Bott Towers

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 19 September 2012

Bott Towers

Let $\ell \in {\mathbb{Z}}_{>0}$ and

$$\text{let}C={\left\{c_{ij}\right\}}_{1\le i<j\le \ell },\text{with}{c}_{ij}\in \mathbb{Z}\text{.}$$

The Bott tower $Y$ is ${({ℂ}^2-\left\{(0,0)\right\})}^{\ell }=({ℂ}^2-\left\{(0,0)\right\})\times \dots \times ({ℂ}^2-\left\{(0,0)\right\})$ with the relation

$$[z_1,w_1,\dots ,z_{\ell },w_{\ell }]=[z_1,w_1,\dots ,z_{i-1},w_{i-1}{z}_{i}t,w_{i}t,z_{i+1},w_{i+1}{t}^{c_{i,i+1}},\dots ,z_{\ell },w_{\ell }{t}^{c_{i,\ell }},$$

for $i=1,2,\dots ,\ell$ and $t\in {ℂ}^{*}\text{.}$ For $S\subseteq \{1,\dots ,\ell \}$ let

$$Y_S=\{[z_1,w_1,\dots ,z_{\ell },w_{\ell }]\mid w_i=0\text{if}i\notin S\text{and}{w}_i\in {ℂ}^{\times }\text{is}i\in S\}\text{.}$$

Then

$$Y=\underset{S\subseteq \{1,\dots ,N\}}{⨆}{Y}_S,\overline{Y_S}=\underset{T\subseteq S}{⨆}{Y}_T,\text{and}{Y}_S\cong {ℂ}^{\text{Card}\left(S\right)}\text{.}$$

The torus $D={\left({ℂ}^{\times }\right)}^{\ell }$ acts on $Y$ by

$$(t_1,t_{\ell })[z_1,w_1,\dots ,z_{\ell },w_{\ell }]=[z_1,t_1{w}_1,\dots ,z_{\ell },t_{\ell }{w}_{\ell }],$$

and the $D$-fixed points in $Y$ are

$$\left[S\right]=[{\sigma }_1,\dots ,{\sigma }_{\ell }]\text{with}{\sigma }_i=\{\begin{array}{ccc}(1,0),& & \text{if}i\notin S,\\ (0,1),& & \text{if}i\in S\text{.}\end{array}$$

Let $P_1,\dots ,P_n$ be the minimal parabolic subgroups $P_i⊉B$ of $G\text{.}$ Let $\ell \in {\mathbb{Z}}_{>0}$ and let $(i_1,\dots ,i_{\ell })$ be a sequence in $\{1,\dots ,n\}$ of length $\ell \text{.}$ The Bott-Samelson variety is

$$\Gamma =P_{i_1}{\times }_B{P}_{i_2}{\times }_B\dots {\times }_B{P}_{i_{\ell }}/B,$$

so that $\Gamma$ is $P_{i_1}\times \dots P_{i_{\ell }}$ with the relations

$$[g_1,\dots ,g_{\ell }b]=[g_1,\dots ,g_{\ell }]\text{and}[g_1,\dots ,g_{i-1},g_{i}b,g_{i+1},\dots ,g_{\ell }]=[g_1,\dots ,g_{i-1},g_i,b{g}_{i+1},\dots ,g_{\ell }],$$

for $b\in B$ and $i=1,2,\dots ,N-1\text{.}$ The torus $T$ acts on $\Gamma$ by

$$t[g_1,\dots ,g_{\ell }b]=[t{g}_1,\dots ,g_{\ell }b],\text{for}t\in T,[g_1,\dots ,g_{\ell }b]\in \Gamma \text{.}$$

Since each component $[z_i,w_i]$ of a point $Y$ lives in ${\mathbb{P}}^1$ and each component $x_{i_j}\left(c_j\right)s_j$ of a point of $\Gamma$ lives in ${\mathbb{P}}^1,$

$$\Gamma \cong Y\text{where}{c}_{jk}=⟨{\alpha }_{i_k},{\alpha }_{ij}^{\vee }⟩,\text{for}1\le j<k\le \ell \text{.}$$

One must be quote careful at this step as this isomorphism is not an isomorphism in the algebraic category (THANKS TO DAVE ANDERSON FOR FLAGGING THIS). See [Wi1, Remark 2.10].

Furthermore the $T$ action on $\Gamma$ is the restriction of the $D$ action on $Y$ to $T$ via the homomorphism

$$\begin{array}{ccc}T& ⟶& D\\ t& ⟼& (X^{{\alpha }_{i_1}}\left(t\right),\dots ,X^{{\alpha }_{i_{\ell }}}\left(t\right))\text{.}\end{array}$$

There is a $T$–equivariant morphism

$$\begin{array}{ccc}{\Gamma }_{i_1,\dots ,i_{\ell }}=P_{i_1}{\times }_B{P}_{i_2}{\times }_B\dots {\times }_B{P}_{i_{\ell }}/B& \stackrel{{\gamma }_{i_1,\dots ,i_{\ell }}}{⟶}& X_w\hookrightarrow G/B\\ (x_{i_1}\left(c_1\right)s_{i_1},\dots ,x_{i_{\ell }}\left(c_{\ell }\right)s_{i_{\ell }}B)& ⟼& x_{i_1}\left(c_1\right)s_{i_1},\dots ,x_{i_{\ell }}\left(c_{\ell }\right)s_{i_{\ell }}B\end{array}$$

where $w=s_{i_1}\dots s_{i_{\ell }}\text{.}$

The "Borel picture" for $Y$ and $\Gamma$ are

$$\begin{array}{ccc}{H}_D^{*}\left(Y\right)& \cong & \frac{ℂ[y_1,\dots ,y_{\ell },x_1,\dots ,x_{\ell }]}{⟨x_j^2=y_j{x}_j-c_{1_j}{x}_1{x}_j-\dots -c_{j-1,j}{x}_{j-1}{x}_j,\text{for}j=1,\dots ,\ell ⟩},\\ H_T^{*}\left(\Gamma \right)& \cong & \frac{ℂ[{\alpha }_1,\dots ,{\alpha }_n,x_1,\dots ,x_{\ell }]}{⟨x_j^2={\alpha }_{i_j}{x}_j-⟨{\alpha }_{i_1},{\alpha }_{i_j}^{\vee }⟩x_1{x}_i-\dots -⟨{\alpha }_{i_{j-1}},{\alpha }_{i_j}^{\vee }⟩x_{j-1}{x}_j,\text{for}j=1,\dots ,\ell ⟩}\text{.}\end{array}$$

The "moment graph picture" is given by Willems, who computes the map $\Phi =\underset{S\subseteq \{1,\dots ,\ell \}}{⨁}{\iota }_S^{*}$ and finds

$$\frac{ℂ[y_1,\dots ,y_{\ell },x_1,\dots ,x_{\ell }]}{⟨x_j^2=y_j{x}_j-c_{1_j}{x}_1{x}_j-\dots -c_{j-1,j}{x}_{j-1}{x}_j,\text{for}j=1,\dots ,\ell ⟩}\stackrel{\Phi }{⟶}\underset{S\subseteq \{1,\dots ,\ell \}}{⨁}{\iota }_S^{*}ℂ[y_1,\dots ,y_{\ell }]$$

is given by $\Phi {\left(x_j\right)}_S=0$ if $j\notin S$ and

$$\Phi {\left(x_j\right)}_S=y_j+\sum _{m\in {\mathbb{Z}}_{>0}}{(-1)}^{m+1}\sum _{\{i_1,\dots ,i_m,j\}\subseteq S}{c}_{i_1{i}_2}\dots c_{i_{m}j}{y}_{i_1},\text{if}j\in S\text{.}$$

Willems also computes the map

$${\gamma }^{*}:H_T(G/B)\to H_T\left(\Gamma \right)\text{and finds}{\gamma }^{*}\left(\left[X_w\right]\right)=\sum _{\genfrac{}{}{0ex}{}{S=(i_1,\dots ,i_r)\subseteq \{1,\dots ,\ell \}}{r=\ell \left(w\right),w=s_{i_1}\dots s_{i_r}}}{x}_{i_1}\dots x_{i_r}\text{.}$$

The favourite basis of $H_D\left(Y\right)$ is

$${\sigma }_{\epsilon }^D=\prod _{{\pi }^{+}\left(\epsilon \right)}{\hat{\sigma }}_i^D=x_{i_1}\dots x_{i_{\ell }},\text{if}\epsilon =\{i_1,\dots ,i_{\ell }\}\text{.}$$

FIX THIS LAST SENTENCE SO THAT IT MAKES SOME SENSE!!!

Example: Bott towers and Bott-Samelson varieties

The Bott tower $Y$ for the data $C=\left\{\begin{array}{cc}{c}_{12},& c_{13}\\ & c_{23}\end{array}\right\}$ has moment graph

$${123} {12} {13} {23} {1} {2} {3} ∅ ε1 ε1 ε1 ε1 ε2 ε2 ε2-c12ε1 ε2-c12ε1 ε3 ε3-c23ε2 ε3-c13ε2 ε3-c13ε1-c23ε2+c12c23ε1$$

and the favourite basis for $h_D\left(Y\right)$ consists of the sections

$$\Phi \left(x_1\right)=\begin{array}{ccc}& y_{{\epsilon }_1}& \\ y_{{\epsilon }_1}& y_{{\epsilon }_1}& 0\\ y_{{\epsilon }_1}& 0& 0\\ & 0& \end{array}$$ $$\Phi \left(x_2\right)=\begin{array}{ccc}& y_{{\epsilon }_2-c_{12}{\epsilon }_1}& \\ y_{{\epsilon }_2-c_{12}{\epsilon }_1}& 0& y_{{\epsilon }_2}\\ 0& y_{{\epsilon }_2}& 0\\ & 0& \end{array}$$ $$\Phi \left(x_3\right)=\begin{array}{ccc}& y_{{\epsilon }_2-c_{13}{\epsilon }_1-c_{23}{\epsilon }_2+c_{12}{c}_{23}{\epsilon }_1}& \\ 0& y_{{\epsilon }_3-c_{13}{\epsilon }_1}& y_{{\epsilon }_3-c_{23}{\epsilon }_2}\\ 0& 0& y_{{\epsilon }_3}\\ & 0& \end{array}$$ $$\Phi \left(x_1{x}_2\right)=\begin{array}{ccc}& y_{{\epsilon }_1}{y}_{{\epsilon }_2-c_{12}{\epsilon }_1}& \\ y_{{\epsilon }_1}{y}_{{\epsilon }_2-c_{12}{\epsilon }_1}& 0& 0\\ 0& 0& 0\\ & 0& \end{array}$$ $$\Phi \left(x_2{x}_3\right)=\begin{array}{ccc}& y_{{\epsilon }_2-c_{12}{\epsilon }_1}{y}_{{\epsilon }_2-c_{13}{\epsilon }_1-c_{23}{\epsilon }_2+c_{12}{c}_{23}{\epsilon }_1}& \\ 0& 0& y_{{\epsilon }_2}{y}_{{\epsilon }_3-c_{23}{\epsilon }_2}\\ 0& 0& 0\\ & 0& \end{array}$$ $$\Phi \left(x_1{x}_3\right)=\begin{array}{ccc}& y_{{\epsilon }_1}{y}_{{\epsilon }_2-c_{13}{\epsilon }_1-c_{23}{\epsilon }_2+c_{12}{c}_{23}{\epsilon }_1}& \\ 0& y_{{\epsilon }_1}{y}_{{\epsilon }_3-c_{13}{\epsilon }_1}& 0\\ 0& 0& 0\\ & 0& \end{array}$$ $$\Phi \left(x_1{x}_2{x}_3\right)=\begin{array}{ccc}& y_{{\epsilon }_1}{y}_{{\epsilon }_2-c_{12}{\epsilon }_1}{y}_{{\epsilon }_2-c_{13}{\epsilon }_1-c_{23}{\epsilon }_2+c_{12}{c}_{23}{\epsilon }_1}& \\ 0& 0& 0\\ 0& 0& 0\\ & 0& \end{array}$$

which provide the relations

$$\begin{array}{ccc}{x}_1^2& =& y_{{\epsilon }_1}{x}_1,\\ x_2^2& =& y_{{\epsilon }_2}{x}_2+\frac{y_{{\epsilon }_2-c_{12}{\epsilon }_1}-y_{{\epsilon }_2}}{y_{{\epsilon }_1}}{x}_1{x}_2,\\ x_3^2& =& y_{{\epsilon }_3}{x}_3+\frac{y_{{\epsilon }_3-c_{23}{\epsilon }_2}-y_{{\epsilon }_3}}{y_{{\epsilon }_2}}{x}_2{x}_3+\frac{y_{{\epsilon }_3-c_{13}{\epsilon }_1}-y_{{\epsilon }_3}}{y_{{\epsilon }_1}}{x}_1{x}_3\\ & & +\frac{y_{{\epsilon }_2}{y}_{{\epsilon }_2-c_{12}{\epsilon }_1}+y_{{\epsilon }_2-c_{12}{\epsilon }_1}{y}_{{\epsilon }_3-c_{23}{\epsilon }_2}-y_{{\epsilon }_2-c_{12}{\epsilon }_1}{y}_{{\epsilon }_3}-y_{{\epsilon }_2-c_{13}{\epsilon }_1-c_{23}{\epsilon }_2+c_{12}{c}_{23}{\epsilon }_1}{y}_{{\epsilon }_2}}{y_{{\epsilon }_1}{y}_{{\epsilon }_2}{y}_{{\epsilon }_2-c_{12}{\epsilon }_1}}{x}_1{x}_2{x}_3\text{.}\end{array}$$

In particular, the "Borel picture" in cohomology is

$$H_D\left(Y\right)=\frac{ℂ[y_1,y_2,y_3,x_1,x_2,x_3]}{⟨\begin{array}{ccc}{x}_1^2& =& y_1{x}_1,\\ x_2^2& =& y_2{x}_2-c_{12}{x}_1{x}_2,\\ x_3^2& =& y_3{x}_3-c_{23}{x}_2{x}_3-c_{13}{x}_1{x}_3\end{array}⟩}$$

Now consider the Bott-Samelson variety $\Gamma =P_1{\times }_B{P}_2{\times }_B{P}_1/B$ for $G=S{L}_3$ for which

$$C=\left\{\begin{array}{cc}{c}_{12}=-1,& c_{13}=2\\ & c_{23}=-1\end{array}\right\}\text{since}\begin{array}{cc}⟨{\alpha }_1,{\alpha }_2^{\vee }⟩=-1,& ⟨{\alpha }_1,{\alpha }_1^{\vee }⟩=2,\\ & ⟨{\alpha }_2,{\alpha }_1^{\vee }⟩=-1\text{.}\end{array}$$

The "Borel picture" in cohomology is

$$H_T\left(\Gamma \right)=\frac{ℂ[{\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3]}{⟨\begin{array}{ccc}{x}_1^2& =& {\alpha }_1{x}_1,\\ x_2^2& =& {\alpha }_2{x}_2+x_1{x}_2,\\ x_3^2& =& {\alpha }_3{x}_3+x_2{x}_3+2x_1{x}_3\end{array}⟩}$$

since $y_1=y_{-{\alpha }_1},y_2=y_{-{\alpha }_2},y_3=y_{-{\alpha }_3}\text{.}$ In terms of moment graphs, the inclusion ${\gamma }^{*}:H_T(G/B)⟶H_T\left(\Gamma \right)$ is given by

$$f1 fs2 fs1 fs1s2 fs1s2 fs1s2s1 y-α2 y-(α1+α2) y-(α1+α2) y-α1 y-α1 y-(α1+α2) y-α2 y-α2 y-α1 f1 fs2 w0fs1s2s1 fs1 fs1s2 fs2s1 fs1s2 fs1s2s1 y1 y-α1 y-α1 y-α1 y-α2 y-α2 y-α1 y-(α1+α2) y-α1 y-s2α1 y-s1α1 y-α2$$

Then

$$\begin{array}{cccc}{\gamma }^{*}:& H_T(G/B)& ⟶& H_T\left(\Gamma \right)\\ & \left[X_{s_1{s}_2{s}_1}\right]& ⟼& 1\\ & \left[X_{s_1{s}_2}\right]& ⟼& x_1+x_3\\ & \left[X_{s_2{s}_1}\right]& ⟼& x_2\\ & \left[X_{s_1}\right]& ⟼& x_1{x}_2\\ & \left[X_{s_2}\right]& ⟼& x_2{x}_3\\ & \left[X_1\right]& ⟼& x_1{x}_2{x}_3\end{array}$$

and the composite

$$H_T(G/B)\stackrel{{\gamma }^{*}}{⟶}{H}_T\left(\Gamma \right)\stackrel{\Phi }{⟶}\underset{S\subseteq \{1,2,3\}}{⨁}{H}_T\left(\text{pt}\right)=\underset{S\subseteq \{1,2,3\}}{⨁}ℂ[y_{{\alpha }_1},y_{{\alpha }_2}]$$

has

$$\Phi \left(x_1\right)=\begin{array}{ccc}& y_{-{\alpha }_1}& \\ y_{-{\alpha }_1}& y_{-{\alpha }_1}& 0\\ y_{-{\alpha }_1}& 0& 0\\ & 0& \end{array}$$ $$\Phi \left(x_2\right)=\begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}& \\ y_{-({\alpha }_1+{\alpha }_2)}& 0& y_{-{\alpha }_2}\\ 0& y_{-{\alpha }_2}& 0\\ & 0& \end{array}=\begin{array}{ccc}& y_{-s_1{\alpha }_2}& \\ y_{-s_1{\alpha }_2}& 0& y_{-{\alpha }_2}\\ 0& y_{-{\alpha }_2}& 0\\ & 0& \end{array}$$ $$\Phi \left(x_3\right)=\begin{array}{ccc}& y_{-{\alpha }_2}& \\ 0& y_{{\alpha }_1}& y_{-({\alpha }_1+{\alpha }_2)}\\ 0& 0& y_{-{\alpha }_1}\\ & 0& \end{array}=\begin{array}{ccc}& y_{-s_1{s}_2{\alpha }_2}& \\ 0& y_{-s_1{\alpha }_1}& y_{-s_2{\alpha }_1}\\ 0& 0& y_{-{\alpha }_1}\\ & 0& \end{array}$$

giving

$$\Phi \left({\gamma }^{*}\left[X_{s_1{s}_2}\right]\right)=\Phi (x_1+x_3)=\begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}& \\ y_{-{\alpha }_1}& 0& y_{-({\alpha }_1+{\alpha }_2)}\\ y_{-{\alpha }_1}& 0& y_{-{\alpha }_1}\\ & 0& \end{array}$$ $$\Phi \left({\gamma }^{*}\left[X_{s_2{s}_1}\right]\right)=\Phi \left(x_2\right)=\begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}& \\ y_{-({\alpha }_1+{\alpha }_2)}& 0& y_{-{\alpha }_2}\\ 0& y_{-{\alpha }_2}& 0\\ & 0& \end{array}$$ $$\Phi \left({\gamma }^{*}\left[X_{s_1}\right]\right)=\Phi \left(x_1{x}_2\right)=\begin{array}{ccc}& y_{-{\alpha }_1}{y}_{-({\alpha }_1+{\alpha }_2)}& \\ y_{-{\alpha }_1}{y}_{-({\alpha }_1+{\alpha }_2)}& 0& 0\\ 0& 0& 0\\ & 0& \end{array}$$ $$\Phi \left({\gamma }^{*}\left[X_{s_2}\right]\right)=\Phi \left(x_2{x}_3\right)=\begin{array}{ccc}& y_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}& \\ 0& 0& y_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}\\ 0& 0& 0\\ & 0& \end{array}$$ $$\Phi \left({\gamma }^{*}\left[X_1\right]\right)=\Phi \left(x_1{x}_2{x}_3\right)=\begin{array}{ccc}& y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}& \\ 0& 0& 0\\ 0& 0& 0\\ & 0& \end{array}$$

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