Partial flag varieties and Bott-Samelson classes $\left[Z_{\overrightarrow{w}}\right]$

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

Last update: 17 February 2013

Partial flag varieties and Bott-Samelson classes $\left[Z_{\overrightarrow{w}}\right]$

In this section we review the formulas for the Bott-Samelson classes as established in, for example, [HKi0903.3926, CPZ0905.1341, BEv0968883, BEv1044959]. Though some of these references are not considering the equivariant case, the same machinery applies to define these classes in ${\Omega }_T(G/B)\text{.}$ In particular, this is the place in the theory where the BGG/Demazure operators are derived from the geometry. These operators play a fundamental role in the combinatorial study of ${\Omega }_T(G/B)\text{.}$

Pushforwards to partial flag varieties: BGG/Demazure operators

Using the notation for parabolic subgroups and partial flag varieties as in (2.3), if $J\subseteq \{1,2,\dots ,n\}$ and

$$\begin{array}{cccc}{\pi }_J:& G/B& \to & G/P_J\\ & gB& \mapsto & g{P}_J\end{array}\text{then}{\pi }_J\left(wB\right)=u{P}_J,\text{where}\hspace{0.17em}w{W}_J=u{P}_J\text{.}$$

Then, in the setting on Theorem 3.1,

$$S{\otimes }_{S_0^W}{S}^{W_J}\cong {\Omega }_T(G/P_J),$$

and ${\pi }_J^{*}:{\Omega }_T(G/P_J)\to {\Omega }_T(G/B)$ and ${\left({\pi }_J\right)}_{!}:{\Omega }_T(G/B)\to {\Omega }_T(G/P_J)$ correspond to

$$\begin{array}{cc}{\pi }_J^{*}:S{\otimes }_{S^{W_0}}{S}^{W_J}\hookrightarrow S{\otimes }_{S^{W_0}}S\text{and}{\left({\pi }_J\right)}_{!}:S{\otimes }_{S^{W_0}}S⟶S{\otimes }_{S^{W_0}}{S}^{W_J}& \text{(4.1)}\end{array}$$

where ${\left({\pi }_J\right)}_{!}$ is given by the operator in the nil affine Hecke algebra given by

$${\left({\pi }_J\right)}_{!}=\left(\sum _{v\in W_J}{t}_v\right)\frac{1}{x_J},\text{where}{x}_J=\prod _{\alpha \in R_J^{+}}{x}_{-\alpha }\text{.}$$

with $R_J^{+}$ the set of positive roots for $P_J\supseteq B\supseteq T\text{.}$ A special case is when $J=\left\{i\right\},$ for which

$$\begin{array}{cc}{W}_J=\{1,s_i\}\text{and}{\pi }_i^{*}{\left({\pi }_i\right)}_{!}=A_i=(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}},& \text{(4.2)}\end{array}$$

is the BGG-Demazure operator (see [BEv0968883, Cor.-Def. 1.9]). The calculus of the operators $A_i$ is controlled via the identities in Section 8.

Bott-Samelson classes

For a sequence $\overrightarrow{w}=(i_1,\dots ,i_{\ell })$ with $1\le i_1,\dots ,i_{\ell }\le n$ define the Bott-Samelson class

$$\begin{array}{cc}\left[Z_{\overrightarrow{w}}\right]=\left[Z_{i_1{i}_2\dots i_{\ell }}\right]=A_{i_1}{A}_{i_2}\text{.}{A}_{i_{\ell }}\left[Z_{\text{pt}}\right],& \text{(4.3)}\end{array}$$

where, in the notation of (3.17),

$$\begin{array}{cc}{\left[Z_{\text{pt}}\right]}_v=\{\begin{array}{cc}{\prod }_{\alpha \in R^{+}}{y}_{-\alpha },& \text{if}\hspace{0.17em}v=1,\\ 0,& \text{if}\hspace{0.17em}v\ne 1\text{.}\end{array}& \text{(4.4)}\end{array}$$

Theorem 4.1. [BEv1044959, Prop. 1], [HKi0903.3926, Prop. 3.1], [KKu0895705, Lemma 3.15], see also [HHH0409305, Proposition 4.1]) The generalized cohomology

$$h_T(G/B)\hspace{0.17em}\text{has}\hspace{0.17em}{h}_T\left(\text{pt}\right)\text{-basis}\{\left[Z_{\overrightarrow{w}}\right]=[{\gamma }_{\overrightarrow{w}}:{\Gamma }_{\overrightarrow{w}}\to G/B]\hspace{0.17em}\mid \hspace{0.17em}w\in W_0\},$$

where, for each $w\in W_0,$ $\overrightarrow{w}=s_{i_1}\dots s_{i_{\ell }}$ is a fixed reduced word for $w\text{.}$

Let us explain where this comes from. Let $X$ be a $T\text{-variety.}$ Following [Ful1644323, Example 1.9.1], or [CGi1433132, §5.5], a cellular decomposition of $X$ is a filtration

$$\varnothing =X_{-1}\subseteq X_0\subseteq X_1\subseteq \dots \subseteq X_d=X$$

by closed subvarieties such that $X_i=X_{i-1}$ are isomorphic to a disjoint union of affine spaces ${𝔸}^{{\ell }_i}$ for $i=1,2,\dots ,d\text{.}$ The "cells" of $X$ are the $X_i-X_{i-1}\text{.}$

Theorem 4.2. (see [Gro1958, Prop. 7]; [Ful1644323, Example 1.9.1] who refers to [Cho0078006]; [CGi1433132, Lemma 5.5.1]; [BEv1044959, Proposition 1]; [HKi0903.3926, Theorem 2.5]) Let $X$ be a $T\text{-variety}$ with a cellular decomposition. Then $h_T\left(X\right)$ has an $h_T\left(\text{pt}\right)\text{-basis}$ given by resolutions of cell closures (choose one resolution for each cell).

For $X=G/B,$ the Bruhat decomposition

$$G=\underset{w\in W_0}{⨆}BwB\text{provides the desired cell decomposition}$$

and the Schubert varieties $X_w=\overline{BwB}$ are the closures of the Schubert cells. Let $P_1,\dots ,P_n$ be the minimal parabolics of $G$ (with $P_i\supseteq B$ and $P_i\ne B\text{)}$ and let $s_1,\dots ,s_n$ be the corresponding simple reflections in $W_0\text{.}$ The group $W_0$ is generated by $s_1,\dots ,s_n\text{.}$ Let $\overrightarrow{w}=s_{i_1}\dots s_{i_{\ell }}$ be a reduced word for $w\text{.}$ Then the Bott-Samelson variety ${\Gamma }_{i_1,\dots ,i_{\ell }}=P_{i_1}{\times }_B{P}_{i_2}{\times }_B\dots {\times }_B{P}_{i_{\ell }}/B$ provides a resolution of $X_w,$

$$\begin{array}{cc}\begin{array}{cccccc}{\gamma }_{i_1,\dots ,i_{\ell }}:& P_{i_1}{\times }_B{P}_{i_2}{\times }_B\dots {\times }_B{P}_{i_{\ell }}{\times }_B\text{pt}& ⟶& X_w& \hookrightarrow & G/B\\ & [g_1,\dots ,g_{\ell }]& ⟼& g_1\dots g_{\ell }B\end{array}& \text{(4.5)}\end{array}$$

Then following, for example, the proof of [BEv1044959, Prop. 2], since the diagram

$$\begin{array}{cc}\begin{array}{ccc}{P}_{i_1}{\times }_B\dots {\times }_B{P}_{i_{\ell }}{\times }_B{P}_{i_{\ell +1}}{\times }_B\text{pt}& \stackrel{{\gamma }_{i_1\dots i_{\ell +1}}}{⟶}& G/B\\ \tau \downarrow & & \downarrow {\pi }_{i_{\ell +1}}\\ P_{i_1}{\times }_B\dots {\times }_B{P}_{i_{\ell }}{\times }_B\text{pt}& \underset{{\gamma }_{i_1\dots i_{\ell }}}{⟶}G/B\underset{{\pi }_{i_{\ell +1}}}{⟶}& G/P_{i_{\ell +1}}\end{array}& \text{(4.6)}\end{array}$$

1. commutes, and
2. has both vertical maps fibrations with fibre $P_{i_{\ell +1}}/B,$

it is a pullback square. Thus

$$\begin{array}{cc}\begin{array}{ccc}{\left({\gamma }_{i_1\dots i_{i+1}}\right)}_{!}\left({\iota }^{*}\left(1\right)\right)& =& {\pi }_{i_{\ell +1}}^{*}{({\pi }_{i_{\ell +1}}\circ {\gamma }_{i_1\dots i_{\ell }})}_{!}\left(1\right)\\ & =& {\pi }_{i_{\ell +1}}^{*}{\left({\pi }_{i_{\ell +1}}\right)}_{!}{\left({\gamma }_{i_1\dots i_{\ell }}\right)}_{!}\left(1\right)& =& A_{i_{\ell +1}}{\left({\gamma }_{i_1\dots i_{\ell }}\right)}_{!}\left(1\right)\text{.}\end{array}& \text{(4.7)}\end{array}$$

The following result then follows by induction.

Theorem 4.3. ([HKi0903.3926, Theorem 3.2], [BEv1044959, Proposition 2]) If $U=(i_1,\dots ,i_{\ell })$ is a sequence in $\{1,\dots ,n\}$ and ${\gamma }_{i_1\dots i_{\ell }}$ is as in (4.5) then

$$\left[Z_{i_1\dots i_{\ell }}\right]=\left[{\left({\gamma }_{i_1\dots i_{\ell }}\right)}_{!}\left(1\right)\right]=A_{i_1}\dots A_{i_{\ell }}\left[Z_{\text{pt}}\right],\text{where}\hspace{0.17em}\left[Z_{\text{pt}}\right]\hspace{0.17em}\text{is the class of a point.}$$

Theorem 4.3 says that the values on the vertices of the element $\left[Z_{i_1\dots i_{\ell }}\right]$ on the moment graph of ${\Gamma }_{i_1,\dots ,i_{\ell }}$ are exactly the coefficients of the $2^{\ell }$ terms in the expansion of

$$A_{i_1}\dots A_{i_{\ell }}=(1+t_{s_{i_1}})\frac{1}{x_{-{\alpha }_{i_1}}}\dots (1+t_{s_{i_{\ell }}})\frac{1}{x_{-{\alpha }_{i_{\ell }}}}\text{.}$$

For example, in type $G{L}_3,$

$$\left[Z_{121}\right]=\left(\begin{array}{c}\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}}1·1·1\\ +\frac{y_{-{\alpha }_2}}{y_{{\alpha }_1}}{t}_{s_1}·1·1+1·t_{s_2}·1+\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}}1·1·t_{s_1}\\ +t_{s_1}·t_{s_2}·1+\frac{y_{-{\alpha }_2}}{y_{{\alpha }_1}}{t}_{s_1}·1·t_{s_1}+1·t_{s_2}·t_{s_1}\\ +t_{s_1}·t_{s_2}·t_{s_1}\end{array}\right)b_1$$

provides the expansion of $\left[Z_{121}\right]=(1+t_{s_1})\frac{1}{x_{-{\alpha }_1}}(1+t_{s_1})\frac{1}{x_{-{\alpha }_1}}(1+t_{s_1})\frac{1}{x_{-{\alpha }_1}}{y}_{R^{-}}{b}_1$ in the basis $\{b_w\hspace{0.17em}\mid \hspace{0.17em}w\in W_0\}\text{.}$ An example of the pushpull in (4.6) in the case of type $G{L}_3$

$$\begin{array}{cc}\begin{array}{ccc}{P}_1{\times }_B{P}_2{\times }_B{P}_1{\times }_B\text{pt}& \stackrel{{\gamma }_{121}}{⟶}& G{L}_3/B\\ \tau \downarrow & & \downarrow {\pi }_1\\ P_1{\times }_B{P}_2{\times }_B\text{pt}& \underset{{\gamma }_{12}}{⟶}G{L}_3/B\underset{{\pi }_1}{⟶}& G{L}_3/P_1\end{array}& \text{(4.8)}\end{array}$$

has moment graphs as in Figure 1, and the computation in (4.7) for this example is

$$\begin{array}{ccccc}\begin{array}{cc}& 1\\ 1& 1& 1\\ 1& 1& 1\\ & 1\end{array}& & \stackrel{{\left({\gamma }_{121}\right)}_{!}}{⟶}& & \begin{array}{cc}& {\Delta }_{121}\\ {\Delta }_{121}& & 1\\ 1& & 1\\ & 1\end{array}\\ \uparrow {\tau }^{*}& & & & \uparrow {\pi }_1^{*}\\ \begin{array}{cc}& 1\\ 1& 1\\ 1\end{array}& \stackrel{{\left({\gamma }_{12}\right)}_{!}}{⟶}& \begin{array}{cc}& y_{-({\alpha }_1+{\alpha }_2)}\\ y_{-{\alpha }_2}& & y_{-({\alpha }_1+{\alpha }_2)}\\ 0& & y_{-{\alpha }_2}\\ & 0\end{array}& \stackrel{{\left({\pi }_1\right)}_{!}}{⟶}& \begin{array}{c}{\Delta }_{121}\\ & 1\\ & 1\end{array}\end{array}$$

where ${\Delta }_{121}=\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}}+\frac{y_{-{\alpha }_2}}{y_{{\alpha }_1}}\text{.}$

Change of groups morphisms across $\iota :B\hookrightarrow P_J$

In the same way that Theorem 3.1 provides $S{\otimes }_S^{W_0}S\cong {\Omega }_T(G/B)$ one can obtain

$$S^{W_J}{\otimes }_{S_0^W}S\cong {\Omega }_{P_J}(G/B),$$

and, if $\iota :B\hookrightarrow P_J$ is the inclusion then the change of group homomorphisms

$${\iota }^J:{\Omega }_{P_J}(G/B)\to {\Omega }_T(G/B)\text{and}{\iota }_J:{\Omega }_T\left(G/B\right)\to {\Omega }_{P_J}(G/B)$$

are given, combinatorially, by

$${\iota }^J:S^{W_J}{\otimes }_{S_0^W}S\hookrightarrow S{\otimes }_{S_0^W}S\text{and}{\iota }_J:S{\otimes }_{S_0^W}S⟶S^{W_J}{\otimes }_{S_0^W}S,$$

with

$${\iota }^J(f\otimes g)=\sum _{w\in W_J}w\left(\frac{1}{y_J}f\right)\otimes g,\text{where}{y}_J=\prod _{\alpha \in R_J^{+}}{y}_{-\alpha },$$

with $R_J^{}$ the set of positive roots for $P_J\supseteq B\supseteq T\text{.}$ The pushforward ${\iota }^J$ is similar to the pushforward operator ${\left({\pi }_J\right)}_{!}$ appearing in (4.1) except acting on the left factor of $S{\otimes }_{S^{W_0}}S$ (see, for example, the definition of ${\delta }_i$ in [Kaj2010, §7]).

$$\begin{array}{c}\begin{array}{ccccc} 1·1·1 s1·1·1 1·s2·1 1·1·s1 s1·s2·1 s1·1·s1 1·s2·s1 s1·s2·s1 y-α1 y-α1 y-α1 y-α1 y-s1α2 y-s1α2 y-α2 y-α2 y-s1s2α1 y-s1α1 y-s2α1 y-α1 & & \stackrel{{\left({\gamma }_{121}\right)}_{!}}{⟶}& & 1 s1 s2 s1s2 s2s1 s1s2s1=s2s1s2 y-α2 y-(α1+α2) y-(α1+α2) y-α1 y-α1 y-(α1+α2) y-α2 y-α2 y-α1 \\ \uparrow {\tau }^{*}& & & & \uparrow {\pi }_1^{*}\\ 1·1 s1·1 1·s2 s1·s2 y-s2α1 y-α2 y-α2 y-α1 & \stackrel{{\left({\gamma }_{12}\right)}_{!}}{⟶}& 1 s1 s2 s1s2 s2s1 s1s2s1=s2s1s2 y-α2 y-(α1+α2) y-(α1+α2) y-α1 y-α1 y-(α1+α2) y-α2 y-α2 y-α1 & \stackrel{{\left({\pi }_1\right)}_{!}}{⟶}& 1 s2 s2s1 y-α2 y-α1 y-(α1+α2) \end{array}\\ \text{Figure 1: An example of the moment graphs for the diagram (4.8)}\end{array}$$

Notes and References

This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.

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