# 7.1 (Friday, September 17) :::{.remark} See Fulton, Young Tableaux. ::: :::{.remark} Given $A$ we produce $\mcg$ a Kac-Moody group, with standard parabolics $P_{\lambda} \subseteq \mcg$. We'll show $G/P_{\lambda} \embeds \PP(V)$ for some projective space over $V$ an integrable highest weight space in $\mods{\lieg}$, which is generally an Ind-variety, and if we show it's closed it will inherit the structure of a projective variety. Write $V = L^{\max}( \lambda) = V_{ \lambda}$ as a highest weight module. Idea: for $m_w: Z_w \to \mcg/B$ for $Z_w$ a BSDH, for any word $w\in \mcw$, if $w\in W_Y'$ is reduced, compose the above map with $\mcg/B \to \mcg/ P_{Y}$ to get a map \[ m_w^Y: Z_w \to \mcg/P_Y .\] We'll show $Z_w$ is projective, which is easier since it's an iterated line bundle. Let $v_0 \in V_\lambda$ (thought of in the finite type case as a highest weight vector in the irreducible, but may generally not coincide) consider the maps \[ \bar\iota_V: \mcg &\to \PP(V) \\ \iota_V: \mcg/P_Y &\to \PP(V) \\ m_w(v_0) = \iota_{v_0} \circ m_w^Y: Z_w &\to \PP(V) .\] ::: :::{.theorem title="?"} \envlist 1. $m_w(v_0)$ a morphism of varieties: easy to believe, hard to show! See the book. 2. \[ \im(m_w(v_0)) = \Union_{v\leq w, v\in W_Y'} BvP_Y/P_Y \subseteq \mcg/P_Y ,\] which is some subvariety of the flag variety which we'll define as the Schubert variety $X_W^Y$. ::: :::{.proposition title="5.1.3"} For $Y \subseteq S, w\in W_Y'$, and let $w = w_1\cdots w_k$ a reduced decomposition, $\ell(w) = \sum \ell(w_i)$. Let $Z_i \subseteq P_{w_i} \da P_{\ts{w_i}}$ be a subset of a simple parabolic such that $Z_i \surjects P_{w_i}/B$. [^see_fulton] Then \[ \im\qty{\prod Z_i \mapsvia{\text{mult}} G \surjects G/P_Y} = \Union_{v\leq w} BvP_Y/P_Y .\] [^see_fulton]: See Fulton for an explicit description, taking a Plucker embedding and studying actual equations. ::: :::{.remark} Where does the additional condition $v\in W_Y'$ come from in the theorem statement? Take a Bruhat decomposition \[ \mcg/B = \Disjoint_{\substack{ v\leq w \\ v\in \dcoset{W}{W_Y}{W_Y} }} P_{Y'}vP_V .\] ::: :::{.example title="?"} Take $G = \GL_n$, then - $\lambda\in X\dual(T)$ - $\lambda(t) = t_1\cdots t_k$ for $1\leq k\leq n$, - $\lambda \in D_\ZZ$ and $V_{\lambda} = \Extalg^k \CC^n$. - $S = \ts{1, \cdots, \ell}$ where $\ell = n-1$. - $G/P_Y \subseteq \PP( \Extalg^k \CC^n)$, - $Y( \lambda) \da \ts{1\leq i \leq \ell \st \lambda(\alpha_i\dual) = 0}$. Then \[ \lambda &\in (1, \cdots_k, 1, 0, \cdots_{n-k}, 0)\\ \alpha_i\dual &= (0, \cdots, 1, -1, 0, \cdots, 0) ,\] so we can write $Y( \lambda) = \ts{1,\cdots, k-1, k+1, \cdots, n-1} = \bar k$. Then set $F^k \in \PP(\Extalg^k \CC^n) = \Gr_k(\CC^n)$, so $0 \subseteq F^k \subseteq \CC^n$, and define the map \[ \iota_{\lambda}(F^k) = [f_1 \wedgeprod f_2 \wedgeprod \cdots \wedgeprod f_k] ,\] where $\ts{f_i}$ is a choice of ordered basis. ::: :::{.fact} Some facts about $Z_w = \prod^B_{1\leq k\leq m} P_{i_k}/B$, recalling the action of $B$ given last time. Set $w= (s_{i_1}, \cdots, s_{i_m}) \in \mcw$. There is a map \[ \varphi: P_{i_1}\mix{B} \cdots \mix{B} P_{i_m} &\to B/B \times G/B \times \cdots \times G/B \\ [p_1,\cdots, p_m] &\mapsto [B/B, p_1 B/B, p_1p_2 B/B, \cdots, p_1\cdots p_m B/B] .\] Showing this is well-defined: follows from universal property of quotients, looking at where point stabilizers are contained. Then \[ \im \varphi = B/B \fiberprod{G/P_{i_1}} G/B \fiberprod{G/P_{i_2}} \times \cdots \fiberprod{G/P_{i_m}} G/B .\] How to define the BSDH: construct a lattice by deleting elements in the sequence of flags corresponding to various words, and take the right-most flag in the result: \begin{tikzcd} & {i_1=n-2} &&&& {i_m = n-2} \\ {\CC^n} && {\CC^{n}} && {\CC^{n}} && {\CC^{n}} &&& {\CC^n} &&&&&&& {\CC^n} \\ {\CC^{n-1}} && {A^{n-1}} && {E^{n-1}} && {F^{n-1}} &&& {\CC^{n-1}} &&&& {E^{n-1}} &&& {E^{n-1}} \\ {\CC^{n-2}} && {A^{n-2}} && {E^{n-2}} && {F^{n-2}} &&& {\CC^{n-2}} && {A^{n-2}} &&& {F^{n-2}} && {F^{n-2}} \\ \vdots && \vdots && \vdots && \vdots && {} & \vdots &&& {C_2} & {D_3} &&& {D^3} \\ {\CC^1} && {A^1} && {E^1} && {F^1} &&& {\CC^{1}} && {B_1} && \ddots &&& {C^2} \\ 0 && 0 && 0 && 0 &&& 0 &&&&&&& {B^1} \\ &&&&&&&&&&&&&&&& 0 \arrow["{\text{can differ}}", from=4-1, to=4-3] \arrow[Rightarrow, from=5-7, to=5-9] \arrow[from=3-10, to=4-12] \arrow[from=4-10, to=4-12] \arrow[from=7-10, to=6-12] \arrow[from=6-12, to=5-10] \arrow[from=6-12, to=5-13] \arrow[from=5-13, to=5-10] \arrow[from=4-12, to=5-14] \arrow[from=5-13, to=5-14] \arrow[from=4-12, to=3-14] \arrow[from=2-10, to=3-14] \arrow[from=3-14, to=4-15] \arrow[from=5-14, to=4-15] \arrow[from=5-13, to=6-14] \arrow["{\text{can differ}}"{description}, from=4-5, to=4-7] \arrow[from=8-17, to=7-17] \arrow[from=7-17, to=6-17] \arrow[from=6-17, to=5-17] \arrow[from=5-17, to=4-17] \arrow[from=4-17, to=3-17] \arrow[from=3-17, to=2-17] \arrow[from=7-3, to=6-3] \arrow[from=7-1, to=6-1] \arrow[from=4-1, to=3-1] \arrow[from=3-1, to=2-1] \arrow[from=3-3, to=2-3] \arrow[from=4-3, to=3-3] \arrow[from=3-5, to=2-5] \arrow[from=4-5, to=3-5] \arrow[from=4-7, to=3-7] \arrow[from=3-7, to=2-7] \arrow[from=3-10, to=2-10] \arrow[from=4-10, to=3-10] \arrow[from=7-10, to=6-10] \arrow[from=7-7, to=6-7] \arrow[from=7-5, to=6-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) Here the word is $(i_{n-2}, i_1, i_2, i_{3}, i_{n-1}, i_{n-2})$. :::