Indexings of canonical bases: Lyndon words, MV polytopes and the path model

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 8 April

Affine Weyl group

Let $$Q=\sum _{i\in I}\mathbb{Z}{\alpha }_i\text{and}{𝔥}^{*}=\sum _{i\in I}ℝ{\alpha }_i$$ with a symmetric bilinear form $⟨,⟩:{𝔥}^{*}\times {𝔥}^{*}\to ℝ$ given by values $⟨{\alpha }_i,{\alpha }_j\in \mathbb{Z}⟩$ so that $$A=\left(⟨{{\alpha }_i}^{\vee },{\alpha }_j⟩\right)\text{with}{{\alpha }_i}^{\vee }=\frac{2{\alpha }_i}{⟨{\alpha }_i,{\alpha }_i⟩}$$ is the Cartan matrix of a symmetrisable Kac-Moody Lie algebra $𝔤.$ Let ${ℛ}^{+}=\left\{\text{positive roots of}𝔤\right\}.$

The Weyl group is $W_0=⟨s_i|i\in I⟩\subseteq \mathrm{GL}\left({𝔥}^{*}\right)$ with $$\begin{array}{rcll}{s}_i:& {𝔥}^{*}& \to & {𝔥}^{*}\\ & \lambda & \mapsto & \lambda -⟨{{\alpha }_i}^{\vee },\lambda ⟩{\alpha }_i\end{array}.$$

The affine Weyl group is $W={W_0}^{*}Q=\left\{w{X}^{\mu }|w\in W_0,\mu \in Q\right\}$ with $$\begin{array}{rcll}{X}^{\mu }:& {𝔥}^{*}& \to & {𝔥}^{*}\\ & \lambda & \mapsto & \lambda +\mu \end{array}.$$

The alcoves are the connected components of $${𝔥}^{*}\backslash \left(\underset{\alpha \in {ℛ}^{+},j\in \mathbb{Z}}{\cup }{𝔥}^{{\alpha }^{\vee }+jd}\right)\text{where}{𝔥}^{{\alpha }^{\vee }+jd}=\left\{\lambda \in {𝔥}^{*}|⟨{\alpha }^{\vee },\lambda ⟩+j=0\right\}.$$

Example: $${𝔥}^{*}=ℝ\text{-span}\left\{{\alpha }_{+},{\alpha }_{-}\right\},\phi ={\alpha }_{+}+{\alpha }_{-}$$ Each alcove has two types of address, $w{X}^{\mu }$ and $s_{i_1},\dots s_{i_l}.$

${ℛ}^{+}=\left\{{\alpha }_{+},{\alpha }_{-},{\alpha }_{+}+{\alpha }_{-}\right\}$ since $G\cap L=\left\{+,-,+-\right\}$ if $+<-.$

$W$ is generated by $s_0$ and $s_i,i\in I$ where $s_0=X^{\phi }{s}_{\phi }$ and $$W\leftrightarrow \left\{\text{alcoves}\right\}$$ $$W_0\leftrightarrow \left\{\text{alcoves in the 0-hexagon}\right\}$$ $$Q\leftrightarrow \left\{\text{hexagons}\right\}.$$

Chevalley groups $G^{\vee }\left(𝔽\right)$

$G^{\vee }\left(𝔽\right)$ is generated by "elementary matrices" $$X_{{\alpha }^{\vee }}\left(f\right)\text{and}{X}_{-{\alpha }^{\vee }}\left(f\right),f\in 𝔽,\alpha \in {ℛ}^{+}$$ with relations (see Steinberg of Parkinson-Schwer-Ram) where $$X_{{\alpha }^{\vee }}\left(f\right)X_{-{\alpha }^{\vee }}\left(-f^{-1}\right)X_{{\alpha }^{\vee }}\left(f\right)=h_{\alpha }\left(f\right)n_{{\alpha }^{\vee }}$$ and $$h_{\lambda }\left(f\right)h_{\mu }\left(f\right)=h_{\lambda +\mu }\left(f\right),\text{for}\lambda ,\mu \in Q.$$

The loop group is $G\left(ℂ\left(\left(t\right)\right)\right)$ where $$ℂ\left(\left(t\right)\right)=\left\{a_{-l}{t}^{-l}+a_{-l+1}{t}^{-l+1}+\dots |a_i\in ℂ,l\in \mathbb{Z}\right\}.$$

Define $$X_{{\alpha }^{\vee }+jd}\left(c\right)=X_{{\alpha }^{\vee }}\left(c{t}^j\right)$$$$t_{\lambda }=h_{\lambda }\left(t^{-1}\right),\text{for}\lambda \in Q,$$ $$n_{{\alpha }^{\vee }+jd}=X_{{\alpha }^{\vee }+jd}\left(1\right)X_{-{\alpha }^{\vee }-jd}\left(-1\right)X_{{\alpha }^{\vee }+jd}\left(1\right).$$

Let $$X_0\left(c\right)=X_{-{\phi }^{\vee }+d}\left(c\right),X_i\left(c\right)=X_{{\alpha }_i^{\vee }}\left(c\right),$$ $$n_0=n_{-{\phi }^{\vee }+d}\left(c\right),n_i=n_{{\alpha }_i^{\vee }}.$$

Let $$ℂ\left[\right[t\left]\right]=\left\{a_0+a_{1}t+a_2{t}^2+\dots |a_i\in ℂ\right\}.$$

Example $$II=\left\{+,-\right\},A=\left(\begin{array}{rc}2& -1\\ -1& 2\end{array}\right).$$

$G^{\vee }\left(𝔽\right)={\mathrm{SL}}_3\left(𝔽\right)$ is generated by $$X_{+}\left(f\right)=\left(\begin{array}{ccc}1& f& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)X_{-}\left(f\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& f\\ 0& 0& 1\end{array}\right)X_{+-}\left(f\right)=\left(\begin{array}{ccc}1& 0& f\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$ $$X_{-{\alpha }_{+}^{\vee }}\left(f\right)=\left(\begin{array}{ccc}1& 0& 0\\ f& 1& 0\\ 0& 0& 1\end{array}\right)X_{-{\alpha }_{-}^{\vee }}\left(f\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ f& 0& 1\end{array}\right)X_{-{\alpha }_{+}^{\vee }-{\alpha }_{-}^{\vee }}\left(f\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ f& 0& 1\end{array}\right)$$ $$Q=\left\{m{\alpha }_{+}+n{\alpha }_{-}|m,n\in \mathbb{Z}\right\}$$ and $$t_{\lambda }=h_{m{\alpha }_{+}+n{\alpha }_{-}}\left(t^{-1}\right)=\left(\begin{array}{ccc}{t}^{-m}& 0& 0\\ 0& t^{m-n}& 0\\ 0& 0& t^n\end{array}\right),\text{if} \lambda =m{\alpha }_{+}+n{\alpha }_{-}.$$ $$X_0\left(c\right)=X_{-{\phi }^{\vee }+d}\left(c\right)=X_{-{\phi }^{\vee }}\left(ct\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ ct& 0& 1\end{array}\right),$$ $$n_0=\left(\begin{array}{ccc}0& 0& -t^{-1}\\ 0& 1& 0\\ t& 0& 1\end{array}\right)n_{+}=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right)n_{-}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right)$$

MV intersections and MV cycles

$$\begin{array}{ccc}{G}^{\vee }=G^{\vee }\left(ℂ\left(\left(t\right)\right)\right)& & \\ \cup |& & \\ K=G^{\vee }\left(ℂ\left[\right[t\left]\right]\right)& \stackrel{t=0}{\to }& G^{\vee }\left(ℂ\right)\\ \cup |& & \cup |\\ I=\text{Iwahori subgroup}& \to & B^{\vee }=⟨X_{{\alpha }^{\vee }}\left(c\right)h_{\lambda }\left(c\right)|\alpha \in {ℛ}^{+},c\in ℂ,\lambda \in Q⟩\end{array}$$

$G/K$ is the loop Grassmannian.

$G/I$ is the affine flag variety.

$$G^{\vee }=\bigsqcup _{w\in W}IwI{G}^{\vee }=\bigsqcup _{\lambda \in {𝔥}_{\mathbb{Z}}^{+}}K{t}_{\lambda }K$$ $$G^{\vee }=\bigsqcup _{v\in W}{U}^{-}vI{G}^{\vee }=\bigsqcup _{\mu \in {𝔥}_{\mathbb{Z}}}{U}^{-}{t}_{\mu }K$$ where $$U^{-}=⟨X_{-{\alpha }^{\vee }}\left(f\right)|\alpha \in {ℛ}^{+}, f\in ℂ\left(\left(t\right)\right)⟩$$ $${𝔥}_{\mathbb{Z}}=\left\{\lambda \in {𝔥}^{*}|⟨\lambda ,{\alpha }_i^{\vee }⟩\in \mathbb{Z}\right\}$$ $${𝔥}_{\mathbb{Z}}^{+}=\left\{\lambda \in {𝔥}^{*}|⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}\right\}$$

The MV-intersections are $$IwI\cap U^{-}vI\text{and}{K}^{\vee }{t}_{\lambda }{K}^{\vee }\cap U^{-}{t}_{\mu }{K}^{\vee }$$ and the MV-cycles are the irreducible components of $$$$K ∨ tλ K ∨∩ U - tμ K ∨ in G ∨ / K ∨

Points in $IwI\cap U^{-}vI$

Let $w\in W$ be an alcove and $w=s_{i_1}\dots s_{i_l}$ be a minimal length walk to $w.$

(Steinberg) The points of $IwI$ are $$X_{i_1}\left(c_1\right){n_{i_1}}^{-1}\dots X_{i_l}\left(c_l\right){n_{i_l}}^{-1}I\text{with}{c}_1,\dots ,c_l\in ℂ.$$

The folding algorithm:

Case 1: $$X_{{\gamma }_1}\left(c_1\text{'}\right)\dots X_{{\gamma }_v}\left(c_v\text{'}\right)n_v{X}_j\left(c\right){n_j}^{-1}b$$ is replaced by $X_{{\gamma }_1}\left(c_1\text{'}\right)\dots X_{{\gamma }_v}\left(c_v\text{'}\right)X_{v{\alpha }_j}\left(c\right)n_{v{s}_j}b\text{'}$

Case 2: $$X_{{\gamma }_1}\left(c_1\text{'}\right)\dots X_{{\gamma }_v}\left(c_v\text{'}\right)n_v{X}_j\left(c\right){n_j}^{-1}b$$ is replaced by $$X_{{\gamma }_1}\left(c_1\text{'}\right)\dots X_{{\gamma }_v}\left(c_v\text{'}\right)X_{-v\alpha j}\left(c^{-1}\right)n_{v}b\text{'}$$

Case 3: $$X_{{\gamma }_1}\left(c_1\text{'}\right)\dots X_{{\gamma }_v}\left(c_v\text{'}\right)n_v{X}_j\left(0\right){n_j}^{-1}b$$ is replaced by $$X_{{\gamma }_1}\left(c_1\text{'}\right)\dots X_{{\gamma }_v}\left(c_v\text{'}\right)X_{-v{\alpha }_j}\left(0\right)n_{v{s}_j}b\text{'}$$

The resulting path (without labels) is a Littelmann path and $$IwI\cap U^{-}vI=\left\{\text{points of}IwI\text{whose folding ends in}v\right\}.$$

The MV-polytope of $IwI\cap U^{-}vI$ is the support of the folded paths in $IwI\cap U^{-}vI.$ (Similarly for $K{t}_{\mu }K\cap U^{-}{t}_{\mu }K$ )

Dual canonical bases

Let $$b_g*=\sum _{h\in I^{*}}{a}_{gh}h$$ and draw the word $h=i_1\dots i_l$ as a path in ${𝔥}^{*}.$

Then the support of ${b_g}^{*}$ is an MV-polytope $$\begin{array}{rcl}\left\{{b_g}^{*}|g\in G\right\}& \to & \left\{\text{MV-polytopes}\right\}\\ {b_g}^{*}& \mapsto & \text{support of}{b_g}^{*}\end{array}$$ is a bijection.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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