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

\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Cx}{\C^\times}
\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 November 11}


\begin{document}
\maketitle
\begin{enumerate}
\item
Let $ \uk= (k_1, \dots, k_m )$ be natural numbers with $ k_1 + \dots + k_m = n $.  Recall the partial flag variety $ Fl(\uk,\C^n) $.

\begin{enumerate}
\item Show that 
$$ T^* Fl(\uk, \C^n) =  \{ (X, V_\bullet) : V_\bullet \in Fl(\uk, V), X \in \mathcal{N}, XV_i \subseteq V_{i-1} \}
$$
\item
Show that $T^* Fl(\uk, \C^n) $ is the resolution of a nilpotent orbit closure.  Which one?
\item
For any $ X \in \mathcal{N} $, let $ Fl(\uk)^X $ denote the fibre of $ T^* Fl(\uk, \C^n) $ over the point $ X $.  Give a bijection between the irreducible components of $ Fl(\uk)^X $ and the set of semistandard Young tableaux of shape $ \lambda$ and content $ \uk$. (Having content $ \uk $ means that the tableau contains $ k_1$ 1s, $k_2$ 2s, \dots, $k_m$ ms.)
\end{enumerate}

\item 
Consider the algebra $ R = \C[x_1, \dots, x_n]/ \langle \C[x_1, \dots, x_n]^{S_n}_+ \rangle $, the quotient of the polynomial ring by the positive degree invariant symmetric polynomials.  As discussed in class $ R \cong H^*(Fl_n) $.  Without using this geometric fact, prove that $ R $ is isomorphic to the regular representation $ \C[S_n] $ of $ S_n $.  (Hint: one way to do this is to compute the character of both representations.)

\item
Let $ \lambda $ be a partition of $ n $ and let $ X $ be the nilpotent Jordan form matrix given by $ \lambda $. In class, for any row-strict tableau $ U $ of shape $\lambda $, we define a point $ E_\bullet^U \in Fl_n^X $.  Also, for any standard Young tableau $ U $, we defined a subset $ Y_U \subset Fl_n^X $.

\begin{enumerate}
 \item For any $ U$, show that there exists a Schubert cell in $ Fl_n $, such that $Y_U $ is the intersection of $ Fl_n^X $ with this Schubert cell.
\item Find all pairs $ U, V $ (where $U $ is row-strict and $ V $ is standard) such that $ E_\bullet^U \in \overline{Y_V}$.
\item For each $ U $ standard, find an action of $ \Cx $ on $ Fl_n^X $ such that $ Y_U $ is the attracting set $E_\bullet^U $ for this action.
\end{enumerate}

\item
For each $ n $, let $ C_n $ denote the set of crossingless matchings of $ n $ points on a line.  There is a diagrammatic algebra, called the Temperley-Lieb algebra $ TL_n $ which has basis $ C_n^2 $ (see for example, the wikipedia page).  The algebra $TL_n $ usually depends on a parameter denoted $ \delta $.  We will be concerned with the case $ \delta = 2 $. (Sometimes it is defined with a parameter $ q $; $ \delta = 2 $ corresponds to $ q = 1 $.)

\begin{enumerate}
\item Show that there is an algebra map $ \C[S_n] \rightarrow TL_n $ given by $ s_i \mapsto U_i - 1 $.

\item Let $ V_n $ be a vector space with basis $ C_n $.  Show that there is a natural action of $ TL_n $ on $ V_n $. By (a), this gives us an action of $ S_n $ on $ V_n $.  Show that $V_n $ is the irreducible representation of $ S_n $ corresponding to the partition $(n,n)$.

\item Let $ F_n = Fl_n^X $ denote the Springer fibre to the $(n,n) $ nilpotent matrix $ X $.  Recall that for each $ U \in C_n$, we have an irreducible component $ \overline{Y_U} \subset F_n $, described by 
\begin{align*}
\overline{Y_U} = \{ &V_\bullet \in Fl_n^X : X^{-k} V_i = V_j 
\text{ whenever region $ i $ and region $ j$} \\ &\text{are connected in $ U $ and $ j = i + 2k$ }\}
\end{align*}
Thus we get a vector space isomorphism $ H_{top}(F_n) \cong V_n $ taking $ [\overline{Y_U}] $ to $ U $.

Recall the isomorphism $ H(Z) = \C[S_n] $ and the action of $ H(Z) $ on $ H_{top}(F_n) $.  Prove by a direct computation that the isomorphism  $ H_{top}(F_n) \cong V_n $ is $ S_n$-equivariant.
\end{enumerate}
  

\end{enumerate}

\end{document}