# Approximate Triangulations of Grassmann Manifolds
## 2 Materials and Methods
### Schubert Cells  
Grassmannians have a well-known cell decompostion into Schubert cells. Consider the sequence of subspaces of \(\mathbb{R}^{n}\): \(\mathbb{R}^{0}\subset\mathbb{R}^{1}\subset\mathbb{R}^{2}\subset\cdots\subset \mathbb{R}^{n}\), where \(\mathbb{R}^{i}\) consists of the vectors of the form \((a_{1},\dots,a_{i},0,\dots,0)\). Any \(k\)-plane \(X\) gives rise to a sequence of integers  
\[0\leq\dim(X\cap\mathbb{R}^{1})\leq\dim(X\cap\mathbb{R}^{2})\leq\cdots\leq \dim(X\cap\mathbb{R}^{n})=k.\]  
Consecutive integers differ by at most 1.  
**Definition 2**.: _A Schubert symbol \(\sigma=(\sigma_{1},\dots,\sigma_{k})\) is a sequence of \(k\) integers satisfying_  
\[1\leq\sigma_{1}<\sigma_{2}<\cdots<\sigma_{k}\leq n.\]  
Given a Schubert symbol \(\sigma\), let \(e(\sigma)\subset G_{k}(\mathbb{R}^{n})\) denote the set of \(k\)-planes \(X\) such that  
\[\dim(X\cap\mathbb{R}^{\sigma_{i}})=i,\dim(X\cap\mathbb{R}^{\sigma_{i}-1})=i-1.\]Each \(X\in G_{k}(\mathbb{R}^{n})\) belongs to precisely one of the sets \(e(\sigma)\).  
Lemma 2: \(e(\sigma)\) _is an open cell of dimension \(d(\sigma)=(\sigma_{1}-1)+(\sigma_{2}-2)+\cdots+(\sigma_{k}-k)\)._  
In terms of matrices, \(X\in e(\sigma)\) if and only if it can be described as the row space of a \(k\times n\) matrix of the form  
\[\left[\begin{array}{ccccccccccccccccc}*&\cdots&*&1&0&\cdots&0&0&0&\cdots&0&0& 0&\cdots&0\\ *&\cdots&*&*&*&\cdots&*&1&0&\cdots&0&0&0&\cdots&0\\ \vdots&&&&&&&&&&&&\vdots\\ *&\cdots&*&*&*&\cdots&*&*&*&\cdots&*&1&0&\cdots&0\end{array}\right]\]  
where the \(i\)-th row has \(\sigma_{i}\)-th entry positive (say equal to \(1\)) and all subsequent entries zero. Equivalently, we could (and do in the sequel) consider the column space of the transpose of this matrix.  
For example, the possible Schubert symbols and cells for \(G_{2}(\mathbb{R}^{4})\) are as follows. Such a symbol has the form \(\sigma=(\sigma_{1},\sigma_{2})\) where \(1\leq\sigma_{1}<\sigma_{2}\leq 4\).  
\[\begin{array}{c|c}\sigma&d(\sigma)\\ \hline(1,2)&0\\ (1,3)&1\\ (1,4)&2\\ (2,3)&2\\ (2,4)&3\\ (3,4)&4\end{array}\]  
Theorem 2: _The \((^{n}_{k})\) sets \(e(\sigma)\) form the cells of a CW-decomposition of \(G_{k}(\mathbb{R}^{n})\)._  
Proposition 1: _The number of \(r\)-cells in \(G_{k}(\mathbb{R}^{n})\) is equal to the number of partitions of \(r\) into at most \(k\) integers, each of which is \(\leq n-k\)._  
The mod 2 homology of \(G_{k}(\mathbb{R}^{n})\) is easily computed from the Schubert cell decomposition: since the induced boundary maps are all either \(0\) or multiplication by \(2\), the mod 2 homology has a basis corresponding to the cells.  
Continuing the example of \(G_{2}(\mathbb{R}^{4})\), we have  
\[H_{i}(G_{2}(\mathbb{R}^{4}),\mathbb{Z}/2)=\begin{cases}\mathbb{Z}/2&i=0\\ \mathbb{Z}/2&i=1\\ \mathbb{Z}/2\oplus\mathbb{Z}/2&i=2\\ \mathbb{Z}/2&i=3\\ \mathbb{Z}/2&i=4\end{cases}\]  
Since we will need it below, we also note the integral homology of \(G_{2}(\mathbb{R}^{4})\)[7]:  
\[H_{i}(G_{2}(\mathbb{R}^{4}),\mathbb{Z})=\begin{cases}\mathbb{Z}&i=0\\ \mathbb{Z}/2&i=1,2\\ 0&i=3\\ \mathbb{Z}&i=4\end{cases}\]  
Using the Universal Coefficient Theorem, one then quickly deduces that the homology groups \(H_{i}(G_{2}(\mathbb{R}^{4}),\mathbb{Z}/3)\) are \(\mathbb{Z}/3\) for \(i=0,4\) and \(0\), otherwise.