Séminaire Lotharingien de Combinatoire 75 (2016), Article B75f

TORUS FIXED POINTS IN SCHUBERT VARIETIES AND NORMALIZED MEDIAN GENOCCHI NUMBERS

XIN FANG AND GHISLAIN FOURIER

Abstract. We give a new proof for the fact that the number of torus fixed points for the degenerate flag variety is equal to the normalized median Genocchi number, using the identification with a certain Schubert variety. We further study the torus fixed points for the symplectic degenerate flag variety and develop a combinatorial model, symplectic Dellac configurations, to parametrize them. The number of these symplectic fixed points is conjectured to be a median Euler number.

Introduction

We consider the Schubert variety X_{τ_{n}} associated to the Weyl group element

τ_{n} := (s_{n}s_{n+1} ⋯ s_{2n−2}) ⋯ (s_{k}s_{k+1} ⋯ s_{2k−2}) ⋯ (s_{3}s_{4})s_{2} ∈ 𝔖_{2n}

in the partial flag variety SL_{2n}/P, where P is the standard parabolic subalgebra associated to the simple roots {α_{1}, α_{3}, …, α_{2n−1}}. Then there is a natural action of a (2n − 1)-dimensional torus T_{2n−1}, and we are mainly interested in the fixed points X_{τ_{n}}^{T_{2n−1}} of this torus action. It is well known that the fixed points are parametrized by Weyl groups elements which are less than or equal to τ_{n} in the Bruhat order (modulo the stabilizer of the parabolic subalgebra; in this case, the subgroup generated by s_{1}, s_{3}, …, s_{2n−1}). Our first result is the following.

Theorem A. There is an explicit bijection b from Dellac configurations DC_{n} (Definition 1) of 2n columns and n rows to X_{τ_{n}}^{T_{2n−1}}. Hence the number of torus fixed points is equal to the normalized median Genocchi number h_{n} (see Section 1 for definition).

Here is an example of the Dellac configuration corresponding to a fixed point for n = 3:

• •
• •  ↦ σ = 124536
•  •

We also consider Schubert varieties of the symplectic flag variety, e.g., the Schubert variety X_{\overline{τ}_{2n}}^{sp} corresponding to the element (of the symplectic Weyl group)

\overline{τ}_{2n} := (r_{2n} ⋯ r_{n+1}) ⋯ (r_{2n}r_{2n−1}r_{2n−2})(r_{2n}r_{2n−1})r_{2n}(r_{n} ⋯ r_{2n−2}) ⋯ (r_{4}r_{5}r_{6})(r_{3}r_{4})r_{2}

in the symplectic partial flag variety. In this case, there is a natural action of T_{2n} on the Schubert variety, and we are again interested in the fixed points of this torus action. To parametrize them similar to the non-symplectic case, we introduce symplectic Dellac configurations (Definition 2). These are Dellac configurations with 4n columns and 2n rows, which are invariant under the involution mapping the i-th row to the (2n−i+1)-st row. Here is our second result.

Theorem B. The torus fixed points in X_{\overline{τ}_{2n}}^{sp} are parametrized by the symplectic Dellac configurations SpDC_{2n}.

We conjecture that the number of symplectic Dellac configurations is equal to a normalized median Euler number (cf. [K97]).

We should explain here why we are interested in these particular Schubert varieties. E. Feigin [Fei11] defined the degenerate flag variety

Fl_{n}^{a} := {(U_{1}, …, U_{n−1}) ∈ ∏_{i=1}^{n−1} Gr_{i}(ℂ^{n}) | pr_{i+1} U_{i} ⊂ U_{i+1}},

where pr_{i} is the endomorphism of ℂ^{n} setting the i-th coordinate to be zero. This is in fact a flat degeneration of the classical flag variety Fl_{n}. Moreover it was shown in [CFR12, CLL15] that there is an action of T_{2n−1} on Fl_{n}^{a}. The symplectic degenerate flag variety (Fl_{2n}^{a})^{sp} has been defined in [FFiL14] in a similar way.

The degenerate flag variety is one of the main objects in the framework of PBW filtrations and degenerations on universal enveloping algebras of simple Lie algebras (see [FFoL11a, FFoL11b, FFoL13, FFR15, Hag14, Fou14, Fou15, CFR12] for various aspects). Here, one obtains degenerate flag varieties Fl^{a}(λ) as highest weight orbits of PBW degenerate modules. In [Fei11, FFiL14], it has been shown that these highest weight orbits have an interpretation as a variety of certain flags.

Recently, it was shown in [CL15] that these degenerate flag varieties are in fact our particular Schubert varieties.

Theorem (Cerulli Irelli–Lanini). (1) In the 𝔰𝔩_{n}-case, the degenerate flag variety Fl_{n}^{a} is isomorphic to the Schubert variety X_{τ_{n}}. Moreover, the isomorphism ζ : Fl_{n}^{a} ⟶^{∼} X_{τ_{n}} is T_{2n−1}-equivariant.

(2) In the 𝔰𝔭_{2n}-case, the degenerate symplectic flag variety is isomorphic to X_{\overline{τ}_{2n}}^{sp}, and again the isomorphism ζ^{sp} : X_{\overline{τ}_{2n}}^{sp} ⟶^{∼} (Fl_{2n}^{a})^{sp} is torus-equivariant.

The torus fixed points of the degenerate flag variety in type A_{n} have been studied in [Fei11]. In that paper, an explicit bijection f with the set of Dellac configurations has been provided. Hence it was shown that the number of torus fixed points is equal to a normalized median Genocchi number.

Combining the theorem by Cerulli Irelli and Lanini with Theorem A, we obtain another proof of this fact, using the classical set up of Schubert varieties only. Moreover, we can show that the following diagram commutes (there, α denotes the natural identification of W_{≤τ_{n}}^{J} with X_{τ_{n}}^{T_{2n−1}})

\xymatrix{(Fl_{n}^{a})^{T_{n}} \ar[r]^{f} \ar[d]^{ζ} & DC_{n} \ar[d]^{b} \\ X_{τ_{n}}^{T_{2n−1}} & W_{≤τ_{n}}^{J} \ar[l]_{α}} .

In the symplectic case, the map f is not present, mainly because the construction of symplectic Dellac configurations has not been seen in the literature before. Nevertheless, we obtain a similar picture, namely the torus fixed points in the symplectic degenerate flag variety are parametrized by SpDC_{2n}. We should mention here that E. Feigin (via the symplectic degenerate flag variety in [FFiL14]) as well as G. Cerulli Irelli (via the quiver Grassmannian in [CFR12]) also conjectured the number of torus fixed points to be a normalized median Euler number.

This paper is organized as follows. In Section 1, we prove our first theorem for 𝔰𝔩_{n}, while, in Section 2, we consider the symplectic case. In Section 3 we relate our results to the framework of degenerate flag varieties.

Acknowledgments. The work of Xin Fang is supported by the Alexander von Humboldt Foundation. The work of Ghislain Fourier is funded by the DFG priority program 1388 ”Representation Theory”. The authors would like to thank Evgeny Feigin and Bruce Sagan for their helpful comments.

1. Symmetric groups and Median Genocchi numbers

1.1. Let W = 𝔖_{2n} be the symmetric group generated by S = {s_{1}, s_{2}, …, s_{2n−1}} where s_{i} = (i, i + 1). Let J = {s_{1}, s_{3}, …, s_{2n−1}} ⊂ S and W_{J} be the subgroup generated by J, and let W^{J} be the set of minimal representatives of right cosets of W_{J} in W. We define

τ_{n} = (s_{n}s_{n+1} ⋯ s_{2n−2}) ⋯ (s_{k}s_{k+1} ⋯ s_{2k−2}) ⋯ (s_{3}s_{4})s_{2} ∈ W.

Then, for t = 1, 2, …, 2n, we have

τ_{n}(t) = \begin{cases}k, & t = 2k − 1; \\ n + k, & t = 2k. \end{cases}  (1.1)

By construction, τ_{n} is a representative of minimal length in W/W_{J}, so τ_{n} ∈ W^{J}. We define

W_{≤τ_{n}} = {w ∈ W | w ≤ τ_{n}}, W_{≤τ_{n}}^{J} = {w ∈ W^{J} | w ≤ τ_{n}},

where ≤ is the Bruhat order.

Definition 1. A Dellac configuration C is a board of 2n columns and n rows with 2n marked cells such that:

(1) each column contains exactly one marked cell;

(2) each row contains exactly two marked cells;

(3) if the (i, j)-cell is marked, then i ≤ j ≤ n + i. Let DC_{n} denote the set of such configurations.

It is worth pointing out that the definition of a Dellac configuration given above differs from that in [Fei11] by a rotation of the board by 90^{∘}.

The cardinality h_{n} of the set DC_{n} is called the n-th normalized median Genocchi number (see [Fei11, Fei12] and the references therein). Consider the polynomials defined recursively by: H_{0}(x) = 1,

H_{n}(x) = \frac{1}{2}(x + 1)((x + 1)H_{n−1}(x + 1) − xH_{n−1}(x)).

Then it is proved in [DR94] that h_{n} = H_{n}(1).

The following theorem is originally proved by Cerulli Irelli and Lanini in [CL15] as a corollary of their main result and a result of Feigin [Fei11] (see Remark 4 for details).

Theorem 1. For any integer n ≥ 1, we have h_{n} = #W_{≤τ_{n}}^{J}.

In this section, we provide a purely combinatorial proof of the theorem, in terms of a bijection.

1.2. Rook configurations. Consider a board of n rows and columns. A rook configuration R is a filling of the cells by n marks such that each row and each column have exactly one mark. Let R_{n} denote the set of all rook configurations. There is a bijection

φ : R_{n} ⟶^{∼} 𝔖_{n}  (1.2)

sending a rook configuration R to the permutation σ_{R} satisfying σ_{R}(i) = j if and only if the cell (i, j) is marked in R, for i = 1, 2, …, n. For σ ∈ 𝔖_{n}, we denote R_{σ} := φ^{−1}(σ).

Let R be a rook configuration. The convex hull of the marked cells in R is the smallest right-aligned skew Ferrers board containing all marks in R.

From now on we consider 𝔖_{2n}: R_{τ_{n}} is a board of 2n columns and rows. A restricted rook configuration with respect to τ_{n} is a rook configuration such that all marked cells in the board are contained in the convex hull (which is called the right hull in [Sjo07]) of the marked cells in R_{τ_{n}}. Let R_{≤τ_{n}} denote the set of all restricted rook configurations with respect to τ_{n}.

Example 1. We consider an example where n = 3. Then τ_{3} = 142536, and the shadowed area is the convex hull of the marked cells in R_{τ_{3}}. We fix σ = 124536. The rook configuration of σ is (given by the dots):

R_{σ} = • ; • ; • ; • ; • ; •

R_{σ} is the restricted rook configuration with respect to τ_{3}.

It is clear that τ_{n} avoids the patterns 4231, 35142, 42513, and 351624. The following result is a special case of Theorem 4 in [Sjo07].

Theorem 2 ([Sjo07]). The restriction of φ on R_{≤τ_{n}} gives a bijection R_{≤τ_{n}} ⟶^{∼} W_{≤τ_{n}}.

1.3. From rook configurations to Dellac configurations. We define two maps m : R_{≤τ_{n}} → DC_{n}, called the melt map, and b : DC_{n} → R_{≤τ_{n}}, called the blow-up map.

Let R ∈ R_{≤τ_{n}} be a restricted rook configuration. Consider a board C_{R} of 2n columns and n rows defined by: the cell (k, l) of C_{R} is marked if and only if either the cell (2k − 1, l) or the cell (2k, l) is marked in R. Intuitively, the k-th row of C_{R} is obtained by merging the (2k − 1)-st and the 2k-th row in R.

Lemma 1. The board C_{R} is a Dellac configuration.

Proof. By the definition of a rook configuration, each row of C_{R} has exactly two marked cells, and each column of C_{R} has exactly one marked cell. Moreover, when R is restricted with respect to τ_{n}, then, by (1.1), C_{R} has the following property: if the cell (r, s) in C_{R} is marked, then r ≤ s ≤ n + r. □

By using the lemma, we obtain a well-defined melt map

m(R) := C_{R}.

Example 2. Let σ = 124536 be the permutation in Example 1. The corresponding Dellac configuration via the melt procedure is given by:

\begin{aligned}• & • \\ & & & • & • \\ & & • & & & • \end{aligned}

Let C ∈ DC_{n} be a Dellac configuration. A board R_{C} of 2n rows and columns is associated to C in the following way: the cells (i, j) and (i, k) with j < k are marked in C if and only if the cells (2i − 1, j) and (2i, k) are marked in R_{C}. Intuitively, the i-th row in C is split into two rows, where the first row gets the first marked point, while the second row gets the second.

Lemma 2. The board R_{C} is a restricted rook configuration with respect to τ_{n}.

Proof. Conditions (1) and (2) in the definition of the Dellac configuration guarantees that R_{C} is a rook configuration. The condition (3) means that R_{C} is restricted with respect to τ_{n}. □

By defining b(C) = R_{C}, the blow-up map is well-defined by Lemma 2.

Lemma 3. The following statements hold:

(1) the map b is injective with im(b) = φ^{−1}(W_{≤τ_{n}}^{J});

(2) we have m ∘ b = id.

Proof. By construction, the only thing to be proved is im(b) = φ^{−1}(W_{≤τ_{n}}^{J}). This relation indeed holds by the following description of W^{J}:

W^{J} = {σ ∈ W | σ(2k − 1) < σ(2k) for 1 ≤ k ≤ n}.  □

As an application of these maps, we provide a bijective proof of Theorem 1.

Proof of Theorem 1. By Lemma 3, the blow-up map b induces a bijection DC_{n} ⟶^{∼} W_{≤τ_{n}}^{J}. Since | DC_{n} | = h_{n}, we proved h_{n} = #W_{≤τ_{n}}^{J}. □

Remark 1. The normalized median Genocchi numbers h_{n} count a combinatorial structure in 𝔖_{2n+2} called normalized Dumont permutations. Although a posteriori there exists a bijection between the normalized Dumont permutations and W_{≤τ_{n}}^{J}, our approach is different from the one in [K97], see also [Fei11].

2. Symplectic case

2.1. Notations. Let W̃ = 𝔖_{4n} be the symmetric group acting on {1, 2, …, 4n}, and J̃ = {s_{1}, s_{3}, …, s_{4n−1}}. Let ι be the involution of W̃ defined by

ι(σ)(k) = 4n + 1 − σ(4n + 1 − k) for σ ∈ W̃ and 1 ≤ k ≤ 4n.

The Weyl group W of the symplectic group Sp_{4n} with generators {r_{1}, r_{2}, …, r_{2n}} can be embedded into W̃ via the map κ : W → W̃, r_{i} ↦ s_{i}s_{4n−i} for 1 ≤ i ≤ 2n − 1 and r_{2n} ↦ s_{2n}. The image of κ consists of the set W̃^{ι} of ι-fixed elements in W. Let J = {r_{1}, r_{3}, …, r_{2n−1}}. We define

\overline{τ}_{2n} = (r_{2n} ⋯ r_{n+1}) ⋯ (r_{2n}r_{2n−1}r_{2n−2})(r_{2n}r_{2n−1})r_{2n}(r_{n} ⋯ r_{2n−2}) ⋯ (r_{4}r_{5}r_{6})(r_{3}r_{4})r_{2} ∈ W.

It is observed in [CLL15] that κ(\overline{τ}_{2n}) = τ_{2n}.

By Corollary 8.1.9 in [GTM05] (notice the differences between the indices here and those in the reference), the restriction of κ to W_{≤\overline{τ}_{2n}} gives a bijection

α : W_{≤\overline{τ}_{2n}} ⟶^{∼} (W̃_{≤τ_{2n}})^{ι}.

By passing to right cosets, α induces a bijection α′ : W_{≤\overline{τ}_{2n}}^{J} ⟶^{∼} (W̃ J̃ ; ≤τ_{2n})^{ι}.

2.2. Symplectic Dellac configurations.

Definition 2. A symplectic Dellac configuration C is a board of 4n columns and 2n rows with 4n marked cells such that:

(1) each column contains exactly one marked cell;

(2) each row contains exactly two marked cells;

(3) if the (i, j)-cell is marked, then i ≤ j ≤ 2n + i;

(4) for 1 ≤ i, j ≤ 2n, the (i, j)-cell is marked if and only if the (2n−i+1, 4n−j +1)- cell is marked.

Let SpDC_{2n} denote the set of such configurations and e_{n} its cardinality.

We have e_{1} = 1, e_{2} = 2, e_{3} = 10, e_{4} = 98, e_{5} = 1594. Consider the sequence of polynomials defined by recursion: E_{0}(x) = 1,

E_{n}(x) = \frac{1}{2}(x + 1)((x + 2)E_{n−1}(x + 2) − xE_{n−1}(x)).

Conjecture 1. For any n ≥ 0, we have e_{n+1} = E_{n}(1).

Remark 2. Giovanni Cerulli Irelli and Evgeny Feigin kindly informed us that they have also a similar conjecture.

If this conjecture were true, these numbers e_{n} coincide with the numbers r_{n} in [RZ96] (see A098279 in OEIS), where the continued fraction expansion of the corresponding generating function is given (Théorème 29 in loc. cit.).

2.3. Main result. The main result of this section is the following.

Theorem 3. For any integer n ≥ 1, we have e_{n} = #W_{≤\overline{τ}_{2n}}^{J}.

Proof. We prove the theorem by establishing a bijection between W_{≤\overline{τ}_{2n}}^{J} and SpDC_{2n}, following the strategy in the proof of Theorem 1.

A symplectic rook configuration C is a board of 4n columns and rows with 4n marked points satisfying:

(1) C is a rook configuration;

(2) for 1 ≤ i ≤ 4n and 1 ≤ j ≤ 2n, the cell (i, j) is marked if and only if the cell

(4n + 1 − i, 4n + 1 − j) is marked. The set of symplectic rook configurations is denoted by SR_{4n}. Similarly to Section 1.2, we can define the restricted symplectic rook configurations with respect to τ_{2n}: SR_{≤τ_{2n}} := SR_{4n} ∩ R_{≤τ_{2n}}.

Consider the bijection φ : R_{4n} ⟶^{∼} 𝔖_{4n} from (1.2).

Lemma 4. (1) The restriction of the map φ induces a bijection φ′ : SR_{4n} ⟶^{∼}

W̃^{ι} = W.

(2) The restriction of the map φ′ induces a bijection ψ : SR_{≤τ_{2n}} ⟶^{∼} (W̃_{≤τ_{2n}})^{ι}.

Proof. (1) Take a board R in SR_{4n}. Condition (2) in its definition implies that φ(R) is invariant under the involution ι. It suffices to show that φ′ is surjective. Let σ ∈ W̃. By definition of ι, the element σ is fixed by the involution ι if and only if σ(4n + 1 − k) = 4n + 1 − σ(k) for all k with 1 ≤ k ≤ 4n, i.e., for all i and j with 1 ≤ i ≤ 4n and 1 ≤ j ≤ 2n, we have σ(i) = j if and only if σ(4n + 1 − i) = 4n + 1 − j. This implies that φ^{−1}(σ) is in SR_{4n}.

(2) Since SR_{≤τ_{2n}} = SR_{4n} ∩ R_{≤τ_{2n}} and (W̃_{≤τ_{2n}})^{ι} = W̃^{ι} ∩ W̃_{≤τ_{2n}}, the bijectivity of ψ follows from (1) and Theorem 2. □

Moreover, consider the restriction of the melt map m : R_{≤τ_{2n}} → DC_{2n} to SR_{≤τ_{2n}}. Since Condition (2) in the definition of the symplectic rook configuration translates into Condition (4) in the definition of the symplectic Dellac configuration under the melt map, m induces a map m′ : SR_{≤τ_{2n}} → SpDC_{2n}.

Example 3. Let us consider an example where n = 2 and the permutation is given by the following rook configuration:

•

•

•

•

•

•

•

•

where the shadowed area is the convex hull of the marked cells in R_{\overline{τ}_{4}}. It is straightforward to see that the rook configuration is fixed by ι and hence symplectic. The corresponding symplectic Dellac configuration via the melt map m is given by

\begin{aligned}• & & & • \\ & • & • \\ & & & & & • & • \\ & & & & • & & & • \end{aligned}

Continuation of the proof of Theorem 3. The restriction of the blow-up map b : DC_{2n} → R_{≤τ_{2n}} to SpDC_{2n} gives a map b′ : SpDC_{2n} → SR_{≤τ_{2n}}. By Lemma 3, b is injective with im(b) = φ^{−1}(W̃ J̃ ; ≤τ_{2n}). This implies that b′ is injective with

im(b′) = φ^{−1}(W̃ J̃ ; ≤τ_{2n} ∩ W̃^{ι}) = ψ^{−1}((W̃ J̃ ; ≤τ_{2n})^{ι})

and m′ ∘ b′ = id.

By the above argument, the blow-up map b′ gives a bijection SpDC_{2n} ⟶^{∼} (W̃ J̃ ; ≤τ_{2n})^{ι}. Composing with (φ′)^{−1}, we get a bijection SpDC_{2n} ⟶^{∼} W_{≤\overline{τ}_{2n}}^{J}. □

3. Application to torus fixed points

We show how the construction in Section 1 is related to the study of the torus fixed points in the degenerate flag variety.

3.1. Schubert varieties. Let σ_{n} ∈ 𝔖_{2n} be the permutation defined by

σ_{n}(r) = \begin{cases}k, & r = 2k; \\ n + 1 + r, & r = 2k + 1. \end{cases}  (3.1)

We see that σ_{n} can be obtained by restricting τ_{n+1} ∈ S_{2n+2} to the set {2, …, 2n + 1}.

We denote by X_{σ_{n}} the Schubert variety corresponding to σ_{n} in the projective variety SL_{n}/P, where P is the standard parabolic subalgebra defined as the stabilizer of the highest weight line of weight ϖ_{1} + ϖ_{3} + ⋯ + ϖ_{2n−1}. The maximal torus T_{2n−1} of SL_{2n} acts naturally on X_{σ_{n}}. Let X_{σ_{n}}^{T_{2n−1}} be the set of torus fixed points.

It is a standard result that the torus fixed points X_{σ_{n}}^{T_{2n−1}} can be identified with the quotient W_{≤σ_{n}}^{J}, where W = 𝔖_{2n} and J = {2, 4, …, 2n − 2}. For τ ∈ W_{≤σ_{n}}^{J}, the corresponding torus fixed point in X_{σ_{n}}^{T_{2n−1}} is

⟨e_{τ(1)}⟩_{ℂ} ⊂ ⟨e_{τ(1)}, e_{τ(2)}, e_{τ(3)}⟩_{ℂ} ⊂ ⋯ ⊂ ⟨e_{τ(1)}, e_{τ(2)}, …, e_{τ(2n−1)}⟩_{ℂ} ∈ X_{σ_{n}},

where e_{1}, e_{2}, …, e_{2n} is a fixed basis of ℂ^{2n}.

3.2. Degenerate flag varieties. We fix a basis {f_{1}, f_{2}, …, f_{n+1}} of ℂ^{n+1}. Let Fl_{n+1}^{a} be the degenerate flag variety of SL_{n+1} (see [Fei11] for details):

Fl_{n+1}^{a} = {(V_{1}, V_{2}, …, V_{n}) ∈ ∏_{i=1}^{n} Gr_{i}(ℂ^{n+1}) | pr_{i+1}(V_{i}) ⊂ V_{i+1} for i = 1, 2, …, n},

where pr_{i} : ℂ^{n+1} → ℂ^{n+1} is the linear projection along the line generated by f_{i}. By [CFR12], the torus T_{2n−1} acts on Fl_{n+1}^{a}. Let (Fl_{n+1}^{a})^{T_{2n−1}} be the corresponding set of torus fixed points.

In [CL15], it is shown that there exists a T_{2n−1}-equivariant isomorphism of projective varieties ζ : Fl_{n+1}^{a} ⟶^{∼} X_{σ_{n}} ⊂ SL_{2n}/P. We are particularly interested in the image of torus fixed points under ζ.

Fix a basis {e_{1}, e_{2}, …, e_{2n}} of ℂ^{2n}. For i = 1, 2, …, n, we write U_{n+i} for the coordinate subspace ⟨e_{1}, e_{2}, …, e_{n+i}⟩ ⊂ W. The surjection π_{i} : U_{n+i} → ℂ^{n+1} is defined by

π_{i}(e_{k}) = \begin{cases}0, & if 1 ≤ k ≤ i − 1; \\ f_{k}, & if i ≤ k ≤ n + 1; \\ f_{k−n−1}, & if n + 2 ≤ k ≤ n + i. \end{cases}  (3.2)

Define ζ_{i} : Gr_{i}(ℂ^{n+1}) → Gr_{2i−1}(ℂ^{2n}) to be the concatenation of the maps

Gr_{i}(ℂ^{n+1}) → Gr_{2i−1}(U_{n+i}) → Gr_{2i−1}(ℂ^{2n}), U ↦ π_{i}^{−1}(U) ↦ π_{i}^{−1}(U).

Then ζ : Fl_{n+1}^{a} → X_{σ_{n}} is given by ∏_{i=1}^{n} ζ_{i} (see Section 2 of [CL15] for details).

It is clear that the torus T_{n} of SL_{n+1} acts naturally on Fl_{n+1}^{a}. By results in Section 7.2 of [CFR12], any T_{2n−1}-fixed point in Fl_{n+1}^{a} is in fact a T_{n}-fixed point. In [Fei11], an explicit bijection f between the T_{2n−1}-fixed points and Dellac configuration is provided.

3.3. A commutative diagram. As a summary, starting with a T_{n}-fixed point in Fl_{n+1}^{a}, there are two ways to obtain a Dellac configuration:

(1) via the bijection f given by [Fei11];

(2) consider this fixed point as a fixed point in the Schubert variety X_{σ_{n}}, hence identify it with an element in W_{≤σ_{n}}^{J}, then melt the corresponding rook configuration to get a Dellac configuration.

It is natural to ask whether the following diagram commutes:

\xymatrix{(Fl_{n+1}^{a})^{T_{2n−1}} = (Fl_{n+1}^{a})^{T_{n}} \ar[r]^{f} \ar[d]^{β} & DC_{n+1} \ar[d]^{b} \\ X_{τ_{n+1}}^{T_{2n+1}} = X_{σ_{n}}^{T_{2n−1}} & W_{≤τ_{n+1}}^{J} \ar[l]_{α}} ,

where the map α is given as follows: for σ ∈ W_{≤τ_{n+1}}^{J}, where W = 𝔖_{2n+2}, we define the map α as follows: α(σ) is the sequence of subspaces W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{n} such that W_{i} is the subspace of ℂ^{2n} generated by e_{\overline{σ}(1)}, e_{\overline{σ}(2)}, …, e_{\overline{σ}(2i−1)}, where \overline{σ} is the (well-defined) restriction of σ to 𝔖_{2n}. We can identify this element in X_{σ_{n}}^{T_{2n−1}} with n subsets J_{1}, …, J_{n} of \{1, 2, …, 2n\} such that J_{i} = \{\overline{σ}(1), \overline{σ}(2), …, \overline{σ}(2i − 1)\}.

It remains to consider the restriction of the map ζ to fixed points. Here we have to include an extra twist, since the definition of the degenerate flag variety is slightly different in [Fei11] and [CL15]: let (V_{1}, V_{2}, …, V_{n}) ∈ (Fl_{n+1}^{a})^{T_{n}}. This flag can be identified (see [Fei11, Corollary 2.11]) with n subsets I_{1}, I_{2}, …, I_{n} of {1, 2, …, n + 1} such that #I_{k} = k and, for k = 1, 2, …, n, I_{k}\{k + 1} ⊂ I_{k+1}.

We let κ = (12 ⋯ n + 1)^{−1} be the inverse of the longest cycle in 𝔖_{n+1}. Suppose that I_{l} = {i_{l,1}, i_{l,2}, …, i_{l,l}}. We write I_{l}^{κ} = {κ(i_{l,1}), κ(i_{l,2}), …, κ(i_{l,l})}. Furthermore, we define a map p_{l} : {1, 2, …, n + l} → {1, 2, …, n + 1} by

p_{l}(s) = \begin{cases}0, & if 1 ≤ s ≤ l − 1; \\ s, & if l ≤ k ≤ n + 1; \\ s − n − 1, & if n + 2 ≤ k ≤ n + l. \end{cases}  (3.3)

Then β((I_{1}, I_{2}, …, I_{n})) = (T_{1}, T_{2}, …, T_{n}), where T_{l} = p_{l}^{−1}(I_{l}^{κ}).

Theorem 4. The diagram above commutes, i.e., ζ = α ∘ b ∘ f.

The proof consists of a case-by-case examination. We only provide a sketch.

Proof. We pick I = (I_{1}, I_{2}, …, I_{n}) ∈ (Fl_{n+1}^{a})^{T_{n+1}}. Recall that the map f is given in [Fei11, Proposition 3.1].

(1) Suppose that l ∉ I_{l−1}. Then I_{l}\I_{l−1} = {j}. We consider the case j > l: in the Dellac configuration f(I), the cells (l, l) and (l, j) are marked. Then, by definition, σ = b(f(I)) satisfies σ(2l − 1) = l and σ(2l) = j. Hence, in α(σ), J_{l}\J_{l−1} = {l − 1, j − 1}.

We compute β(I): it is clear that I_{l}^{κ}\I_{l−1}^{κ} = {j−1}. Then p_{l}^{−1}(I_{l}^{κ})\p_{l−1}^{−1}(I_{l−1}^{κ}) = p_{l}^{−1}({l−1, j −1}) = {l−1, j −1}. Therefore T_{l}\T_{l−1} = {l−1, j −1}, i.e., J_{l} = T_{l}.

The case j < l can be dealt with similarly.

(2) Suppose that l ∈ I_{l−1} and l ∈ I_{l}. Then I_{l}\I_{l−1} = {j}. We study the case j < l: in the corresponding Dellac configuration, the cells (l, l+n+1) and (l, j +n+1) are marked. The associated permutation σ = b(f(I)) satisfies σ(2l − 1) = j + n + 1 and σ(2l) = l + n + 1. Hence, in α(σ), J_{l}\J_{l−1} = {j + n, l + n}.

For β(I): l ∈ I_{l−1} ∩ I_{l} and I_{l}\I_{l−1} = {j} imply that l − 1 ∈ I_{l−1}^{κ} ∩ I_{l}^{κ} and I_{l}^{κ}\I_{l−1}^{κ} = {κ(j)}. Notice that, no matter whether j = 1 or j > 1, p_{l}^{−1}(κ(j)) = j + n. By the assumption j < l, we have

p_{l}^{−1}(I_{l}^{κ})\p_{l−1}^{−1}(I_{l−1}^{κ}) = p_{l}^{−1}({l − 1, κ(j)}) = {j + n, l + n},

which establishes J_{l} = T_{l}.

The case where j > l can be dealt with similarly.

(3) Suppose that l ∈ I_{l−1} and l ∉ I_{l}. Then there exist j_{1} and j_{2} such that I_{l}\I_{l−1} = {j_{1}, j_{2}}. We assume that j_{1} < l and j_{2} > l. In the corresponding Dellac configuration, the cells (l, j_{1}+n+1) and (l, j_{2}) are marked. Hence, in α(b(f(I))), J_{l}\J_{l−1} = {j_{1} + n, j_{2} − 1}.

For β(I), we have

p_{l}^{−1}(I_{l}^{κ})\p_{l−1}^{−1}(I_{l−1}^{κ}) = p_{l}^{−1}({κ(j_{1}), j_{2} − 1}) = {j_{1} + n, j_{2} − 1},

therefore J_{l} = T_{l}.

All other cases can be proved in the same way. □

Remark 3. A similar diagram without the map f exists in the symplectic case by changing

(1) the degenerate flag variety to the symplectic degenerate flag variety (see [FFiL14]);

(2) the Schubert variety of SL_{2n} by the Schubert variety in the symplectic group (see [CL15]); (3) the Dellac configuration by the symplectic Dellac configuration;

(4) the set W_{≤τ_{n+1}}^{J} by W_{≤\overline{τ}_{2n+2}}^{J}.

Remark 4. The original proof of Theorem 1 is given by showing that the composition α^{−1} ∘ β ∘ f^{−1} is a bijection: that f is a bijection is shown in [Fei11]; by the main theorem of [CL15], β is a bijection; α is a well-known bijection. Our proof of the theorem uses the intuitive map b to avoid the geometrical proof.

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Xin Fang: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931, Köln, Germany

  E-mail address: xinfang.math@gmail.com

12 XIN FANG AND GHISLAIN FOURIER

School of Mathematics and Statistics, University of Glasgow

E-mail address: ghislain.fourier@glasgow.ac.uk