## Picture Library

Last update: 22 January 2014

The aim of this page is to collect in one place the SVG pictures used in these notes, as a handy reference library for future notes/pages.

## Picture of the action of $W$ on $\frac{{𝔥}^{*}}{ℂS}$ for ${\stackrel{^}{\mathrm{𝔰𝔩}}}_{2}$

The Tits cone is ${\stackrel{\circ }{𝔥}}_{ℝ}^{*}+{ℝ}_{>0}{\Lambda }_{0}$ (the upper half plane in this picture.

## From "Double affine braid groups and Hecke algebras of classical type"

The following diagram is from section 5.2 of the original notes this is based on, and represents the "full twist". $σ=$

## From "Spectral subalgebras"

$= = = =$

## From "The affine Weyl group"

$Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 = ω2 ω1 ε1 0 Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 ε1 ρ The set P+ The set P++$

## From "The ring $ℤ{\left[P\right]}^{W}$"

$Hα1+α2 Hα1 Hα2 Hα1+2α2 Hα1+α2-δ Hα1-δ Hα1-δ Hα2-δ Hα1+2α2-δ C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C λ1 λs1 λs2s1 λs2 λs1s2 λs1s2s1 λs2s1s2 λs1s2s1s2 The arrangement 𝒜-$

## From "The affine Hecke algebra"

$Hα1 Hα1+α2 Hα2 H α1+2α2 Hα0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2$ $Tw-1-1 = A wA 2 1 2 1 2 1 2 1 0 0 0 0 0 0 0 0 Xλ = WA λ+WA$

## Dynkin Diagrams

 $1$ $2$ $n-1$ $1$ $2$ $3$ $n$ $1$ $2$ $3$ $n$ $1$ $2$ $3$ $4$ $n$ $1$ $2$ $3$ $4$ $5$ $6$ $0$ $1$ $2$ $3$ $4$ $5$

## From "Weight lattices"

$𝔥ℤ* {\epsilon }_{2} {\epsilon }_{1}$ $ε2+ε3 ε1+ε3 ε1+ε2 ε1+ε2+ε3 ε2 ε3 ε1$

## From "The positive formula"

 $1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 4 3 3 3 4 4 4 5 4 5 5 6 6 7 7$ $1 1 2 2 1 2 3 3 4 4 2 ⟼ 1 1 2 2 1 2 3 2 3 4 4 ⟼ 1 1 2 2 1 2 3 2 3 4 4 ⟼ 1 1 2 2 1 2 3 2 3 4 4 ⟼ 1 1 2 2 1 2 3 2 3 4 4 ⟼ 1 1 2 2 1 2 3 4 2 3 4 ⟼ 1 1 2 2 1 2 3 4 2 3 4 ⟼ 1 1 2 2 1 2 3 4 2 3 4 ⟼ 1 1 2 2 1 2 3 4 2 3 4 ⟼ 1 1 2 2 1 2 3 4 2 3 4 ⟼ 1 1 1 2 2 2 3 4 2 3 4 ⟼ 1 1 1 2 2 2 2 3 4 3 4 ⟼ 1 1 1 2 2 2 2 3 4 3 4 ⟼ 1 1 1 2 2 2 2 3 4 3 4$

## From "Representation Theory, Reflection groups and Groups of Lie Type"

$N P N P M=N⊕P 0⟶P⟶M⟶N⟶0 butM≠N⊕P$

$𝔥α1∨ 𝔥α2∨ 𝔥α3∨ s1 s2 α2 α1 s1s2 s2s1 s1s2s1=s2s1s2 C0$

## From "Representations of the symmetric group"



$Board Beads$

$if then or$

Young's lattice

$-2 -1 0 1 2 3$ $∅ 0 -1 1 -2 2 -1 1 -3 3 -1 1 -2 2 0$

## From "The Weyl Character Formula"

$C0 s1C0 s2C0 ρ s1s2C0 s2s1C0 s1s2s1C0=s2s1s2C0$

## From "Crystals from paths and MV polytopes"

$𝔥αi p f∼ip pos.side (towardsC0) 𝔥αi p f˜ip=0$

$C0 𝔥α1∨ 𝔥α2∨$

$f∼1 f∼2 f∼1 f∼1 f∼2 f∼2 f∼2 f∼1 f∼2 f∼2 f∼2$

## From "Crystals from KLR and preprojective algebras"

$Q‾= a1 a2 an-2 a1* a2* an-2*$

## From "Link invariants from quantum groups"

$knot (unknot) knot (trefoil) knot (Borromean rings)$

## From "The structure of local regions"

$Hα1 Hα2 Hα1+2α2 Hα1+α2+δ Hα1+α2 Hα1+2α2+δ Hα1+2α2-δ Hα2+δ Hα2-δ Hα1-δ Hα1+δ Hα1+α2-δ γ s2s1s2γ s1s2γ s2γ Hα1 Hα2 Hα1+2α2 Hα1+α2 J={α2,α1+α2} J={α2} J=∅ C s2s1s2C s2s1C s2C$

## From "The connection to standard Young tableaux"

$⋮ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 … 10 12 13 14 6 8 11 5 7 9 4 2 3 1 0 1 -2 -1 -1 0 -1 1 -1 0 1 -1 0 0 -2 0 1 Page0 Page12$

## From "Skew shapes, ribbons, conjugation, etc. in type A"

$a b c a b c a b i-1 i i+1 i-1 i i+1 i-1 i i j k ℓ -1 0 1 -1 1 -1 0$

## From "The type A, root of unity case"

$10 11 8 10 1 5 11 8 … … 12 1 5 9 13 2 6 12 14 3 9 13 2 6 4 7 14 3 4 7$

## From "Standard tableaux for type C in terms of boxes"

$12 -32 -12 32 -52 -12 32 52 -72 -32 -12 12 32 72 -52 -32 12 52 -32 12 32 -12$

## From "Type ${G}_{2}$"

$α1 α2 The typeG2root system$ $ω2 ω1 0 Hα1 Hα2$ $Hα1 H2α1+α2 H3α1+α2 Hα2 Hα1+α2 H3α1+2α2+δ H3α1+2α2 Hα1+α2+δ Hα2+δ H3α1+α2+δ H2α1+α2+δ Hα1+δ$ $C C s2C s2s1C t1,1,q2≠1 t1,-1,q2≠±1$ $C s1C C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C t13,1,q2 ≠1,q6≠1 tz,1,q2 ≠1,zgeneric$

## From "The Pieri-Chevalley formula"

$ω2 α2 α1 ω2 α2 α1$ $ω2 α2 α1$

## From "Classification and construction of calibrated representations"

$Case (1) Case (2) Case (3) Case (4)$

## From ""Garnir relations” and an analogue of Young’s natural basis"

$sdL= a ^ b ^ ⋮ ^ c ^ d > d+1 ^ e ^ ⋮ ^ f sdL= d+1 d Case (1) Case (2)$

## From "Type ${A}_{1}$"

$Hα-δ Hα Hα+δ tq-1 tq0 tq tq2 tq3 Characters tqx, generic q.$ $Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα tq0 tq1 tq2 tq3 tq0 tq tq2 tq3 tq0 Characters tqx, q4=1.$ $Hα Hα Hα Hα Hα Hα Hα Hα Hα tq0 tq tq0 tq tq0 tq tq0 tq tq0 Characters tqx, q2=1.$ $t1, t-1, q2≠1 t1, t-1, q2=1$ $tq, t-q, q2≠1 tz,z≠ q2,1$

## From "Type ${A}_{2}$"

$α1 α2 α1+α2 ω2 ω1 The type A2 root system$ $Hα1 Hα2 ω1 ω2 0 The weight lattice P$ $Hα1 Hα2 Hα1+α2 TQ$ $Hα1 Hα2 Hα1+α2 Hα1+α2+δ Hα1+δ Hα2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z Figure 1: Representatives of some central characters of modules over H∼, with general q. Hα1 Hα1+δ Hα1-δ Hα1 Hα2-δ Hα2+δ Hα2 Hα1+α2 Hα1+α2-δ Hα1+α2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z Figure 2: q2 a primitive third root of unity. Hα1 Hα1±δ Hα2±δ Hα2 Hα1+α2 Hα1+α2±δ t1,1 t1,q2 t1,z tz,w tq2,z Figure 3: q2-1. Hα1 Hα2$