2.1 Recollections on $\mathrm {Fl}_{n}$

We work over $\mathbb C$ and denote by $T_{n},B_{n},B_{n}^-\subset \mathrm {GL}_{n}$ the subsets of diagonal, upper triangular, and lower triangular invertible $n\times n$ matrices. When there is no ambiguity we write T, B, and $B^-$ for $T_{n}$ , $B_{n}$ , and $B^{-}_{n}$ . We will denote by $\chi _i$ the i-th standard character of $T_n$ , corresponding to the i-th entry along the diagonal, so that for $a = \mathrm {diag}(a_{1}, \ldots , a_{n}) \in T$ we have $\chi _{i}(a) = a_{i}$ . For $\chi $ a character of T, we will denote by $\mathbb {C}_{\chi }$ the corresponding $1$ -dimensional representation.

We identify the complete flag variety

$$\begin{align*}\mathrm{Fl}_{n} := \text{flags of subspaces }\big(\{0\}=\mathcal{F}_0\subsetneq \mathcal{F}_1\subsetneq \mathcal{F}_2\subsetneq \cdots \subsetneq \mathcal{F}_{n-1}\subsetneq \mathcal{F}_{n}=\mathbb C^{n}\big). \end{align*}$$

with $\mathrm {GL}_{n}/B_{n}$ via the transitive action of $\mathrm {GL}_{n}$ with $B_{n}$ the stabilizer of the $\color {blue}{{standard\, coordinate\, flag}}$ with $\mathcal {F}_i=\langle e_1,\ldots ,e_i\rangle$ for $1\leq i\leq n$ . Via this identification the i-th subspace in the flag associated to $hB_{n}$ is the column span of the first i columns of h for $1\leq i\leq n$ .

For $w \in S_{n}$ , we will denote the Schubert cycles in $\mathrm {Fl}_{n}$ by $X^v=\overline {BvB}$ , the opposite Schubert cycles by $X_u=\overline {B^-uB}$ , and for $u\le v$ in the Bruhat order the Richardson varieties $X^v_u:= X^v\cap X_u=\overline {BvB}\cap \overline {B^-uB}$ . The Bruhat decomposition

(2.1) $$ \begin{align} \mathrm{Fl}_{n}=\bigsqcup_{w\in S_n}BwB \qquad \text{ with } \qquad BwB\cong \mathbb{A}^{\ell(w)}, \end{align} $$

gives an affine paving if the $BwB$ are ordered via any linear extension of the Bruhat order on $S_n$ ; see Section 9 for more details.

For $w \in S_{n}$ and a dominant weight $\lambda =(\lambda _1\ge \cdots \ge \lambda _n)\in \mathbb {Z}^n$ , we have a $\color {blue}{{Pl\unicode{x00FC}cker\, function}}$ defined for $h\in \mathrm {GL}_n$ by

$$\begin{align*}\mathrm{Pl}_{\lambda, w}(h) = (\det h_{w(1)})^{\lambda_1-\lambda_2}(\det h_{w(1),w(2)})^{\lambda_2-\lambda_3}\cdots (\det h_{w(1),\ldots,w(n)})^{\lambda_n}, \end{align*}$$

where $h_{i_1,\ldots ,i_k}$ is the submatrix of h with columns $1,\ldots ,k$ and rows $i_1,\ldots ,i_k$ . These functions together define a map $\mathrm {Pl}_{\lambda }:\mathrm {Fl}_{n}\to \mathbb {P}^{n!-1}$ , which we write simply $\mathrm {Pl}$ if $\lambda $ is the fundamental dominant weight ${(n,n-1,\ldots ,1)}$ . Then $\mathrm {Pl}_{\lambda }$ is $T_n$ -equivariant with respect to the $T_n$ -action on $\mathbb {P}^{n!-1}$ where $\mathrm {diag}(a_1,\ldots ,a_n)\in T_n$ acts in the w-coordinate by $a_{w(1)}^{\lambda _1}a_{w(2)}^{\lambda _2}\cdots a_{w(n)}^{\lambda _n}=a_1^{\lambda _{w^{-1}(1)}}\cdots a_n^{\lambda _{w^{-1}(n)}}$ .

The moment polytope of $\mathrm {Fl}_{n}$ under $\mathrm {Pl}_{\lambda }$ is the generalized permutahedron

$$\begin{align*}\operatorname{Perm}(\lambda) =\operatorname{conv}\{u\cdot {\lambda}\;|\; u\in S_n\} \subseteq \mathbb R^{n}, \end{align*}$$

where $u\cdot {\lambda } = (\lambda _{u^{-1}(1)},\ldots ,\lambda _{u^{-1}(n)})$ and $\operatorname {conv}$ denotes taking the convex hull. If $X\subset \mathrm {Fl}_{n}$ is a $T_n$ -invariant subvariety, then its moment polytope under $\mathrm {Pl}_{\lambda }$ is

$$\begin{align*}P(X;{\lambda})=\operatorname{conv}\{u\cdot {\lambda} \;|\; u\in X^T\} \subseteq \operatorname{Perm}(\lambda). \end{align*}$$

The moment polytope of a $T_{n}$ -orbit closure X is always a $\color {blue}{{flag\ matroid\ polytope}}$ (see [Reference Gelfand, Goresky, MacPherson and Serganova22]), meaning its vertices are contained in the vertices of $\operatorname {Perm}(\lambda )$ and all edges are parallel to the type $A_{n-1}$ roots $e_i-e_j$ for various $i,j$ .

2.2 Noncrossing partitions

A $\color {blue}{{combinatorial\, noncrossing\, partition}}$ is a partition $A_1\sqcup \cdots \sqcup A_k=[n]$ such that for $i\ne j$ , distinct elements $a,b\in A_i$ , and distinct elements $c,d\in A_j$ , we never have $a<c<b<d$ . We depict combinatorial noncrossing partitions as noncrossing arc diagrams; for example we draw $\{1,2,3,6\}\sqcup \{4,5\}=[6]$ as

A $\color {blue}{{backwards\, cycle}}$ is a cycle $(b_1\,b_2\,\cdots \, b_r)$ with $b_1>b_2>\cdots >b_r$ . An $\color {blue}{{algebraic}} \color {blue}{{noncrossing\, partition}}$ is defined to be a permutation w whose disjoint cycle decomposition $\operatorname {Cyc}(w):= C_{1}C_{2} \cdots C_{k}$ consists of backwards cycles whose underlying sets define a combinatorial noncrossing partition. For example, $w = (6321)(54)$ is the algebraic noncrossing partition associated to the arc diagram above. We denote

$$\begin{align*}\operatorname{NC}_{n} := \{\text{algebraic noncrossing partitions of } [n]\}\subset S_n. \end{align*}$$

The $\color {blue}{{Kreweras\, order}}$ on $\operatorname {NC}_n$ is defined by setting $u\le _K v$ if the combinatorial noncrossing partition associated to u refines the combinatorial noncrossing partition associated to v. For example, $(63)(21)(54)\le _K (6321)(54)$ . The Kreweras order is a lattice, and the lattice structure is induced by the lattice of set partitions under refinement [Reference Kreweras32].

Going forward, we will identify combinatorial and algebraic noncrossing partitions, using “combinatorial” and “algebraic” only when disambiguation is needed.

We say that permutations $u,w\in S_n$ are $\color {blue}{{adjacent}}$ if $w=(i\,j)u$ for some transposition $(i\,j)$ . Let $\operatorname {Cayley}(S_n)$ be the $\color {blue}{{Cayley\, graph}}$ on $S_n$ in which adjacent permutations are connected by an edge. Then $u,w\in \operatorname {NC}_n$ are adjacent in the Kreweras lattice (meaning one element covers the other) if and only if $u,w$ are adjacent in $\operatorname {Cayley}(S_n)$ . In this way, we may identify the Hasse diagram of the Kreweras lattice with the induced subgraph of $\operatorname {Cayley}(S_n)$ on $\operatorname {NC}_n$ as was first observed by Biane [Reference Biane9]. We denote this induced subgraph by $\operatorname {Cayley}(\operatorname {NC}_{n})$ .

2.3 Binary trees and nested forests

A $\color {blue}{{planar\, binary\, tree}}$ is a rooted tree T in which each node v is either an $\color {blue}{{internal\, node}}$ with exactly $2$ children $v_L$ and $v_R$ (the left and right children), or v is a $\color {blue}{{leaf}}$ with zero children. Let $\operatorname {IN}(T)$ denote the set of internal nodes in T. We allow for the possibility that $|\operatorname {IN}(T)|=0$ , in which case the unique node is both a root and a leaf.

A $\color {blue}{{nested\, forest\, supported\, on\, [n]}}$ is a family $\widehat {F} = (T_{C})_{C \in \operatorname {Cyc}(w)}$ of binary trees $T_{C}$ indexed by the disjoint cycles of a noncrossing partition $w = C_{1} C_{2} \cdots C_{k} \in \operatorname {NC}_{n}$ such that each $T_{C}$ has $|C|$ leaves. We identify the leaves of each $T_{C}$ with the underlying set of C in increasing order and depict $\widehat {F}$ by drawing the $T_{C}$ in the upper half plane so that the set $[n]$ of all leaves appears in increasing order along the horizontal axis.

The internal nodes of $\widehat {F}$ are $\operatorname {IN}(\widehat {F}) = \bigsqcup _{C \in \operatorname {Cyc}(w)} \operatorname {IN}(T_{C})$ . The $\color {blue}{{canonical\, label}}$ of each $v \in \operatorname {IN}(\widehat {F})$ is the value of the rightmost leaf descendant of $v_L$ . See, for example, Figure 2 (left).

Figure 2

A nested forest with its canonical labeling in red (left) and the underlying noncrossing partition in cycle notation (right).

We define a map $\operatorname {NCPerm}: \operatorname {NestFor}_{n} \to \operatorname {NC}_{n}$ that sends a nested forest $(T_{C})_{C \in \operatorname {Cyc}(w)}$ to its underlying noncrossing partition $w=\prod _{C\in \operatorname {Cyc}(w)}C$ . See, for example, Figure 2 (right).

3.1 $\Psi _i^-$ and $\Psi _i^+$

We recall certain pattern maps that were studied by Bergeron–Sottile [Reference Bergeron and Sottile7] and summarize their essential properties. We refer the reader to [Reference Billey and Braden13, Reference Billey and Postnikov11] for a more general perspective. Throughout, let $\operatorname {Mat}_{a\times b}$ be the set of $a\times b$ matrices and, for a property P, define

$$ \begin{align*}\delta_P=\begin{cases}1&\text{if } P \text{ is true, and}\\0&\text{if } P \text{ is false.}\end{cases}\end{align*} $$

Definition 3.1. Let $\Psi _{i,j}\colon \operatorname {Mat}_{m-1\times m-1}\to \operatorname {Mat}_{m\times m}$ be the operation

$$ \begin{align*}(\Psi_{i,j}M)_{k,\ell}=\begin{cases}M_{k-\delta_{k<i},\ell-\delta_{\ell<j}}&k\ne i\text{ and }\ell\ne j\\ 1&(k,\ell)=(i,j)\\0&\text{otherwise.}\end{cases}\end{align*} $$

The map $\Psi _{1,i}$ was crucial to the construction in [Reference Nadeau, Spink and Tewari38, §5]. In contrast, the following two pattern maps will be important to us:

$$\begin{align*}\Psi_i^-:= \Psi_{i,i}\text{ and }\Psi_i^+:= \Psi_{i,i+1}. \end{align*}$$

For example, when $m=4$ we have

$$ \begin{align*}\Psi_{2}^-\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=\begin{bmatrix}a&0&b&c\\0&1&0&0\\d&0&e&f\\g&0&h&i\end{bmatrix}\text{ and }\Psi_2^+\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=\begin{bmatrix}a&b&0&c\\0&0&1&0\\d&e&0&f\\g&h&0&i\end{bmatrix}.\end{align*} $$

By restricting to invertible matrices, the pattern maps descend to closed embeddings

$$\begin{align*}\Psi_{i,j}:\mathrm{Fl}_{m-1}\hookrightarrow \mathrm{Fl}_{m}. \end{align*}$$

Define $\gamma _i:T_m\to T_{m-1}$ to be the map $\mathrm {diag}(a_1,\ldots ,a_{m})\mapsto \mathrm {diag}(a_1,\ldots ,a_{i-1},a_{i+1},\ldots ,a_{m})$ .

Write $\epsilon _{i}:\mathbb {C}^{m-1}\hookrightarrow \mathbb {C}^{m}$ for the inclusion $(x_1,\ldots ,x_{m-1})\mapsto (x_1,\ldots ,x_{i-1},0,x_i,\ldots ,x_{m-1})$ . Then $\epsilon _i$ is a $T_m$ -equivariant inclusion if we give $\mathbb {C}^{m-1}$ the action of $T_m$ induced by $\gamma _i$ , and we have

$$ \begin{align*} \Psi_i^-\mathcal{F}&=\{0\}\subset \epsilon_{i}(\mathcal{F}_1)\subset \cdots \subset \epsilon_{i}(\mathcal{F}_i)\subset \epsilon_{i}(\mathcal{F}_{i})\oplus \langle e_i\rangle \subset \epsilon_{i}(\mathcal{F}_{i+1})\oplus \langle e_i\rangle\subset \cdots\\ \Psi_i^+\mathcal{F}&=\{0\}\subset \epsilon_{i}(\mathcal{F}_1)\subset \cdots \subset \epsilon_{i}(\mathcal{F}_i)\subset \epsilon_{i}(\mathcal{F}_{i+1})\subset \epsilon_{i}(\mathcal{F}_{i+1})\oplus \langle e_i\rangle\subset \cdots. \end{align*} $$

3.2 Building $\mathbb P^1$ -bundles with $\mathbb P_i$

We use the subbundle convention for relative projectivization, so that for $\mathcal {V}$ a vector bundle on a variety X we have $\mathsf {Proj}(\mathcal {V})_X:=(\mathcal {V}\setminus \{0\})/\mathbb {C}^*$ . Consider the sequence of maps

$$\begin{align*}\mathrm{Fl}_{m-1}^{\gamma_i}\overset{\Psi_i^{\pm}}{\rightrightarrows}\mathrm{Fl}_{m}\xrightarrow{\pi_i}\mathrm{GL}_{m}/P_i, \end{align*}$$

where $P_i$ is the minimal parabolic subgroup of $\mathrm {GL}_{m}$ containing $s_i$ , and $\pi _i$ is the projection map.

The space $\mathrm {GL}_m/P_i$ is typically identified with the variety of partial flags $\{0\}\subset \mathcal {F}_1\subset \cdots \subset \mathcal {F}_{i-1}\subset \mathcal {F}_{i+1}\subset \cdots \subset \mathcal {F}_m=\mathbb {C}^m$ with $\dim \mathcal {F}_j=j$ . Under this identification $\pi _i$ becomes the map that forgets the i-th subspace of a complete flag.

We now study how the $\mathbb P^1$ -bundle $\pi _i$ interacts with the maps $\Psi _i^{\pm }$ . Since $s_{i}P_{i} = P_{i}$ , we have the equality $\pi _i\Psi _i^-=\pi _i\Psi _i^+.$ Consequently we write

$$ \begin{align*} \pi_i\Psi_i:= \pi_i\Psi_i^+=\pi_i\Psi_i^- \end{align*} $$

to emphasize that this composite does not depend on $\pm $ . The map $\pi _i\Psi _i$ is given by

$$ \begin{align*}\pi_i\Psi_i(\mathcal{F})_j=\begin{cases}\epsilon_i(\mathcal{F}_j)&j<i\\ \epsilon_i(\mathcal{F}_{j-1})\oplus \langle e_i\rangle&j>i,\end{cases}\end{align*} $$

and is a closed $T_m$ -equivariant embedding $\mathrm {Fl}_{m-1}^{\gamma _i}\hookrightarrow \mathrm {GL}_m/P_i$ with image

$$ \begin{align*}\pi_i\Psi_i(\mathrm{Fl}_{m-1})=\big\{\{\mathcal{F}_j\}_{j\in [m]\setminus i}: \mathcal{F}_{i-1}\subset \{x_i=0\}\text{ and }e_i\in \mathcal{F}_{i+1}\big\}\subset \mathrm{GL}_m/P_i. \end{align*} $$

For $Z\subset \mathrm {Fl}_{m-1}$ , we define a map

$$\begin{align*}\mathbb{P}_i:\{\text{subsets of } \mathrm{Fl}_{m-1}\}\to \{\text{subsets of } \mathrm{Fl}_{m}\}\quad\text{by}\quad \mathbb P_iZ:= \pi_i^{-1}\pi_i\Psi_i Z\subset \mathrm{Fl}_{m}. \end{align*}$$

We now describe $\mathbb {P}_iZ$ using matrix representatives. For an $(m-1)\times (m-1)$ matrix M, let $\mathbb {G}_iM$ be the set of all $m \times m$ matrices obtained from $\Psi _i^+M$ by replacing the $0$ in entry $(i,i)$ with any value $+ \in \mathbb {C}^*$ . The construction of $\mathbb {G}_i$ commutes with forward column operations and therefore descends to a map $\{\text {subsets of } \mathrm {Fl}_{m-1}\}\to \{\text {subsets of } \mathrm {Fl}_{m}\}$ , and $\mathbb {P}_iZ=(\Psi _i^+Z)\sqcup (\mathbb {G}_iZ)\sqcup (\Psi _i^-Z)$ . For example, when $m=4$ we have

$$\begin{align*}\mathbb P_{2}\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}B_3=\left\{\begin{bmatrix} a&b&0&c\\ 0&0 &1&0\\ d&e&0&f\\ g&h&0&i \end{bmatrix}B_4\right\}\sqcup \left\{\begin{bmatrix} a&b&0&c\\ 0&+ &1&0\\ d&e&0&f\\ g&h&0&i \end{bmatrix}B_4:+\in \mathbb{C}^*\right\} \sqcup \left\{\begin{bmatrix} a&0&b&c\\ 0&1&0&0\\ d&0&e&f\\ g&0&h&i\end{bmatrix}B_4\right\}. \end{align*}$$

4.1 Combinatorics of bicolored nested forests

Let $\mathrm {RESeq}$ be the set of words from the alphabet

$$\begin{align*}\bigcup_{i=1}^{\infty} \{\mathsf{r}_{i}^-,\mathsf{r}_{i}^+,\mathsf{e}_{i}\}, \end{align*}$$

and for $\Omega \in \mathrm {RESeq}$ define $|\Omega |$ to be the number of $\mathsf {e}_{i}$ letters in $\Omega $ . For example, $\mathsf {r}_{2}^-\mathsf {r}_{3}^+\mathsf {e}_{2}\in \mathrm {RESeq}$ and $|\mathsf {r}_{2}^-\mathsf {r}_{3}^+\mathsf {e}_{2}|=1$ . We define a distinguished subset $\mathrm {RESeq}_n\subset \mathrm {RESeq}$ by

$$\begin{align*}\mathrm{RESeq}_n:= \{\mathsf{x}_{1}\cdots \mathsf{x}_{n}\in \mathrm{RESeq}\;|\; \mathsf{x}_{i}\in \{\mathsf{r}_{1}^-,\ldots,\mathsf{r}_{i}^-,\mathsf{r}_{1}^+,\ldots,\mathsf{r}_{i-1}^+,\mathsf{e}_{1},\ldots,\mathsf{e}_{i-1}\}\text{ for all } i\}. \end{align*}$$

Note that every word of $\mathrm {RESeq}_n$ begins with $\mathsf {r}_{1}^-$ . We will write either $\Omega _1 \Omega _2$ or $\Omega _1\cdot \Omega _2$ to describe the concatenation of $\Omega _1$ and $\Omega _2$ .

Definition 4.1. Let $\operatorname {BNestFor}_n$ be the quotient of $\mathrm {RESeq}_n$ by the local relations

$$ \begin{align*} & \mathsf{e}_{i} \mathsf{e}_{j}=\mathsf{e}_{j} \mathsf{e}_{i+1}\text{ for }i> j\\ \mathsf{r}_{i}^- \mathsf{r}_{j}^-=\mathsf{r}_{j}^- \mathsf{r}_{i+1}^-\text{ for } & i\ge j, \qquad \qquad \mathsf{r}_{i}^+ \mathsf{r}_{j}^+=\mathsf{r}_{j}^+ \mathsf{r}_{i+1}^+\text{ for }i>j,\\ \mathsf{r}_{i}^+ \mathsf{r}_{j}^-=\mathsf{r}_{j}^- \mathsf{r}_{i+1}^+\text{ for } & i\ge j, \qquad \qquad \mathsf{r}_{i}^- \mathsf{r}_{j}^+=\mathsf{r}_{j}^+ \mathsf{r}_{i+1}^-\text{ for } i>j,\\ \mathsf{e}_{i} \mathsf{r}_{j}^-=\mathsf{r}_{j}^- \mathsf{e}_{i+1}\text{ for } & i\ge j, \qquad \qquad \mathsf{r}_{i}^- \mathsf{e}_{j}=\mathsf{e}_{j} \mathsf{r}_{i+1}^-\text{ for }i >j\\ \mathsf{e}_{i} \mathsf{r}_{j}^+=\mathsf{r}_{j}^+ \mathsf{e}_{i+1}\text{ for } & i>j, \qquad \qquad \mathsf{r}_{i}^+ \mathsf{e}_{j}=\mathsf{e}_{j}\mathsf{r}_{i+1}^+\text{ for }i>j. \end{align*} $$

This is well-defined as the relations all preserve $\mathrm {RESeq}_n$ .

We recall from [Reference Bergeron, Gagnon, Nadeau, Spink and Tewari6, Proposition 9.4] how the equivalence classes in $\operatorname {BNestFor}_{n}$ can be represented diagrammatically. A $\color {blue}{{bicolored\, nested\, forest}}$ is a nested forest in which each internal node has been colored with either black ( ) or white ( ). We now define a map from $\mathrm {RESeq}_{n}$ to bicolored nested forests with support in $[n]$ .

Example 4.3. We demonstrate the recursive process for $\Omega =\mathsf {r}_{1}^-\mathsf {r}_{1}^{+}\mathsf {r}_{2}^{-}\mathsf {e}_{1}\mathsf {e}_{3}\mathsf {r}_{2}^+$ :

Going forward, we identify elements of $\operatorname {BNestFor}_{n}$ with the associated bicolored nested forest, so that we can write $\widehat {K} = \widehat {H} \cdot \mathsf {x}_{i}$ to mean $\widehat {H} = \widehat {F}(\Omega )$ and $\widehat {K} = \widehat {F}(\Omega \cdot \mathsf {x}_{i})$ for some $\Omega \in \mathrm {RESeq}_n$ . The defining relations of $\operatorname {BNestFor}_n$ preserve $|\Omega |$ , so it makes sense to discuss $|\widehat {F}|$ for $\widehat {F}\in \operatorname {BNestFor}_{n}$ . Diagrammatically, $|\widehat {F}|$ is the number of black nodes in $\widehat {F}$ . See Example 4.3 for a bicolored nested forest built from an $\Omega \in \mathrm {RESeq}_6$ with $|\Omega |=2$ .

See the examples given in Figure 3. We note that by definition, $\operatorname {Tree}_{n} \subseteq \operatorname {\mathsf {Forest}}_n \subseteq \operatorname {BNestFor}_{n}$ .

Figure 3

Examples of Definition 4.5: an element $T = \mathsf {r}_{1}^-\mathsf {e}_{1}\mathsf {e}_{1}\mathsf {e}_{1}\mathsf {e}_{3} $ of $\operatorname {Tree}_{5}$ (left) and an element $F = (\mathsf {r}_{1}^{-})^{3} \mathsf {e}_{1}\mathsf {e}_{4}\mathsf {e}_{5}$ of $\operatorname {\mathsf {Forest}}_{6} \setminus \operatorname {Tree}_{6}$ (right).

Figure 3 Long description

The figure consists of two panels.

Left panel: Labeled T equals. It features a horizontal line with five points marked with an x and numbered 1 through 5 from left to right. Above this baseline, a binary tree structure rises. Points 1 and 2 connect to a single node above them. Points 3 and 4 connect to another node above them. The node above 1 and 2 connects to a higher node, which also connects to the node above 3 and 4. Finally, this high node and point 5 connect at the highest peak node of the tree.

Right panel: Labeled F equals. It features a horizontal line with six points marked with an x and numbered 1 through 6 from left to right. This panel shows a forest of two separate trees. The first tree is formed by points 1 and 2 connecting to a single node above. Point 3 remains isolated on the baseline. The second tree starts with points 5 and 6 connecting to a node above them. This node and point 4 then connect to a higher peak node, forming a larger triangular structure.

4.2 Combinatorics of building operations

Lemma 4.6. We have the relations

$$ \begin{align*} & \mathbb P_{j}\mathbb P_{i}=\mathbb P_{i+1}\mathbb P_{j} \text{ for }i> j\\ \Psi_{j}^-\Psi_{i}^-= \Psi_{i+1}^-\Psi_{j}^-\text{ for } & i\ge j, \qquad \qquad \Psi_{j}^+\Psi_{i}^+= \Psi_{i+1}^+\Psi_{j}^+\text{ for }i>j,\\ \Psi_{j}^-\Psi_{i}^+= \Psi_{i+1}^+\Psi_{j}^-\text{ for } & i\ge j, \qquad \qquad \Psi_{j}^+\Psi_{i}^-= \Psi_{i+1}^-\Psi_{j}^+\text{ for } i>j,\\ \Psi_{j}^-\mathbb P_{i}= \mathbb P_{i+1}\Psi_{j}^-\text{ for } & i\ge j, \qquad \qquad \mathbb P_{j}\Psi_{i}^-=\Psi_{i+1}^-\mathbb P_{j} \text{ for }i >j\\ \Psi_{j}^+\mathbb P_{i}=\mathbb P_{i+1}\Psi_{j}^+ \text{ for } & i>j, \qquad \qquad \mathbb P_{j}\Psi_{i}^+=\Psi_{i+1}^+\mathbb P_{j}\text{ for }i>j. \end{align*} $$

Moreover, these relations hold when all $\mathbb {P}$ ’s are replaced by $\mathbb {G}$ ’s.

Example 4.8. The following computation witnesses the relation $\mathbb {G}_1\mathbb {G}_2=\mathbb {G}_3\mathbb {G}_1$ :

$$\begin{align*}\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} \xrightarrow{\mathbb{G}_2} \begin{bmatrix}a&b&0&c\\0&+&1&0\\d&e&0&f\\g&h&0&i\end{bmatrix} \xrightarrow{\mathbb{G}_1} \begin{bmatrix}+&1&0&0&0\\a&0&b&0&c\\0&0&+&1&0\\d&0&e&0&f\\g&0&h&0&i\end{bmatrix} \xleftarrow{\mathbb{G}_3} \begin{bmatrix}+&1&0&0\\a&0&b&c\\d&0&e&f\\g&0&h&i\end{bmatrix} \xleftarrow{\mathbb{G}_1} \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}. \end{align*}$$

Since the relations of Lemma 4.6 are opposite to the defining relations of $\operatorname {BNestFor}_n$ , the following is well-defined.

In the next section, we will show that $X^{\circ }(F)$ is a T-orbit, whose closure is $X(F)$ .

Example 4.10. If $\widehat {F}=\widehat {F}(\mathsf {r}_{1}^{-}\mathsf {e}_{1}\mathsf {e}_{2})$ then $X(\widehat {F})=\mathbb {P}_2\mathbb {P}_1[1]$ is the union of $9$ subsets of $\mathrm {Fl}_{3}$ depicted below, with each $+ \in \mathbb C^{*}$ . The T-orbit $X^{\circ }(\widehat {F})=\mathbb {G}_2\mathbb {G}_1[1]$ appears in the second row and column.

$$\begin{align*}\begin{array}{ccccc} \Psi_2^-\Psi_1^-[1]=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}B_{3} &\quad& \Psi_2^-\mathbb{G}_1[1]=\begin{bmatrix}+&0&1\\0&1&0\\1&0&0\end{bmatrix}B_{3} &\quad& \Psi_2^-\Psi_1^+[1]=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}B_{3} \\[5ex] \mathbb{G}_2\Psi_1^-[1]=\begin{bmatrix}1&0&0\\0&+&1\\0&1&0\end{bmatrix}B_{3} && \mathbb{G}_2\mathbb{G}_1[1]=\begin{bmatrix}+&1&0\\0&+&1\\1&0&0\end{bmatrix}B_{3} && \mathbb{G}_2\Psi_1^+[1]=\begin{bmatrix}0&1&0\\0&+&1\\1&0&0\end{bmatrix} B_{3} \\[5ex] \Psi_2^+\Psi_1^-[1]=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}B_{3} && \Psi_2^+\mathbb{G}_1[1]=\begin{bmatrix}+&1&0\\0&0&1\\1&0&0\end{bmatrix}B_{3} && \Psi_2^+\Psi_1^+[1]=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}B_{3} \end{array} \end{align*}$$

8.1 Colored Tamari equivalence and normal forms

Every torus orbit closure in $\mathrm {QFl}_{n}$ is by definition contained in $X(T)$ for some $T \in \mathrm {Tree}_{n}$ . In Section 5, we showed every torus orbit closure in $X(T)$ is of the form $X(\widehat {F})$ for a bicolored nested forest $\widehat {F} \le _{re} T$ . Thus the torus orbit closures in $\mathrm {QFl}_{n}$ can be parametrized by $\operatorname {BNestFor}_n$ . However, there is some redundancy in this parametrization as is apparent from Figure 6. This is explained by the following two additional relations that the building operations satisfy.

Lemma 8.2. For all $1 \le i < n$ we have the relations

$$\begin{align*}\Psi_{i+1}^+\mathbb P_{i}= \mathbb P_{i}\Psi_{i}^+\quad \text{ and }\quad \Psi_{i+1}^+\Psi_{i}^+=\Psi_{i}^+\Psi_{i}^+. \end{align*}$$

In the correspondence between words in $\mathrm {RTSeq}_n$ and compositions of building operations, these relations correspond to $\mathsf {e}_{i} \mathsf {r}_{i+1}^+=\mathsf {r}_{i}^+ \mathsf {e}_{i}$ and $\mathsf {r}_{i}^+ \mathsf {r}_{i+1}^+=\mathsf {r}_{i}^+ \mathsf {r}_{i}^+$ . Considering the relations at the level of binary forests leads to the following definition.

Definition 8.3. We say that $\widehat {F}, \widehat {G}\in \operatorname {BNestFor}_n$ are $\color {blue}{{colored\, Tamari\, equivalent}}$ , denoted by $\widehat {F}\overset {c}{\sim } \widehat {G}$ , if one can be transformed into the other by a sequence of $\color {blue}{{colored\, Tamari\, rotations}}$ shown below.

By the preceding discussion, we get the following result.

Every element of $\operatorname {BNestFor}_n$ can be transformed to some element of $\operatorname {BNestFor}^{\operatorname {nf}}_n$ by applying colored Tamari rotations repeatedly. We will prove that this normal form is unique in the next section. Figure 7 depicts a bicolored tree as well as its colored Tamari equivalent normal form.

Remark 8.6. Proposition 6.3 relates bicolored nested forests to factorizations of noncrossing partitions, where the colored Tamari equivalence allows certain relations $(i\,k)(j\,k)=(j\,k)(i\,j)$ .

Figure 7

A bicolored tree which is not in normal form (left), its Tamari-equivalent normal form (right), and the associated factorizations of $\boldsymbol {c}$ for each tree (below).

Figure 7 Long description

The image contains two panels.

Left panel: A bicolored tree with a root at the top represented by a white circle. The tree branches down to the right to another white circle, which then branches to a black circle. From this black circle, branches lead to another black circle and then a white circle. The leaves are marked with crosses on a horizontal line numbered 1 through 7. The structure is skewed heavily to the right. Below this tree is the factorization: (1 7) (3 7) bold (6 7) (4 6) (5 6) bold (2 3).

Right panel: The Tamari-equivalent normal form tree. It has a more balanced, symmetric structure. The root is a white circle at the top. It branches left to a white circle and right to a black circle. The left branch further divides into a black circle. The right branch further divides into a white circle and then a black circle. The leaves are similarly marked with crosses on a horizontal line numbered 1 through 7. Below this tree is the factorization: (3 7) (1 3) bold (6 7) (5 6) (4 5) bold (2 3).

In both trees, nodes are either hollow white circles or solid black circles, and the leaf nodes 1 through 7 serve as the base anchors.

8.2 The Bruhat maximal element of $X(\widehat {F})$ and uniqueness of normal form

We will need to understand the Bruhat order on $I_{\widehat {F}}$ . Since $X(\widehat {F})$ is a torus orbit closure in $\mathrm {Fl}_{n}$ , we have that $I_{\widehat {F}}$ is a flag matroid [Reference Gelfand, Goresky, MacPherson and Serganova22]. Hence we have the following fact.

In order to describe this distinguished element, we define a new map. For $\widehat {F}$ in $\operatorname {BNestFor}^{\operatorname {nf}}_n$ , let $\text{ RC } (\widehat {F})$ denote the set of internal nodes that are right children. By definition of normal form, we note that all nodes in $\text{ RC } (\widehat {F})$ are black. We define a map

$$\begin{align*}\begin{array}{rccc} \operatorname{ForToNC}: &\operatorname{BNestFor}^{\operatorname{nf}}_n&\to& \operatorname{NC}_{n} \\ & \widehat{F} & \mapsto & \operatorname{NCPerm}\Big(\widehat{F} \setminus \text{ RC }(\widehat{F})\Big) \end{array} \end{align*}$$

where $\widehat {F} \setminus \text{ RC } (\widehat {F})$ denotes the bicolored nested forest obtained by left edge deletion for each node $v \in \text{ RC } (\widehat {F})$ and contracting it in the sense of Proposition 5.4. This is a natural extension of the map on $\operatorname {\mathsf {Forest}}_n$ which we defined in [Reference Bergeron, Gagnon, Nadeau, Spink and Tewari6, Section 7.2] using the same notation.

Example 8.8. We have

Lemma 8.9. For $\widehat {F} = \widehat {G} \cdot \mathsf {x}_{i} \in \operatorname {BNestFor}^{\operatorname {nf}}_{n}$ with $\widehat {G}\in \operatorname {BNestFor}^{\operatorname {nf}}_{n-1}$ , we have

$$\begin{align*}\operatorname{ForToNC}(\widehat{F} ) = \begin{cases} \Psi_i^-\operatorname{ForToNC}(\widehat{G})&\text{if }\ \mathsf{x}_{i} = \mathsf{r}_{i}^{-} \text{ or leaf } i \text{ is a right child in } \widehat{G}\\ \Psi_i^+\operatorname{ForToNC}(\widehat{G})&\text{otherwise.}\end{cases} \end{align*}$$

Proof. From the definitions we immediately verify

$$\begin{align*}\widehat{F} \setminus \text{ RC }(\widehat{F}) = \begin{cases} (\widehat{G} \setminus \text{ RC }(\widehat{G})) \cdot \mathsf{r}_{i}^{-} &\text{if } \mathsf{x}_{i} = \mathsf{r}_{i}^{-} \text{ or leaf } i \text{ is a right child in } \widehat{G}\\ (\widehat{G} \setminus \text{ RC }(\widehat{G})) \cdot \mathsf{x}_{i} &\text{otherwise}\end{cases} \end{align*}$$

after which the formulas (6.1) complete the proof.

Proof. We proceed by induction on n. For $n = 1$ , $\operatorname {ForToNC}(\widehat {F})$ is the only element of $I_{\widehat {F}}$ . For $n> 1$ , we have $\widehat {F} = \widehat {G} \cdot \mathsf {x}_{i}$ for $\mathsf {x}_{i} \in \{\mathsf {r}_{i}^{\pm }, \mathsf {e}_{i}\}$ and $\widehat {G}\in \operatorname {BNestFor}^{\operatorname {nf}}_{n-1}$ . Setting $w = \operatorname {ForToNC}(\widehat {F})$ and $u = \operatorname {ForToNC}(\widehat {G})$ , Lemma 8.9 states that $w = \Psi ^{\epsilon }_{i}(u)$ for $\epsilon \in \{+, -\}$ .

Figure 8

An example of the construction in Proposition 8.11 for $n=12$ (left) and all elements $\widehat {(F,S)}\in \operatorname {BNestFor}^{\operatorname {nf}}_4$ for $F = \widehat {F}(\mathsf {r}_{1}^{-}\mathsf {e}_{1}\mathsf {e}_{1}\mathsf {e}_{3}) \in \operatorname {\mathsf {Forest}}_{4}$ (right).

Figure 8 Long description

The left panel displays a vertical mapping. At the bottom is a forest structure labeled (F, S) consisting of two blue-lined trees over a horizontal axis marked with x symbols. Five nodes in this forest are circled in red. An upward arrow with a bar at the base points to the top structure labeled (F hat, S). In this top version, the red circles are removed, and some solid black nodes have been replaced with hollow white nodes.

The right panel is a Hasse diagram showing a lattice of eight individual forest elements.

* At the top level is a single forest with two hollow white nodes.

* The second level down contains three forests connected to the top by lines, showing different combinations of one solid black node and one hollow white node.

* The third level down contains three forests, showing further variations with more solid black nodes.

* At the bottom level is a single forest where the primary nodes are solid black.

Each forest in the lattice is composed of blue lines connecting nodes over a base of four x marks on a horizontal line.

For $\mathsf {x}_{i} = \mathsf {r}_{i}^{\epsilon }$ , we note that both $\Psi _{i}^{-}$ and $\Psi _{i}^{+}$ preserve the Bruhat order. This follows, for instance, from the tableau criterion [Reference Björner and Brenti14, Theorem 2.6.3]. Thus, using our inductive hypothesis on u, we have that $w = \Psi ^{\epsilon }_{i}(u)$ is the Bruhat maximum of $I_{\widehat {F}} = \Psi ^{\epsilon }_{i}(I_{\widehat {G}})$ .

Now suppose that $\mathsf {x}_{i} = \mathsf {e}_{i}$ . By Fact 8.7, it suffices to show that w is greater than all adjacent fixed points in $I_{\widehat {F}}$ . As $I_{\widehat {F}}$ is a combinatorial cube and $\Psi ^{\epsilon }_{i}$ is a face inclusion, all but one of these adjacent elements are contained in $\Psi ^{\epsilon }_{i}(I_{\widehat {G}})$ and therefore covered by the previous argument. The remaining adjacent fixed point is $ws_{i}$ , so what remains is to show that $w(i)> w(i+1)$ . Let v be the (black) node associated to $\mathsf {x}_{i}$ . If v is a right child in $\widehat {F}$ , then $w(i) = i$ and $w(i+1) < i+1$ . If v is not a right child, then i is the smallest element of its cycle, which must also contain $i+1$ , so again $w(i)> w(i+1)$ .

We need an alternative construction related to the map $\operatorname {ForToNC}$ . Let $(F,S)$ be a pair consisting of an indexed forest $F\in \operatorname {\mathsf {Forest}}_n$ and a subset $S \subseteq \operatorname {IN}(F)$ . We define

$$ \begin{align*}\widehat{(F,S)}\in \mathrm{Face}(F)\subset \operatorname{BNestFor}^{\operatorname{nf}}_n\end{align*} $$

by performing left-edge deletion on all vertices in $S\cap \text{ RC } (F)$ , and coloring the remaining vertices of S white. An example is shown in Figure 8 (left).

We are now ready to prove Theorem 8.1.

8.3 Combinatorics of torus orbit closures

In this section, we describe several enumerative aspects of the complex $\mathrm {Complex}(\mathrm {QFl}_{n})$ . We moreover show in Proposition 8.15 that the number of such orbits for $n=1,2,3,\ldots $ is given by generalized Catalan numbers [Reference Sloane42, A064062], while counting them according to dimension gives a refinement known as $\color {blue}{{Borel's\, triangle}}$ [Reference Sloane42, A234950].

We now describe the cell structure of $\mathrm {Complex}(\mathrm {QFl}_{n})$ . In general the intersection of two torus orbit closures under inclusion is a union of one or more torus orbit closures. As shown on the right in Figure 6, the intersection of the top-dimensional orbit closures in $\mathrm {Complex}(\mathrm {QFl}_{3})$ is the (nondisjoint) union of two intervals. Theorem 8.1 shows that faces of $\mathrm {Complex}(\mathrm {QFl}_{n})$ are in bijection with the fixed point sets $I_{\widehat {F}} \subseteq \operatorname {NC}_{n}$ for $\widehat {F}\in \operatorname {BNestFor}_n$ , with inclusion of faces corresponding to inclusion of sets. The faces are also in bijection with normal form forests in $\operatorname {BNestFor}^{\operatorname {nf}}_n$ , but inclusion is harder to compute with these objects; in particular it is strictly stronger than the restriction of the order $\le _{re}$ .

Proposition 8.15. Let $G(z,u)=\sum _{n,k\geq 0}f_{n,k}z^nu^k$ , where $f_{n,k}$ is the number of torus orbits in $\mathrm {QFl}_n$ of dimension k. Then

(8.1) $$ \begin{align} G(z,u)=\frac{1+2u-\sqrt{1-4(u+1)z}}{2(z+u)}. \end{align} $$

Proof. We have that $f_{n,k}$ is the number of $\widehat {F}\in \operatorname {BNestFor}^{\operatorname {nf}}_n$ with k black nodes. First, let $G_{\boldsymbol {c}}(z,u)$ be the generating function for trees $T \in \operatorname {BNestFor}^{\operatorname {nf}}_n$ , so that $\operatorname {NCPerm}(T)=\boldsymbol {c}$ . Decomposition at the root gives a quadratic functional equation for $G_{\boldsymbol {c}}(z,u)$ that has the solution

(8.2) $$ \begin{align} G_{\boldsymbol{c}}(z,u)=\frac{1+2u-z-\sqrt{1+z^2-2(2u+1)z}}{2u}. \end{align} $$

For the general case, note that each $\widehat {F} \in \operatorname {BNestFor}^{\operatorname {nf}}_n$ is given by the choice of an element $w\in \operatorname {NC}_{n}$ and for each cycle $C = (c_{1}\,\cdots \,c_{k})$ of w, a tree in $\operatorname {BNestFor}^{\operatorname {nf}}_k$ . As in [Reference Chapoton and Nadeau18, §2.2], we can therefore apply the R-transform from free probability to obtain the equation $G(z,u)=G_{\boldsymbol {c}}(zG(z,u),u)$ . This can be solved using the quadratic equation for $G_{\boldsymbol {c}}(z,u)$ , from which we obtain the desired result.

The expression (8.1) is the generating function for Borel’s triangle [Reference Sloane42, A234950] whose entries have the explicit closed form

$$\begin{align*}f_{n,k}=\frac{1}{n}\binom{2n}{n-k-1}\binom{n+k-1}{k}.\end{align*}$$

From the expression (8.2), it follows that the enumeration for trees is given by the classical $\color {blue}{{large\, Schr\unicode{x00F6}der\, numbers}}$ [Reference Sloane42, A006318] and the refined version according to $|\widehat {F}|$ is [Reference Sloane42, A088617].

9.1 The paving

For each $F\in \operatorname {\mathsf {Forest}}_n$ , the Bott manifold structure of $X(F)$ from Section 5 gives an affine chart $C(F)$ around the T-fixed point $\operatorname {ForToNC}(F)\in X(F)$ . Explicitly, $C(F)$ is isomorphic to an affine space $\mathbb {A}^{|F|}$ of dimension $|F|$ and decomposes into sub-torus-orbits of $X(F)$ as

(9.1) $$ \begin{align} C(F) = \bigsqcup_{\substack{\widehat{H} \in \mathrm{Face}(F) \\ \operatorname{ForToNC}(F) \in I_{\widehat{H}}}} X^{\circ}(\widehat{H}). \end{align} $$

Theorem 9.1. The affine charts form a partition of $\mathrm {QFl}_n$ :

(9.2) $$ \begin{align} \mathrm{QFl}_n = \bigsqcup_{F\in\operatorname{\mathsf{Forest}}_n} C(F). \end{align} $$

Moreover, for any total ordering $F_1,F_2,\ldots ,F_{|\operatorname {NC}_{n}|}$ of $\operatorname {\mathsf {Forest}}_n$ that extends the pullback of the Bruhat order via $\operatorname {ForToNC}$ , we have $\bigsqcup _{i=1}^{k} C(F_{i})=\bigcup _{i=1}^k X(F_i)$ .

We prove the theorem after the following remark and lemma.

Remark 9.2. The decomposition in Theorem 9.1 can be interpreted as a partition of $\mathrm {Complex}(\mathrm {QFl}_{n})$ by associating each $C(F)$ with the half-open subspace of the moment polytope for $X(F)$ around the vertex $\operatorname {ForToNC}(F)$ ; see Figure 9.

Figure 9

The decomposition of $\mathrm {Complex}(\mathrm {QFl}_{3})$ induced by our affine paving of $\mathrm {QFl}_{3}$ as described in Remark 9.2; compare to Figure 6.

Figure 9 Long description

The diagram consists of four distinct geometric components arranged horizontally.

1. Far-left panel: A large orange-shaded triangle and a separate diagonal line segment below it. The triangle has a solid top-right vertex labeled with a blue square and red text (3 2 1). Its interior is labeled C followed by a tree graph with a root node branching into two nodes. The line segment below it has a blue square vertex at the top-left labeled (2 1)(3) and is labeled C followed by a tree graph where the root is on the left.

2. Center-left panel: A vertical line segment with a solid blue square at the top labeled (1)(3 2) and an open square at the bottom. It is labeled C followed by a tree graph where the root is on the right.

3. Center-right panel: A large light-blue shaded triangle. Its rightmost vertex is a solid blue square labeled (3 1)(2). The interior is labeled C followed by a tree graph with a root node branching into two nodes, mirrored from the first triangle. The left edge of this triangle is a dashed line with open square vertices.

4. Far-right panel: A single solid blue square point at the bottom, labeled (1)(2)(3). It is labeled C followed by a horizontal line with three x marks and no branching tree structure.

Proof of Theorem 9.1

By Proposition 8.14, we have

(9.3) $$ \begin{align} \mathrm{QFl}_n=\bigsqcup_{\widehat{F}\in\operatorname{BNestFor}^{\operatorname{nf}}_n}X^{\circ}(\widehat{F}). \end{align} $$

By Lemma 9.3 and Equation (9.1), we have

(9.4) $$ \begin{align} C(F) = \bigsqcup_{\substack{\widehat{H} \in \operatorname{BNestFor}^{\operatorname{nf}} \\ \operatorname{ForToNC}(\widehat{H})=\operatorname{ForToNC}(F)}} X^{\circ}(\widehat{H}). \end{align} $$

As $\operatorname {ForToNC}$ is surjective when restricted to $\operatorname {\mathsf {Forest}}_{n}$ , it follows immediately that Equation (9.4) coarsens the partition in Equation (9.3) into the one in Equation (9.2).

We now show that $\bigcup _{i=1}^k C(F_i)=\bigcup _{j=1}^k X(F_j)$ for any k. Since $C(F)\subset X(F)$ for any $F\in \operatorname {\mathsf {Forest}}$ , we only have to show that any $\mathcal {F}\in X(F_j)$ for some $j\leq k$ is included in $C(F_i)$ for some $i\leq k$ . We have ${\mathcal {F}\in X^{\circ }(\widehat {G})}$ for some $\widehat {G}\in \operatorname {BNestFor}^{\operatorname {nf}}_n$ such that $X^{\circ }(\widehat {G})\subset X(F_j)$ . By Theorem 8.1, this implies that ${I_{\widehat {G}}\subset I_{F_j}}$ . In particular, using the characterization of Proposition 8.10, we have $\operatorname {ForToNC}(\widehat {G})\le \operatorname {ForToNC}(F_j)$ in Bruhat order. By our choice of total order we then have that $\operatorname {ForToNC}(\widehat {G})=\operatorname {ForToNC}(F_i)$ for some $i\leq j$ . Because of (9.4) we then have $X^{\circ }(\widehat {G})\subset C(F_i)$ , and thus $\mathcal {F}\in C(F_i)$ , which concludes the proof.

Remark 9.4 (Points over $\mathbb {F}_q$ )

The definition of $\mathrm {QFl}_n$ and all arguments used so far make sense over any field, not just $\mathbb {C}$ , and so using (9.2) and (9.3) we may count the number of points of $\mathrm {QFl}_n$ over a finite field $\mathbb {F}_q$ . In this case $C(F)\simeq \mathbb {F}_q^{\operatorname {IN}(F)}$ has cardinality $q^{|F|}$ . Summing over all $F\in \operatorname {\mathsf {Forest}}_n$ we get

$$\begin{align*}\#\mathrm{QFl}_n(\mathbb{F}_q)=\sum_{F\in\operatorname{\mathsf{Forest}}_n}q^{|F|}=\sum_{k=0}^{n-1}c_{n,k}q^k\end{align*}$$

where $c_{n,k}=\frac {n-k}{n+k}\binom {n+k}{k}$ [Reference Nadeau and Tewari41]. Using (9.3) we get the alternative expression

$$\begin{align*}\#\mathrm{QFl}_n(\mathbb{F}_q)=\sum_{F\in\operatorname{BNestFor}_n}(q-1)^{|F|}=\sum_{k=0}^{n-1}f_{n,k}(q-1)^k\end{align*}$$

where the numbers $f_{n,k}$ were introduced in Section 8.3. These two expressions are polynomial in q, and thus one can extract $f_{n,k}=\sum _{m=k}^n\binom {m}{k}c_{n,m}$ . This gives another proof that the numbers $f_{n,k}$ are given by Borel’s triangle [Reference Sloane42, A234950] as seen in Section 8.3.

9.2 Paving with noncrossing Bruhat cells

We now give a combinatorial description of our affine paving for $\mathrm {QFl}_{n}$ . Throughout, we use the convention that a matrix M with entries in $\mathbb C \cup \{\ast , +\}$ represents the set of all matrices whose entries are, depending on the corresponding entry of M, either a particular complex number (if in $\mathbb C$ ) or taken freely from either $\mathbb C$ (if $\ast $ ) or $\mathbb C^{\times }$ (if $+$ ).

We first recall the combinatorial construction of Bruhat cells in $\mathrm {Fl}_{n}$ . The $\color {blue}{{inversion\, set}}$ of $w \in S_{n}$ is $\operatorname {Inv}(w)=\{(i,j) \;|\; i<j \text { and } w(i)>w(j)\}$ . For $w \in S_{n}$ , let $\color {blue}{M(w)}$ be the matrix with $1$ ’s in positions $(w(i), i)$ , $\ast $ ’s in positions $(w(j),i)$ for $(i, j) \in \operatorname {Inv}(w)$ , and $0$ ’s elsewhere; see, for example, Figure 10. Then $M(w)$ is isomorphic to an affine space where each $\ast $ represents a coordinate, and this gives a complete set of representatives for the Bruhat cell $BwB \subseteq \mathrm {Fl}_{n}$ . In order to reproduce the standard action on $\mathrm {Fl}_{n}$ , we have T act on elements of $M(w)$ by scaling the $k, \ell $ entry by the character $\chi _{k} \chi _{w(\ell )}^{-1}$ . Thus as a T-representation

(9.5) $$ \begin{align} M(w)\cong \bigoplus_{(i,j)\in \operatorname{Inv}(w)}\mathbb{C}_{\chi_{w(j)}\chi_{w(i)}^{-1}}. \end{align} $$

In order to state a similar result for $\mathrm {QFl}_{n}$ , we introduce an important subset of the inversion set of a noncrossing partition.

Definition 9.5. The $\color {blue}{{noncrossing\, inversion\, set}}$ of $w \in \operatorname {NC}_{n}$ is

$$\begin{align*}\operatorname{Inv}_{\operatorname{NC}}(w):= \{(i,j)\in \operatorname{Inv}(w):w(i,j)\in \operatorname{NC}_n\}. \end{align*}$$

Definition 9.6. The $\color {blue}{{noncrossing\, Bruhat\, cell}}$ for $w\in \operatorname {NC}_n$ is the set represented by the matrix $M_{\operatorname {NC}}(w)$ with entries $1$ in position $(w(i), i)$ for each $i \in [n]$ , $\ast $ in position $(w(j),i)$ for each $(i,j)\in \operatorname {Inv}_{\operatorname {NC}}(w)$ , and $0$ elsewhere. For $F\in \operatorname {\mathsf {Forest}}_n$ , we define

$$\begin{align*}M_{\operatorname{NC}}(F) := M_{\operatorname{NC}}(\operatorname{ForToNC}(F)). \end{align*}$$

Figure 10

The Bruhat cell and noncrossing Bruhat cell for $w = 612543$ .

For $w = \operatorname {ForToNC}(F)$ , we have a canonical identification

$$\begin{align*}M_{\operatorname{NC}}(w)\cong \bigoplus_{(i,j)\in \operatorname{Inv}_{\operatorname{NC}}(w)}\mathbb{C}_{\chi_{w(j)}\chi_{w(i)}^{-1}}\subset \bigoplus_{(i,j)\in \operatorname{Inv}(w)}\mathbb{C}_{\chi_{w(j)}\chi_{w(i)}^{-1}}\cong M(w). \end{align*}$$

See, for example, Figure 10. We now state the main result of the section.

Theorem 9.7. For $w\in \operatorname {NC}_n$ we have $\mathrm {QFl}_n\cap BwB=M_{\operatorname {NC}}(w)B$ and

(9.6) $$ \begin{align} \mathrm{QFl}_n = \bigsqcup_{w\in \operatorname{NC}_n} M_{\operatorname{NC}}(w)B. \end{align} $$

Moreover, for any total ordering $w_1,w_2,\ldots ,w_{|\operatorname {NC}_{n}|}$ of $\operatorname {NC}_n$ that extends the Bruhat order we set

$$\begin{align*}X_k = \bigcup_{i=1}^{k} M_{\operatorname{NC}}(w_{i})B, \end{align*}$$

then each $X_k$ is closed, $X_{1} \subseteq X_{2} \subseteq \cdots \subseteq X_{|\operatorname {NC}_{n}|} = \mathrm {QFl}_{n}$ , and $X_{k+1}\setminus X_k=M_{\operatorname {NC}}(w_{k+1})B$ .

We will prove this result at the end of the section after some preparation.

Our first step is to extend a combinatorial characterization of the noncrossing inversion set stated in [Reference Bergeron, Gagnon, Nadeau, Spink and Tewari6, Remark 8.18]. For $\widehat {F} \in \operatorname {BNestFor}_{n}$ , the $\color {blue}{{spread}}$ of $v \in \operatorname {IN}(\widehat {F})$ is the pair $(i, j)$ consisting of the leftmost leaf descendant i of v and the rightmost leaf descendant j of v. Figure 11 depicts a bicolored nested forest in which each internal node is labeled by its spread. Note that the spread of an internal node v is not necessarily the transposition $\tau _v$ assigned to each $v \in \operatorname {IN}(\widehat {F})$ earlier in Section 6.

Figure 11

A bicolored nested forest with spreads recorded for each internal node.

Figure 11 Long description

A tree diagram sits above a horizontal baseline marked with integers 1 through 12, each indicated by an x symbol. The tree consists of black filled nodes and white hollow nodes, each labeled with a red coordinate pair representing a spread.

At the top, the root node is a black dot labeled (1, 12). It branches left to a white node (1, 9) and right to a black node (10, 12).

From the white node (1, 9), the tree branches left to a black node (1, 3) and right to a white node (4, 9).

From the black node (1, 3), it branches to leaf nodes 1 and 2, with an intermediate black node (2, 3) branching to leaf nodes 2 and 3.

From the white node (4, 9), it branches left to a black node (4, 5) which connects to leaf nodes 4 and 5, and right to a black node (8, 9) which connects to leaf nodes 8 and 9. Between these, a white node (6, 7) connects to leaf nodes 6 and 7.

On the far right, the black node (10, 12) branches down to leaf nodes 10 and 12, with an intermediate connection to leaf node 11.

Proposition 9.9. If $\widehat {F} \in \operatorname {BNestFor}^{\operatorname {nf}}_{n}$ and $w = \operatorname {ForToNC}(\widehat {F})$ , then

$$\begin{align*}\{(i, j) \in \operatorname{Inv}_{\operatorname{NC}}(w) \;|\; w(i, j) \in I_{\widehat{F}} \} = \{\text{spreads } (i, j) \text{ of black nodes } v \in \operatorname{IN}(\widehat{F})\}. \end{align*}$$

In particular, if ${F} \in \operatorname {\mathsf {Forest}}_{n}$ , then $\operatorname {Inv}_{\operatorname {NC}}(w)$ is the set of all spreads for $v \in \operatorname {IN}({F})$ .

We next give a recursive characterization of the charts in $\mathrm {Fl}_{n}$ determined by the $M_{\operatorname {NC}}(F)$ using the building operations from Section 3.

Proposition 9.10. The representatives of the noncrossing Bruhat cells in $\mathrm {Fl}_{n}$ are characterized recursively by $M_{\operatorname {NC}}(\mathsf {r}_{1}^{-})B = \mathrm {Fl}_{1}$ and for $F = G \cdot \mathsf {e}_{i} \in \operatorname {\mathsf {Forest}}_{n}$ ,

$$\begin{align*}M_{\operatorname{NC}}(F)B = \begin{cases} \mathbb{G}_{i}M_{\operatorname{NC}}(G)B \sqcup \Psi_i^{-}M_{\operatorname{NC}}(G)B & \text{if leaf } i \text{ is a right child in } G \\ \mathbb{G}_{i}M_{\operatorname{NC}}(G)B \sqcup \Psi_i^{+}M_{\operatorname{NC}}(G)B & \text{otherwise.} \end{cases} \end{align*}$$

We prove the proposition using a technical argument involving a modified version of the $\mathbb {G}_{i}$ operation. Given a matrix M with column i equal to the jth basis vector for $j < i$ , let $\mathbb {G}_i'M$ be obtained from $\Psi _i^-M$ by setting the $j,i$ -entry to $+$ .

Example 9.11. We have

$$\begin{align*}\mathbb{G}_2\begin{bmatrix}a&1&b\\c&0&d\\e&0&f\end{bmatrix}=\begin{bmatrix}a&1&0&b\\0&+&1&0\\c&0&0&d\\e &0&0&f\end{bmatrix} \qquad\text{and}\qquad \mathbb{G}^{\prime}_2\begin{bmatrix}a&1&b\\c&0&d\\ e &0&f\end{bmatrix}=\begin{bmatrix}a&+&1&b\\0&1&0&0\\c&0&0&d\\e &0&0&f\end{bmatrix} \end{align*}$$

Proof of Proposition 9.10

Let $w = \operatorname {ForToNC}(F)$ and $u = \operatorname {ForToNC}(G)$ . By Lemma 8.9, we have $w = \Psi _{i}^{\pm }(u)$ . Further, by Proposition 9.9, we have

$$\begin{align*}\operatorname{Inv}_{\operatorname{NC}}(w) = \{ (j + \delta_{j \ge i}, k + \delta_{k \ge i}) \;|\; (j, k) \in \operatorname{Inv}_{\operatorname{NC}}(u)\} \cup \{(i, i+1)\}. \end{align*}$$

Thus, if we write $M_{\operatorname {NC}}(w; x)$ for the matrix obtained from $M_{\operatorname {NC}}(w)$ by setting the $(w(i+1), i)$ entry equal to x, then we have

$$\begin{align*}M_{\operatorname{NC}}(w; 0) = \begin{cases} \Psi^{-}_{i}M_{\operatorname{NC}}(u) & \text{if leaf } i \text{ is a right child in }G \\ \Psi^{+}_{i}M_{\operatorname{NC}}(u) & \text{otherwise.} \end{cases} \end{align*}$$

Moreover, $M_{\operatorname {NC}}(w) = M_{\operatorname {NC}}(w; 0) \sqcup M_{\operatorname {NC}}(w; +)$ , so we complete the proof by showing that

$$\begin{align*}\mathbb{G}_{i}M_{\operatorname{NC}}(u)B = M_{\operatorname{NC}}(w; +)B. \end{align*}$$

If leaf i is not a right child in G, then $w(i)> w(i+1)$ and we have a direct equality $M_{\operatorname {NC}}(w; +) = \mathbb {G}_{i}M_{\operatorname {NC}}(u)$ . If leaf i is a right child in G, then $w(i) < w(i+1)$ and $M_{\operatorname {NC}}(w; +) = \mathbb {G}_{i}'M_{\operatorname {NC}}(u)$ , so we must use Lemma 9.12 to relate $\mathbb {G}_{i}'$ and $\mathbb {G}_{i}$ . Indeed, as a right child i can never be the left endpoint of an internal node, Proposition 9.9 implies that u has no noncrossing inversion ending in i. Thus $M_{\operatorname {NC}}(G)$ has no $\ast $ ’s in column i and the hypotheses of Lemma 9.12 are satisfied, giving $M_{\operatorname {NC}}(w; +)B = \mathbb {G}_{i}M_{\operatorname {NC}}(u)B$ .

Proposition 9.13. Let $w \in \operatorname {NC}_{n}$ , and let F be the unique forest in $\operatorname {\mathsf {Forest}}_n$ such that $w=\operatorname {ForToNC}(F)$ . We have $M_{\operatorname {NC}}(F)B = C(F)$ , and as a consequence there is a T-equivariant isomorphism

$$\begin{align*}C(F) \cong M_{\operatorname{NC}}(w) \cong \bigoplus_{(i,j)\in \operatorname{Inv}_{\operatorname{NC}}(w)}\mathbb{C}_{\chi_{w(j)}\chi_{w(i)}^{-1}}. \end{align*}$$

Proof. We show that $C(F)$ agrees with the recursive characterization of $M_{\operatorname {NC}}(F)B$ given in Proposition 9.10. As $C((\mathsf {r}_{1}^{-})^{n-k}) = \operatorname {id}_{S_{n-k}}\in \mathrm {Fl}_{n-k}$ , this amounts to showing that for $F = G \cdot \mathsf {e}_{i} \in \operatorname {\mathsf {Forest}}_{n}$ ,

$$\begin{align*}C(F)=\begin{cases} \mathbb{G}_iC(G) \sqcup \Psi_i^-C(G)&\text{if leaf } i \text{ is a right child in } G\\ \mathbb{G}_iC(G) \sqcup \Psi_i^+C(G)&\text{otherwise.}\end{cases} \end{align*}$$

By the definition of $\mathrm {Face}(\widehat {F})$ and the $\le _{re}$ order, we have

$$\begin{align*}\mathrm{Face}(\widehat{F}) = \{\widehat{H}'\cdot \mathsf{x}_{i} \;|\; \mathsf{x}_{i} \in \{\mathsf{r}_{i}^{-}, \mathsf{r}_{i}^{+}, \mathsf{e}_{i}\} \text{ and } \widehat{H}' \in \mathrm{Face}(G)\}. \end{align*}$$

If we furthermore write $w = \operatorname {ForToNC}(F)$ and $u = \operatorname {ForToNC}(G)$ , then by Lemma 8.9 we have $w = \Psi ^{\epsilon }_{i}(u)$ for $\epsilon \in \{+, -\}$ . Thus

$$\begin{align*}\{\widehat{H} \in \operatorname{BNestFor}_{n} \;|\; w \in I_{\widehat{H}} \} = \{\widehat{H}'\cdot \mathsf{x}_{i} \;|\; \mathsf{x}_{i} \in \{\mathsf{r}_{i}^{\epsilon}, \mathsf{e}_{i}\}, \widehat{H}' \in \operatorname{BNestFor}_{n-1}, \text{ and } u \in I_{\widehat{H}'}\}. \end{align*}$$

By intersecting the sets above, we arrive at the following description of $C(F)$ :

$$ \begin{align*} C(F) & = \bigsqcup_{\substack{ \widehat{H}' \le_{re} G \\ u \in I_{\widehat{H}}}} \Big( X^{\circ}(\widehat{H} \cdot \mathsf{e}_{i}) \sqcup X^{\circ}(\widehat{H} \cdot \mathsf{r}_{i}^{\epsilon})\Big) \\ & = \bigsqcup_{\substack{ \widehat{H}' \le_{re} G \\ u \in I_{\widehat{H}}}} \Big( \mathbb{G}_{i} X^{\circ}(\widehat{H}) \sqcup \Psi_{i}^{\epsilon} X^{\circ}(\widehat{H})\Big) = \mathbb{G}_{i}C(G) \sqcup \Psi_{i}^{\epsilon}C(G) \end{align*} $$

This completes the proof, as by Lemma 8.9 $\epsilon = -$ if leaf i is a left child and $+$ otherwise.

We can now complete the proof of Theorem 9.7.

10.1 Characterization with Plücker functions

This section proves Theorem B, which states that

$$\begin{align*}\mathrm{QFl}_n=\hspace{-0.75em}\bigcap_{w\in S_n\setminus \operatorname{NC}_n}\hspace{-0.75em}\{\mathrm{Pl}_{w} =0\}. \end{align*}$$

The proof is given at the end of the section. We begin with a classical observation, which can be found for instance in [Reference Lee, Masuda and Park35, Proposition 2.6].

Every element of $BwB$ has a unique representative h in the set $M(w)$ defined in Section 9. Thus when restricted to $BwB$ , we can view the Plücker functions as polynomials in the matrix entries which are not uniformly $0$ or $1$ across all of $M(w)$ , namely the $h_{w(j), i}$ for $(i, j) \in \operatorname {Inv}(w)$ . Going forward, we define a $\mathbb Z^{n}$ -grading on such polynomials by setting

$$\begin{align*}\operatorname{degree}(h_{w(j), i}) = e_{w(j)} - e_{w(i)}, \end{align*}$$

where $e_{k}$ denotes the kth standard basis vector. This grading is the weight of the character of the T-action on each entry of $M(w)$ under the action of T as described in Section 9.