\documentclass{amsart}
\usepackage{amsmath, amssymb, enumitem}

\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}}
\renewcommand{\P}{{\mathbb P}}
\renewcommand{\S}{\operatorname{Sym}}
\renewcommand{\O}{{\mathcal O}}
\newcommand{\uk}{\underline{k}}

\newtheorem{Lemma}{Lemma}

\title{Flag Varieties \\ Assignment 2 \\ Due Friday October 14}


\begin{document}
\maketitle
\begin{enumerate}


 \item
Let $ \uk= (k_1, \dots, k_m )$ be natural numbers with $ k_1 + \dots + k_m = n $.  For any vector space $ V $ of dimension $ n$, let $ Fl(\uk,V) $ denote the partial flag variety
$$
\{ 0 \subseteq V_1 \subseteq \cdots \subseteq V_m = V : \dim V_i = \dim V_{i-1} + k_i \}
$$
\begin{enumerate}
 \item How many $ B$-orbits are there on $ Fl(\uk, \C^n) $?  (As usual $ B $ denotes the subgroup of upper triangular matrices.)  As usual, these $B$-orbits are called Schubert cells.
\item Find the dimension of the Schubert cells.
\item Describe the Schubert cells in terms of intersections with the standard flag.
\item Recall that if $ \lambda = (\lambda_1, \dots, \lambda_n) \in \R^n $, then $ H_\lambda $ denotes the set of Hermitian matrices with eigenvalues $ \lambda_1,\dots, \lambda_n $.  Recall that if $ \lambda_i $ are distinct, then we constructed an isomorphism between $ H_\lambda $ and the full flag variety.  For each $ \uk $, find some $ \lambda$, such that there is a natural bijection $ H_\lambda \cong Fl(\uk, \C^n) $.
\item A special case of a partial flag variety is a Grassmannian.  This corresponds to the choice $ \uk = (k,n-k) $.  In this case, show that there are $ \binom{n}{k} $ Schubert cells.
\end{enumerate}

\item Let $\lambda \in \Z^n_+ $ be a dominant weight and let $V(\lambda) = \Gamma(Fl_n, \O(\lambda))^* $.  Recall that in class we showed that $ V(\lambda) $ is an irreducible representation of $ GL_n$.

\begin{enumerate}
 \item In class, we constructed a non-zero $N$-invariant section $ s \in \Gamma(Fl_n, \O(\lambda)) $.  Show that this section has weight $ (-\lambda_n, \dots, -\lambda_1) $.  
\item If $ V $ is any representation of $ GL_n $ and $ \mu $ is any weight, use the embedding $ S_n \subset GL_n $ to prove that $ V_\mu \cong V_{w \mu} $ for any $ w \in S_n $.
\item Use the first two parts to deduce that $ V(\lambda)_{w\lambda} $ is 1-dimensional for each $ w \in S_n $.
\end{enumerate}
Recall that we write 
$$ \lambda \ge \mu, \ \text{ if } \lambda - \mu \in Q_+ = \{ \sum_{i=1}^{n-1} p_i \alpha_i : p_i \in \N \} $$
(where $ \alpha_i = (0, \dots, 0, 1, -1, 0, \dots, 0) $).  The following Lemma from representation theory is helpful.
\begin{Lemma}
 If $ \lambda, \mu \in \Z^n_+ $ and $ \mu \le \lambda $, then $ V(\lambda)_\mu \ne 0 $.
\end{Lemma}
\begin{enumerate}[resume]
\item Use the lemma to find all $ \lambda \in \Z^n_+$ such that the restriction map $ \Gamma(Fl_n, \O(\lambda)) \rightarrow \Gamma(Fl^T_n, \O(\lambda)) $ is an isomorphism.
\end{enumerate}


\item In this question, we consider $ Fl_n \rightarrow \P(\C^n) \rightarrow \P(\S^k \C^n) $, where the first map takes $ V_\bullet $ to $ V_1 $ and the second map is the Veronese.
\begin{enumerate}
\item Show that the pullback of $\O(1) $ to $ Fl_n $ is the line bundle $ \O(k\omega_1)$.
\item Show that $ \S^k \C^n $ is an irreducible representation of $ Fl_n$ and that $ \S^k \C^n \cong V(k\omega_1) $. (You can use the Lemma mentioned above.)
\item Use our results from class regarding line bundles on flag varieties to show that the homogeneous coordinate ring of $ \P(\C^n) $ inside $ \P(\S^k \C^n) $ is $$ \bigoplus_{r=0}^\infty (\S^{rk} \C^n)^* $$
\item Deduce that the ideal of $ \P(\C^n) $ in $ \P(\S^k \C^n) $ is generated by 
$$
\frac{\binom{n+k-1}{k}\left( \binom{n+k-1}{k} +1\right)}{2} - \binom{n+2k-1}{2k}
$$
quadratic equations.
\item In the case when $ n = 2 $, show that this simplifies to $ k(k-1)/2 $. Find these equations explicitly in coordinates. 
\item Can you find the equations in coordinates for general $n$?
\end{enumerate}

\item In this question, we consider the flag variety $ Fl(V) $ where $ V $ is an $n$-dimensional vector space over a finite field $k$ with $ q $ elements.  This flag variety is a finite set.
\begin{enumerate}
\item Using the decomposition of $ Fl(V)$ into Schubert cells, find $ |Fl(V)|$.  (Your answer should be a sum over the symmetric group.)
\item For any $m$-dimensional vector space $ W $ over $k$, show that $$ |\P(W)| = \frac{q^m - 1}{ q-1} $$  We define $ [m]_q :=  q^m - 1/ q-1$
\item Use the previous question to show that 
$$|Fl(V) | = [n]_q [n-1]_q \cdots [1]_q 
$$
This number is often called the $q$-analog of $ n! $.
\end{enumerate}


\end{enumerate}

\end{document}