Picture Library

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

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.

General templates

Commutative square

A B C D α β γ δ

Commutative diamond

A B C D α β γ δ

Big commutative diagram template

A B C D E F αa αb βa βb γa γb δa δb

Lattice template #1

Lattice template #2

Lattice template #3

MV Polytope template

Template for use in building column strict tableau

11 12 13 14 15 16 17 18 19 110 111 112 113 114 115 116 117 118 119 120 21 22 23 24 25 26 27 28 29 210 211 212 213 214 215 216 217 218 219 220 31 32 33 34 35 36 37 38 39 310 311 312 313 314 315 316 317 318 319 320 41 42 43 44 45 46 47 48 49 410 411 412 413 414 415 416 417 418 419 420 51 52 53 54 55 56 57 58 59 510 511 512 513 514 515 516 517 518 519 520 61 62 63 64 65 66 67 68 69 610 611 612 613 614 615 616 617 618 619 620 71 72 73 74 75 76 77 78 79 710 711 712 713 714 715 716 717 718 719 720 81 82 83 84 85 86 87 88 89 810 811 812 813 814 815 816 817 818 819 820 91 92 93 94 95 96 97 98 99 910 911 912 913 914 915 916 917 918 919 920 101 102 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 111 112 113 114 115 116 117 118 119 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 121 122 123 124 125 126 127 128 129 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 131 132 133 134 135 136 137 138 139 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 141 142 143 144 145 146 147 148 149 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 151 152 153 154 155 156 157 158 159 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 161 162 163 164 165 166 167 168 169 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 171 172 173 174 175 176 177 178 179 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 181 182 183 184 185 186 187 188 189 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 191 192 193 194 195 196 197 198 199 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 201 202 203 204 205 206 207 208 209 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

Template for folding diagrams

Template for curved operator arrows

Template for affine braid group diagrams

Template for elements of the symmetric group

Example Riemann surface

Picture of the action of W on 𝔥*S for 𝔰𝔩^2

𝔥α1 𝔥-(φ+δ)=𝔥α0 s0ρ=2Λ0 ω 3ω ρ 2s1γ s1γ Λ0=s0γ γ 2γ -2α -α -ω ω α 2α
Λ0 𝔥α level 2 level 1 level 0 𝔥*+2Λ0 𝔥*+Λ0 𝔥* 𝔥α+α,αc ω 2ω 3ω
The Tits cone is 𝔥*+>0Λ0 (the upper half plane in this picture.

From "Morphisms and products"

X0 X1 Y g0 g1
X0 X1 Y Z X0×Y X1 g0 g1 f0 f1 π0 π1
X0 X1 Y X0×Y X1 g0 g1 π0 π1

From "The folding algorithm and the intersections U-vI IwI"

Hα2+δ Hα2 H-α2+δ H-α2+3δ H-α2+5δ H-φ+4δ H-φ+3δ H-φ+2δ Hα0 Hφ Hφ+δ Hφ+2δ Hφ+3δ Hφ+4δ H-α1+δ Hα1 Hα1+δ Hα1+3δ Hα1+5δ - + - + - + - + - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - 1 s0 w0 w0s0 s1 s2 s0s1 s0s2 s1s0 s2s0
1 w Hβ2 Hβ4 Hβ1 Hβ3 Hβ5
Hvαj v vsj

From "Root operators"

Hαi,-1 Hαi t sip p h
Hαi p Hαi di+(p) di-(p) pαi Hαi f˜ip Hαi f˜pαi

From "An Introduction to Tor"

0 M' M M'' 0 0 M' M'M'' M'' 0 k 1 p
P M M'' 0 g h f

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

gi gj
gi gj
gi gj
T0 T1 T2 Tn-2 Tn-1 Tn T0 T1 T2 Tn-2 Tn-1 Tn T0 T1 T2 Tn-2 Tn-1 Tn

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

From "Reflection groups and Braid groups"

with product { , , , , , , , } with product g1g2= g1 g2 so that = . S3= { , , , , , , , } S2= { , } and S1= { } x θ 1 iy x+iy ereiθ eiθ /5= 1 ξ4 ξ3 ξ2 ξ=e2πi/5
, , x 0 = x 0 = [0,1]= and x2 is a path. x 0 = = ( ) ( ) ϕ =

From "Spectral subalgebras"

= = = =

From "Presenting the Affine Hecke algebra"

ω1 ω2 𝔥θ 𝔥α1 𝔥α2

From "A folding example"

From "Covers, subgroups and ramifications"

ramified here y ramified point X f Y degf=2 x1 x2 X Y deg(f)=4 = #   of sheets 2 sheets come together here e1=2 2 sheets come together here e2=2

From "The affine Weyl group"

Hα1 Hφ = Hα1+α2 H α1+α2, -1 H α1+α2, -2 H α1+α2, -3 H α1+α2, -4 H α1+α2, -5 H α2,5 H α2,4 H α2,3 H α2,2 H α2,1 H α2,0 = Hα2 H α1+2α2, 3 H α1+2α2, 2 H α1+2α2, 1 H α1+2α2, 0 = H α1+2α2 H α1+2α2, -1 H α1+2α2, -2 H α1+α2, 1 = Hφ,1 = Hα0 H α1+2α2, -3 α2 s2 s1A s0A s2A sφA A φ ε1 α1
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
Hα1 Hα1+α2 Hα2 H α1+2α2 Hα0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0
λ λ+s1A λ+A λ+s1s2A λ+s2A λ+s1s2s1A λ+s2s1A λ+w0A λ+s2s1s2A
Hα1+α2 Hα1 Hα2 Hα1+2α2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C α1+2α2 ε2 α2 α1+α2 ε1 α1
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 [P]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"

i - +
i - +
i - +
i - +
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"

𝔥* ε2 ε1 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=NP 0PMN0 butMNP

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

C0 𝔥α1 𝔥α2

f1 f2 f1 f1 f2 f2 f2 f1 f2 f2 f2

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 -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 G2"

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,q21 t1,-1,q2±1 C s1C C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C t13,1,q2 1,q61 tz,1,q2 1,zgeneric C s2C s2s1C s2s1s2C s2s1s2s1C s2s1s2s1s2C C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C t1,z,q2 1,zgeneric tz,wz,w generic C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C tq2,z,q2 1,zgeneric tz,q21, zgeneric C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C tq2,q2, qgeneric tq2,q2, qa primitive twelfth root of unity C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C t13,q2 ,q2 1,qgeneric tq2,-q-2 qgeneric orq2a primitive third or fifth root of unity orq2a primitive fourth or fifth root of unity C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C tq2,q2,q2 a primitive sixth root of unity C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C s2s1s2s1C s1s2s1s2s1C s2s1s2s1s2C s1s2s1s2s1s2C t1,1,q2=1 C s2C s2s1C s2s1s2C s2s1s2s1C s2s1s2s1s2C M N C s2C s2s1C M N t1,q2, q2-1 t1,q2, q2=-1 C s2C s2s1C s2s1s2C s2s1s2s1C s2s1s2s1s2C C s2C s2s1C s2s1s2C s2s1s2s1C s2s1s2s1s2C t1,q2,q2 a primitive third root of unity t1,q2,q2 not a primitive third root of unity,q2-1 C s1C s1s2C C s2C s2s1C s2s1s2C s1C s1s2C M N t-1,1,q2=1 t±q-1,q2 ,q21 C s1C M N C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C M N tq2,1,q2 a primitive third root of unity tq2,1,q2 a primitive fourth root of unity C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C M N tq2,1,q2 not a primitive third or fourth root of unity C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C M N C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C tα2,1,q2 not a primitive third or fourth root of unity tq2,1,q2 a primitive fourth root of unity C s1C s2C s1s2C s2s1C s1s2s1C M N C s1C s2C s1s2C s2s1C s1s2s1C M N tq2,±q-3 ,P(t)={α1} tq2,q2, q10=1 C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C tq2,1,q2 a primitive fourth root of unity

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 A1"

Hα-δ Hα Hα+δ tq-1 tq0 tq tq2 tq3 Characterstqx, genericq. Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα tq0 tq1 tq2 tq3 tq0 tq tq2 tq3 tq0 Characterstqx,q4=1. Hα Hα Hα Hα Hα Hα Hα Hα Hα tq0 tq tq0 tq tq0 tq tq0 tq tq0 Characterstqx,q2=1. t1, t-1, q21 t1, t-1, q2=1 tq, t-q, q21 tz,z q2,1

From "Type A2"

α1 α2 α1+α2 ω2 ω1 The typeA2root system Hα1 Hα2 ω1 ω2 0 The weight latticeP 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 overH, with generalq. 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:q2a 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