The Flag variety

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 16 June 2007. This page is the result of joint work with James Parkinson and Christoph Schwer.

Chevalley group relations

A Chevalley group is a group in which row reduction works. This means that it is a group with a special set of generators (the "elementary matrices") and relations which are generalizations of the usual row reduction operations.

The Chevalley group $G$ is the group given by generators

$\begin{array}{rl}\text{(R1)}& x_{\alpha }\left(f_1\right)x_{\alpha }\left(f_2\right)=x_{\alpha }(f_1+f_2),\\ \text{(R2)}& x_{\alpha }\left(f_1\right)x_{\beta }\left(f_2\right)=x_{\beta }\left(f_2\right)x_{\alpha }\left(f_1\right)x_{\alpha +\beta }\left(c_{11}{f}_1{f}_2\right)x_{2\alpha +\beta }\left(c_{21}{f}_1^2{f}_2\right)x_{\alpha +2\beta }\left(c_{12}{f}_1{f}_2^2\right)\cdots ,\end{array}$

For $\alpha \in R_{\mathrm{re}}$ and $f\in k^{\times }$ define

$\begin{array}{rl}\text{(R4)}& n_{\alpha }\left(t\right)x_{\beta }\left(u\right)n_{\alpha }(t{)}^{–1}=x_{s_{\alpha }\beta }(\epsilon t^{–⟨\beta ,{\alpha }^{\vee }⟩}u)\\ \text{(R5)}& n_{\alpha }\left(t\right)n_{\beta }\left(u\right)n_{\alpha }(t{)}^{–1}=n_{s_{\alpha }\beta }(\epsilon t^{–⟨\beta ,{\alpha }^{\vee }⟩}u)\\ \text{(R6)}& n_{\alpha }\left(t\right)h_{\alpha }\left(u\right)n_{\alpha }(t{)}^{–1}=h_{s_{\alpha }\beta }(\epsilon t^{–⟨\beta ,{\alpha }^{\vee }⟩}u\left)h_{s_{\alpha }\beta }\right(\epsilon t^{–⟨\beta ,{\alpha }^{\vee }⟩}{)}^{–1}\\ \text{(R7)}& h_{{\alpha }^{\vee }}\left(t\right)x_{\beta }\left(u\right)h_{{\alpha }^{\vee }}(t{)}^{–1}=x_{\beta }(t^{⟨\beta ,{\alpha }^{\vee }⟩}u)\\ \text{(R8)}& h_{{\alpha }^{\vee }}\left(t\right)n_{\beta }\left(u\right)h_{{\alpha }^{\vee }}(t{)}^{–1}=n_{\beta }(t^{⟨\beta ,{\alpha }^{\vee }⟩}u)\\ \text{(R9)}& h_{{\alpha }^{\vee }}\left(t\right)h_{{\beta }^{\vee }}\left(u\right)h_{{\alpha }^{\vee }}(t{)}^{–1}=h_{{\beta }^{\vee }}(t^{–⟨\beta ,{\alpha }^{\vee }⟩}u\left)h_{{\beta }^{\vee }}\right(t^{–⟨\beta ,{\alpha }^{\vee }⟩}{)}^{–1}\\ \text{(R10)}& h_{{\lambda }^{\vee }}\left(g\right)h_{{\mu }^{\vee }}\left(g\right)=h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(g\right),\end{array}$

Note that (R5-9) all follow easily from (R4) and the definitions of $n_{\alpha }\left(g\right)$ and $h_{{\alpha }^{\vee }}\left(g\right)$. Let

$\begin{array}{rcl}{h}_{{\lambda }^{\vee }}\left(f\right)h_{{\mu }^{\vee }}\left(g\right)& =& h_{{\mu }^{\vee }}\left(f^{–1}\right)h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(f\right)h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(g\right)h_{{\lambda }^{\vee }}\left(g^{–1}\right)\\ & =& h_{{\mu }^{\vee }}\left(f^{–1}\right)h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(fg\right)h_{{\lambda }^{\vee }}{g}^{–1})\\ & =& h_{{\mu }^{\vee }}\left(f^{–1}\right)h_{{\mu }^{\vee }}\left(fg\right)h_{{\lambda }^{\vee }}\left(fg\right)h_{{\lambda }^{\vee }}\left(g^{–1}\right)\\ & =& h_{{\mu }^{\vee }}\left(g\right)h_{{\lambda }^{\vee }}\left(f\right),\end{array}$

if ${\omega }_1^{\vee },\dots ,{\omega }_n^{\vee }$ is a $\mathbb{Z}$-basis of $P^{\vee }$ then

$h_{{\lambda }^{\vee }}\left(g\right)=h_{{\omega }_1^{\vee }}\left(g^{{\lambda }_1}\right)\cdots h_{{\omega }_n^{\vee }}\left(g^{{\lambda }_n}\right)\text{,}$ for ${\lambda }^{\vee }={\lambda }_1{\omega }_1^{\vee }+\cdots +{\lambda }_n{\omega }_n^{\vee }$.

The Borel subgroup

The flag variety is $G/B$ where

$B=langle{x}_{\alpha }\left(c\right),h_{{\lambda }^{\vee }}\left(d\right)\mid \alpha \in R^{+},{\lambda }^{\vee }\in P^{\vee }c\in k,d\in k^{\times }\mathrm{rangle}$.

If ${\beta }_1,{\beta }_2,\dots ,{\beta }_N$ is an ordering of $R^{+}$then $B={𝔛}_{{\beta }_1}{𝔛}_{{\beta }_2}\cdots {𝔛}_{{\beta }_N}\cdot H.$ More precisely, if $b\in B$ then there are unique $d_1,\dots ,d_n\in k^{\times }$ and $c_1,\dots ,c_N\in k$ such that

$b=x_{{\beta }_1}\left(c_1\right)\cdots x_{{\beta }_N}\left(c_N\right)h_{{\omega }_1^{\vee }}\left(d_1\right)\cdots h_{{\omega }_{\mathrm{n}}^{\vee }}\left(d_n\right)$

Let $w\in W$. The inversion set of $w$ is

$R\left(w\right)=\{\alpha \in R^{+}\mid w\alpha \notin R^{+}\}\text{and}\ell \left(w\right)=\mathrm{Card}\left(R\right(w\left)\right)$

is the length of $w$. If $n_w$ is a representative of $w$ in $N$ then

$BwB=\left(\prod _{\alpha \in R^{+}}{𝔛}_{\alpha }\right)n_{w}B=n_w\left(\prod _{w^{–1}\alpha \notin R^{+}}{𝔛}_{w^{–1}\alpha }\right)\left(\prod _{w^{–1}\alpha \in R^{+}}{𝔛}_{w^{–1}\alpha }\right)B=\left(\prod _{\alpha \in R\left(w\right)}{𝔛}_{\alpha }\right)n_{w}B$

so that if $R\left(w\right)=\{{\beta }_1,\dots ,{\beta }_{\ell }\}$ then

$BwB=\left\{x_{{\beta }_1}\right(c_1\left)\cdots x_{{\beta }_{\ell }}\right(c_{\ell })n_{w}B\mid c_1,\dots ,c_{\ell }\in k\}.$

The easiest way to check that if $w\ne v$ then $BwB\ne BvB$, and that the set $\left\{x_{{\beta }_1}\right(c_1\left)\cdots x_{{\beta }_{\ell }}\right(c_{\ell })n_w\mid c_1,\dots ,c_{\ell }\in k\}$ is a set of coset representatives of the $B$-cosets in $BwB$ is to use representation theory and to think about the images of points in a representation of highest weight.

The simple reflections $s_1,\dots ,s_n$ are the elements of length $1$ in $W$. Let

$n_i=n_{{\alpha }_i}\left(1\right),\text{where}R\left(s_i\right)=\left\{{\alpha }_i\right\}\text{.}$

If $w=s_{i_1}\cdots s_{i_{\mathrm{ell}}}$ is a reduced word then

$R\left(w\right)=\{{\beta }_1,\dots ,{\beta }_{\ell }\},\text{where}{\beta }_j=s_{i_1}\cdots s_{i_{j–1}}{\alpha }_{i_j},$

and, by (R4),

$x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell }}{(c}_{\ell })n_{i_{\ell }}=x_{{\beta }_1}(c_1\left)x_{{\beta }_1}\right({\epsilon }_1{c}_2\left)\cdots x_{{\beta }_{\ell }}\right({\epsilon }_{\ell –1}{c}_{\ell })n_{i_1}\cdots n_{i_{\ell }}$

where ${\epsilon }_1,\dots ,{\epsilon }_{\ell –1}\in \{1,–1\}$. Hence

$BwB=\left\{x_{{\beta }_1}\right(c_1\left)\cdots x_{{\beta }_{\ell }}\right(c_{\ell })n_{w}B\mid c_1,\dots ,c_{\ell }\in k\}=\left\{x_{i_1}\right(c_1)n_{i_1}\cdots x_{i_{\ell }}{(c}_{\ell })n_{i_{\ell }}B\mid c_1,\dots ,c_{\ell }\in k\}\text{.}$

Identify elements of the group algebra $ℂG$ with functions on $G$ by

$f=\sum _{g\in G}f\left(g\right)g\text{,}\text{for}f:G\to ℂ$

and use the linear extension of the group multiplication to define the convolution product on functions. The Hecke algebra of $(G,B)$ is

$H=C(B\setminus G/B)=\{f:G\to ℂ\mid f(b_{1}g{b}_2)=f(g),\text{ for all }{b}_1,b_2\in B\}$.

and

$1_B^G=C(G/B)=\{f:G\to ℂ\mid f(gb)=f(g),\text{ for all }b\in B\}$.

is a right $H$-module.

If $\gamma \in W$ and $\ell \left(\gamma \right)=0$ then

$B\gamma B=\gamma \left({\gamma }^{–1}\mathrm{B\gamma }\right)B=\gamma BB=\gamma B$,

and if $s_i\in W$ and $\ell \left(s_i\right)=1$ then

${Bs}_{i}B=s_i\left(s_{i}B{s}_i\right)B=s_i{𝔛}_{{–\alpha }_i}\left(\prod _{\alpha \in R^{+}–\left\{{\alpha }_i\right\}}{𝔛}_{\alpha }\right)B=s_i{𝔛}_{{–\alpha }_i}B={𝔛}_{{\alpha }_i}{s}_{i}B=\left\{x_i\right(c)s_{i}B\mid c\in k\}$.

If $x_i\left(c_1\right)s_{i}B=x_i\left(c_2\right)s_{i}B$ then $x_i\left(c_1\right)s_i=x_i\left(c_2\right)s_{i}b$, for some $b\in B$, and

$b=s_i{x}_i(–c_2)x_i\left(c_1\right)s_i=x_{–{\alpha }_i}(c_1–c_2)$ which forces $c_1–c_2=0$ and $b=1$.

Thus $\left\{x_i\right(c)s_i\mid c\in k\}$ is a set of coset representatives of the $B$-cosets in $B{s}_{i}B$. Since

$x_i\left(c\right)s_i=x_i\left(c\right)x_i(–c)x_{–{\alpha }_i}\left(c^{–1}\right)x_i(–c)=x_{–{\alpha }_i}\left(c^{–1}\right)x_i\left(c\right)$,

and so the set {$s_i\}\cup \{x_{–{\alpha }_i}\left(c\right)\mid c\in k^{\times }\}$ is another set of coset representatives of the $B$-cosets in $B{s}_{i}B$. Since

$B{s}_{i}B\cdot B{s}_{i}B=\left\{x_i\right(c_1)s_{i}B{s}_{i}B\mid c_1\in k\}=\left\{x_i\right(c_1\left)s_i{x}_i\right(c_2)s_{i}B\mid c_1,c_2\in k\}$

and, if $c_2=0$ then

$x_i\left(c_1\right)s_i{x}_i\left(c_2\right)s_{i}B=x_i\left(c_1\right)s_i{s}_{i}B=x_i\left(c_1\right)B$,

and, if $c_2\ne 0$ then

$x_i\left(c_1\right)s_i{x}_i\left(c_2\right)s_{i}B=x_i\left(c_1\right)x_{–{\alpha }_i}\left(c_2\right)B=x_i(c_1+c_2^{–1})x_{{\alpha }_i}(–c_2^{–1})x_{–{\alpha }_i}\left(c_2\right)B=x_i(c_1+c_2^{–1})s_i{x}_{{\alpha }_i}\left(c_2^{–1}\right)B$

and so

$B{s}_{i}B\cdot B{s}_{i}B=B\cup B{s}_{i}B$.

The Hecke algebra

Let $q=\mathrm{Card}\left(k\right)$. Then the characteristic functions $\{T_w\mid w\in W\}$ of the double cosets $\mathrm{BwB}$ are a basis of the Hecke algebra $H=C(B\setminus G/B)$ and

$T_w{T}_{s_j}=\{\begin{array}{ll}{T}_{w{s}_j},& \text{if }\ell \left(w{s}_j\right)>\ell \left(w\right),\\ q{T}_{w{s}_j}+(q–1)T_w,& \text{if }\ell \left(w{s}_j\right)<\ell \left(w\right).\end{array}$

The product $\mathrm{BwB}\cdot B{s}_{j}B$ contains the points

If $\ell \left(w{s}_j\right)>\ell \left(w\right)$ then

$Bw{s}_{j}B=BwB\cdot B{s}_{j}B=\left\{x_{i_1}\right(c_1)n_{i_1}\cdots x_{i_{\ell }}(c_{\ell }\left)n_{i_{\ell }}{x}_j\right(c)s_{j}B\mid c_1,\dots ,c_{\ell },c\in k\}.$

If $\ell \left(w{s}_j\right)<\ell \left(w\right)$ and $c\ne 0$ then

$\begin{array}{l}{x}_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_{i_{\ell }}{(c}_{\ell }\left)n_{i_{\ell }}{x}_j\right(c)n_{j}B\\ \phantom{x_{i_1}\left(c_1\right)n_{i_1}}=x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_j\left(c_{\ell }\right)n_{j_{}}{x}_j\left(c\right)n_{j}B\\ \phantom{x_{i_1}\left(c_1\right)n_{i_1}}=x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_{{\alpha }_j}{(c}_{\ell }\left)x_{{–\alpha }_j}\right(c)B\\ \phantom{x_{i_1}\left(c_1\right)n_{i_1}}=x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_{{\alpha }_j}{(c}_{\ell }+c^{–1}\left)x_{{\alpha }_j}\right(–c^{–1}\left)x_{{–\alpha }_j}\right(c\left)x_{{\alpha }_j}\right(–c^{–1})B\\ \phantom{x_{i_1}\left(c_1\right)n_{i_1}}=x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_{{\alpha }_j}(c_{\ell }+c^{–1})n_{j}B\text{.}\end{array}$

If $\ell \left(w{s}_j\right)<\ell \left(w\right)$ and $c=0$ then

$x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_{i_{\ell }}{(c}_{\ell }\left)n_{i_{\ell }}{x}_j\right(0)n_{j}B=x_{i_1}(c_1)n_{i_1}\cdots x_{i_{\ell –1}}(c_{\ell –1})n_{i_{\ell –1}}$B.

Hence, in the product (1.1) the coset

$x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}{x}_{{\alpha }_j}\left(a\right)n_{j}B$

appears once for each choice of $c_{\ell }\in k$ and $c\in k^{\times }$ such that $c_{\ell }+c^{–1}=a$, a total of $q–1$ times. The coset

$x_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell –1}}\left(c_{\ell –1}\right)n_{i_{\ell –1}}B$

appears once for each choice of $c_{\ell }\in k$, a total of $q$ times. Thus

$p_{\overrightarrow{w}}(c_1,\dots ,c_{\ell })\text{ and }{T}_w\text{are the characteristic functions of}{x}_{i_1}\left(c_1\right)n_{i_1}\cdots x_{i_{\ell }}{(c}_{\ell })n_{i_{\ell }}\text{ and }BwB\text{,}$

respectively. If $\ell \left(w{s}_j\right)>\ell \left(w\right)$ then

${\displaystyle p_{\overrightarrow{w}}(c_1,\dots ,c_{\ell })T_{s_j}=\sum _{c\in k}{p}_{\overrightarrow{w}{s}_j}(c_1,\dots ,c_{\ell },c)}\text{and}{T}_w{T}_{s_j}=\sum _{c_1,\dots ,c_{\ell }}{p}_{\overrightarrow{w}}(c_1,\dots ,c_{\ell })T_{s_j}$

${\displaystyle =\sum _{c_1,\dots ,c_{\ell },c\in k}{p}_{\overrightarrow{w}{s}_j}(c_1,\dots ,c_{\ell },c)=T_{w{s}_j}.}$

and if $\ell \left(w{s}_j\right)<\ell \left(w\right)$ then

${\displaystyle p_{\overrightarrow{w}}(c_1,\dots ,c_{\ell })T_{s_i}=p_{\overrightarrow{w}{s}_j}(c_1,\dots ,c_{\ell –1})+\sum _{a\in k,a\ne c_{\ell }}{p}_{\overrightarrow{w}}(c_1,\dots ,c_{\ell –1},a)}$

and

${\displaystyle T_w{T}_{s_j}=\sum _{c_1,\dots ,c_{\ell }\in k}{p}_{\overrightarrow{w}}(c_1,\dots ,c_{\ell })T_{s_j}=\sum _{c_1,\dots ,c_{\ell }}{p}_{\overrightarrow{w}{s}_j}(c_1,\dots ,c_{\ell –1})+\sum _{\genfrac{}{}{0ex}{}{c_1,\dots ,c_{\ell },a}{a\ne c_{\ell }}}{p}_{\overrightarrow{w}}(c_1,\dots ,c_{\ell },a)}$

$=q{T}_{w{s}_j}+(q–1)T_w.$

This argument has shown that

${\displaystyle G=\bigsqcup _{w\in W}BwB}$

since the right hand side contains a set of generators of $G$ and is closed under multiplication.

By induction on $\ell$, the product $BwB\cdot B{s}_{j}B$ contains the points

$y_{i_1}\left(c_1\right)\cdots y_{i_{\ell }}\left(c_{\ell }\right)B\cdot B{s}_{j}B=y_{i_1}\left(c_1\right)\cdots y_{i_{\ell }}\left(c_{\ell }\right)B{s}_{j}B=\left\{y_{i_1}\right(c_1)\cdots y_{i_{\ell }}(c_{\ell }\left)y_j\right(c)B\mid c\in k\}\text{.}$

If $y_{i_1}\left(c_1\right)\cdots y_{i_{\ell }}\left(c_{\ell }\right)b=y_{i_1}\left(c_1^{\prime }\right)\cdots y_{i_{\ell }}\left(c_{\ell }^{\prime }\right)b^{\prime }$ then

${y_{i_1}(c_1^{\prime }{)}^{–1}y}_{i_1}\left(c_1\right)\cdots y_{i_{\ell }}\left(c_{\ell }\right)b=y_{i_2}\left(c_2^{\prime }\right)\cdots y_{i_{\ell }}\left(c_{\ell }^{\prime }\right)b^{\prime }$.

Then $y_{i_1}\left(c_1^{\prime }{)}^{–1}{y}_{i_1}\right(c_1)\in B{s}_{i_1}B\cdot B{s}_{i_1}B=B\cup B{s}_{i_1}B$ but, since the right hand side is in $B{s}_{i_1}wB$, we have

References

[St] R. Steinberg, Lecture Notes on Chevalley groups, Yale University, 1967.