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

## Affine Weyl group

Let $Q= ∑ i∈ I ℤ αi and 𝔥 * = ∑ i∈ I ℝ αi$ with a symmetric bilinear form $⟨,⟩:{𝔥}^{*}×{𝔥}^{*}\to ℝ$ given by values $⟨{\alpha }_{i},{\alpha }_{j}\in ℤ⟩$ so that $A= α i ∨ α j with α i ∨ = 2 αi αi αi$ 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 ${W}_{0}=⟨{s}_{i}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}i\in I⟩\subseteq \mathrm{GL}\left({𝔥}^{*}\right)$ with $s i : 𝔥 * → 𝔥 * λ ↦ λ- αi ∨ λ αi .$

The affine Weyl group is $W={{W}_{0}}^{*}Q=\left\{w{X}^{\mu }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}w\in {W}_{0},\mu \in Q\right\}$ with $X μ : 𝔥 * → 𝔥 * λ ↦ λ+ μ .$

The alcoves are the connected components of $𝔥 * \ ∪ α∈ ℛ + ,j ∈ ℤ 𝔥 α∨ + jd where 𝔥 α∨ + jd = λ∈ 𝔥 * | α∨ λ + j=0 .$

Example: $𝔥 * =ℝ -span α+ α- , φ = α+ + α-$ 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 $W↔ alcoves$ $W 0 ↔ alcoves in the 0-hexagon$ $Q↔ hexagons .$

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

${G}^{\vee }\left(𝔽\right)$ is generated by "elementary matrices" $X α∨ f and X - α∨ f ,f∈ 𝔽,α∈ ℛ +$ with relations (see Steinberg of Parkinson-Schwer-Ram) where $X α∨ f X - α∨ - f -1 X α∨ f = h α f n α∨$ and $h λ f h μ f = h λ+ μ f , for λ,μ ∈ Q.$

The loop group is $G\left(ℂ\left(\left(t\right)\right)\right)$ where $ℂ t = a -l t -l + a -l+ 1 t -l+ 1 + … | a i ∈ ℂ,l∈ ℤ .$

Define $X α∨ + jd c = X α∨ c t j$$t λ = h λ t -1 , for λ∈ Q,$ $n α∨ + jd = X α∨ + jd 1 X - α∨ - jd -1 X α∨ + jd 1 .$

Let $X0 c = X - φ ∨ + d c , X i c = X αi ∨ c ,$ $n 0 = n - φ ∨ + d c , n i = n αi∨ .$

Let $ℂ[[t]] = a 0 + a 1 t+ a 2 t 2 + … | a i ∈ ℂ .$

Example $I I = + - , A= 2 - 1 -1 2 .$

${G}^{\vee }\left(𝔽\right)={\mathrm{SL}}_{3}\left(𝔽\right)$ is generated by $X + f = 1 f 0 0 1 0 0 0 1 X - f =1 0 0 0 1 f 0 0 1 X + - f =1 0 f 0 1 0 0 0 1$ $X - α+ ∨ f =1 0 0 f 1 0 0 0 1 X - α- ∨ f =1 0 0 0 1 0 f 0 1 X - α+ ∨ - α- ∨ f =1 0 0 0 1 0 f 0 1$ $Q= m α+ + n α- | m,n∈ ℤ$ and $X 0 c = X - φ ∨ + d c = X - φ ∨ ct =1 0 0 0 1 0 ct0 1 ,$ $n 0 = 0 0 - t -1 0 1 0 t0 1 n + =0 1 0 -1 0 0 0 0 1 n - =1 0 0 0 0 1 0 -1 0$

## MV intersections and MV cycles

$G ∨ = G ∨ ℂ t ∪| K= G ∨ ℂ[[t]] → t=0 G ∨ ℂ ∪| ∪| I= Iwahori subgroup → B ∨ = X α∨ c hλ c | α∈ ℛ + ,c∈ ℂ,λ∈ Q$

$G/K$ is the loop Grassmannian.

$G/I$ is the affine flag variety.

$G∨ = ⊔ w∈ W I w I G∨ = ⊔ λ∈ 𝔥 ℤ+ K t λ K$ $G ∨ = ⊔ v∈ W U - v I G ∨ = ⊔ μ ∈ 𝔥 ℤ U - t μ K$ where $𝔥 ℤ = λ∈ 𝔥 * | λ αi∨ ∈ ℤ$ $𝔥 ℤ + = λ∈ 𝔥 * | λ αi∨ ∈ ℤ ≥ 0$

The MV-intersections are $IwI∩ U - vI and K ∨ tλ K ∨ ∩ U - tμ K ∨$ 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 $X i1 c1 n i1 -1 … X il cl n il -1 I with c1 ,… , cl ∈ ℂ.$

The folding algorithm:

Case 1: $X γ 1 c 1 ' … X γ v c v ' n v X j c n j -1 b$ 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: $X γ 1 c 1 ' … X γ v c v ' n v X j c n j -1 b$ is replaced by $X γ 1 c 1 ' … X γ v c v ' X -vαj c -1 n v b '$

Case 3: $X γ 1 c 1 ' … X γ v c v ' n v X j 0 n j -1 b$ is replaced by $X γ 1 c 1 ' … X γ v c v ' X -v αj 0 n v sj b'$

The resulting path (without labels) is a Littelmann path and $IwI∩ U - v I= points of IwI whose folding ends in v .$

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 $b g * = ∑ h∈ I * a gh h$ 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 $b g * | g∈ G → MV-polytopes b g * ↦ support of b g *$ is a bijection.