j} − 1 if j = 1, \\ l_{j} = i_{j} − i_{j−1}, \end{matrix}

so h_{j} ⩽ l_{j+1} − 1 and the equality holds if j ≠ 1. Furthermore the shape of M_{j} is l^{a_{j,0}}dl^{a_{j,1}}d … dl^{a_{j,h_{j}}}, where a_{j,i} ∈ ℕ, 0 ⩽ i ⩽ h_{j}. We then have that

(3)  l_{j} = ∑_{i=0}^{h_{j}} a_{j,i}.

In the above example, we have M_{3} = dldl^{3}dl, M_{2} = lddldl^{2}d and M_{1} = ddl.

We shall now generate a Dyck path step by step from the M_{j}. We call a peak of a Dyck path, an occurrence of ud in the corresponding Dyck word.

First, let D_{k+1} be the Dyck path of length 2(ℓ + 1 − i_{k}) containing ℓ + 1 − i_{k} peaks. Next, we have M_{k} = l^{a_{k,0}}dl^{a_{k,1}}d … dl^{a_{k,h_{k}}}. We insert a_{k,0} peaks on the first peak of the already existing Dyck path D_{k+1}, then a_{k,1} peaks on the second peak, and so on. We call D_{k} the new Dyck path obtained. Observed that the highest peaks of D_{k} are exactly those newly inserted, so there are exactly l_{k}. Since h_{k−1} ⩽ l_{k} − 1, the procedure can then be iterated by inserting peaks only on highest peaks. Each intermediate Dyck path obtained after using the word M_{j} is denoted by D_{j}. At the end, we obtain a Dyck path D_{λ} of length 2ℓ + 2.

For example, let us consider ℓ = 7 and λ = (5, 3, 1, 1, 1, 0, 0):

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[figure: x + y = ℓ + 1]
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Figure 3

We have n(λ) = k = 2, i_{2} = 5 and i_{1} = 1. Then D_{3} is the following Dyck path:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0]

We have M_{2} = l^{2}dl^{2}d, so we first insert 2 peaks on the first peak of D_{3}, then again two peaks on the second one. We obtain D_{2}:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0]

Finally, M_{1} = dl so we insert a_{1,0} = 0 peak on the first highest peak of D_{2} and a_{1,1} = 1 peak on the second highest peak. We obtain D_{λ}:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 1; 2; 3]

By [AKOP], we have the following proposition.

Proposition 3.1. The map D : λ ↦ D_{λ} defines a bijection between the set of ℓ-partitions and the set of Dyck paths of length 2ℓ + 2.

4. Dyck path and number of occurrences of “udu”

Let λ = (λ_{1}, …, λ_{ℓ}) be an ℓ-partition such that n(λ) = k. Let D_{λ} be the Dyck path obtained from λ as described in Section 3. We shall see how to count the number of occurrences of “udu” contained in D_{λ}.

A peak could be followed by a “u”, a “d” or nothing in the Dyck word. If it is followed by a “u”, we call it a u-peak. Each u-peak will give an “udu” and vice versa.

Let 1 ⩽ j ⩽ k + 1. Let u_{j} be the number of u-peaks in the Dyck path D_{j}. For example, D_{k+1} contains ℓ − λ_{1} + 1 = ℓ − i_{k} + 1 peaks, so it is easy to see that u_{k+1} = ℓ − λ_{1}.

To construct D_{j−1} from D_{j}, we add some peaks on the highest peaks of D_{j}. Then, one must understand how the insertion of p peaks on a highest peak modifies the number of occurrences of “udu”. Consider a peak P of maximal height on a Dyck path. If we add p peaks, the part of the Dyck word which corresponds to P (which was ud) becomes uudud … udd (with p ud), so we obtain p − 1 occurrences of udu. If P is a u-peak, then we also “destroy” the udu given by P. So at the end, we only add p − 2 occurrences of udu. For example, let us consider the following Dyck path which contains 2 occurrences of udu:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0]

Figure 4

If we add 2 peaks on the first highest peak, we add 2−2 = 0 occurrences of udu. So we obtain the following Dyck path with still 2 occurrences of udu:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0]

If P is not a u-peak, then we do not “destroy” a udu, so we indeed add p − 1 occurrences of “udu”. For example, if we add 2 peaks on the second highest peak of Figure 4, we add 2 − 1 = 1 occurrence of udu, so we obtain 3 occurrences of udu at the end:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0]

Set a_{k+1,0} = ℓ − i_{k} + 1, M_{k+1} = l^{a_{k+1,0}}, and h_{k+1} = 0. We have seen that each word M_{j} is in the form l^{a_{j,0}}dl^{a_{j,1}}d … dl^{a_{j,h_{j}}}. Let

A_{j} = {(j, t); t ∈ {0, …, h_{j}}; a_{j,t} ≠ 0},

A = ⋃_{j=1}^{k} A_{j}.

Recall from the construction that the number of highest peaks in D_{j} is

(4)  ∑_{t=0}^{h_{j}} a_{j,i} = l_{j}.

Observe that a highest peak is a u-peak if it is not the last one of a consecutive group of highest peaks. Hence, the q-th peak of D_{j} is not a u-peak if and only if there exists r ∈ {0, …, h_{j}} such that q = ∑_{s=0}^{r} a_{j,s}. Set

L_{p} = {(p, t); there exists 0 ⩽ r ⩽ h_{p+1}; t + 1 = ∑_{q=0}^{r} a_{p+1,q}} ,

U_{p} = A_{p} \ L_{p}, L = ⋃_{p=1}^{k} L_{p}, U = ⋃_{p=1}^{k} U_{p}.

Thus L_{j} corresponds exactly to the set of highest peaks in D_{j} which are not u-peaks and where we insert new peaks. It follows that

u_{j−1} = u_{j} + ∑_{(j−1,t)∈U_{j−1}}(a_{j−1,t} − 2) + ∑_{(j−1,t)∈L_{j−1}}(a_{j−1,t} − 1).

At the end of the construction, the number of occurrences of “udu” in D_{λ} is u_{1}. By induction, we have

u_{1} = ℓ − λ_{1} + ∑_{(j,t)∈U}(a_{j,t} − 2) + ∑_{(j,t)∈L}(a_{j,t} − 1).

Since ∑_{(j,t)∈A} a_{j,t} = λ_{1}, we obtain the following proposition.

Proposition 4.1. Let λ be an ℓ-partition. Then, the number of occurrences of “udu” in D_{λ} is ℓ − 2♯U − ♯L.

To illustrate this, we could follow again the construction of the Dyck path which corresponds to λ = (5, 3, 1, 1, 1, 0, 0). We first have the Dyck path D_{3} in Section 3, with n − λ_{1} + 1 = 3 peaks, and u_{3} = 2. Then we use the word M_{2} = l^{2}dl^{2}d = l^{a_{2,0}}dl^{a_{2,1}}d, where a_{2,0}, a_{2,1} ∈ L_{2}, so we add a_{2,0} − 2 + a_{2,1} − 2 = 0 peak. So u_{2} = 2. Then we use the word M_{1} = dl = l^{a_{1,0}}dl^{a_{1,1}}, where a_{1,1} ∈ U_{1}, so we add a_{1,1} − 1 = 0 peak. Hence, u_{1} = 2.

5. Ad-nilpotent ideals of a parabolic subalgebra and Dyck paths

Let I ⊂ Π and 𝔦 be an ad-nilpotent ideal of 𝔭_{I}. We set

Φ_{𝔦} = {α ∈ Δ^{+} \ Δ_{I}; 𝔤_{α} ⊆ 𝔦}.

Then 𝔦 = ⨁_{α∈Φ_{𝔦}} 𝔤_{α} and if α ∈ Φ_{𝔦}, β ∈ Δ^{+} ∪ Δ_{I} are such that α + β ∈ Δ^{+}, then α + β ∈ Φ_{𝔦}.

Conversely, set

F_{I} = {Φ ⊂ Δ^{+}\Δ_{I}; if α ∈ Φ, β ∈ Δ^{+}∪Δ_{I}, α+β ∈ Δ^{+}, then α+β ∈ Φ}.

Then for Φ ∈ F_{I}, 𝔦_{Φ} = ⨁_{α∈Φ} 𝔤_{α} is an ad-nilpotent ideal of 𝔭_{I}. We obtain therefore a bijection

{ad-nilpotent ideals of 𝔭_{I}} → F_{I}, 𝔦 ↦ Φ_{𝔦}.

Recall the following partial order on Δ^{+}: α < β if β − α is a sum of positive roots. Then it is easy to see that Φ ∈ F_{∅} if and only if for all α ∈ Φ, β ∈ Δ^{+}, such that α < β, then β ∈ Φ.

Let Φ ∈ F_{∅}. Set

Φ_{min} = {β ∈ Φ; β − α ∉ Φ, for all α ∈ Δ^{+}}.

Then, Φ_{min} is an antichain of Δ^{+} with respect to the above partial order. Conversely, if we consider an antichain Γ, then, the set of roots which are bigger than any one of the elements of Γ is an element of F_{∅}.

As in [CP], we display the positive roots Δ^{+} in the Ferrers diagram T_{ℓ} of (ℓ, ℓ − 1, …, 1) as follows: we assign to each box in the i-th row and the j-th column, labelled (i, j) in T_{ℓ}, a positive root t_{i,j} = α_{i} + ⋯ + α_{ℓ−j+1}, 1 ⩽ i, j ⩽ ℓ.

For example, for ℓ = 5, we have

\overbrace{t_{1,1} t_{1,2} t_{1,3} t_{1,4} t_{1,5} ; t_{2,1} t_{2,2} t_{2,3} t_{2,4} ; t_{3,1} t_{3,2} t_{3,3} ; t_{4,1} t_{4,2} ; t_{5,1}}^{ℓ}

Observe that given two positive roots α and β, α is bigger than or equal to β if the box corresponding to α is in the quadrant north-west of the box corresponding to β. It follows easily that the map which sends an element Φ ∈ F_{∅} to the subdiagram of T_{ℓ} consisting of the boxes corresponding to the roots of Φ defines a bijection between F_{∅} and the set of northwest flushed subdiagrams of T_{ℓ}, i.e with the set of subdiagrams which contain the quadrant north-west of their boxes. Hence, by Section 2, we obtain a bijection σ from F_{∅} to the set of ℓ-partitions.

By Proposition 3.1, D ∘ σ is a bijection from F_{∅} to the set of Dyck paths of length 2ℓ + 2.

For Φ ∈ F_{∅}, set

I_{Φ} = {α ∈ Π; Φ ∈ F_{{α}}}.

It is the maximal element of {I ⊂ Π; Φ ∈ F_{I}} with respect to inclusion order. We shall see how to link the number of occurrences of “udu” of the Dyck path (D ∘ σ)(Φ) and the cardinality of I_{Φ}.

Set α_{i,j} = α_{i} + ⋯ + α_{j}, for all 1 ⩽ i ⩽ j ⩽ ℓ. We have easily the following lemma.

Lemma 5.1. Let I ⊂ Π. An element Φ ∈ F_{∅} is an element of F_{I} if and only if for all α_{i,j} ∈ Φ_{min}, we have α_{i}, α_{j} ∉ I.

It follows from Lemma 5.1 that

I_{Φ} = Π \ {α_{i} ∈ Π; there exists α_{i,j} or α_{k,i} ∈ Φ_{min}}.

The problem is not to count the same root twice. For example, in A_{7}, for Φ_{min} = {α_{1,3}, α_{2,5}, α_{5,7}}, we have Π \ I_{Φ} = {α_{1}, α_{2}, α_{3}, α_{5}, α_{7}} but we find α_{5} in the beginning or in the end of the support of two roots in Φ_{min}. So if we set

L = {α_{i,j} ∈ Φ_{min}; there exists a root of shape α_{p,i} ∈ Φ_{min}},

U = Φ_{min} \ L,

we obtain that

(5)  ♯I_{Φ} = l − 2♯U − ♯L.

Let λ = σ(Φ), F its Ferrers diagram and D_{λ} = D(λ) be the Dyck path which corresponds to λ via the AKOP-bijection. Let α_{i,j} ∈ Φ_{min}. Then the cell (i, ℓ + 1 − j) = (i, λ_{i}) of α_{i,j} in F is a south-east corner of the diagram and two cases are possible: there exists a rectangle R_{p} such that (i, λ_{i}) ∈ R_{p} or (i, λ_{i}) is not in any rectangle. If the latter case occurs, then (i, ℓ + 1 − j) is above a rectangle R_{p}. For example, if λ = (5, 3, 1, 1, 1, 0, 0), we have that α_{2,5}, α_{5,7} are in the first case and α_{1,3} is in the second case.

[figure: α_{1,3}]
[figure: α_{2,5}; x + y = ℓ + 1]
[figure: R_{2}]
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[figure: α_{5,7}]
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[figure: R_{1}]
[figure]

If α_{i,j} is in the rectangle R_{p}, then the cell (i, λ_{i}) = (i, ℓ−j +1) which corresponds to α_{i,j} in F satisfies

(6)  ℓ − i_{p+1} + 2 < i < ℓ − i_{p} + 2,
(7)  i_{p−1} < λ_{i} ⩽ i_{p},

and so we have

(8)  ℓ − i_{p} + 1 ⩽ j < ℓ − i_{p−1} + 1.

If α_{i,j} is above the rectangle R_{p}, then the cell (i, ℓ − j + 1) which corresponds to α_{i,j} in F satisfies

(9)  (i, ℓ − j + 1) = (ℓ − i_{p+1} + 2, i_{p}).

Define the map r from Φ_{min} to {1, …, k} which associates to α_{i,j} the integer r(α_{i,j}) = p such that α_{i,j} is in or immediately above the rectangle R_{p}.

Let α_{i,j} ∈ Φ_{min} and p = r(α_{i,j}). Since the cell (i, ℓ − j + 1) which contains α_{i,j} in T_{ℓ} is a south-east corner, there is a horizontal line under this cell. If c = (i, ℓ − j + 1) is in the rectangle R_{p}, then it is at the row q = i − (ℓ − i_{p+1} + 2) of R_{p} and the line under c correspond to the part l^{a_{p,q}} in M_{p}. Furthermore (p, q) ∈ A_{p}.

If c is immediately above the rectangle R_{p}, then the line under c corresponds to l^{a_{p,0}} in M_{p} and (p, 0) ∈ A_{p}. Since in this case, by (9) we have (i, ℓ−j +1) = (ℓ−i_{p+1}+2, i_{p}), we obtain that i−(ℓ−i_{p+1}+2) = 0. We can define in any case the map s from Φ_{min} to ℕ by

(10)  s(α_{i,j}) = i − (ℓ − i_{r(α_{i,j})+1} + 2).

Furthermore, in both cases, the line under the cell which contains α_{i,j} is the part l^{a_{r(α_{i,j}),s(α_{i,j})}} in M_{r(α_{i,j})} and (r(α_{i,j}), s(α_{i,j})) ∈ A_{r(α_{i,j})}.

Conversely, let (p, q) ∈ A_{p}. Then, there is a horizontal line under the row i = q − ℓ − i_{p+1} + 2 of F which is under a south-east corner of F. This south-east corner is a cell (i, λ_{i}) which corresponds to a root α_{i,j}, where ℓ − j + 1 = λ_{i}. So we have a bijection

\begin{aligned}Ψ & : & Φ_{min} & → & A \\ & & α_{i,j} & ↦ & (r(α_{i,j}), s(α_{i,j})). \end{aligned}

Lemma 5.2. We have Ψ(U) = U and Ψ(L) = L.

Proof. Since L = Φ_{min} \ U and L = A \ U, it suffices to prove that Ψ(L) = L.

Let α_{i,j} ∈ L. Set p = r(α_{i,j}), q = s(α_{i,j}) and let c = (i, λ_{i}) be the cell which corresponds to α_{i,j} in F.

First assume that i = j. Then, we have c = (i, ℓ − i + 1). If c ∈ R_{p}, then by (6) and (8), we have

i = ℓ − i_{p} + 1,

so by (10), we have that q = i_{p+1} − i_{p} − 1 so by (3), a_{p,q} ∈ L_{p}.

If c is above R_{p}, then by (9), we have c = (i, ℓ−i+1) = (ℓ−i_{p+1}+2, i_{p}), so q = 0 and i_{p+1} − i_{p} = 1, hence by (3) we also have a_{p,q} ∈ L_{p}.

Now assume that i ≠ j and there exists a root of shape α_{m,i} ∈ Φ_{min}. Set t = r(α_{m,i}). Let (m, λ_{m}) = (m, ℓ − i + 1) be the cell which corresponds to α_{m,i} in λ. If c ∈ R_{p}, then by (6), we have

i_{p} ⩽ λ_{m} ⩽ i_{p+1} − 2.

So either (m, λ_{m}) ∈ R_{p+1} or (m, λ_{m}) = (ℓ − i_{p+1} + 2, i_{p}).

If (m, λ_{m}) ∈ R_{p+1}, then between the columns i_{p+1} and λ_{m} = ℓ − i + 1, we have i_{p+1} − (ℓ − i + 1) columns, so there exists n such that ∑_{u=0}^{n} a_{p+1,u} = i_{p+1} − (ℓ − i + 1). Furthermore, by (10), we have q = i − (ℓ − i_{p+1} + 2), hence a_{p,q} ∈ L_{p}.

If (m, λ_{m}) = (ℓ − i_{p+1} + 2, i_{p}), then i = ℓ − i_{p} + 1 and by (10), we have that

q = (ℓ − i_{p} + 1) − (ℓ − i_{p+1} + 2) = i_{p+1} − i_{p} − 1.

Hence, by (3), we have a_{p,q} ∈ L_{p}.

Conversely, let a_{p,q} ∈ L_{p}, then there exists 0 ⩽ t ⩽ h_{p+1} such that q +1 = ∑_{f=0}^{t} a_{p+1,f}. There also exists α_{i,j} ∈ Φ_{min} such that r(α_{i,j}) = p and s(α_{i,j}) = q. By (10), we have that

q = i − (ℓ − i_{p+1} + 2).

Observe that for all 0 ⩽ j ⩽ h_{p+1}, there exists a south-east corner (n_{j}, λ_{n_{j}}) in or above the rectangle R_{p+1} such that

λ_{n_{j}} = i_{p+1} − ∑_{f=0}^{j} a_{p+1,f}.

So there exists a south-east corner (n_{j}, λ_{n_{j}}) such that

λ_{n_{j}} = i_{p+1} − (q + 1) = ℓ − i + 1.

The element of Φ_{min} which corresponds to the cell (n_{j}, λ_{n_{j}}) is α_{n_{j},i}, so we have α_{i,j} ∈ L. □

It follows by Proposition 4.1 and Equation (5) that we have the following theorem.

Theorem 5.3. There is a bijection between the elements Φ ∈ F_{∅} such that ♯I_{Φ} = r and the Dyck paths of length 2ℓ + 2 having r occurrences of “udu”.

Since the number of Dyck paths having a fixed number of occurrences of udu is calculated in Theorem 2.1 of [Sun], we have the following corollary.

Corollary 5.4. The number of elements of Φ ∈ F_{∅} such that ♯I_{Φ} = r is

( \begin{matrix}ℓ \\ r \end{matrix} ) ∑_{k=0}^{[ℓ−r/2]} ( \begin{matrix}ℓ − r \\ 2k \end{matrix} ) C_{k}

where C_{k} denotes the k-th Catalan number.

Example 5.5. Let N_{r}^{ℓ} be the number of elements Φ ∈ F_{∅} such that ♯I_{Φ} = r. We have by Corollary 5.4:

r N_{r}^{1} N_{r}^{2} N_{r}^{3} N_{r}^{4} N_{r}^{5}
0 1 2 4 9 21
1 1 2 6 16 45
2  1 3 12 40
3  1 4 20
4  1 5
5  1

6. Duality

We shall construct a duality between the elements of F_{∅} such that ♯Φ_{min} = p and those such that ♯Φ_{min} = ℓ − p.

Proposition 6.1. Let Φ ∈ F_{∅}. Let N be the number of peaks in (D ∘ σ)(Φ), then we have

♯Φ_{min} = ℓ − (N − 1).

Proof. Let λ = σ(Φ) be the corresponding ℓ-partition. Recall that the construction of D(λ) is iterative. At each step, when we add a_{p,q} peaks to a highest peak, for (p, q) ∈ A_{p}, we also “destroy” the initial highest peak. So, we add only a_{p,q} − 1 peaks. At the end of the construction we have

ℓ − λ_{1} + 1 + ∑_{p=1}^{k} ∑_{(p,q)∈A_{p}}(a_{p,q} − 1)

peaks. Since ∑_{p=1}^{k} ∑_{(p,q)∈A_{p}} a_{p,q} = ∑_{(p,q)∈A} a_{p,q} = λ_{1} and A is in bijection with Φ_{min} by Section 5, we obtain the result. □

Proposition 6.2. Let Φ ∈ F_{∅} and p be the number of peaks in (P ∘ σ)(Φ), then we have

♯Φ_{min} = p − 1.

Proof. The result is clear by the construction of (P ∘ σ)(Φ) defined in Section 2. □

Theorem 6.3. The map σ^{−1} ∘ P^{−1} ∘ D ∘ σ induces a bijection from F_{∅} to F_{∅} which sends Φ ∈ F_{∅} such that ♯Φ_{min} = p to Ψ ∈ F_{∅} such that ♯Ψ_{min} = ℓ − p.

For example, in sl_{4}(ℂ), the element Φ = {θ} ∈ F_{∅} corresponds to the partition λ = (1, 0, 0), and the Dyck path D_{λ} is:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; 0]

Then, P^{−1}(D_{λ}) = (3, 2, 0) which is the partition which corresponds to Ψ such that Ψ_{min} = {α_{1}, α_{2}}.

Remark 6.4. It was proved in [Pa] that when 𝔤 is a simple Lie algebra of type A or C, the number of elements Φ ∈ F_{∅} such that ♯Φ_{min} = p is the same as the number of elements Φ ∈ F_{∅} such that ♯Φ_{min} = ℓ − p. But the duality of [Pa] is not the same as the one defined above. For example, in sl_{4}(ℂ), if we consider Φ = {θ} like above, the dual ideal defined by [Pa] is Ψ where Ψ_{min} = {α_{1} + α_{2}, α_{3}}.

References

[AKOP] G. E. Andrews, C. Krattenthaler, L. Orsina and P. Papi. Adnilpotent 𝔟-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration. Trans. Amer. Math. Soc. 354 (2002), 3835–3853.

[CP] P. Cellini, P. Papi. Ad-nilpotent ideals of a Borel subalgebra. J. Algebra 225 (2000), 130–140.

[Pa] D.I. Panyushev. Ad-nilpotent ideals of a Borel subalgebra: generators and duality. J. Algebra 274 (2004), 822–846.

[R] C. Righi. Ad-nilpotent ideals of a parabolic subalgebra, J. Algebra 319 (2008), 1555–1584.

[Sun] Y. Sun. The statistic ”number of udu’s” in Dyck paths. Discrete Math. 287 (2004), 177–186.

UMR 6086 CNRS, Département de Mathématiques, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France

  E-mail address: celine.righi@math.univ-poitiers.fr