# Schubert cells of mixed type in complex Lagrangian Grassmannians

Hyunmoon Kim

###### Abstract

We describe CW decompositions of complex Lagrangian Grassmannians, that
contain as subcomplexes, CW decompositions of real Lagrangian Grassmannians by
Schubert-Arnol’d cells. The degrees of attaching maps are explicitly computed
in terms of quantities that can be read off from the corresponding shifted
Young diagrams of mixed type. The signs are determined by a choice of
lexicographical ordering on coordinates. As an immediate consequence, we
obtain the homotopy extension property for real and complex Lagrangian
Grassmannians. We also show some torsion classes in the integral homology of
the real Lagrangian Grassmannian are contractible inside the complex
Lagrangian Grassmannian.

##  1 Introduction

This paper solves a problem in algebraic topology motivated by geometric
quantization. A key ingredient in geometric quantization is given by
polarizations, which can loosely be described as some information to keep
track of choices of observables that (Poisson) commute. At a linear level,
these choices can be viewed as points on complex (or real) Lagrangian
Grassmannians. As these Grassmannians have nontrivial topology, topologically
nontrivial families of such choices exist. A famous example is the Maslov
cycle. Its relevance in quantization was first observed by Maslov (cf Section
II.2.2 [Mas72]) and understood by Arnol’d in [Arn67] in terms of the geometry
of the real Lagrangian Grassmannian.

For real Lagrangian Grassmannians, higher dimensional homology classes were
described by Arnol’d in his seminar (see footnote in Fuks [Fuk68]) and the
cohomology ring structures were studied by Borel [Bor51] (Proposition 31.4,
for the oriented Lagrangian Grassmannian), [Bor53] (Theorem 12.1), and Fuks
[Fuk68]. The torsion classes are all $2$-torsion, the $\mathbb{Z}_{2}$
cohomology ring is an exterior algebra, and the integral cohomology ring
modulo torsion is another exterior algebra (see Section 22 of Vassilyev
[Vas88] for an exposition). The cellular decompositions into Schubert-Arnol’d
cells were described Fuchs [Fuc04], with the incidence coefficients specified
up to sign. The signs were recently computed explicitly in Rabelo [Rab16] from
Lie theoretic techniques.

The cohomology ring of complex Lagrangian Grassmannians were also studied by
Borel [Bor51] (Theorem 26.1). Complex Lagrangian Grassmannians also have
cellular decompositions into complex Schubert cells. These cells appear in
only even degrees, so all attaching maps have degree zero, and there are no
torsion classes in integral homology. These structures were described by Borel
in [Bor54] (Theorem 3), Bernstein-Gel’fand-Gel’fand in [BGG73] (Theorem 2) and
revisited by Pragacz in [Pra91] in the language of shifted Young diagrams,
together with a purely combinatorial description of the integral cohomology
ring.

While these CW complexes are concrete combinatorial models for real and
complex Lagrangian Grassmannians, they are insufficient to describe the
induced map of chain complexes for the natural embedding of real Lagrangian
Grassmannians inside complex Lagrangian Grassmannians. We construct CW
decompositions of complex Lagrangian Grassmannians, that contain the CW
complexes of real Lagrangian Grassmannians by real Schubert cells (Schubert-
Arnol’d cells) as subcomplexes. They are subdivisions of the CW structure
given by complex Schubert cells. Here are precise statements.

######  Theorem.

There is a partition of the complex Lagrangian Grassmannian of
$\mathbb{R}^{2n}$

$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})=\bigsqcup_{\lambda,\mu:\mu\leq\lambda}C_{\lambda+i\mu}$

where $\lambda$, $\mu$ are shifted Young diagrams associated to subsets of
$\\{1,2,...,n\\}$ such that the following holds:

  1. 1.

Each $C_{\lambda+i\mu}$ is diffeomorphic to a product of finitely many copies
of $\mathbb{R}$, $\mathbb{R}^{\times}$, and $\mathbb{C}$, of real dimension
$|\lambda|+|\mu|$.

  2. 2.

The partition is a stratification with the following frontier condition:

$C_{\lambda+i\mu}\subseteq\overline{C_{\lambda^{\prime}+i\mu^{\prime}}}\iff
C_{\lambda+i\mu}\cap\overline{C_{\lambda^{\prime}+i\mu^{\prime}}}\neq\phi\iff\lambda\leq\lambda^{\prime}\text{
and }\mu\leq\mu^{\prime}.$

  3. 3.

When $\mu$ is the empty diagram, $C_{\lambda+i\mu}$ is a real Schubert cell
$C_{\lambda}^{\mathbb{R}}$ of the embedded image of
$\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})$ in
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$ by
$\cdot\otimes_{\mathbb{R}}\mathbb{C}$.

  4. 4.

Each complex Schubert cell $C_{\lambda}$ of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$ partitions into
$\bigsqcup_{\mu:\mu\leq\lambda}C_{\lambda+i\mu}$.

######  Theorem.

Let $\\{C_{\lambda+i\mu,\varepsilon}\\}_{\varepsilon\in\mathcal{E}(\mu)}$ be
the connected components of $C_{\lambda+i\mu}$. There is a CW decomposition of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$ having the
$C_{\lambda+i\mu,\varepsilon}$’s as cells and a closed formula for the degrees
for the attaching maps $\partial C_{\lambda+i\mu,\varepsilon}\to
C_{\lambda^{\prime}+i\mu^{\prime},\varepsilon^{\prime}}$ (equations (14),
(15), (16)).

By a CW complex, we refer to the definitions of Sections 4, 5 of Whitehead
[Whi49]. In particular, these definitions require attaching maps (or
characteristic maps) to be realized by actual maps on the boundaries of balls.
We note that in the literature the terms cell, cellular, CW complexes are
often used in a weaker sense (eg Borel [Bor54]), or only the incidence
coefficients are provided (eg Fuchs [Fuc04], Rabelo [Rab16]). We provide an
explicit construction of the attaching maps in Proposition 8.

We call the cells of this complex _Schubert cells of mixed type._ They are
indexed by shifted Young diagrams of Pragacz [Pra91] with additional labels on
the boxes. The degrees of the attaching maps are obtained using a
straightforward, but involved computation in row reduction. It agrees up to
sign with Fuchs [Fuc04] and Rabelo [Rab16] (and has a different sign
convention from [Rab16]). Some low dimensional examples are given in Figures 1
and 2.

An immediate consequence of this realization is that we have the homotopy
extension property for the pair
$(\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n}),(\cdot\otimes_{\mathbb{R}}\mathbb{C})(\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})))$.
This construction is compatible with adding more variables (Corollary 20). In
fact, we can realize $\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})$ as a
subcomplex of any $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2(n+m)})$, and
with all inclusion maps being compatible.

This construction is relevant in geometric quantization for the following
reason. Geometric quantization requires a passage to the complex numbers, as
observables are quantized from real valued smooth functions on symplectic
manifolds to complex linear Hermitian operators on a Hilbert space. Locally,
this task is performed by tensoring with $\mathbb{C}$. Our construction
enables us to understand how some topologically nontrivial families of real
Lagrangian subspaces behave after complexification. We refer to Corollary 21
for some examples.

In Section 2 we revisit the classical Schubert (Schubert-Arnol’d) calculus for
real and complex Lagrangian Grassmannians in the language of shifted Young
diagrams. In Section 3 we describe the Schubert calculus for Schubert cells of
mixed type.

###  1.1 Notation

Let $\omega$ be a nondegenerate antisymmetric $\mathbb{R}$-bilinear form on
$\mathbb{R}^{2n}$, and we will identify
$\mathbb{R}^{2n}\otimes_{\mathbb{R}}\mathbb{C}\cong\mathbb{C}^{2n}$ with
$\mathbf{v}\otimes_{\mathbb{R}}a\mapsto a\mathbf{v}$. We will denote the
$\mathbb{C}$ bilinear extension of $\omega$ to $\mathbb{C}^{2n}$ by
$\omega^{\mathbb{C}}$. A real Lagrangian subspace $L$ of $\mathbb{R}^{2n}$ is
an $n$ dimensional real vector subspace of $\mathbb{R}^{2n}$ such that the
restriction of $\omega$ to $L\times L$ vanishes. A complex Lagrangian subspace
$\Gamma$ of $\mathbb{C}^{2n}$ is an $n$ dimensional complex vector subspace of
$\mathbb{C}^{2n}$ such that the restriction of $\omega^{\mathbb{C}}$ to
$\Gamma\times\Gamma$ vanishes. We will refer to the Grassmannian manifold (cf
Section 3.3 of Arnol’d [Arn67]) consisting of all real Lagrangian subspaces of
$\mathbb{R}^{2n}$ by $\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})$ and
the Grassmannian manifold consisting of all complex Lagrangian subspaces of
$\mathbb{C}^{2n}$ as $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$.

centertableaux, boxsize = 5pt

Figure 1: $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2})$ can be identified
with the complex projective line. In this case the Schubert cells of mixed
type are $C_{\ytableaushort{+}}=\\{[x+iy:1]:y>0\\}$,
$C_{\ytableaushort{-}}=\\{[x+iy:1]:y<0\\}$,
$C_{\ytableaushort{\circ}}=\\{[x:1]\\}$, and $C_{\phi}=\\{[1:0]\\}$. Figure 2:
A diagrammatic description of the CW decomposition of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{4})$ by Schubert cells of mixed
type. An empty box, a box with a circle, a box with a $\pm$ denotes
respectively, a copy of $\mathbb{C}$, a copy of $\mathbb{R}$, and a copy of
$\mathbb{R}\times i\mathbb{R}^{\pm}$. Single headed arrows represent attaching
maps of degree $\pm 1$, double headed arrows represent attaching maps of
degree $\pm 2$, and filled, dashed arrows represent, respectively attaching
maps with positive, negative signs. No arrows means either the cells are not
incident or that they attach with degree $0$. The CW decomposition of
$\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{4})$ is the subcomplex consisting
of rightmost diagrams in every row except the first row, and the CW
decomposition of $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2})$ is in the
bottom right corner.

##  2 Review of Schubert cells

###  2.1 Shifted Young diagrams of Schubert cells

For the real Lagrangian Grassmannian, their Schubert stratification has been
developed independently by Arnol’d (cf [Fuc04]). We will review the Schubert
cells of the complex Lagrangian Grassmannian following Pragacz [Pra91], with
their indexing by shifted Young diagrams.

There are $2^{n}$ Schubert cells of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$, corresponding to the
$2^{n}$ subsets $J$ of $[n]:=\\{1,2,\cdots,n\\}$.

The reduced row echelon forms of these Schubert cells are described as
follows. We will fix a Darboux basis of $\mathbb{R}^{2n}$,
$\\{\mathbf{e}_{1},\cdots,\mathbf{e}_{n},\mathbf{f}_{1},\cdots,\mathbf{f}_{n}\\}$,
so that

$\omega(\mathbf{e}_{j},\mathbf{e}_{k})=\omega(\mathbf{f}_{j},\mathbf{f}_{k})=0\quad\text{and}\quad\omega(\mathbf{e}_{j},\mathbf{f}_{k})=\begin{cases}1&\mbox{if
}j=k\\\ 0&\mbox{otherwise.}\end{cases}$

We will denote

$\mathbf{v}=\sum_{j=1}^{n}q_{j}\mathbf{e}_{j}+\sum_{k=1}^{n}p_{k}\mathbf{f}_{k}\in\mathbb{C}^{2n}$

as a row vector, with the $p_{k}$’s in reverse order:

$[\mathbf{v}]_{\\{\mathbf{e},\mathbf{f}\\}}:=\left[\begin{array}[]{ccc|ccc}q_{1}&\cdots&q_{n}&p_{n}&\cdots&p_{1}\end{array}\right].$

If $\Gamma$ is a complex Lagrangian subspace of $\mathbb{C}^{2n}$, there is a
unique $n\times 2n$ complex matrix in reduced row echelon form corresponding
to $\Gamma$ in the fixed Darboux basis. Since $\Gamma$ is Lagrangian,
$\mathbf{e}_{k}$ and $\mathbf{f}_{k}$ cannot both be pivot locations for two
different row vectors of the reduced row echelon form. Let $I:=[n]\backslash
J$. Then denote by $j_{k}$ the $k$th smallest element of $J$ and by $i_{\ell}$
the $\ell$th smallest element of $I$ so that $J=\\{j_{k}\\}_{k=1}^{|J|}$ and
$I=\\{i_{\ell}\\}_{\ell=1}^{|I|}$. Then the reduced row echelon form
corresponding to $J$ is such that the $k$th row has pivots at the $k$th
element of

$\mathbf{e}_{i_{1}},\cdots,\mathbf{e}_{i_{|I|}},\mathbf{f}_{j_{|J|}},\cdots\mathbf{f}_{j_{1}},$

with additional relations (equations (1), (2)) imposed on the entries by the
Lagrangian condition.

More precisely, an $n\times 2n$ matrix with complex entries, in the reduced
row echelon form of corresponding to $J$ has its $k$th row vector as
$[\mathbf{v}_{k}]_{\\{\mathbf{e},\mathbf{f}\\}}$ if $1\leq k\leq|I|$ and as
$[\mathbf{w}_{n-k+1}]_{\\{\mathbf{e},\mathbf{f}\\}}$ if $|I|<k\leq n$, where

$\displaystyle\mathbf{v}_{a}$ $\displaystyle:=$
$\displaystyle\sum_{\begin{subarray}{c}1\leq i<i_{a}\\\ i\notin
I\end{subarray}}r_{ai}\mathbf{e}_{i}+\mathbf{e}_{i_{a}}\quad 1\leq a\leq|I|$
$\displaystyle=:$ $\displaystyle\sum_{\begin{subarray}{c}1\leq i\leq n\\\
i\notin I\end{subarray}}r_{ai}\mathbf{e}_{i}$ $\displaystyle\mathbf{w}_{b}$
$\displaystyle:=$ $\displaystyle\sum_{\begin{subarray}{c}1\leq j\leq n\\\
j\notin
I\end{subarray}}q_{bj}\mathbf{e}_{j}+\sum_{\begin{subarray}{c}j_{b}<\ell\leq
n\\\ \ell\notin
J\end{subarray}}p_{b\ell}\mathbf{f}_{\ell}+\mathbf{f}_{j_{b}}\quad 1\leq
b\leq|J|$ $\displaystyle=:$ $\displaystyle\sum_{\begin{subarray}{c}1\leq j\leq
n\\\ j\notin
I\end{subarray}}q_{bj}\mathbf{e}_{j}+\sum_{\begin{subarray}{c}1\leq\ell\leq
n\\\ \ell\notin J\end{subarray}}p_{b\ell}\mathbf{f}_{\ell}.$

The additional relations imposed by the isotropy condition are the following

$\displaystyle\omega^{\mathbb{C}}(\mathbf{v}_{a},\mathbf{w}_{b})=$
$\displaystyle r_{aj_{b}}+p_{bi_{a}}$ $\displaystyle=0\quad\text{if
}i_{a}>j_{b}$ (1)
$\displaystyle\omega^{\mathbb{C}}(\mathbf{w}_{a},\mathbf{w}_{b})=$
$\displaystyle q_{aj_{b}}-q_{bj_{a}}$ $\displaystyle=0.$ (2)

We can assign and count the number of independent parameters is as follows.
All the coefficients of $\mathbf{v}_{a}$’s are determined by the coefficients
of $\mathbf{w}_{b}$’s, and among the $(2n-j_{b})-(|I|+(|J|-b))$ possibly
nonzero, nonpivot coefficients of $\mathbf{w}_{b}$, $b-1$ are determined by
the coefficients of $\mathbf{w}_{b^{\prime}}$ for $b^{\prime}<b$. So each
$\mathbf{w}_{b}$ contributes $j_{b}^{\vee}:=n-j_{b}+1$ additional independent
parameters, and there are a total of $\sum_{b=1}^{|J|}j_{b}^{\vee}$
independent parameters.

The sequence $j_{1}^{\vee}>\cdots>j_{|J|}^{\vee}$ (or equivalently,
$J^{\vee}:=\\{j_{b}^{\vee}\\}_{b=1}^{|J|}$) determines a shifted Young diagram

$\lambda_{J^{\vee}}:=\\{(k,\ell)\in\mathbb{Z}\times\mathbb{Z}:1\leq
k\leq|J|,\quad k\leq\ell\leq j_{k}^{\vee}+k-1\\}$

and we will identify $\lambda_{J^{\vee}}$ with the box diagram corresponding
to it. Following the indexing conventions of entries of matrices, we will take
the first component of $\mathbb{Z}\times\mathbb{Z}$ to denote the vertical
position of the boxes, and the second component of
$\mathbb{Z}\times\mathbb{Z}$ to denote the horizontal position of the boxes.
The cardinality of $\lambda_{J^{\vee}}$ is
$|\lambda_{J^{\vee}}|=\sum_{b=1}^{|J|}j_{b}^{\vee}$. We will denote an element
of $\\{\lambda_{J^{\vee}}\\}_{J\subseteq[n]}$ by $\lambda$ to refer to a
shifted Young diagram without having to specify its shape. We will denote both
the empty subset of $[n]$ and the empty shifted Young diagram by $\phi$,
condoning the notation $\lambda_{\phi}=\phi$.

Let

$z_{k\ell}:=\begin{cases}q_{\ell j_{k}}&\text{if}\quad 1\leq k\leq|J|,\quad
k\leq\ell\leq|J|\\\ -r_{n+1-\ell,j_{k}}&\text{if}\quad 1\leq
k\leq|J|,\quad|J|\leq\ell\leq n.\end{cases}$

Suppressing the dependence on the fixed choice of Darboux basis, let
$C_{\lambda_{J^{\vee}}}$ (or $C_{\lambda}$) be the set of complex Lagrangian
subspaces of $\mathbb{C}^{2n}$ consisting of row spaces of $n\times 2n$
complex matrices in the reduced row echelon form corresponding to $J$ (or
$\lambda$).

Denote by
$\mathbb{C}_{(k,\ell)}:=\mathbb{C}\times\\{(k,\ell)\\}\subseteq\mathbb{C}\times\mathbb{Z}^{2}$.
Then we have homeomorphisms

$z^{\lambda}:C_{\lambda}\cong\prod_{(k,\ell)\in\lambda}\mathbb{C}_{(k,\ell)}.$

When $\lambda=\phi$ we identify $\prod_{(k,\ell)\in\phi}\mathbb{C}_{(k,\ell)}$
with $\\{0\\}$. We will assume the lexicographic order for the elements of
$\lambda$, so that $(k,\ell)<(k^{\prime},\ell^{\prime})$ if either
$k<k^{\prime}$, or both $k=k^{\prime}$ and $\ell<\ell^{\prime}$. With this
ordering we will denote again

$z^{\lambda}:C_{\lambda}\cong\mathbb{C}^{|\lambda|}.$

From the existence and uniqueness of reduced row echelon forms, we have

$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})=\bigsqcup_{\lambda\subseteq\lambda_{[n]}}C_{\lambda}.$

######  Remark 1 (Arnol’d stratification).

If $\Gamma_{\phi}$ is the unique element of $C_{\phi}$, then
$\dim_{\mathbb{C}}(\Gamma\cap\Gamma_{\phi})=k$ if and only if $\Gamma\in
C_{\lambda_{J^{\vee}}}$ and $|I|=k$ (cf 3.2.0 of Arnol’d [Arn67] ). Suppose
$\overline{j}_{b}^{\vee}:=j_{|J|+1-b}^{\vee}$, and
$\operatorname{pr}_{\\{\mathbf{f}\\}}$ denotes the projection from
$\mathbb{C}^{2n}$ to

$\operatorname{span}_{\mathbb{C}}\\{\mathbf{f}_{1},\cdots,\mathbf{f}_{n}\\}$

along

$\operatorname{span}_{\mathbb{C}}(\mathbf{e}_{1},\cdots,\mathbf{e}_{n}).$

Then $\Gamma\in C_{\lambda_{J^{\vee}}}$ if and only if

$\operatorname{pr}_{\\{\mathbf{f}\\}}\Gamma\cap\operatorname{span}\\{\mathbf{f}_{n},\cdots,\mathbf{f}_{n+1-\overline{j}}\\}=b\quad\text{for}\quad\overline{j}_{b}^{\vee}\leq\overline{j}<\overline{j}_{b+1}^{\vee}.$

For real coefficients, the linear extension of
$\mathbf{e}_{i}\mapsto\mathbf{f}_{n+1-i}$,
$\mathbf{f}_{i}\mapsto-\mathbf{e}_{n+1-i}$ maps $C_{\lambda_{J^{\vee}}}$ to
$e\\{\overline{j}_{1}^{\vee},\cdots\overline{j}_{|J|}^{\vee}\\}$ in 2.2 of
Fuchs [Fuc04].

######  Remark 2 (Maslov cycles and Maslov-Arnol’d cycles).

$C^{\mathbb{R}}_{\lambda_{[n]}}$ is the contractible open dense subset of all
the Lagrangian subspaces transverse to the unique Lagrangian subspace of
$C^{\mathbb{R}}_{\phi}$. So
$\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})\setminus
C^{\mathbb{R}}_{\lambda_{[n]}}=\bigsqcup_{\lambda\subsetneq\lambda_{[n]}}C_{\lambda}^{\mathbb{R}}$
is the Maslov cycle (Sections 3.5, 3.6 of Arnold [Arn67], Section 2 of Robbin-
Salamon [RS93]). To compare with the Maslov-Arnol’d homology cycles $A_{k}$,
$B_{k}$, and $\Lambda_{n}^{k}$ of Fuks [Fuk68], we note that
$\dim_{\mathbb{C}}(\Gamma\cap\operatorname{span}_{\mathbb{C}}\\{\mathbf{e}_{1},\cdots,\mathbf{e}_{k}\\})=\ell$
if and only if $\Gamma\in C_{\lambda_{J^{\vee}}}$ where $|I\cap[k]|=\ell$.

######  Example 3.

If $J=\\{1,3,4\\}$ and $I=\\{2,5\\}$, the pivot locations are at
$\mathbf{e}_{2}$, $\mathbf{e}_{5}$, $\mathbf{f}_{4}$, $\mathbf{f}_{3}$,
$\mathbf{f}_{1}$, and the matrix is of the form

$\left[\begin{array}[]{ccccc|ccccc}\ast&1&0&0&0&0&0&0&0&0\\\
\ast&0&\ast&\ast&1&0&0&0&0&0\\\ \ast&0&\ast&\ast&0&\ast&1&0&0&0\\\
\ast&0&\ast&\ast&0&\ast&0&1&0&0\\\
\ast&0&\ast&\ast&0&\ast&0&0&\ast&1\end{array}\right].$

Since the pivot locations are at $r_{ai_{a}}=p_{bj_{b}}=1$ (or
$p_{aj_{a}}=p_{bj_{b}}=1$), the position of the entries
$r_{aj_{b}},r_{ai_{a}},p_{bj_{b}},p_{bj_{a}}$ (or
$q_{aj_{b}},p_{aj_{a}},p_{bj_{b}},q_{bj_{a}}$) in the matrix are vertices of a
parallelogram.

Equations (1) are $r_{11}=-p_{12}$, $r_{21}=-p_{15}$, $r_{23}=-p_{25}$, and
$r_{24}=-p_{35}$ so that the matrix form is

$\left[\begin{array}[]{ccccc|ccccc}-p_{12}&1&0&0&0&0&0&0&0&0\\\
-p_{15}&0&-p_{25}&-p_{35}&1&0&0&0&0&0\\\ \ast&0&\ast&\ast&0&p_{35}&1&0&0&0\\\
\ast&0&\ast&\ast&0&p_{25}&0&1&0&0\\\
\ast&0&\ast&\ast&0&p_{15}&0&0&p_{12}&1\end{array}\right]$

and equations (2) are $q_{31}=q_{14}$, $q_{33}=q_{24}$, and $q_{21}=q_{13}$

$\left[\begin{array}[]{ccccc|ccccc}\ast&1&0&0&0&0&0&0&0&0\\\
\ast&0&\ast&\ast&1&0&0&0&0&0\\\ q_{14}&0&q_{24}&q_{34}&0&\ast&1&0&0&0\\\
q_{13}&0&q_{23}&q_{24}&0&\ast&0&1&0&0\\\
q_{11}&0&q_{13}&q_{14}&0&\ast&0&0&\ast&1\end{array}\right].$

$J^{\vee}=\\{5,3,2\\}$, so the corresponding shifted Young diagram is

$\lambda_{J^{\vee}}=\ytableausetup{centertableaux,boxsize=5pt}\ytableau\none&\none\none\\\
\none\none\\\ .$

The independent parameters are labelled by

$\left[\begin{array}[]{ccccc|ccccc}-z_{15}&1&0&0&0&0&0&0&0&0\\\
-z_{14}&0&-z_{24}&-z_{34}&1&0&0&0&0&0\\\
z_{13}&0&z_{23}&z_{33}&0&\ast&1&0&0&0\\\
z_{12}&0&z_{22}&\ast&0&\ast&0&1&0&0\\\
z_{11}&0&\ast&\ast&0&\ast&0&0&\ast&1\end{array}\right],$

where, as in Cartesian coordinates, the horizontal location is given by the
first index. Equivalently, the independent parameters are labelled by

$\left[\begin{array}[]{ccccc|ccccc}\ast&1&0&0&0&0&0&0&0&0\\\
\ast&0&\ast&\ast&1&0&0&0&0&0\\\ \ast&0&\ast&z_{33}&0&z_{34}&1&0&0&0\\\
\ast&0&z_{22}&z_{23}&0&z_{24}&0&1&0&0\\\
z_{11}&0&z_{12}&z_{13}&0&z_{14}&0&0&z_{15}&1\end{array}\right],$

where, as in matrix component notation, the horizontal location is given by
the second index. Asterisks denote dependent parameters. The labels for the
independent parameters can be read off from their corresponding locations in
the shifted Young diagrams (Figure 3).

Figure 3: The labels for the independent parameters can be read off from their
corresponding locations in the shifted Young diagrams.

###  2.2 Attaching maps of Schubert cells

We will denote by $\lambda\leq\lambda^{\prime}$ if $\lambda$ is contained in
$\lambda^{\prime}$, or equivalently, if there exist $J,J^{\prime}\subseteq[n]$
such that $\lambda=\lambda_{J^{\vee}}$,
$\lambda^{\prime}=\lambda_{(J^{\prime})^{\vee}}$, $|J|\leq|J^{\prime}|$ and
$j_{k}^{\vee}\leq(j_{k}^{\prime})^{\vee}$ for all $1\leq k\leq|J|$. Since
every nonempty $\lambda$ contains $(1,1)$, the containment of shifted Young
diagrams assumes they are aligned on the bottom left corner. For a comparison
with the Bruhat ordering, we refer the reader to Proposition 6.1 of Ikeda and
Naruse [IN09].

Every $C_{\lambda}$ is homeomorphic to $\mathbb{C}^{|\lambda|}$, so when
tubular neighborhoods of $C_{\lambda}$ inside some
$\overline{C_{\lambda^{\prime}}}$ exist, they are homeomorphic to trivial
bundles over $C_{\lambda}$. We will construct such homeomorphisms in detail
when $\lambda$, $\lambda^{\prime}$ are such that $\lambda\leq\lambda^{\prime}$
and $|\lambda^{\prime}|=|\lambda|+1$ to obtain information about degrees of
attaching maps.

There are two cases to consider, i) when $\lambda^{\prime}$ has more rows than
$\lambda$, and ii) when $\lambda$ and $\lambda^{\prime}$ have the same number
of rows. In both cases we setup some notation, that
$\lambda=\lambda_{J^{\vee}}$ and
$\lambda^{\prime}=\lambda_{(J^{\prime})^{\vee}}$, and that
$\lambda^{\prime}\setminus\lambda=\\{(\star,\star+(j_{\star}^{\prime})^{\vee}-1)\\}$.

If $|J^{\prime}|>|J|$ then $\star=|J^{\prime}|=|J|+1$, and
$(j_{\star}^{\prime})^{\vee}=1$. So $j_{\star}^{\prime}=n$. Since all
$j_{k}^{\vee}=(j_{k}^{\prime})^{\vee}$ for $1\leq k\leq|J|$,
$j_{\star}^{\prime}\notin J$, so $j_{\star}^{\prime}\in I$, and
$I^{\prime}=I\setminus\\{j_{\star}^{\prime}\\}$,
$J^{\prime}=J\sqcup\\{j_{\star}^{\prime}\\}$, and
$i_{|I|}=j_{\star}^{\prime}=n$. Suppose the reduced row echelon form of
$\Gamma\in C_{\lambda}$ has row vectors
$\\{\mathbf{v}_{a},\mathbf{w}_{b}\\}_{1\leq a\leq|I|,1\leq b\leq|J|}$. Let

$\mathbf{w}^{\prime}_{t}:=\mathbf{v}_{|I|}+t\mathbf{f}_{j_{\star}^{\prime}}\quad
t\in\mathbb{C}.$

Then
$\Gamma_{t}:=\operatorname{span}_{\mathbb{C}}\\{\mathbf{v}_{a},\mathbf{w}^{\prime}_{t},\mathbf{w}_{b}\\}_{1\leq
a\leq|I|-1,1\leq b\leq|J|}$ is always complex Lagrangian, because

$\displaystyle\omega^{\mathbb{C}}(\mathbf{w}_{t}^{\prime},\mathbf{v}_{a})$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\omega^{\mathbb{C}}(\mathbf{w}_{t}^{\prime},\mathbf{w}_{b})$
$\displaystyle=$ $\displaystyle-tq_{b,j_{\star}^{\prime}}=0.$ (3)

When $t\neq 0$, we can row reduce these vectors as

$\displaystyle\mathbf{v}^{\prime}_{a}(t)$ $\displaystyle:=$
$\displaystyle\mathbf{v}_{a}\quad\quad\quad\quad\quad\quad
i_{a}<j_{\star}^{\prime}=n$ $\displaystyle\mathbf{w}^{\prime}_{\star}(t)$
$\displaystyle:=$ $\displaystyle t^{-1}\mathbf{w}^{\prime}_{t}$
$\displaystyle\mathbf{w}^{\prime}_{b}(t)$ $\displaystyle:=$
$\displaystyle\mathbf{w}_{b}-t^{-1}z_{b,\star}\mathbf{w}_{t}^{\prime}\quad
j_{b}<j_{\star}^{\prime}=n.$ (4)

So $\Gamma_{t}$ is represented by the $n\times 2n$ complex matrix with $k$th
row $[\mathbf{v}^{\prime}_{k}(t)]_{\\{\mathbf{e},\mathbf{f}\\}}$ for $1\leq
k\leq|I|-1$, $[\mathbf{w}_{\star}^{\prime}(t)]_{\\{\mathbf{e},\mathbf{f}\\}}$
for $k=|I|$, and
$[\mathbf{w}^{\prime}_{n-k+1}(t)]_{\\{\mathbf{e},\mathbf{f}\\}}$ for $1\leq
k\leq|J|$. It is in reduced row echelon form with pivots at

$\mathbf{e}_{i_{1}},\cdots,\mathbf{e}_{i_{|I|-1}},\mathbf{f}_{j_{\star}^{\prime}},\mathbf{f}_{j_{|J|}},\cdots\mathbf{f}_{j_{1}}.$

So $\Gamma_{t}\in C_{\lambda_{(J^{\prime})^{\vee}}}$ whenever
$t\in\mathbb{C}^{\times}$. By construction $\Gamma_{0}=\Gamma$, so

$\varphi_{\lambda}^{\lambda^{\prime}}(\Gamma,t):=\Gamma_{t}$

is a local homeomorphism from $C_{\lambda}\times\mathbb{C}^{\times}$ to
$C_{\lambda^{\prime}}$.

If $|J|=|J^{\prime}|$, then $(j_{k}^{\prime})^{\vee}=j_{k}^{\prime}$ for all
$1\leq k\leq|J|$, except when $k=\star$, in which case
$(j_{\star}^{\prime})^{\vee}=j_{\star}^{\vee}+1$, so
$j_{\star}^{\prime}+1=j_{\star}$. This also implies that
$(j_{\star}^{\prime})^{\vee}\notin J^{\vee}$ and
$j_{\star}^{\vee}\notin(J^{\prime})^{\vee}$. So
$I^{\prime}=I\sqcup\\{j_{\star}\\}\setminus\\{j_{\star}^{\prime}\\}$ and
$J^{\prime}=J\sqcup\\{j_{\star}^{\prime}\\}\setminus\\{j_{\star}\\}$. Suppose
the reduced row echelon form of $\Gamma\in C_{\lambda}$ has row vectors
$\\{\mathbf{v}_{a},\mathbf{w}_{b}\\}_{1\leq a\leq|I|,1\leq b\leq|J|}$. Let
$\circledast$ be such that $i_{\circledast}=j_{\star}^{\prime}$, and

$\displaystyle\mathbf{v}^{\prime}_{t}$ $\displaystyle:=$
$\displaystyle\mathbf{v}_{\circledast}-t\mathbf{e}_{j_{\star}}$
$\displaystyle\mathbf{w}^{\prime}_{t}$ $\displaystyle:=$
$\displaystyle\mathbf{w}_{\star}+t\mathbf{f}_{j_{\star}^{\prime}}\quad
t\in\mathbb{C}.$

Then
$\Gamma_{t}:=\operatorname{span}_{\mathbb{C}}\\{\mathbf{v}_{a},\mathbf{v}^{\prime}_{t},\mathbf{w}^{\prime}_{t},\mathbf{w}_{b}\\}_{i_{a}\in
I\setminus\\{j_{\star}^{\prime}\\},j_{b}\in J\setminus\\{j_{\star}\\}}$ is
complex Lagrangian for all $t\in\mathbb{C}$, because

$\displaystyle\omega^{\mathbb{C}}(\mathbf{v}_{t}^{\prime},\mathbf{w}_{b})=$
$\displaystyle-t\cdot p_{b,j_{\star}}$ $\displaystyle=0$
$\displaystyle\omega^{\mathbb{C}}(\mathbf{w}_{t}^{\prime},\mathbf{w}_{b})=$
$\displaystyle t\cdot q_{b,j_{\star}^{\prime}}$ $\displaystyle=0$
$\displaystyle\omega^{\mathbb{C}}(\mathbf{w}_{t}^{\prime},\mathbf{v}_{a})=$
$\displaystyle t\cdot r_{a,j_{\star}^{\prime}}$ $\displaystyle=0$
$\displaystyle\omega^{\mathbb{C}}(\mathbf{v}_{t}^{\prime},\mathbf{w}_{t}^{\prime})=$
$\displaystyle-t+t$ $\displaystyle=0.$ (5)

When $t\neq 0$, we can row reduce these vectors as follows

$\mathbf{v}^{\prime}_{a}(t):=\begin{cases}\mathbf{v}_{a}&\text{ if
}i_{a}<j_{\star}^{\prime}\\\ -t^{-1}\mathbf{v}_{t}^{\prime}&\text{ if
}i_{a}=j_{\star}^{\prime}\\\
\mathbf{v}_{a}-t^{-1}z_{\star,n+1-a}\mathbf{v}_{t}^{\prime}&\text{ if
}i_{a}>j_{\star}^{\prime}.\end{cases}$ (6)

and

$\mathbf{w}^{\prime}_{b}(t):=\begin{cases}\mathbf{w}_{b}+t^{-1}z_{\star,b}\mathbf{v}_{t}^{\prime}&\text{
if }j_{b}>j_{\star}\\\
t^{-1}\mathbf{w}_{t}^{\prime}+t^{-2}z_{\star,\star}\mathbf{v}_{t}^{\prime}&\text{
if }j_{b}=j_{\star}\\\
\mathbf{w}_{b}+t^{-1}z_{b,\star}\mathbf{v}_{t}^{\prime}-z_{b,\star+j_{\star}^{\vee}}\mathbf{w}_{\star}^{\prime}(t)&\text{
if }j_{b}<j_{\star}.\end{cases}$ (7)

Then the $n\times 2n$ complex matrix with $k$th row
$[\mathbf{v}^{\prime}_{k}(t)]_{\\{\mathbf{e},\mathbf{f}\\}}$ for $1\leq
a\leq|I|$, $[\mathbf{w}^{\prime}_{n-k+1}(t)]_{\\{\mathbf{e},\mathbf{f}\\}}$
for $1\leq k\leq|J|$ is in reduced row echelon form with pivots at

$\mathbf{e}_{i_{1}},\cdots,\mathbf{e}_{j_{\star}},\cdots,\mathbf{e}_{i_{|I|}},\mathbf{f}_{j_{|J|}},\cdots,\mathbf{f}_{j_{\star-1}},\mathbf{f}_{j_{\star}^{\prime}},\mathbf{f}_{j_{\star+1}},\cdots,\mathbf{f}_{j_{1}}.$

Thus $\Gamma_{t}\in C_{\lambda^{\prime}}$ when $t\in\mathbb{C}^{\times}$.

By construction $\Gamma_{0}=\Gamma$, and again
$\varphi_{\lambda}^{\lambda^{\prime}}(\Gamma,t):=\Gamma_{t}$ is a local
homeomorphism from $C_{\lambda}\times\mathbb{C}^{\times}$ to
$C_{\lambda^{\prime}}$.

We can obtain charts of $\overline{C_{\lambda^{\prime}}}$ in the neighborhood
of points in $C_{\lambda}$ from local inverses of
$\varphi_{\lambda}^{\lambda^{\prime}}$. From the row reduction prescription of
equations (4), (6), and (7) we can explicitly compute the transition map of
$C_{\lambda^{\prime}}$

$z^{\lambda^{\prime}}\circ\varphi_{\lambda}^{\lambda^{\prime}}\circ((z^{\lambda})^{-1}\times\mathrm{Id}_{\mathbb{C}^{\times}}):\mathbb{C}^{|\lambda|}\times\mathbb{C}^{\times}\to\mathbb{C}^{|\lambda^{\prime}|}.$

If $|J^{\prime}|>|J|$, they are

$\displaystyle z_{k\ell}^{\prime}=$ $\displaystyle
z_{k\ell}+t^{-1}z_{k,\star}z_{\ell,\star}$ if $\ell<\star$ (8a) $\displaystyle
z_{k\ell}^{\prime}=$ $\displaystyle-t^{-1}z_{k\ell}$ if $\ell<k=\star$ (8b)
$\displaystyle z_{k\ell}^{\prime}=$ $\displaystyle z_{k\ell}$ if $\ell>\star$
(8c) $\displaystyle z_{k\ell}^{\prime}=$ $\displaystyle t^{-1}$ if
$k=\ell=\star.$ (8d)

If $|J^{\prime}|=|J|$, they are

$\displaystyle z_{k\ell}^{\prime}=$ $\displaystyle
z_{k\ell}-t^{-1}z_{k,\star}z_{\ell,\star+j_{\star}^{\vee}}-t^{-1}z_{\ell,\star}z_{k,\star+j_{\star}^{\vee}}$
$\displaystyle z_{k\ell}^{\prime}=$
$\displaystyle\quad+t^{-2}z_{\star,\star}z_{k,\star+j_{\star}^{\vee}}z_{\ell,\star+j_{\star}^{\vee}}$
if $k\leq\ell<\star$ (9a) $\displaystyle z_{k\ell}^{\prime}=$ $\displaystyle
z_{k\ell}+t^{-1}z_{\star,\ell}z_{k,\star+j_{\star}^{\vee}}$ if
$k<\star<\ell<\star+j_{\star}^{\vee}$ (9b) $\displaystyle z_{k\ell}^{\prime}=$
$\displaystyle
t^{-1}z_{k\ell}-t^{-2}z_{\star,\star}z_{k,\star+j_{\star}^{\vee}}$ if
$k<\star=\ell$ (9c) $\displaystyle z_{k\ell}^{\prime}=$ $\displaystyle
t^{-1}z_{k\ell}$ if $k=\star<\ell<\star+j_{\star}^{\vee}$ (9d) $\displaystyle
z_{k\ell}^{\prime}=$ $\displaystyle-t^{-1}z_{k\ell}$ if
$k<\star,\ell=\star+j_{\star}^{\vee}$ (9e) $\displaystyle z_{k\ell}^{\prime}=$
$\displaystyle t^{-2}z_{k\ell}$ if $k=\ell=\star$ (9f) $\displaystyle
z_{k\ell}^{\prime}=$ $\displaystyle t^{-1}$ if
$k=\star,\ell=\star+j_{\star}^{\vee}$ (9g) $\displaystyle z_{k\ell}^{\prime}=$
$\displaystyle z_{k\ell}$ if $k>\star\text{ or }\ell>\star+j_{\star}^{\vee}$
(9h)

for $(k,\ell)\in\lambda^{\prime}$. We refer to Figure 4, 6, and Remark 6 for a
diagrammatic description of the nine cases in terms of shifted Young diagrams.

Figure 4: Terms appearing in some equations of (9h):
$z_{12}^{\prime}=z_{12}-t^{-1}z_{2,\star}z_{1,\star+j_{\star}^{\vee}}-t^{-1}z_{1,\star}z_{2,\star+j_{\star}^{\vee}}+t^{-2}z_{\star,\star}z_{1,\star+j_{\star}^{\vee}}z_{2,\star+j_{\star}^{\vee}}$,
$z_{35}^{\prime}=z_{35}-t^{-1}z_{\star,5}z_{37}$, $z_{56}^{\prime}=z_{56}$,
$z_{19}^{\prime}=z_{19}$. Additional terms are in terms of variables
succeeding in the lexicographical order.

Because of the shape of the row reduced echelon form, and the fact that we
only subtract rows above from rows below, additional terms are only dependent
on variables succeeding in the lexicographical order. So the complex Jacobian
determinant of the transition map is upper triangular.

We can compute the diagonal entries. When $|J^{\prime}|>|J|$, they are

$\displaystyle\frac{\partial z_{k\ell}^{\prime}}{\partial z_{k\ell}}$
$\displaystyle=$ $\displaystyle\begin{cases}1&\text{ if
}(k,\ell)\in\lambda_{J^{\vee}},\ell\neq\star\\\ -t^{-1}&\text{ if
}(k,\ell)\in\lambda_{J^{\vee}},\ell=\star\end{cases}$
$\displaystyle\frac{\partial z_{\star\star}^{\prime}}{\partial t}$
$\displaystyle=$ $\displaystyle-t^{-2}.$

When $|J^{\prime}|=|J|$, they are

$\frac{\partial z_{k\ell}^{\prime}}{\partial z_{k\ell}}=\begin{cases}1&\text{
if }k\neq\star,\ell\neq\star,\star+j_{\star}^{\vee}\\\ t^{-1}&\text{ if
}k\neq\star,\ell=\star\text{ or
}k=\star,\ell\neq\star,\star+j_{\star}^{\vee}\\\ -t^{-1}&\text{ if
}k\neq\star,\ell=\star+j_{\star}^{\vee}\\\ t^{-2}&\text{ if
}k=\ell=\star\end{cases}$

for $(k,\ell)\in\lambda_{J^{\vee}}$ and

$\frac{\partial z_{\star,\star+j_{\star}^{\vee}}^{\prime}}{\partial
t}=-t^{-2}.$

Finally, since the new variable
$z_{\star,\star+j_{\star}^{\vee}}^{\prime}=t^{-1}$ its position in the
lexicographic order does not affect the upper triangularity of the complex
Jacobian matrix, but contributes an overall sign of
$(-1)^{\sum_{k=\star+1}^{|J|}j_{k}^{\vee}}$ to the complex determinant.

So the complex Jacobian determinant of the transition map is equal to

$\begin{cases}(-1)^{\star}\cdot t^{-\star-1}&\text{ if }|J^{\prime}|>|J|\\\
(-1)^{\star+\sum_{k=\star+1}^{|J^{\prime}|}j_{k}^{\vee}}\cdot t^{-2\star-
j_{\star}^{\vee}-1}&\text{ if }|J^{\prime}|=|J|.\end{cases}$ (10)

We will denote this function as
$T_{\lambda}^{\lambda^{\prime}}(z^{\lambda},t)=T_{\lambda}^{\lambda^{\prime}}(t)$.

######  Example 4 ($|J^{\prime}|>|J|$ case).

Suppose $I=\\{3\\}$, $J=\\{1,2\\}$, $I^{\prime}=\phi$,
$J^{\prime}=\\{1,2,3\\}$. Then $\Gamma_{t}$ is the complex row space of some

$\left[\begin{array}[]{ccc|ccc}-z_{13}&-z_{23}&1&t&0&0\\\
z_{12}&z_{22}&0&z_{23}&1&0\\\ z_{11}&z_{12}&0&z_{13}&0&1\end{array}\right].$

This matrix row reduces to

$\left[\begin{array}[]{ccc|ccc}-t^{-1}z_{13}&-t^{-1}z_{23}&t^{-1}&1&0&0\\\
z_{12}+t^{-1}z_{13}z_{23}&z_{22}+t^{-1}z_{23}^{2}&-t^{-1}z_{23}&0&1&0\\\
z_{11}+t^{-1}z_{13}^{2}&z_{12}+t^{-1}z_{13}z_{23}&-t^{-1}z_{13}&0&0&1\end{array}\right].$

The complex Jacobian matrix of
$z^{\lambda^{\prime}}\circ\varphi_{\lambda}^{\lambda^{\prime}}\circ((z^{\lambda})^{-1}\times\mathrm{Id}_{\mathbb{C}^{\times}})$
is

$\frac{\partial(z_{11}^{\prime},z_{12}^{\prime},z_{13}^{\prime},z_{22}^{\prime},z_{23}^{\prime},z_{33}^{\prime})}{\partial(z_{11},z_{12},z_{13},z_{22},z_{23},t)}=\begin{pmatrix}1&0&2t^{-1}z_{13}&0&0&-t^{-2}z_{13}^{2}\\\
0&1&t^{-1}z_{23}&0&t^{-1}z_{13}&-t^{-2}z_{13}z_{23}\\\
0&0&-t^{-1}&0&0&t^{-2}z_{13}\\\ 0&0&0&1&2t^{-1}z_{23}&-t^{-2}z_{23}^{2}\\\
0&0&0&0&-t^{-1}&t^{-2}z_{23}\\\ 0&0&0&0&0&-t^{-2}\end{pmatrix}.$

This matrix has complex determinant $(-1)^{3}\cdot t^{-4}$. The diagonal
entries can be schematically represented as 5.

Figure 5: Diagonal entries of Jacobians of some transition maps

######  Example 5 ($|J^{\prime}|=|J|$ case).

Suppose $I=\\{2,3\\}$, $J=\\{1,4,5\\}$, $I^{\prime}=\\{2,4\\}$, and
$J^{\prime}=\\{1,3,5\\}$. Then $\Gamma_{t}$ is the complex row space of some

$\left[\begin{array}[]{ccccc|ccccc}-z_{15}&1&0&0&0&0&0&0&0&0\\\
-z_{14}&0&1&-t&0&0&0&0&0&0\\\ z_{13}&0&0&z_{23}&z_{33}&1&0&0&0&0\\\
z_{12}&0&0&z_{22}&z_{23}&0&1&t&0&0\\\
z_{11}&0&0&z_{12}&z_{13}&0&0&z_{14}&z_{15}&1\end{array}\right].$

After row reduction, the transition map
$z^{\lambda^{\prime}}\circ\varphi_{\lambda}^{\lambda^{\prime}}\circ((z^{\lambda})^{-1}\times\mathrm{Id}_{\mathbb{C}^{\times}})$
is the following

$\displaystyle z_{11}^{\prime}$ $\displaystyle=$ $\displaystyle
z_{11}-2t^{-1}z_{12}z_{14}+t^{-2}z_{22}z_{14}^{2}$ $\displaystyle
z_{12}^{\prime}$ $\displaystyle=$ $\displaystyle
t^{-1}z_{12}-t^{-2}z_{22}z_{14}$ $\displaystyle z_{13}^{\prime}$
$\displaystyle=$ $\displaystyle z_{13}-t^{-1}z_{23}z_{14}$ $\displaystyle
z_{14}^{\prime}$ $\displaystyle=$ $\displaystyle-t^{-1}z_{14}$ $\displaystyle
z_{15}^{\prime}$ $\displaystyle=$ $\displaystyle z_{15}$ $\displaystyle
z_{22}^{\prime}$ $\displaystyle=$ $\displaystyle t^{-2}z_{22}$ $\displaystyle
z_{23}^{\prime}$ $\displaystyle=$ $\displaystyle t^{-1}z_{23}$ $\displaystyle
z_{33}^{\prime}$ $\displaystyle=$ $\displaystyle z_{33}$ $\displaystyle
z_{24}^{\prime}$ $\displaystyle=$ $\displaystyle t^{-1}$

The complex Jacobian matrix is upper triangular in the lexicographic ordering,
and the diagonal entries can be represented schematically in Figure 5. The
product is $(-1)^{2+1}\cdot t^{-7}$.

######  Remark 6 (Arches, columns, and roofs).

When
$\lambda^{\prime}\setminus\lambda=\\{(\star,\star+(j^{\prime}_{\star})^{\vee}-1)\\}$,
let

$\displaystyle\alpha_{\star}$ $\displaystyle:=$
$\displaystyle\\{(k,\star),(k,\star+(j_{\star}^{\prime})^{\vee}-1):1\leq
k\leq\star\\}\cup\\{(\star,\ell):\star\leq\ell\leq\star+(j_{\star}^{\prime})^{\vee}-1\\}.$
$\displaystyle c_{\star}$ $\displaystyle:=$
$\displaystyle\\{(k,\star+(j_{\star}^{\prime})^{\vee}-1):1\leq k\leq\star\\}.$
$\displaystyle\rho_{\star}$ $\displaystyle:=$
$\displaystyle\\{(k,\ell)\in\lambda:k>\star\\}.$

In this notation, the complex Jacobian determinants of equation (10) are

$T_{\lambda}^{\lambda^{\prime}}(t)=\begin{cases}(-1)^{|c_{\star}|}\cdot
t^{-|c_{\star}|-1}&\text{ if }|J^{\prime}|>|J|\\\
(-1)^{|c_{\star}|+|\rho_{\star}|}\cdot t^{-|\alpha_{\star}|-2}&\text{ if
}|J^{\prime}|=|J|.\end{cases}$ (11)

Figure 6: $\alpha_{\star}$, $c_{\star}$, and $\rho_{\star}$ are shaded for
$\lambda_{\\{7,5,3,2,1\\}}\leq\lambda_{\\{7,5,4,2,1\\}}$

######  Proposition 7 (Frontier condition).

The following are equivalent

  1. 1.

$C_{\lambda}\cap\overline{C_{\lambda^{\prime}}}\neq\phi$.

  2. 2.

$C_{\lambda}\subseteq\overline{C_{\lambda^{\prime}}}$.

  3. 3.

$\lambda\leq\lambda^{\prime}$.

###### Proof.

$(3)\implies(2)$: There exists an increasing sequence
$\lambda_{1}\leq\lambda_{2}\leq\cdots\lambda_{\ell}$ such that
$\lambda_{1}=\lambda$, $\lambda_{\ell}=\lambda^{\prime}$, and
$|\lambda_{k+1}|=|\lambda_{k}|+1$ for $1\leq
k<\ell=|\lambda^{\prime}|-|\lambda|$, and a sequence of embeddings
$\\{\varphi_{\lambda_{k}}^{\lambda_{k+1}}\\}_{k=1}^{\ell}$. For each
$\Gamma\in C_{\lambda}$, let

$\Gamma_{t_{1},\cdots,t_{\ell}}:=\varphi_{\lambda_{\ell-1}}^{\lambda_{\ell}}(\cdots(\varphi_{\lambda_{2}}^{\lambda_{3}}(\varphi_{\lambda_{1}}^{\lambda_{2}}(\Gamma,t_{1}),t_{2}),\cdots),t_{\ell})\quad
t_{1},\cdots,t_{\ell}\in\mathbb{C}$

where if $t_{j}=0$ for some $j\in\\{1,\cdots,\ell\\}$, then $t_{k}=0$ for all
$k\geq j$. If $t_{1},\cdots,t_{\ell}$ are all nonzero, then
$\Gamma_{t_{1},\cdots,t_{\ell}}\in C_{\lambda^{\prime}}$ and
$\Gamma_{t_{1},\cdots,t_{\ell}}\to\Gamma$ as $(t_{1},\cdots,t_{\ell})\to 0$.
$(2)\implies(1)$ is immediate, as $C_{\lambda}$ is nonempty. $(1)\implies(3)$:
Suppose $\lambda=\lambda_{J^{\vee}}$ and
$\lambda^{\prime}=\lambda_{(J^{\prime})^{\vee}}$. If
$\lambda\nleq\lambda^{\prime}$, then there exists a $1\leq b\leq|J|$ such that
$j_{b}^{\vee}>(j_{b}^{\prime})^{\vee}$. Then $p_{bj_{b}}(\Gamma^{\prime})=0$
for all $\Gamma^{\prime}\in C_{\lambda^{\prime}}$ and $p_{bj_{b}}(\Gamma)=1$
for all $\Gamma\in C_{\lambda}$. So
$C_{\lambda}\cap\overline{C_{\lambda^{\prime}}}=\phi$.∎

We construct the attaching maps following the proof of Theorem 3.2.3 of
Tajakka’s thesis [Taj15] (cf Proposition 1.17 of Hatcher’s book [Hat17]) for
ordinary Grassmannians. To do this we set up some notation.

Let $\mathfrak{J}$ be the compatible linear complex structure on
$\mathbb{R}^{2n}$ given by the linear extension of
$\mathbf{e}_{i}\mapsto\mathbf{f}_{i}$, $\mathbf{f}_{i}\mapsto-\mathbf{e}_{i}$
for $1\leq i\leq n$. Denote again by $\mathfrak{J}$ its $\mathbb{C}$-linear
extension to $\mathbb{C}^{2n}$. Take the hermitian inner product
$\langle\mathbf{v},\mathbf{w}\rangle:=\omega^{\mathbb{C}}(\mathbf{v},\mathfrak{J}\overline{\mathbf{w}})$
of $\mathbb{C}^{2n}$. Let $\operatorname{Sp}(n)$ be the group of complex
linear transformations on $\mathbb{C}^{2n}$ that preserves this hermitian
inner product and $\omega^{\mathbb{C}}$. Given a unitary basis
$\\{\mathbf{u}_{1},\cdots,\mathbf{u}_{n}\\}$ of a complex Lagrangian subspace
$\Gamma$,
$\\{\mathfrak{J}\overline{\mathbf{u}}_{1},\cdots,\mathfrak{J}\overline{\mathbf{u}}_{n}\\}$
is a unitary basis of the complex Lagrangian subspace
$\mathfrak{J}\overline{\Gamma}$. $\Gamma$ and $\mathfrak{J}\overline{\Gamma}$
are always orthogonal, so
$\\{\mathbf{u}_{1},\cdots,\mathbf{u}_{n},\mathfrak{J}\overline{\mathbf{u}}_{1},\cdots,\mathfrak{J}\overline{\mathbf{u}}_{n}\\}$
is a unitary Darboux basis of $\mathbb{C}^{2n}$. Sending
$\\{\mathbf{e}_{1},\cdots,\mathbf{e}_{n},\mathbf{f}_{1},\cdots,\mathbf{f}_{n}\\}$
to this basis determines a unique element of $\operatorname{Sp}(n)$.

Let $\mathcal{F}_{n}$ be the set of $n$-tuples
$\mathfrak{u}:=(\mathbf{u}_{1},\cdots,\mathbf{u}_{n})$ such that
$\\{\mathbf{u}_{1},\cdots,\mathbf{u}_{n}\\}$ is a unitary basis of some
complex Lagrangian subspace of $\mathbb{C}^{2n}$. Taking each such $n$-tuple
to the spanning space of its elements gives a principal $U(n)$-bundle
$\pi:\mathcal{F}_{n}\to\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$.

For each shifted Young diagram $\lambda=\lambda_{J^{\vee}}$, let $E_{\lambda}$
be the set of $(\mathbf{u}_{1},\cdots,\mathbf{u}_{n})$ in $\mathcal{F}_{n}$
such that

$\mathbf{u}_{k}\in\begin{cases}\operatorname{span}_{\mathbb{C}}\\{\mathbf{e}_{1},\cdots,\mathbf{e}_{i_{k}}\\}&\text{
if }1\leq k\leq|I|\\\
\operatorname{span}_{\mathbb{C}}\\{\mathbf{e}_{1},\cdots,\mathbf{e}_{n},\mathbf{f}_{n},\cdots,\mathbf{f}_{j_{n+1-k}}\\}&\text{
if }|I|<k\leq n.\end{cases}$

Let $E(\lambda)\subseteq E_{\lambda}$ be the elements such that all pivot
coefficients (i.e. $q_{i_{k}}(\mathbf{u}_{k})$ for $1\leq k\leq|I|$ and
$p_{j_{n+1-k}}(\mathbf{u}_{k})$ for $|I|<k\leq n$) are nonnegative real
valued. If $\lambda\leq\lambda^{\prime}$ then $E_{\lambda}\subseteq
E_{\lambda^{\prime}}$ and $E(\lambda)\subseteq E(\lambda^{\prime})$.

We can apply the Gram-Schmidt algorithm on the row vectors of the reduced row
echelon forms of $\Gamma$’s in $C_{\lambda}$ such that pivot coefficients take
values in nonnegative real numbers. Then we obtain unique _unitary row echelon
forms_ corresponding to each $\Gamma\in C_{\lambda}$. We obtain continuous
sections of $\pi:\pi^{-1}(\overline{C}_{\lambda})\to\overline{C}_{\lambda}$.

We will refer to the set
$\\{(\xi_{1},\cdots,\xi_{k})\in\mathbb{C}^{k}:\sum_{\ell=1}^{k}|\xi_{\ell}|^{2}=1,\xi_{k}\in[0,1]\\}$
as the _closed upper hemisphere of $S^{2k-1}$_.

######  Proposition 8 (Construction of attaching maps).

$E(\lambda)$ is homeomorphic to a closed $2|\lambda|$-ball. Consequently, the
precomposition of this homeomorphism with the restriction of $\pi$ to
$E(\lambda)\setminus\pi^{-1}(C_{\lambda})$ is the attaching map of
$C_{\lambda}$ in the CW decomposition of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$.

###### Proof.

Suppose $\lambda=\lambda_{J^{\vee}}$ for some $J\subseteq[n]$. We will prove
by induction on $|J|$. If $J=\\{k\\}$, $E(\lambda_{\\{k\\}})$ consists of
$(\mathbf{e}_{1},\cdots,\mathbf{e}_{k-1},\mathbf{e}_{k+1},\cdots,\mathbf{e}_{n-1},\mathbf{u}_{n})$,
where $\mathbf{u}_{n}$ takes values in the closed upper hemisphere of

$S^{2k}\subseteq\operatorname{span}_{\mathbb{C}}\\{\mathbf{e}_{k},\mathbf{f}_{n},\mathbf{f}_{n-1},\cdots,\mathbf{f}_{n-k+2}\\}\oplus\operatorname{span}_{\mathbb{R}}\\{\mathbf{f}_{n-k+1}\\}.$

So $E(\lambda_{\\{k\\}})$ is homeomorphic to a closed $2|\lambda|$-ball.

If $|J|>1$, let
$\operatorname{pr}((\mathbf{u}_{1},\cdots,\mathbf{u}_{n})):=\mathbf{u}_{n}$.
Then $\mathfrak{u}\in\operatorname{pr}^{-1}(\mathbf{f}_{j_{1}})$ can be
represented by $n\times 2n$ matrix with $k$th row equal to
$[\mathbf{u}_{k}]_{\\{\mathbf{e},\mathbf{f}\\}}$ has only zeros on the
$j_{1}$st and $2n-j_{1}+1$st columns and zeros on the $n$th row except at the
‘pivot’ location $\mathbf{f}_{j_{1}}$. So
$\operatorname{pr}^{-1}(\mathbf{f}_{j_{1}})$ is homeomorphic to
$E(\lambda_{\\{j_{2}^{\vee},\cdots,j_{n}^{\vee}\\}})\subseteq\mathcal{F}_{n-1}$.

For $\mathfrak{u}=(\mathbf{u}_{1},\cdots,\mathbf{u}_{n})\in E(\lambda)$,
obtain $(\hat{\mathbf{u}}_{1},\cdots,\hat{\mathbf{u}}_{n-1})$ by applying
Gram-Schmidt to
$\\{\mathbf{u}_{k}-\omega^{\mathbb{C}}(\mathbf{u}_{k},\mathbf{f}_{j_{1}})\mathbf{e}_{j_{1}}\\}_{1\leq
k\leq n-1}$, and let $\hat{\mathbf{u}}_{n}:=\mathbf{f}_{j_{1}}$. Then
$\\{\hat{\mathbf{u}}_{1},\cdots,\hat{\mathbf{u}}_{n}\\}$ is a unitary basis of
a complex Lagrangian subspace. Let $S_{\mathfrak{u}}$ be the
$\mathbb{C}$-linear extension of $\mathbf{u}_{k}\mapsto\hat{\mathbf{u}}_{k}$,
$\mathfrak{J}\overline{\mathbf{u}}_{k}\mapsto\mathfrak{J}\overline{\hat{\mathbf{u}}}_{k}$
for $1\leq k\leq n$. Then $S_{\mathfrak{u}}\in\operatorname{Sp}(n)$.

If $i_{k}<j_{1}$, then
$\omega^{\mathbb{C}}(\mathbf{u}_{k},\mathbf{f}_{j_{1}})=0$, so sending
$\mathfrak{u}$ to $\hat{\mathfrak{u}}$ is equivalent to multiplying the
$n\times 2n$ matrix with $k$th row as
$[\mathbf{u}_{k}]_{\\{\mathbf{e},\mathbf{f}\\}}$, on the left by a lower
triangular $n\times n$ matrix. So
$(\hat{\mathbf{u}}_{1},\cdots,\hat{\mathbf{u}}_{n})\in\operatorname{pr}^{-1}(\mathbf{f}_{j_{1}})\subseteq
E(\lambda)$. Assigning $S_{\mathfrak{u}}$ to $\mathfrak{u}$ also depends
continuously on $\mathfrak{u}$. Let $E(\lambda)^{\prime}$ be the image
$(\operatorname{pr},S_{\cdot})(E(\lambda))\subseteq
S^{2(2n-j_{1}+1)-1}\times\operatorname{pr}^{-1}(\mathbf{f}_{j_{1}})$. Consider
the projection onto the second factor. The fiber above
$(\hat{\mathbf{u}}_{1},\cdots,\hat{\mathbf{u}}_{n-1},\mathbf{f}_{j_{1}})$
consists of all $\mathbf{u}_{n}$ in the intersection of the closed upper
hemisphere of $S^{2(2n-j_{1}+1)-1}$ with the orthogonal complement of

$\operatorname{span}_{\mathbb{R}}\\{\mathbf{u}_{1},\mathfrak{J}\overline{\mathbf{u}}_{1},\cdots,\mathbf{u}_{n-1},\mathfrak{J}\overline{\mathbf{u}}_{n-1},\mathbf{f}_{1},i\mathbf{f}_{1}\cdots,\mathbf{f}_{j_{1}-1},i\mathbf{f}_{j_{1}-1},i\mathbf{f}_{j_{1}}\\}$

which is a closed hemisphere of real dimension
$(4n-(2(n-1)+2j_{1}+1))-1=2j_{1}$. So $E(\lambda)\cong
E(\lambda)^{\prime}\to\operatorname{pr}^{-1}(\mathbf{f}_{j_{1}})\cong
E(\lambda_{\\{j_{2}^{\vee},\cdots,j_{|J|}^{\vee}\\}})$ is a trivial bundle
with fibers homeomorphic to the closed $2j_{1}$-ball. By inductive hypothesis,
the base is homeomorphic to a closed $2|\lambda|-2j_{1}$-ball. The total space
$E(\lambda)$ is thus homeomorphic to a closed $2|\lambda|$-ball. ∎

######  Remark 9 (Integral homology of real and complex Lagrangian
Grassmannians).

From Proposition 8,

$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})=\bigsqcup_{\lambda}C_{\lambda}$

is a CW decomposition. Since all Schubert cells are even dimensional, all
attaching maps have degree $0$, and the integral homology groups can computed
by counting how many shifted Young diagrams of a particular size are allowed:

$\displaystyle
H_{2k}(\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n});\mathbb{Z})$
$\displaystyle\cong$ $\displaystyle\mathbb{Z}^{|\\{\lambda:|\lambda|=k\\}|}$
$\displaystyle
H_{2k+1}(\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n});\mathbb{Z})$
$\displaystyle\cong$ $\displaystyle\\{0\\}.$

Equations (10) and Propositions 7, 8 also hold when all parameters are real
numbers.

$\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})=\bigsqcup_{\lambda}C_{\lambda}^{\mathbb{R}}$

is also a CW decomposition, and the degree of the attaching map
$g_{\lambda}^{\lambda^{\prime}}:\partial C_{\lambda^{\prime}}^{\mathbb{R}}\to
C_{\lambda}^{\mathbb{R}}$ can be computed. By Proposition 7 $\deg
g_{\lambda}^{\lambda^{\prime}}=0$ if $\lambda\nleq\lambda^{\prime}$, and for
dimensional reasons $\deg g_{\lambda}^{\lambda^{\prime}}=0$ if
$\lambda\leq\lambda^{\prime}$ and $|\lambda^{\prime}|>|\lambda|+1$. When
$\lambda_{J^{\vee}}\leq\lambda_{(J^{\prime})^{\vee}}$ and
$|\lambda_{(J^{\prime})^{\vee}}|=|\lambda_{J^{\vee}}|+1$, then from equation
(10) we have

$\displaystyle\deg g_{\lambda_{J^{\vee}}}^{\lambda_{(J^{\prime})^{\vee}}}=$
$\displaystyle(-1)^{\star}\cdot(1-(-1)^{\star+1})$ if $|J^{\prime}|>|J|$ (12a)
$\displaystyle\deg g_{\lambda_{J^{\vee}}}^{\lambda_{(J^{\prime})^{\vee}}}=$
$\displaystyle(-1)^{\star+\sum_{k=\star+1}^{|J|}(j_{k}^{\prime})^{\vee}}\cdot(1-(-1)^{(j_{\star}^{\prime})^{\vee}})$
if $|J^{\prime}|=|J|$ (12b)

since $C_{\lambda_{J^{\vee}}}^{\mathbb{R}}\times\\{t\\}$ and
$C_{\lambda_{J^{\vee}}}^{\mathbb{R}}\times\\{-t\\}$ have opposite
orientations. This agrees with [Fuc04] and [Rab16] up to sign. Integral
homology can be computed algorithmically.

######  Example 10.

The eight Schubert cells of $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{6})$
are shown in Table 1.

centertableaux, boxsize = 5pt  $J$ $I$ RREF Diagram $J^{\vee}$ $\\{1,2,3\\}$
$\phi$ $\left[\begin{array}[]{ccc|ccc}\ast&\ast&z_{33}&1&0&0\\\
\ast&z_{22}&z_{23}&0&1&0\\\ z_{11}&z_{12}&z_{13}&0&0&1\end{array}\right]$ 2 +
1, 1+2, 3 $\\{3,2,1\\}$ $\\{1,2\\}$ $\\{3\\}$
$\left[\begin{array}[]{ccc|ccc}\ast&\ast&1&0&0&0\\\
\ast&z_{22}&0&z_{23}&1&0\\\ z_{11}&z_{12}&0&z_{13}&0&1\end{array}\right]$ 1+2,
3 $\\{3,2\\}$ $\\{1,3\\}$ $\\{2\\}$
$\left[\begin{array}[]{ccc|ccc}\ast&1&0&0&0&0\\\ \ast&0&z_{22}&1&0&0\\\
z_{11}&0&z_{12}&0&z_{13}&1\end{array}\right]$ 1 + 1, 3 $\\{3,1\\}$ $\\{2,3\\}$
$\\{1\\}$ $\left[\begin{array}[]{ccc|ccc}1&0&0&0&0&0\\\ 0&\ast&z_{23}&1&0&0\\\
0&z_{11}&z_{12}&0&1&0\end{array}\right]$ 1 + 1, 2 $\\{2,1\\}$ $\\{1\\}$
$\\{2,3\\}$ $\left[\begin{array}[]{ccc|ccc}\ast&1&0&0&0&0\\\ \ast&0&1&0&0&0\\\
z_{11}&0&0&z_{12}&z_{13}&1\end{array}\right]$ 3 $\\{3\\}$ $\\{2\\}$
$\\{1,3\\}$ $\left[\begin{array}[]{ccc|ccc}1&0&0&0&0&0\\\ 0&\ast&1&0&0&0\\\
0&z_{11}&0&z_{12}&1&0\end{array}\right]$ 2 $\\{2\\}$ $\\{3\\}$ $\\{1,2\\}$
$\left[\begin{array}[]{ccc|ccc}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&z_{11}&1&0&0\end{array}\right]$ 1 $\\{1\\}$ $\phi$ $\\{1,2,3\\}$
$\left[\begin{array}[]{ccc|ccc}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{array}\right]$ $\phi$ $\phi$

Table 1: The eight complex Schubert cells of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{6})$

The integral homology groups are then

$H_{k}(\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{6});\mathbb{Z})\cong\begin{cases}\mathbb{Z}&\text{
if }k=0,2,4,8,10,12\\\ \mathbb{Z}^{2}&\text{ if }k=6\\\ 0&\text{
otherwise.}\end{cases}$

For $\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{6})$ the CW structure is
shown in Figure 7.

Figure 7: A diagrammatic description of the CW structure of
$\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{6})$ by real Schubert cells. Sign
conventions of attaching maps differ from [Rab16] Example 4.3.

The integral homology groups are then

$H_{k}(\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{6});\mathbb{Z})\cong\begin{cases}\mathbb{Z}&\text{
if }k=0,1,5,6\\\ \mathbb{Z}_{2}&\text{ if }k=2,3\\\ 0&\text{
otherwise.}\end{cases}$

##  3 Schubert cells of mixed type

###  3.1 Shifted Young diagrams of Schubert cells of mixed type

Identify $\mathbb{C}_{(k,\ell)}$ with $\mathbb{R}_{(k,\ell)}\times
i\mathbb{R}_{(k,\ell)}$ with coordinates $z_{k\ell}=x_{k\ell}+iy_{k\ell}$. We
give the lexicographical order on the Cartesian product
$\lambda\times\\{x,y\\}$, from the lexicographical order on $\lambda$, and
$x<y$. So $((k,\ell),t)\leq((k^{\prime},\ell^{\prime}),t^{\prime})$ if either
$(k,\ell)<(k^{\prime},\ell^{\prime})$ or both
$(k,\ell)=(k^{\prime},\ell^{\prime})$ and $t\leq t^{\prime}$. Then we identify
$x_{k\ell}$ with $((k,\ell),x)$ and $y_{k\ell}$ with $((k,\ell),y)$.

Let

$\mathring{\mu}:=\bigcap_{\begin{subarray}{c}\mu^{\prime}\leq\mu\\\
|\mu^{\prime}|=|\mu|-1\end{subarray}}\mu^{\prime}.$

and $C_{\lambda+i\mu}:=(z^{\lambda})^{-1}(U_{\lambda+i\mu})$ where

$U_{\lambda+i\mu}:=\prod_{(k,\ell)\in\mathring{\mu}}\mathbb{C}_{(k,\ell)}\times\prod_{(k,\ell)\in\mu\setminus\mathring{\mu}}(\mathbb{R}_{(k,\ell)}\times
i\mathbb{R}_{(k,\ell)}^{\times})\times\prod_{(k,\ell)\in\lambda\setminus\mu}\mathbb{R}_{(k,\ell)}.$

Let
$\mathcal{E}(\mu):=\\{\varepsilon:\mu\setminus\mathring{\mu}\to\\{+1,-1\\}\\}$,
and let

$U_{\lambda+i\mu,\varepsilon}:=\prod_{(k,\ell)\in\mathring{\mu}}\mathbb{C}_{(k,\ell)}\times\prod_{(k,\ell)\in\mu\setminus\mathring{\mu}}(\mathbb{R}_{(k,\ell)}\times\varepsilon(k,\ell)i\mathbb{R}_{(k,\ell)}^{+})\times\prod_{(k,\ell)\in\lambda\setminus\mu}\mathbb{R}_{(k,\ell)}$

and
$C_{\lambda+i\mu,\varepsilon}:=(z^{\lambda})^{-1}(U_{\lambda+i\mu,\varepsilon})$.
Then $C_{\lambda+i\mu,\varepsilon}$ are the connected components of
$C_{\lambda+i\mu}$.

The shifted Young diagrams of mixed type $\lambda+i\mu$ can be decorated by
adding labels on $\lambda\setminus\mathring{\mu}$. We will label copies of
$\mathbb{C}$ with $\ytableaushort{\,}$, copies of $\mathbb{R}\times
i\mathbb{R}^{\times}$ with $\ytableaushort{\times}$, copies of $\mathbb{R}$
with $\ytableaushort{\circ}$, copies of $\mathbb{R}\times i\mathbb{R}^{+}$ by
$\ytableaushort{+}$, and copies of $\mathbb{R}\times-i\mathbb{R}^{+}$ by
$\ytableaushort{-}$.

######  Example 11.

baseline, boxsize = 7pt Suppose

$\lambda=\ytableau\none&\none\\\ \none\\\
,\quad\mu=\ytableau\none&\none\none\\\ \none\none\\\ .$

The shifted Young diagram of $\lambda+i\mu$ is denoted as

$\ytableau\none&\none\circ\\\ \none\times\circ\\\ \times.$

The shifted Young diagrams of the four connected components of
$C_{\lambda+i\mu}$ are

$\ytableau\none&\none\circ\\\ \none+\circ\\\
+,\quad\ytableau\none&\none\circ\\\ \none+\circ\\\
-,\quad\ytableau\none&\none\circ\\\ \none-\circ\\\
+,\quad\ytableau\none&\none\circ\\\ \none-\circ\\\ -.$

###  3.2 Attaching maps of Schubert cells of mixed type

######  Lemma 12.

If $\mu\nleq\mu^{\prime}$, then there exists an element in
$\mu\setminus\mathring{\mu}$ not in $\mu^{\prime}$.

###### Proof.

$\mu^{\prime}\cap\mu<\mu$, so there exists a $\mu^{\prime\prime}$ with
$|\mu^{\prime\prime}|=|\mu|-1$ such that
$\mu^{\prime}\cap\mu\leq\mu^{\prime\prime}<\mu$. Then the unique square in
$\mu\setminus\mu^{\prime\prime}$ is contained in $\mu\setminus\mathring{\mu}$
but is not contained in $\mu^{\prime}$. ∎

Suppose $\mu\neq\mu^{\prime}$, and without loss of generality
$\mu\nleq\mu^{\prime}$. Then the square given by Lemma 12 has the coordinate
$(k,\ell)$. Then $y_{k\ell}^{\lambda}(\Gamma)$ is nonzero for $\Gamma\in
C_{\lambda+i\mu}$ and zero for $\Gamma\in C_{\lambda+i\mu^{\prime}}$. So the
$C_{\lambda+i\mu}$’s are disjoint. Moreover, $\mathbb{C}^{|\lambda|}$ is
partitioned by how many of the last $k$ coordinates are real. So we have the
partitions

$C_{\lambda}=\bigsqcup_{\mu:\mu\leq\lambda}C_{\lambda+i\mu},\quad\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})=\bigsqcup_{\begin{subarray}{c}\lambda,\mu\\\
\mu\leq\lambda\end{subarray}}C_{\lambda+i\mu}.$

######  Lemma 13 (Frontier condition of $C_{\lambda}$).

The following are equivalent

  1. 1.

$C_{\lambda+i\mu}\cap\overline{C_{\lambda+i\mu^{\prime}}}\cap
C_{\lambda}\neq\phi$.

  2. 2.

$C_{\lambda+i\mu}\subseteq\overline{C_{\lambda+i\mu^{\prime}}}\cap
C_{\lambda}$.

  3. 3.

$\mu\leq\mu^{\prime}$.

###### Proof.

$(2)\implies(1)$ is immediate, as $C_{\lambda+i\mu}$ is nonempty.
$(3)\implies(2)$:
$U_{\lambda+i\mu}\subseteq\overline{U_{\lambda+i\mu^{\prime}}}$, so
$(z^{\lambda})^{-1}(U_{\lambda+i\mu})\subseteq(z^{\lambda})^{-1}(\overline{U_{\lambda+i\mu^{\prime}}})$.
$(1)\implies(3)$: If $\mu\nleq\mu^{\prime}$, by Lemma 12, there exists a
$(k,\ell)\in\mu\setminus\mathring{\mu}$ not in $\mu^{\prime}$. Then
$y_{k\ell}^{\lambda}(\Gamma_{k})=0$ for any sequence $\Gamma_{k}\in
C_{\lambda+i\mu^{\prime}}$, but $y_{k\ell}^{\lambda}(\Gamma)\neq 0$ for all
$\Gamma\in C_{\lambda+i\mu}$. ∎

######  Remark 14.

Immediately, we can verify $C_{\lambda+i\phi}\cong C_{\lambda}^{\mathbb{R}}$,
$C_{\phi+i\phi}=C_{\phi}=C_{\phi}^{\mathbb{R}}$, $C_{\lambda+i\lambda}$ is
open dense in $C_{\lambda}$, and $\overline{C_{\lambda+i\mu^{\prime}}}\cap
C_{\lambda}=\bigsqcup_{\mu:\mu\leq\mu^{\prime}}C_{\lambda+i\mu}$.

######  Proposition 15.

If $\lambda\leq\lambda^{\prime}$, $|\lambda^{\prime}|=|\lambda|+1$, and
$\mu\leq\lambda$

$\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu}\times\mathbb{R}^{\times})\subseteq
C_{\lambda^{\prime}+i\mu}.$

Moreover, suppose $\lambda=\lambda_{J}$ and
$\lambda^{\prime}=\lambda_{(J^{\prime})^{\vee}}$, and denote the real Jacobian
determinant of the transition map

$z^{\lambda^{\prime}}\circ\varphi_{\lambda}^{\lambda^{\prime}}\circ((z^{\lambda})^{-1}\times\mathrm{Id}_{\mathbb{R}^{\times}}):U_{\lambda+i\mu}\times\mathbb{R}^{\times}\to
U_{\lambda^{\prime}+i\mu}$

of $C_{\lambda^{\prime}+i\mu}$ by $T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}$.
Then $T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}$ is a function of $t$ only, and
$T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}(t)$ is equal to

$\begin{cases}(-1)^{|c_{\star}|+|c_{\star}\cap\mu|}\cdot
t^{-|c_{\star}|-|c_{\star}\cap\mu|-1}&\text{ if }|J^{\prime}|>|J|\\\
(-1)^{|c_{\star}|+|c_{\star}\cap\mu|+|\rho_{\star}|+|\rho_{\star}\cap\mu|}\cdot
t^{-|\alpha_{\star}|-|\alpha_{\star}\cap\mu|-|\mu\cap\\{(\star,\star)\\}|-2}&\text{
if }|J^{\prime}|=|J|\end{cases}$ (13)

where $t$ is the coordinate of $\mathbb{R}^{\times}$.

###### Proof.

We examine equations (8d) and (9h). When $(k,\ell)\in\mathring{\mu}$,
$z^{\prime}_{k\ell}$ is complex valued. When
$(k,\ell)\in\mu\setminus\mathring{\mu}$, $z^{\prime}_{k\ell}$ has nonvanishing
imaginary part because $z_{k\ell}$ has nonvanishing imaginary part, and if
either $k^{\prime}>k$ or $\ell^{\prime}>\ell$, $z_{k^{\prime},\ell^{\prime}}$
is real valued for $\Gamma\in C_{\lambda+i\mu}$. Similarly, when
$(k,\ell)\in\lambda\setminus\mu$, $z^{\prime}_{k\ell}$ has vanishing imaginary
part because $z_{k\ell}$ has vanishing imaginary part, and if either
$k^{\prime}>k$ or $\ell^{\prime}>\ell$, $z_{k^{\prime},\ell^{\prime}}$ is real
valued.

The ordering of the variables
$\\{x_{k\ell}\\}_{(k,\ell)\in\lambda}\cup\\{y_{k\ell}\\}_{(k,\ell)\in\mu}$
ensures that the real Jacobian determinant of the transition map is upper
triangular. In equations (8b), (8d), (9c), (9d), (9e), (9f), and (9g), we
double count contributions of $\mp t^{-1}$ (and $\mp t^{-2}$) when
$(k,\ell)\in\mu$. We also double count the sign contribution due to
rearranging the order of $x_{\star+j_{\star}^{\vee},\star}$ from
$|\rho_{\star}|$ to $|\rho_{\star}|+|\rho_{\star}\cap\mu|$. ∎

######  Proposition 16 (Frontier condition).

The following are equivalent

  1. 1.

$C_{\lambda+i\mu}\cap\overline{C_{\lambda^{\prime}+i\mu^{\prime}}}\neq\phi$.

  2. 2.

$C_{\lambda+i\mu}\subseteq\overline{C_{\lambda^{\prime}+i\mu^{\prime}}}$.

  3. 3.

$\lambda\leq\lambda^{\prime}$ and $\mu\leq\mu^{\prime}$.

###### Proof.

$(2)\implies(1)$ is immediate as $C_{\lambda+i\mu}$ is nonempty.
$(3)\implies(2)$: There exists an increasing sequence
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{\ell}$ such that
$\lambda_{1}=\lambda$, $\lambda_{\ell}=\lambda^{\prime}$, and
$|\lambda_{k+1}|=|\lambda_{k}|+1$ for $1\leq
k<\ell=|\lambda^{\prime}|-|\lambda|$, and a sequence of embeddings
$\\{\varphi_{\lambda_{k}}^{\lambda_{k+1}}\\}_{k=1}^{\ell}$. For each
$\Gamma\in C_{\lambda+i\mu}$, let

$\Gamma_{t_{1},\cdots,t_{\ell}}:=\varphi_{\lambda_{\ell-1}}^{\lambda_{\ell}}(\cdots(\varphi_{\lambda_{2}}^{\lambda_{3}}(\varphi_{\lambda_{1}}^{\lambda_{2}}(\Gamma,t_{1}),t_{2}),\cdots),t_{\ell})\quad
t_{1},\cdots,t_{\ell}\in\mathbb{R}$

where if $t_{j}=0$ for some $j\in\\{1,\cdots,\ell\\}$, then $t_{k}=0$ for all
$k\geq j$. Then by equations 3, 5 and Proposition 15,
$\Gamma_{t_{1},\cdots,t_{\ell}}\in
C_{\lambda^{\prime}+i\mu}\subseteq\overline{C_{\lambda^{\prime}+i\mu^{\prime}}}$,
and

$\lim_{t_{1},\cdots,t_{\ell}\to 0}\Gamma_{t_{1},\cdots,t_{\ell}}=\Gamma.$

$(1)\implies(3)$: If $\lambda\nleq\lambda^{\prime}$, then
$C_{\lambda}\cap\overline{C_{\lambda^{\prime}}}=\phi$, which contradicts the
assumption. So $\lambda\leq\lambda^{\prime}$. By assumption there exists a
sequence $\Gamma^{\prime}_{m}\to\Gamma$ with $\Gamma^{\prime}_{m}\in
C_{\lambda^{\prime}+i\mu^{\prime}}$. By the computation of Jacobian
determinants of equation (10), $\varphi_{\lambda_{k}}^{\lambda_{k+1}}$ are
local homeomorphisms from
$C_{\lambda_{k}+i\mu^{\prime}}\times\mathbb{R}^{\times}$ to
$C_{\lambda_{k+1}+i\mu^{\prime}}$. By induction, we get a local homeomorphism
$C_{\lambda+i\mu^{\prime}}\times(\mathbb{R}^{\times})^{|\lambda^{\prime}|-|\lambda|}$
to $C_{\lambda^{\prime}+i\mu^{\prime}}$. Let
$(\Gamma_{m},(t^{m}_{1},\cdots,t^{m}_{\ell}))$ be the image
$\Gamma_{m}^{\prime}$ under these local homeomorphisms, which exist for $m$
sufficiently large. Then if $\mu\nleq\mu^{\prime}$, there exists
$(a,b)\in\mu\setminus\mathring{\mu}$ not in $\mu^{\prime}$. Then
$y_{ab}^{\lambda}(\Gamma_{m})=0$ but $y_{ab}^{\lambda}(\Gamma)\neq 0$, which
is a contradiction. So $\mu\leq\mu^{\prime}$. ∎

We obtain attaching maps of Schubert cells of mixed type by restricting the
attaching maps of Proposition 8.

######  Theorem 17 (Attaching maps of Schubert cells of mixed type).

If $|\lambda|+|\mu|+1=|\lambda^{\prime}|+|\mu^{\prime}|+1$, let

$g_{\lambda+i\mu,\varepsilon}^{\lambda^{\prime}+i\mu^{\prime},\varepsilon^{\prime}}:\partial
C_{\lambda^{\prime}+i\mu^{\prime},\varepsilon^{\prime}}\to\overline{C_{\lambda+i\mu,\varepsilon}}.$

  1. 1.

If $\lambda\nleq\lambda^{\prime}$ or $\mu\nleq\mu^{\prime}$, then

$\deg
g_{\lambda+i\mu,\varepsilon}^{\lambda^{\prime}+i\mu^{\prime},\varepsilon^{\prime}}=0.$
(14)

  2. 2.

If $\mu=\mu^{\prime}$ and $\lambda\leq\lambda^{\prime}$ let

$\varepsilon^{\pm}(k,\ell):=\begin{cases}\pm\varepsilon(k,\ell)&\text{ if
}(k,\ell)\in((\mu\setminus\mathring{\mu})\cap(\alpha_{\star}\setminus
c_{\star}))\setminus\\{(\star,\star)\\}\\\ \mp\varepsilon(k,\ell)&\text{ if
}(k,\ell)\in(\mu\setminus\mathring{\mu})\cap c_{\star}\\\
\varepsilon(k,\ell)&\text{ otherwise,}\end{cases}$

where $\alpha_{\star}$ is empty if $\lambda^{\prime}$ has more rows than
$\lambda$. Moreover, let

$\delta^{\pm}(\varepsilon,\varepsilon^{\prime}):=\begin{cases}1&\text{ if
}\varepsilon^{\prime}=\varepsilon^{\pm}\\\ 0&\text{ otherwise.}\end{cases}$

Then

$\deg
g_{\lambda+i\mu,\varepsilon}^{\lambda^{\prime}+i\mu,\varepsilon^{\prime}}=T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}(+1)\cdot\delta^{+}(\varepsilon,\varepsilon^{\prime})-T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}(-1)\cdot\delta^{-}(\varepsilon,\varepsilon^{\prime}),$
(15)

where $T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}$ is as in equation (13).

  3. 3.

If $\lambda=\lambda^{\prime}$ and $\mu\leq\mu^{\prime}$ then

$\deg
g_{\lambda+i\mu,\varepsilon}^{\lambda^{\prime}+i\mu^{\prime},\varepsilon^{\prime}}=\begin{cases}-(-1)^{|\lambda\setminus\mu^{\prime}|}\cdot\varepsilon^{\prime}(\star_{\mu})&\text{
if }\varepsilon^{\prime}=\varepsilon\text{ on
}(\mu^{\prime}\setminus\mathring{\mu}^{\prime})\cap(\mu\setminus\mathring{\mu})\\\
0&\text{ otherwise.}\end{cases}$ (16)

where $\star_{\mu}$ is the unique element of $\mu^{\prime}\setminus\mu$.

###### Proof.

If $\lambda\nleq\lambda^{\prime}$ or $\mu\nleq\mu^{\prime}$, then the degree
of
$g_{\lambda+i\mu,\varepsilon}^{\lambda^{\prime}+i\mu^{\prime},\varepsilon^{\prime}}$
is zero by the frontier condition.

If $\mu=\mu^{\prime}$ and $\lambda\leq\lambda^{\prime}$, we look at Equations
(8d) and (9h). When $(k,\ell)\in\mathring{\mu}$, both $z_{k\ell}$ and
$z_{k\ell}^{\prime}$ are complex valued with no restrictions, and when
$(k,\ell)\in\lambda\setminus\mu$, both $z_{k\ell}$ and $z_{k\ell}^{\prime}$
are real valued with no restrictions. When
$(k,\ell)\in\mu\setminus\mathring{\mu}$, the coefficient of $z_{k\ell}$ is
$t^{-1}$ if $(k,\ell)\in(\alpha_{\star}\setminus
c_{\star})\setminus\\{(\star,\star)\\}$ and $-t^{-1}$ if $(k,\ell)\in
c_{\star}$
($(\lambda^{\prime}\setminus\lambda)\cap(\mu\setminus\mathring{\mu})$ is
empty), and either $1$ or $t^{-2}$ otherwise. So when $t>0$,
$\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu,\varepsilon)}\times\mathbb{R}^{+})\subseteq
C_{\lambda^{\prime}+i\mu,\varepsilon^{\prime}}$ if and only if
$\varepsilon^{\prime}=\varepsilon^{+}$, and if $t<0$,
$\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu,\varepsilon}\times\mathbb{R}^{-})\subseteq
C_{\lambda^{\prime}+i\mu,\varepsilon^{\prime}}$ if and only if
$\varepsilon^{\prime}=\varepsilon^{-}$.

If $\lambda=\lambda^{\prime}$ and $\mu\leq\mu^{\prime}$, then
$-\varepsilon(\star_{\mu})$ points in the gradient direction of
$y_{\star_{\mu}}$, and the sign correction due to the ordering of $y_{k\ell}$
is $(-1)^{|\lambda\setminus\mu^{\prime}|}$.

∎

######  Example 18.

Suppose $\lambda=\lambda_{\\{4,2\\}}$, $\lambda^{\prime}=\lambda_{\\{4,3\\}}$,
and $\mu=\lambda_{\\{4,2\\}}$. Then

$\lambda+i\mu=\ytableaushort{\none\,\times\none,\,\,\,\times}\text{ and
}\lambda^{\prime}+i\mu=\ytableaushort{\none\,\times{*(lightgray)\circ},\,\,\,\times}.$

Then $\mu\setminus\mathring{\mu}=\\{(1,4),(2,3)\\}$,

$((\mu\setminus\mathring{\mu})\cap(\alpha_{\star}\setminus
c_{\star}))\setminus\\{(\star,\star)\\}=\\{(2,3)\\}$

and

$(\mu\setminus\mathring{\mu})\cap c_{\star}=\\{(1,4)\\}.$

Denote $\varepsilon$ by an ordered pair $(\varepsilon(1,4),\varepsilon(2,3))$.
Since there are no restrictions on the values of $z_{11}^{\prime}$,
$z_{12}^{\prime}$, $z_{13}^{\prime}$, $z_{22}^{\prime}$, and

$\displaystyle z_{14}^{\prime}$ $\displaystyle=$ $\displaystyle-t^{-1}z_{14}$
$\displaystyle z_{23}^{\prime}$ $\displaystyle=$ $\displaystyle t^{-1}z_{23},$

so

$\displaystyle\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu,(\pm
1,+1)}\times\mathbb{R}^{+})$ $\displaystyle\subseteq$ $\displaystyle
C_{\lambda^{\prime}+i\mu,(\mp 1,+1)}$
$\displaystyle\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu,(\pm
1,-1)}\times\mathbb{R}^{+})$ $\displaystyle\subseteq$ $\displaystyle
C_{\lambda^{\prime}+i\mu,(\mp 1,-1)}$

and attach with degree
$T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}(+1)=-(+1)^{-9}=-1$, and

$\displaystyle\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu,(+1,\pm
1)}\times\mathbb{R}^{-})$ $\displaystyle\subseteq$ $\displaystyle
C_{\lambda^{\prime}+i\mu,(+1,\mp 1)}$
$\displaystyle\varphi_{\lambda}^{\lambda^{\prime}}(C_{\lambda+i\mu,(-1,\pm
1)}\times\mathbb{R}^{-})$ $\displaystyle\subseteq$ $\displaystyle
C_{\lambda^{\prime}+i\mu,(-1,\mp 1)}$

and attach with degree
$-T_{\lambda+i\mu}^{\lambda^{\prime}+i\mu}(-1)=-(-1)^{-9}=+1$. The attaching
maps are shown in Figure 8.

Figure 8: A diagrammatic description of the attaching maps $\partial
C_{\lambda_{\\{4,3\\}}+i\lambda_{\\{4,2\\}}}\to
C_{\lambda_{\\{4,2\\}}+i\lambda_{\\{4,2\\}}}$. Arrows represent attaching maps
of degree $+1$ and dashed arrows represent attaching maps of degree $-1$.

######  Example 19.

Suppose $\lambda=\lambda^{\prime}=\lambda_{\\{4,3\\}}$,
$\mu^{\prime}=\lambda_{\\{4,2\\}}$, $\mu=\lambda_{\\{4,1\\}}$, so that

$\lambda+i\mu^{\prime}=\ytableaushort{\none\,{*(lightgray)\times}\circ,\,\,\,\times}\text{
and
}\lambda+i\mu=\ytableaushort{\none\times{*(lightgray)\circ}\circ,\,\,\,\times}.$

The attaching maps are shown in Figure 9.

Figure 9: A diagrammatic description of the attaching maps $\partial
C_{\lambda_{\\{4,3\\}}+i\lambda_{\\{4,2\\}}}\to
C_{\lambda_{\\{4,3\\}}+i\lambda_{\\{4,1\\}}}$. Arrows represent attaching maps
of degree $+1$ and dashed arrows represent attaching maps of degree $-1$.

##  4 Applications

If $n,m$ are positive integers, let
$\iota_{n,m}:\mathbb{C}^{2n}\hookrightarrow\mathbb{C}^{2(n+m)}$ be an
embedding given by the linear extension of
$\mathbf{e}_{i}\mapsto\mathbf{e}_{i+m}$,
$\mathbf{f}_{i}\mapsto\mathbf{f}_{i+m}$. $\iota_{n,m}$ induces an embedding of
complex Lagrangian Grassmannians
$I_{n,m}:\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})\hookrightarrow\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2(n+m)})$
as

$I_{n,m}(\Gamma):=\operatorname{span}_{\mathbb{C}}\\{\mathbf{e}_{1},\cdots,\mathbf{e}_{m}\\}\oplus\iota_{n,m}(\Gamma).$

If $M_{\Gamma}$ is the $n\times 2n$ matrix representing the reduced row
echelon form of $\Gamma\in\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$,
then the $(n+m)\times 2(n+m)$ matrix representing the reduced row echelon form
of $I_{n,m}(\Gamma)$ is

$\begin{bmatrix}\operatorname{Id}_{m,m}&0_{m,2n}&0_{m,m}\\\
0_{n,m}&M_{\Gamma}&0_{m,m}\end{bmatrix}.$

$I_{n,m}$ maps each Schubert cell of mixed type $C_{\lambda+i\mu,\varepsilon}$
of $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$ to the Schubert cell of
mixed type $C_{\lambda+i\mu,\varepsilon}$ of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2(n+m)})$. $I_{n,m}$ is
continuous, so it preserves all incidence relations. Moreover,
$\varphi_{\lambda}^{\lambda^{\prime}}\circ
I_{n,m}=I_{n,m}\circ\varphi_{\lambda}^{\lambda^{\prime}}$, so it preserves all
degrees of attaching maps as well. So the
$I_{m,n}(\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n}))$ can be realized as
a subcomplex of $\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2(n+m)})$.

Similarly, let
$\iota_{m,n}^{\mathbb{R}}:\mathbb{R}^{2m}\hookrightarrow\mathbb{R}^{2n}$ and
$I_{n,m}^{\mathbb{R}}:\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2m})\hookrightarrow\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})$
be the corresponding maps for real coefficients. Suppressing these
identifications, we will regard
$\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n})$ as subcomplexes of
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2(n+m)})$.

######  Corollary 20 (Homotopy extension property).

$(\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2(n+m)}),\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n}))$
satisfies the homotopy extension property.

######  Corollary 21.

If $0<k\leq n$ and $k$ is even then
$\overline{C}_{\lambda_{\\{k\\}}}^{\mathbb{R}}$ defines a nontrivial torsion
class in $H_{k}(\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n});\mathbb{Z})$,
and is homotopic to a $2k$-sphere inside
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$.

###### Proof.

By equation (12b), $\overline{C}_{\lambda_{\\{k\\}}+i\phi}$ defines a homology
class in $H_{k}(\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n});\mathbb{Z})$.
By induction on $k$, $\overline{C}_{\lambda_{\\{k-1\\}}+i\lambda_{\\{1\\}},+}$
is contractible. Taking the quotient of this subcomplex inside
$\overline{C}_{\lambda_{\\{k\\}}+i\lambda_{\\{1\\}},+}$ we get a homotopy
equivalence between $\overline{C}_{\lambda_{\\{k\\}}+i\lambda_{\\{1\\}},+}$
and the closed $2k+1$-ball. This equivalence identifies the subcomplex
$\overline{C}_{\lambda_{\\{k\\}}+i\phi}$ with its boundary. ∎

######  Remark 22 ($k=2$ case).

If $k=2\leq n$, $\overline{C}_{\lambda_{\\{2\\}}}^{\mathbb{R}}$ represents the
generator of
$H_{2}(\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n});\mathbb{Z})\cong\mathbb{Z}_{2}$.
Topologically, this set is homeomorphic to a pinched torus. One way to see
this is by doing row reductions to

$\left[\begin{array}[]{cc|cc}\cos\theta&\sin\theta&0&0\\\
-\cos\psi\sin\theta&\cos\psi\cos\theta&-\sin\psi\cos\theta&\sin\psi\sin\theta\end{array}\right]\in
E(\lambda_{\\{2\\}})$

at various values of $\theta,\psi\in[0,\pi]$ (cf Liu [Liu18]). So the
generator of
$H_{2}(\operatorname{Lag}^{\mathbb{R}}(\mathbb{R}^{2n});\mathbb{Z})$ is
spherical, which is something we cannot conclude from the Hurewicz theorem. By
Corollary 21 this pinched torus is homotopic to a $2$-sphere inside
$\operatorname{Lag}^{\mathbb{C}}(\mathbb{R}^{2n})$.

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