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

Commutative diamond

Big commutative diagram template

Lattice template #1

Lattice template #2

Lattice template #3

MV Polytope template

Template for use in building column strict tableau

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 $\frac{{𝔥}^{*}}{ℂS}$ for ${\widehat{\mathrm{𝔰𝔩}}}_2$

The Tits cone is ${\stackrel{\circ }{𝔥}}_{ℝ}^{*}+{ℝ}_{>0}{\Lambda }_0$ (the upper half plane in this picture.

From "Morphisms and products"

From "The folding algorithm and the intersections $U^{-}vI\cap IwI$"

From "Root operators"

From "An Introduction to Tor"

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". $$\sigma =\begin{array}{c} \end{array}$$

From "Reflection groups and Braid groups"

From "Spectral subalgebras"

$$\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}$$

From "Presenting the Affine Hecke algebra"

From "A folding example"

From "Covers, subgroups and ramifications"

From "The affine Weyl group"

$$\begin{array}{cc}\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ s_1{s}_2{s}_1{s}_{2}C \] \[ {\epsilon }_2={\omega }_2 \] \[ {\omega }_1 \] \[ {\epsilon }_1 \] \[ 0 \]\end{array}& \begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ s_1{s}_2{s}_1{s}_{2}C \] \[ {\epsilon }_2 \] \[ {\epsilon }_1 \] \[ \rho \]\end{array}\\ The\; set{P}^{+}& The\; set{P}^{++}\end{array}$$

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

$$\begin{array}{c}\begin{array}{c} \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_1+{\alpha }_2-\delta } \] \[ H_{{\alpha }_1-\delta } \] \[ H_{{\alpha }_1-\delta } \] \[ H_{{\alpha }_2-\delta } \] \[ H_{{\alpha }_1+2{\alpha }_2-\delta } \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ {\lambda }_1 \] \[ {\lambda }_{s_1} \] \[ {\lambda }_{s_2{s}_1} \] \[ {\lambda }_{s_2} \] \[ {\lambda }_{s_1{s}_2} \] \[ {\lambda }_{s_1{s}_2{s}_1} \] \[ {\lambda }_{s_2{s}_1{s}_2} \] \[ {\lambda }_{s_1{s}_2{s}_1{s}_2} \]\end{array}\\ The\; arrangement{𝒜}^{-}\end{array}$$

From "The affine Hecke algebra"

$$\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_0} \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \]\end{array}$$ $$T_{w^{-1}}^{-1}=\begin{array}{c} \[ A \] \[ wA \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \]\end{array}{X}^{\lambda }=\begin{array}{c} \[ WA \] \[ \lambda +WA \]\end{array}$$

Dynkin Diagrams

From "Weight lattices"

$$\begin{array}{cc}{𝔥}_{\mathbb{Z}}^{*}& ε2 ε1 \end{array}$$ $$ε2+ε3 ε1+ε3 ε1+ε2 ε1+ε2+ε3 ε2 ε3 ε1$$

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$$ $$\begin{array}{cccc}& 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 \end{array}$$

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

$$\begin{array}{cc} N P & N P \\ M=N\oplus P& \begin{array}{c}0⟶P⟶M⟶N⟶0\\ \text{but}M\ne N\oplus P\end{array}\end{array}$$

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

From "Representations of the symmetric group"

$$$$

$$\begin{array}{cc} & \\ \text{Board}& \text{Beads}\end{array}$$

$$\begin{array}{cccccc}\text{if}& & \text{then}& & \text{or}& \end{array}$$

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"

$$\begin{array}{cc}\bar{Q}=& a1 a2 an-2 a1* a2* an-2* \end{array}$$

From "Link invariants from quantum groups"

$$\begin{array}{ccc} & & \\ \text{knot (unknot)}& \text{knot (trefoil)}& \text{knot (Borromean rings)}\end{array}$$

From "The structure of local regions"

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

From "The connection to standard Young tableaux"

$$\begin{array}{c} ⋮ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 … \\ 10 12 13 14 6 8 11 5 7 9 4 2 3 1 \\ 0 1 -2 -1 -1 0 -1 1 -1 0 1 -1 0 0 -2 0 1 Page0 Page12 \end{array}$$

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

$$\begin{array}{ccc} a b c & a b c & a b \\ i-1 i i+1 & i-1 i i+1 & i-1 i \\ & i j k ℓ \\ & -1 0 1 -1 1 -1 0 \end{array}$$

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 $G_2$"

$$\begin{array}{c} α1 α2 \\ \text{The type}{G}_2\text{root system}\end{array}$$ $$ω2 ω1 0 Hα1 Hα2$$ $$Hα1 H2α1+α2 H3α1+α2 Hα2 Hα1+α2 H3α1+2α2+δ H3α1+2α2 Hα1+α2+δ Hα2+δ H3α1+α2+δ H2α1+α2+δ Hα1+δ$$ $$\begin{array}{cc} C & C s2C s2s1C \\ t_{1,1},q^2\ne 1& t_{1,-1},q^2\ne \pm 1\end{array}$$ $$\begin{array}{cc} C s1C & C s1C s1s2C s1s2s1C s1s2s1s2C s1s2s1s2s1C \\ t_{\sqrt[3]{1},1},q^2\ne 1,q^6\ne 1& t_{z,1},q^2\ne 1,z\text{generic}\end{array}$$

From "The Pieri-Chevalley formula"

$$\begin{array}{cc} ω2 α2 α1 & ω2 α2 α1 \end{array}$$ $$ω2 α2 α1$$

From "Classification and construction of calibrated representations"

$$\begin{array}{cccc} & & & \\ \text{Case (1)}& \text{Case (2)}& \text{Case (3)}& \text{Case (4)}\end{array}$$

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

$$\begin{array}{cccc}{s}_{d}L=& a ^ b ^ ⋮ ^ c ^ d > d+1 ^ e ^ ⋮ ^ f & s_{d}L=& d+1 d \\ & \text{Case (1)}& & \text{Case (2)}\end{array}$$

From "Type $A_1$"

$$\begin{array}{c} Hα-δ Hα Hα+δ tq-1 tq0 tq tq2 tq3 \\ \text{Characters}\hspace{0.17em}{t}_{q^x}\text{, generic}\hspace{0.17em}q\text{.}\end{array}$$ $$\begin{array}{c} Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα tq0 tq1 tq2 tq3 tq0 tq tq2 tq3 tq0 \\ \text{Characters}\hspace{0.17em}{t}_{q^x},\hspace{0.17em}{q}^4=1\text{.}\end{array}$$ $$\begin{array}{c} Hα Hα Hα Hα Hα Hα Hα Hα Hα tq0 tq tq0 tq tq0 tq tq0 tq tq0 \\ \text{Characters}\hspace{0.17em}{t}_{q^x},\hspace{0.17em}{q}^2=1\text{.}\end{array}$$ $$\begin{array}{ccc} & & \\ t_1,t_{-1},q^2\ne 1& & t_1,t_{-1},q^2=1\end{array}$$ $$\begin{array}{ccc} & & \\ t_q,t_{-q},q^2\ne 1& & t_z,z\ne q^2,1\end{array}$$

From "Type $A_2$"

$$\begin{array}{c} α1 α2 α1+α2 ω2 ω1 \\ \text{The type}\hspace{0.17em}{A}_2\hspace{0.17em}\text{root system}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 ω1 ω2 0 \\ \text{The weight lattice}\hspace{0.17em}P\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 \\ T_Q\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 Hα1+α2+δ Hα1+δ Hα2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z \\ \text{Figure 1: Representatives of some central characters of modules over}\hspace{0.17em}\stackrel{\sim }{H}\text{, with general}\hspace{0.17em}q\text{.}\\ \hspace{0.17em}\\ Hα1 Hα1+δ Hα1-δ Hα1 Hα2-δ Hα2+δ Hα2 Hα1+α2 Hα1+α2-δ Hα1+α2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z \\ \text{Figure 2:}\hspace{0.17em}{q}^2\hspace{0.17em}\text{a primitive third root of unity.}\\ \hspace{0.17em}\\ Hα1 Hα1±δ Hα2±δ Hα2 Hα1+α2 Hα1+α2±δ t1,1 t1,q2 t1,z tz,w tq2,z \\ \text{Figure 3:}\hspace{0.17em}{q}^2-1\text{.}\\ \hspace{0.17em}\\ Hα1 Hα2 Hα1+α2 t1,1 tz,w tq2,z \\ \text{Figure 4:}\hspace{0.17em}{q}^2=1\text{.}\end{array}$$ $$\begin{array}{ccc} (M(t)) tgen =M(t) & (M(t)) tgen (M(t)) s2tgen (M(t)) s1s2tgen & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t \\ t_{1,1},q^2\ne 1& t_{1,z},q^2\ne 1& t_{z,w}\end{array}$$ $$\begin{array}{ccc} (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t \\ t_{q^2,q^2},q^4\ne 1,q^6\ne 1& t_{q^2,q^2},q^2\hspace{0.17em}\text{a primitive third root of unity}& t_{q^2,z},q^2\ne 1\end{array}$$ $$\begin{array}{ccc} (M(t))t (M(t))s2t (M(t))s1s2t & (M(t))t (M(t))s2s1t (M(t))s1t & (M(t))t (M(t))s2t (M(t))s1s2t \\ t_{1,q^2},q^4\ne 1& t_{q^2,1},q^4\ne 1& t_{1,q^2},q^2=-1\end{array}$$ $$\begin{array}{cc} (M(t))t (M(t))s2t (M(t))s1s2t & (M(t))t (M(t))s2s1t (M(t))s1t \\ t_{1,q^2},q^4\ne 1& t_{q^2,1},q^4\ne 1\end{array}$$ $$\begin{array}{c} (M(t))t (M(t))s2t (M(t))s1s2t \\ t_{1,q^2},q^2=-1\end{array}$$ $$\begin{array}{ccc} (M(t))t=M(t) & (M(t)) t (M(t)) s2t (M(t)) s1s2t & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s2s1s2t (M(t))s2s1t (M(t))s1t \\ t_{1,1},q^2=1& t_{1,z},q^2=1& t_{z,w},q^2=1\end{array}$$

From "Type $C_2$"

$$\begin{array}{c} α1 α1+α2 2α1+α2 α2 \\ \text{The type}\hspace{0.17em}{C}_2\hspace{0.17em}\text{root system}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 ω1 ω2 \\ \text{The weight lattice}\hspace{0.17em}P\text{.}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 H2α1+α2 Hα1+α2 Hα1+α2+δ H2α1+α2+δ Hα1+δ Hα2+δ \\ \text{The torus}\hspace{0.17em}{T}_Q\end{array}$$ $$\begin{array}{cc} Hα1 Hα1 Hα2 Hα1+δ & Hα1 Hα1 Hα2 Hα1+δ \\ \text{Case 1:}\hspace{0.17em}\mid Z\left(t\right)\mid \ge 2\text{, so that}\hspace{0.17em}t& \text{Case 2:}\hspace{0.17em}\mid Z\left(t\right)\mid =1\text{, so that}\hspace{0.17em}t\\ \text{lies on at least two hyperplanes}\hspace{0.17em}{H}_{\alpha }\text{.}& \text{lies on exactly one hyperplane}\hspace{0.17em}{H}_{\alpha }\text{.}\\ \\ Hα1 Hα1 Hα2 Hα1+δ & Hα1 Hα1 Hα2 Hα1+δ \\ \text{Case 3:}\hspace{0.17em}\mid Z\left(t\right)\mid =\varnothing ,{\alpha }_1\in P\left(t\right)\text{, so that}\hspace{0.17em}t& \text{Case 3:}\hspace{0.17em}\mid Z\left(t\right)\mid =\varnothing ,{\alpha }_1\notin P\left(t\right)\text{, so that}\hspace{0.17em}t\\ \text{lies on}\hspace{0.17em}{H}_{{\alpha }_1\pm \delta }\text{, but not on any}\hspace{0.17em}{H}_{\alpha }\text{.}& \text{does not lie on}\hspace{0.17em}{H}_{{\alpha }_1\pm \delta }\hspace{0.17em}\text{or any}\hspace{0.17em}{H}_{\alpha }\text{.}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 H2α1+α2 t1,1 t1,q2 t1,z tq2,q2 tq,1 tq2,z tz,w tz,q2 tq2,1 t-q,1 \\ \text{Figure 5: Representatives of some possible central characters of}\hspace{0.17em}\stackrel{\sim }{H}\text{-modules with generic}\hspace{0.17em}q\text{.}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 H2α1+α2 t1,1 t1,q2 t1,z tq2,q2 tq,1 tq2,z tz,w tz,q2 tq2,1 t-1,1 tz,1 \\ \text{Figure 6: Representatives of the possible central characters of modules over}\hspace{0.17em}\stackrel{\sim }{H}\text{, with}\hspace{0.17em}q\hspace{0.17em}\text{a primitive eighth root of unity.}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 H2α1+α2 t1,1 t1,q2 t1,z tq2,q2 t±q,1 tq2,z tz,w tz,q2 tq2,1 t-1,1 tz,1 \\ \text{Figure 7: Representatives of the possible central characters of modules over}\hspace{0.17em}\stackrel{\sim }{H}\text{, with}\hspace{0.17em}{q}^2\hspace{0.17em}\text{a primitive third root of unity.}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 H2α1+α2 t1,1 tq2,1 t1,z tq,1 tq2,z tz,w tz,q2 t-1,1 tz,1 \\ \text{Figure 8: Representatives of the possible central characters of modules over}\hspace{0.17em}\stackrel{\sim }{H}\text{, with}\hspace{0.17em}{q}^2=-1\text{.}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 H2α1+α2 t1,1 t1,z tz,w t-1,1 tz,1 \\ \text{Figure 9: Representatives of the possible central characters of modules over}\hspace{0.17em}\stackrel{\sim }{H}\text{, with}\hspace{0.17em}{q}^2=1\text{.}\end{array}$$ $$\begin{array}{ccc} M(t)=M(t)tgen & M(t)tgen M(t)s2tgen M(t)s1s2tgen M(t)s2s1s2tgen & M(t)tgen M(t)s1tgen M(t)s2s1tgen M(t)s1s2s1tgen \\ t_{1,1},q^2\ne 1& t_{1,z},q^2\ne 1& t_{z,1},q^2\ne 1\end{array}\begin{array}{cc} M(t)tgen M(t)s1tgen & M(t)t M(t)s1t M(t)s2s1t M(t)s1s2s1t M(t)s2t M(t)s1s2t M(t)s2s1s2t M(t)s2s1s2s1t \\ t_{-1,1},q\ne \pm i& t_{z,w}\end{array}$$ $$\begin{array}{ccc} M(t)t M(t)s1t M(t)s2s1t M(t)s1s2s1t M(t)s2t M(t)s1s2t M(t)s2s1s2t M(t)s2s1s2s1t & M(t)t M(t)s1t M(t)s2s1t M(t)s1s2s1t M(t)s2t M(t)s1s2t M(t)s2s1s2t M(t)s2s1s2s1t & M(t)t M(t)s1t M(t)s2s1t M(t)s1s2s1t M(t)s2t M(t)s1s2t M(t)s2s1s2t M(t)s2s1s2s1t \\ t_{q^2,z},q^2\ne 1& t_{z,q^2},q^2\ne 1& t_{-1,q^2},q^4\ne 1,q^8\ne 1\end{array}$$

$$\begin{array}{cc} M(t)t M(t)s1t M(t)s2s1t M(t)s1s2s1t M(t)s2t M(t)s1s2t M(t)s2s1s2t M(t)s2s1s2s1t & M(t)t M(t)s1t M(t)s2s1t M(t)s1s2s1t M(t)s2t M(t)s1s2t M(t)s2s1s2t M(t)s2s1s2s1t \\ t_{q^2,q^2},\hspace{0.17em}q\hspace{0.17em}\text{generic}& t_{q^2,q^2},\hspace{0.17em}q\hspace{0.17em}\text{a primitive eighth root of unity}\end{array}$$ $$\begin{array}{c} M(t)=M(t)t \\ t_{1,1},q^2=1\end{array}$$ $$\begin{array}{cc} C s1C s1s2C s1s2s1C M N & C s1C M N \\ t_{q^2,1},q^2\ne \pm 1& t_{q^2,1},q\hspace{0.17em}\text{a primitive fourth root of unity}\end{array}$$ $$\begin{array}{cc} C s1C s1s2C s1s2s1C & C s1C s1s2C s1s2s1C \\ t_{q^2,1},q^2\hspace{0.17em}\text{a primitive third root of unity}& t_{q^2,1},q\hspace{0.17em}\text{generic}\end{array}$$ $$\begin{array}{c} C s2C s2s1C s2s1s2C M N \\ t_{1,q^2},q\hspace{0.17em}\text{generic}\end{array}$$ $$\begin{array}{cc} M(t)tgen M(t)s2t M(t)s1s2t M(t)s2s1s2tgen & M(t)tgen M(t)s2t M(t)s1s2t M(t)s2s1s2tgen \\ t_{1,q^2},q\hspace{0.17em}\text{a primitive fourth root of unity}& t_{1,q^2},q\hspace{0.17em}\text{generic}\end{array}$$ $$\begin{array}{c} M N \\ t_{\pm q,1}\hspace{0.17em}{q}^2\ne 1,\\ \text{(excluding}\hspace{0.17em}{t}_{-q,1}\hspace{0.17em}\text{when}\hspace{0.17em}q\hspace{0.17em}\text{is a primitive sixth root of unity, and}\\ t_{q,1}\hspace{0.17em}\text{when}\hspace{0.17em}q\hspace{0.17em}\text{is a primitive third root of unity.)}\end{array}$$

From "Geometry of Type $C_2$"

$$\begin{array}{cc} B,xα1(c)w1B w1w2B w1w2w1B & B,xα1(c)w1B w2B,w1w2B \\ (s_{t_{1,q^2}},e_{{\alpha }_2})& (s_{t_{1,q^2}},e_{{\alpha }_1+{\alpha }_2})\end{array}$$ $$\begin{array}{c} B,xα2(c)w2B w2w1B w2w1w2C \\ (s_{t_{q^2,1}},e_{{\alpha }_1})\end{array}$$ $$\begin{array}{c} B,xα2(c)w2B w1B,xα2(c)w2w1B \\ (s_{t_{\pm q,1}},e_{2{\alpha }_1+{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B w2B w2w1B w2w1w2B & B w1B w1w2B w1w2w1B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_1})& (s_{t_{q^2,q^2}},e_{{\alpha }_2})\end{array}\begin{array}{c} B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_1+{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B w1B w1w2B w1w2w1B & B w1B w2B w1w2B \\ (s_{t_{-1,q^2}},e_{{\alpha }_2})& (s_{t_{-1,q^2}},e_{2{\alpha }_1+{\alpha }_2})\end{array}\begin{array}{c} B w1B \\ (s_{t_{-1,q^2}},e_{{\alpha }_2}+e_{2{\alpha }_1+{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B w2B w2w1B w2w1w2B & B w1B w1w2B w1w2w1B \\ (s_{t_{q^2,z}},e_{{\alpha }_1})& (s_{t_{z,q^2}},e_{{\alpha }_2})\end{array}$$ $$\begin{array}{cc} w1w2w1B w1w2B B w1B & B w1B \\ (s_{t_{q^2,q^2}},e_{-2{\alpha }_1-{\alpha }_2})& (s_{t_{q^2,q^2}},e_{{\alpha }_2}+e_{-2{\alpha }_1-{\alpha }_2})\end{array}\begin{array}{c} B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_1}+e_{-2{\alpha }_1-{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B,xα2(c)w2B w2w1B w2w1w2B & w1B,w1xα2(c)w2B B,w1xα2(c)w2w1B \\ (s_{t_{q^2,1}},e_{{\alpha }_1})& (s_{t_{q^2,1}},e_{-2{\alpha }_1-{\alpha }_2})\end{array}\begin{array}{c} B \\ (s_{t_{q^2,1}},e_{{\alpha }_1}+e_{-2{\alpha }_1-{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B,xα1(c)w1B w2B,w1w2B & B,xα1(c)w1B w2B,w1w2B \\ (s_{t_{1,q^2}},e_{{\alpha }_1+{\alpha }_2})& (s_{t_{1,q^2}},e_{-{\alpha }_1-{\alpha }_2})\\ B & B \\ (s_{t_{1,q^2}},e_{{\alpha }_2}+e_{-{\alpha }_1-{\alpha }_2})& (s_{t_{1,q^2}},e_{{\alpha }_1+{\alpha }_2}+e_{-{\alpha }_2})\end{array}$$ $$\begin{array}{c} B,xα2(c)w2B w2w1B,w2w1xα2(c)w2B \\ (s_{t_{q^2,1}},e_{{\alpha }_1})\end{array}$$ $$\begin{array}{cc} B,xα2(c)w2B w1B,w1xα2(c)w2B & B,xα2(c)w2B w1B,xα2(c)w2w1B \\ (s_{t_{\pm q,1}},e_{-2{\alpha }_1-{\alpha }_2})& (s_{t_{\pm q,1}},e_{2{\alpha }_1+{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B w2B w2w1B w2w1w2B & B w2B w2w1B w2w1w2B \\ (s_{t_{q^2,z}},e_{{\alpha }_1})& (s_{t_{q^2,z}},e_{-{\alpha }_1})\end{array}$$ $$\begin{array}{cc} B w1B w1w2B w1w2w1B & B w1B w1w2B w1w2w1B \\ (s_{t_{z,q^2}},e_{{\alpha }_2})& (s_{t_{z,q^2}},e_{-{\alpha }_2})\end{array}$$

From "Geometry of Type $G_2$"

$$\begin{array}{cc} B,xα1(c)w1B w1w2B w1w2w1B w1w2w1w2B w1w2w1w2w1B & B,xα1(c)w1B w2B,w1w2B w1w2w1B w1w2w1w2B \\ (s_{t_{1,q^2}},e_{{\alpha }_2})& (s_{t_{1,q^2}},e_{{\alpha }_1+{\alpha }_2})\\ \multicolumn{2}{c}{ B,xα1(c)w1B xα1(ζi)w1w2B }\\ \multicolumn{2}{c}{(s_{t_{1,q^2}},e_{{\alpha }_2}+e_{3{\alpha }_1+{\alpha }_2})}\end{array}$$ $$\begin{array}{cc} B,xα2(c)w2B w2w1B w2w1w2B w2w1w2w1B w2w1w2w1w2B & B,xα1(c)w1B w2B,xα1(c)w1w2B w2w1B,xα1(c)w1w2w1B \\ (s_{t_{q^2,1}},e_{{\alpha }_1})& (s_{t_{1,\pm q}},e_{3{\alpha }_1+{\alpha }_2})\\ \multicolumn{2}{c}{ B,xα2(c)w2B w1B,xα2(c)w2w1B w1w2B,xα2(c)w2w1w2B }\\ \multicolumn{2}{c}{(s_{t_{1,\pm q}},e_{{\alpha }_2}+e_{3{\alpha }_1+{\alpha }_2})}\end{array}$$ $$\begin{array}{c} B,xα2(c)w2B w1B,xα2(c)w2w1B w2w1w2B w2w1w2w1B \\ (s_{t_{q^{2/3},1}},e_{{\alpha }_2})\end{array}$$ $$\begin{array}{cc} B w2B w2w1B w2w1w2B w2w1w2w1B w2w1w2w1w2B & w1w2w1B w1w2B w1B B w2B w2w1B \\ (s_{t_{q^2,-q^{-2}}},e_{{\alpha }_1})& (s_{t_{q^2,-q^{-2}}},e_{3{\alpha }_1+2{\alpha }_2})\\ \multicolumn{2}{c}{ B w2B w2w1B }\\ \multicolumn{2}{c}{(s_{t_{q^2,-q^{-2}}},e_{{\alpha }_1}+e_{3{\alpha }_1+2{\alpha }_2})}\end{array}$$ $$\begin{array}{cc} w1w2w1w2w1B w1w2w1w2B w1w2w1B w1w2B w1B B & w1B B w2B w2w1B w2w1w2B w2w1w2w1B \\ (s_{t_{1^{1/3},q^2}},e_{{\alpha }_2})& (s_{t_{1^{1/3},q^2}},e_{3{\alpha }_1+{\alpha }_2})\\ \multicolumn{2}{c}{ w1B B }\\ \multicolumn{2}{c}{(s_{t_{1^{1/3},q^2}},e_{{\alpha }_2}+e_{3{\alpha }_1+{\alpha }_2})}\end{array}$$ $$\begin{array}{cc} w1w2w1w2w1B w1w2w1w2B w1w2w1B w1w2B w1B B & B w2B w2w1B w2w1w2B w2w1w2w1B w2w1w2w1w2B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_2})& (s_{t_{q^2,q^2}},e_{{\alpha }_1})\\ \multicolumn{2}{c}{ B }\\ \multicolumn{2}{c}{(s_{t_{q^2,q^2}},e_{{\alpha }_1}+e_{{\alpha }_2})}\end{array}$$ $$\begin{array}{cc} w1w2w1w2w1B w1w2w1w2B w1w2w1B w1w2B w1B B & B w2B w2w1B w2w1w2B w2w1w2w1B w2w1w2w1w2B \\ (s_{t_{z,q^2}},e_{{\alpha }_2})& (s_{t_{z,q^2}},e_{{\alpha }_1})\end{array}$$ $$\begin{array}{cc} w1w2w1B w1w2w1w2B w1w2w1w2w1B B w1B w1w2B & B w1B w1w2B \\ (s_{t_{q^2,q^2}},e_{-3{\alpha }_1-2{\alpha }_2}),q^{12}=1& (s_{t_{q^2,q^2}},e_{{\alpha }_1}+e_{-3{\alpha }_1-2{\alpha }_2}),q^{12}=1\\ B w2B w2w1B w2w1w2B w2w1w2w1B w2w1w2w1w2B & B w1B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_1}),q^{12}=1& (s_{t_{q^2,q^2}},e_{{\alpha }_2}+e_{-3{\alpha }_1-2{\alpha }_2}),q^{12}=1\\ w1w2w1w2w1B w1w2w1w2B w1w2w1B w1w2B w1B B & B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_2}),q^{12}=1& (s_{t_{q^2,q^2}},e_{{\alpha }_1}+e_{{\alpha }_2}),q^{12}=1\end{array}$$ $$\begin{array}{cc} w1w2w1xα2(c)w2B w1w2w1B, w1w2w1xα2(c)w2w1B w1w2B, w1B B & B,xα2(c)w2B w1B,xα2(c)w2w1B w2w1w2B w2w1w2w1B \\ (s_{t_{q^{-4},1}},e_{-3{\alpha }_1-{\alpha }_2}),q^{10}=1& (s_{t_{q^{-4},1}},e_{2{\alpha }_1+{\alpha }_2}),q^{10}=1\\ \multicolumn{2}{c}{ B }\\ \multicolumn{2}{c}{(s_{t_{q^{-4},1}},e_{2{\alpha }_1+{\alpha }_2}+e_{-3{\alpha }_1-{\alpha }_2}),q^{10}=1}\end{array}$$ $$\begin{array}{cc} B w2B w2w1B w2w1w2B w2w1w2w1B, w2w1w2w1xα2(c)w2B & w1B,w1xα2(c)w2B B,w1w2w1B w1w2w1w2B w1w2w1w2w1B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_1}),q^8=1& (s_{t_{q^2,q^2}},e_{{\alpha }_2}),q^8=1\\ B & B w2B w2w1B \\ (s_{t_{q^2,q^2}},e_{{\alpha }_1}+e_{{\alpha }_2}),q^8=1& (s_{t_{q^2,q^2}},e_{{\alpha }_1}+e_{3{\alpha }_1+2{\alpha }_2}),q^8=1\end{array}$$ $$\begin{array}{c} \\ e_{-3{\alpha }_1-2{\alpha }_2},q^6=1\end{array}$$ $$\begin{array}{ccc} triv sign & triv sign & triv sign \\ e_{{\alpha }_2},q^6=1& e_{{\alpha }_1+{\alpha }_2},q^6=1& e_{{\alpha }_2}+e_{3{\alpha }_1+{\alpha }_2},q^6=1\\ \\ sign & sign & sign \\ e_{{\alpha }_2}+e_{-3{\alpha }_1-2{\alpha }_2},q^6=1& e_{{\alpha }_1+{\alpha }_2}+e_{-3{\alpha }_1-2{\alpha }_2},q^6=1& e_{-3{\alpha }_1-2{\alpha }_2},q^6=1\end{array}$$ $$\begin{array}{cc} B,xα1(c)w1B xα2(ζi)w1w2B & B \\ (s_{t_{1,q^2}},e_{{\alpha }_2}+e_{3{\alpha }_1+{\alpha }_2}),q^2=-1& (s_{t_{1,q^2}},e_{{\alpha }_2}+e_{-{\alpha }_1-{\alpha }_2}),q^2=-1\\ \multicolumn{2}{c}{ w1w2B B,xα2(c)w2B }\\ \multicolumn{2}{c}{(s_{t_{1,q^2}},e_{{\alpha }_2}+e_{-2{\alpha }_1-{\alpha }_2}),q^2=-1}\end{array}$$ $$\begin{array}{cc} B,xα1(c)w1B w2B,xα1(c)w1w2B w2w1B,xα1(c)w1w2w1B & w2w1B, xα1(c)w1w2w1B w2B, xα1(c)w1w2B B,xα1(c)w1B \\ (s_{t_{1,q}},e_{3{\alpha }_1+2{\alpha }_2}),q^2=-1& (s_{t_{1,q}},e_{-3{\alpha }_1-2{\alpha }_2}),q^2=-1\end{array}$$ $$\begin{array}{cc} B,xα2(c)w2B w1B, xα2(c)w2w1B w1w2B, xα2(c)w2w1w2B & w1w2B, xα2(c)w2w1w2B w1B, xα2(c)w2w1B B,xα2(c)w2B \\ (s_{t_{q,1}},e_{2{\alpha }_1+{\alpha }_2}),q^2=-1& (s_{t_{q,1}},e_{-2{\alpha }_1-{\alpha }_2}),q^2=-1\end{array}$$ $$\begin{array}{cc} B,xα2(c)w2B w1B, xα2(c)w2w1B w2w1w2B w2w1w2w1B & w2w1w2w1B w2w1w2B w1B, xα2(c)w2w1B B,xα2(c)w2B \\ (s_{t_{-1^{1/3},1}},e_{3{\alpha }_1+{\alpha }_2}),q^2=-1& (s_{t_{-1^{1/3},1}},e_{-3{\alpha }_1-{\alpha }_2}),q^2=-1\end{array}$$ $$\begin{array}{cc} B w2B w2w1B w2w1w2B w1w2w2w1B w2w1w2w1w2B & B w2B w2w1B w2w1w2B w2w1w2w1B w2w1w2w1w2B \\ (s_{t_{q^2,z}},e_{-{\alpha }_1}),q^2=-1& (s_{t_{q^2,z}},e_{{\alpha }_1}),q^2=-1\end{array}$$ $$\begin{array}{cc} w1w2w1w2w1B w1w2w1w2B w1w2w1B w1w2B w1B B & w1w2w1w2w1B w1w2w1w2B w1w2w1B w1w2B w1B B \\ (s_{t_{z,q^2}},e_{{\alpha }_2}),q^2=-1& (s_{t_{z,q^2}},e_{-{\alpha }_2}),q^2=-1\end{array}$$

From "Classification for $A_1$"

$$\begin{array}{c} to ta tb Hα1-δ Hα1 Hα1+δ \\ \text{Figure 2.1.}\hspace{0.17em}\text{Real parts of central characters in Table 2.1}\end{array}$$ $$\begin{array}{c} s1ta to \\ \text{Figure 2.2.}\hspace{0.17em}\text{Real parts of weights of tempered representations}\end{array}$$ $$\begin{array}{ccc} s1ta ta & s1tb tb & to \\ \multicolumn{3}{c}{\text{Figure 2.3.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 2.1}}\end{array}$$

From "Classification for $A_2$"

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 a b c d e f g o \\ \text{Figure 4.1.}\hspace{0.17em}\text{Real parts of central characters in Table 4.1}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 to s1t t s2t s2s1s2ta \\ \text{Figure 4.2.}\hspace{0.17em}\text{Real parts of weights of tempered representations}\end{array}$$ $$\begin{array}{c}\begin{array}{ccc} s1ta ta s2ta s1s2ta s2s1s2ta s2s1ta & s1tb tb s2tb s1s2tb s2s1s2tb s2s1tb & s1tc tc s2tc s1s2tc s2s1s2tc s2s1tc \end{array}\\ \begin{array}{ccc} s1td td s2td s1s2td s2s1s2td s2s1td & s1te te s2te s1s2te s2s1s2te s2s1te & s1tf tf s2tf s1s2tf s2s1s2tf s2s1tf \end{array}\\ \begin{array}{cc} s1tg tg s2tg s1s2tg s2s1s2tg s2s1tg & s1to to s2to s1s2to s2s1s2to s2s1to \end{array}\\ \text{Figure 4.3.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 4.1}\end{array}$$

From "Classification for $C_2$"

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 Hα1+2α2 a b,c d e f g \\ \\ \text{Figure 5.1.}\hspace{0.17em}\text{Real parts of central characters in Table 5.1}\end{array}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+α2 Hα1+2α2 to s1e s1b s2s1e s1s2s1s2a s1s2s1b,s1s2s1s2c \\ \\ \text{Figure 5.2.}\hspace{0.17em}\text{Real parts of weights of tempered representations}\end{array}$$ $$\begin{array}{ccc} ta s2ta s1s2ta s2s1s2ta s1s2s1s2ta s1s2s1ta s2s1ta s1ta & tb s2tb s1s2tb s2s1s2tb s1s2s1s2tb s1s2s1tb s2s1tb s1tb & tc s2tc s1s2tc s2s1s2tc s1s2s1s2tc s1s2s1tc s2s1tc s1tc \\ td s2td s1s2td s2s1s2td s1s2s1s2td s1s2s1td s2s1td s1td & te s2te s1s2te s2s1s2te s1s2s1s2te s1s2s1te s2s1te s1te & tf s2tf s1s2tf s2s1s2tf s1s2s1s2tf s1s2s1tf s2s1tf s1tf \\ & tg s2tg s1s2tg s2s1s2tg s1s2s1s2tg s1s2s1tg s2s1tg s1tg & \\ \\ \multicolumn{3}{c}{\text{Figure 5.3.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 5.1}}\end{array}$$

From "Classification for $G_2$"

$$\begin{array}{c} Hα1 Hα2 Hα1+3α2 H2α1+3α2 Hα1+α2 Hα1+2α2 a b c,d,e f g h i j \\ \text{Figure 6.1.}\hspace{0.17em}\text{Real parts of central characters in Table 6.1}\end{array}$$ $$\begin{array}{cc} ta s2ta s1s2ta s2s1s2ta s1s2s1s2ta s2s1s2s1s2ta s1s2s1s2s1s2ta s1s2s1s2s1ta s2s1s2s1ta s1s2s1ta s2s1ta s1ta & tb s2tb s1s2tb s2s1s2tb s1s2s1s2tb s2s1s2s1s2tb s1s2s1s2s1s2tb s1s2s1s2s1tb s2s1s2s1tb s1s2s1tb s2s1tb s1tb \\ tc s2tc s1s2tc s2s1s2tc s1s2s1s2tc s2s1s2s1s2tc s1s2s1s2s1s2tc s1s2s1s2s1tc s2s1s2s1tc s1s2s1tc s2s1tc s1tc & td s2td s1s2td s2s1s2td s1s2s1s2td s2s1s2s1s2td s1s2s1s2s1s2td s1s2s1s2s1td s2s1s2s1td s1s2s1td s2s1td s1td \\ te s2te s1s2te s2s1s2te s1s2s1s2te s2s1s2s1s2te s1s2s1s2s1s2te s1s2s1s2s1te s2s1s2s1te s1s2s1te s2s1te s1te & tf s2tf s1s2tf s2s1s2tf s1s2s1s2tf s2s1s2s1s2tf s1s2s1s2s1s2tf s1s2s1s2s1tf s2s1s2s1tf s1s2s1tf s2s1tf s1tf \\ tg s2tg s1s2tg s2s1s2tg s1s2s1s2tg s2s1s2s1s2tg s1s2s1s2s1s2tg s1s2s1s2s1tg s2s1s2s1tg s1s2s1tg s2s1tg s1tg & th s2th s1s2th s2s1s2th s1s2s1s2th s2s1s2s1s2th s1s2s1s2s1s2th s1s2s1s2s1th s2s1s2s1th s1s2s1th s2s1th s1th \\ ti s2ti s1s2ti s2s1s2ti s1s2s1s2ti s2s1s2s1s2ti s1s2s1s2s1s2ti s1s2s1s2s1ti s2s1s2s1ti s1s2s1ti s2s1ti s1ti & tj s2tj s1s2tj s2s1s2tj s1s2s1s2tj s2s1s2s1s2tj s1s2s1s2s1s2tj s1s2s1s2s1tj s2s1s2s1tj s1s2s1tj s2s1tj s1tj \\ \multicolumn{2}{c}{\text{Figure 6.2.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 6.1}}\end{array}$$

From "The Partition monoid"

$$\begin{array}{cc} 1 2 3 4 5 6 7 8 1′ 2′ 3′ 4′ 5′ 6′ 7′ 8′ & \text{represents}\hspace{0.17em}\{\{1,2,4,2\prime ,5\prime \},\left\{3\right\},\{5,6,7,3\prime ,4\prime ,6\prime ,7\prime \},\{8,8\prime \},\left\{1\prime \right\}\},\end{array}$$ $$\begin{array}{c}\begin{array}{ccccc}\text{if}\hspace{0.17em}{d}_1=& & \text{and}\hspace{0.17em}{d}_2=& & \text{then}\end{array}\\ \begin{array}{ccccc}{d}_1\circ d_2=& & =& & \text{.}\end{array}\end{array}$$ $$\begin{array}{cccc} & \in I_7,& & \in I_{6+\frac{1}{2}},\\ & \in P_7,& & \in P_{6+\frac{1}{2}},\\ & \in B_7,& & \in T_7,\\ \multicolumn{4}{c}{\begin{array}{cc} & \in S_7\text{.}\end{array}}\end{array}$$ $$\begin{array}{cccccc} & \leftrightarrow & & \leftrightarrow & & ,\end{array}$$ $$\begin{array}{cccccc} & \leftrightarrow & & \leftrightarrow & & \text{.}\end{array}$$ $$\begin{array}{cc}\begin{array}{cccc} & =& & \text{.}\end{array}& \text{(1.6)}\end{array}$$ $$\begin{array}{cc}\begin{array}{ccccc}{p}_{i+\frac{1}{2}}=& i i+1 ⋯ ⋯ & ,p_j=& j ⋯ ⋯ & ,\\ e_i=& i i+1 ⋯ ⋯ & ,s_i=& i i+1 ⋯ ⋯ & \text{.}\end{array}& \text{(1.10)}\end{array}$$ $$\begin{array}{cccccccc}t=& & =& & =& & =& \end{array}$$ $$\begin{array}{cccccc}t=& & =& & =& \begin{array}{c}\left(p_{2\frac{1}{2}}{p}_{3\frac{1}{2}}{p}_{6\frac{1}{2}}\right)\left(p_3{p}_4{p}_6{p}_7\right)\tau \\ ·p_{2\frac{1}{2}}{p}_2{p}_{3\frac{1}{2}}{p}_3{p}_{5\frac{1}{2}}{p}_5{p}_{4\frac{1}{2}}{p}_4\\ ·\left(p_2{p}_3{p}_4{p}_7\right)\left(p_{1\frac{1}{2}}{p}_{2\frac{1}{2}}\right)\text{.}\end{array}\end{array}$$ $$\begin{array}{ccc}t& =& AAAA⏞T1 AA⏞T2 AAAAAA⏟B1 AAAA⏟B2 \end{array}$$ $$\begin{array}{ccccccc}{\sigma }_1=& & \text{so that}{\sigma }_{1}t=& & =& AA⏞T2′ AAAA⏞T1′ AAAAAA⏟B1 AAAA⏟B2 & =t\prime ,\end{array}$$ $$\begin{array}{cccccc}{\sigma }_1=& & \text{so that}{\sigma }_{1}t={\sigma }_2=t\prime {\sigma }_2=& & =& AA⏞T2′ AAAA⏞T1′ AAAA⏟B2′ AAAAAA⏟B1′ \end{array}$$ $$\begin{array}{ccccccc} t π b T B1 B2 & =& γ1-1 t′ γ2-1 π σ1-1 b′ σ2-1 T B1 B2 & =& γ1-1 t′ b′ σ2-1 γ2-1πσ1-1 T B1 B2 & =& γ1-1 t′ b′′ σ2-1 γ2-1πσ1-1 T B1 B2 \end{array}$$

From "Partition algebras"

$$\begin{array}{cc}\begin{array}{c}\begin{array}{ccccc}\text{if}{d}_1=& & \text{and}{d}_2=& & \text{then}\end{array}\\ \begin{array}{ccccc}{d}_1{d}_2=& & =n^2& & ,\end{array}\end{array}& \text{(2.1)}\end{array}$$ $$\begin{array}{cc}\begin{array}{ccc}ℂA_{k-\frac{1}{2}}& \hookrightarrow & ℂA_k\\ 1 k d & ⟼& 1 k d \end{array}\text{and}\begin{array}{ccc}ℂA_{k-1}& \hookrightarrow & ℂA_{k-\frac{1}{2}}\\ 1 k-1 d & ⟼& 1 k d \end{array}& \text{(2.2)}\end{array}$$ $$\begin{array}{cccc}{\epsilon }_{\frac{1}{2}}:& ℂA_k& ⟶& ℂA_{k-\frac{1}{2}}\\ & 1 k d & ⟼& 1 k d \end{array}\text{and}\begin{array}{cccc}{\epsilon }^{\frac{1}{2}}:& ℂA_{k-\frac{1}{2}}& ⟶& ℂA_{k-1}\\ & 1 k d & ⟼& 1 k-1 d \end{array}$$ $$\begin{array}{cc}\begin{array}{ccccc}{\epsilon }_{\frac{1}{2}}& (∑A ∑A) & =& & ,\end{array}& \begin{array}{ccccc}{\epsilon }^{\frac{1}{2}}& (∑A ∑A) & =& & ,\end{array}\end{array}$$ $$\begin{array}{cc}\begin{array}{ccccc}{\epsilon }_{\frac{1}{2}}& (∑A ∑A) & =& & ,\end{array}& \begin{array}{ccccc}{\epsilon }^{\frac{1}{2}}& (∑A ∑A) & =n& & \text{.}\end{array}\end{array}$$ $$\begin{array}{cc}\begin{array}{cccc}{\epsilon }_1:& ℂA_k& ⟶& ℂA_{k-1}\\ & 1 k d & ⟼& 1 k-1 d & \text{.}\end{array}& \text{(2.4)}\end{array}$$ $$\begin{array}{cc}\begin{array}{ccc}{\text{tr}}_k\left(d\right)=& d & ,\text{for}\hspace{0.17em}d\in A_k\text{.}\end{array}& \text{(2.9)}\end{array}$$ $$∅ k=0: k=1: k=2: k=3: k=4:$$ $$∅ ∅ ∅ ∅ ∅ ∅ ∅ k=0: k=0+12: k=1: k=1+12: k=2: k=2+12: k=3:$$ $$\begin{array}{c}\tau =(2,7,8,12,9,16,14,4,15,10,18,6)(3,11)(5,17),\text{since}\\ \begin{array}{ccccc}\tau ·& 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 & =& 1 7 11 15 17 2 8 12 16 18 3 9 13 4 10 14 5 6 & \text{.}\end{array}\end{array}$$ $$\begin{array}{ccc}{A}_{\mid \lambda \mid ,\mathbb{Z}}& ⟶& A_{k,\mathbb{Z}}\\ b & ⟼& b ⋯ ⋯ ...........................⏟k-|λ| & ,\text{if}\hspace{0.17em}k\hspace{0.17em}\text{is an integer, and}\\ \\ A_{\mid \lambda \mid +\frac{1}{2},\mathbb{Z}}& ⟶& A_{k,\mathbb{Z}}\\ b & ⟼& b ⋯ ⋯ ...........................⏟k-|λ|-12 & ,\text{if}\hspace{0.17em}k-\frac{1}{2}\hspace{0.17em}\text{is an integer.}\end{array}$$

From "Schur–Weyl duality for partition algebras"

$$\begin{array}{c} i1 i2 i3 i4 i5 i6 i7 i8 i1′ i2′ i3′ i4′ i5′ i6′ i7′ i8′ \end{array}={\delta }_{i_1{i}_2}{\delta }_{i_1{i}_4}{\delta }_{i_1{i}_{2\prime }}{\delta }_{i_1{i}_{5\prime }}{\delta }_{i_5{i}_6}{\delta }_{i_5{i}_7}{\delta }_{i_5{i}_{3\prime }}{\delta }_{i_5{i}_{4\prime }}{\delta }_{i_5{i}_{6\prime }}{\delta }_{i_5{i}_{7\prime }}{\delta }_{i_8{i}_{8\prime }}\text{.}$$ $$k=0: k=0+12: k=1: k=1+12: k=2: k=2+12: k=3:$$ $$14 12 11 8 7 6 5 4 3 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24$$ $$\begin{array}{cc}{\text{tr}}^{\varnothing }\left(n\right)=1,& {\text{tr}}_{\frac{1}{2}}^{\varnothing }\left(n\right)=n,\\ {\text{tr}}^{ }\left(n\right)=n-1,& {\text{tr}}_{\frac{1}{2}}^{ }\left(n\right)=n(n-2),\\ {\text{tr}}^{ }\left(n\right)=\frac{1}{2}n(n-3),& {\text{tr}}_{\frac{1}{2}}^{ }\left(n\right)=\frac{1}{2}n(n-1)(n-4),\\ {\text{tr}}^{ }\left(n\right)=\frac{1}{2}(n-1)(n-2),& {\text{tr}}_{\frac{1}{2}}^{ }\left(n\right)=\frac{1}{2}n(n-2)(n-3),\\ {\text{tr}}^{ }\left(n\right)=\frac{1}{6}n(n-1)(n-5),& {\text{tr}}_{\frac{1}{2}}^{ }\left(n\right)=\frac{1}{6}n(n-1)(n-2)(n-6),\\ {\text{tr}}^{ }\left(n\right)=\frac{1}{6}n(n-2)(n-4),& {\text{tr}}_{\frac{1}{2}}^{ }\left(n\right)=\frac{1}{6}n(n-1)(n-3)(n-5),\\ {\text{tr}}^{ }\left(n\right)=\frac{1}{6}(n-1)(n-2)(n-3),& {\text{tr}}_{\frac{1}{2}}^{ }\left(n\right)=\frac{1}{6}n(n-2)(n-3)(n-4),\end{array}$$ $$\begin{array}{ccccc}{b}_S=& & \text{and}{d}_{I\subseteq S}=& & \text{.}\end{array}$$ $$\begin{array}{c}{Z}_0=1,Z_{\frac{1}{2}}=1,Z_1=\underset{\underset{p_{\left\{1\right\}}}{⏟}}{\begin{array}{c} \end{array}},\\ Z_{1\frac{1}{2}}=\begin{array}{c} \end{array}+\underset{\underset{Z_1}{⏟}}{\begin{array}{c} \end{array}}-\underset{\underset{{\stackrel{\sim }{p}}_{\{1,2\}}}{⏟}}{\begin{array}{c} \end{array}-\begin{array}{c} \end{array}}+n\underset{\underset{b_{\{1,2\}}}{⏟}}{\begin{array}{c} \end{array}},\text{and}\\ Z_2=\begin{array}{c} \end{array}+\underset{\underset{p_{\left\{1\right\}}}{⏟}}{\begin{array}{c} \end{array}}+\underset{\underset{p_{\left\{2\right\}}}{⏟}}{\begin{array}{c} \end{array}}-\underset{\underset{p_{\{1,2\}}}{⏟}}{\begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}}+n\underset{\underset{b_{\{1,2\}}}{⏟}}{\begin{array}{c} \end{array}},\end{array}$$ $$\begin{array}{ccc}{M}_0& =& 1,M_{\frac{1}{2}}=1,M_1=\begin{array}{c} \end{array}-\begin{array}{c} \end{array},M_{1\frac{1}{2}}=\begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}+n\begin{array}{c} \end{array},\\ M_2& =& \begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array},\text{and}\\ M_{2\frac{1}{2}}& =& 2\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+(n-1)\begin{array}{c} \end{array}+(n-1)\begin{array}{c} \end{array}\\ & & +\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}-\begin{array}{c} \end{array}\\ & & +\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}-n\begin{array}{c} \end{array}\end{array}$$

From "The basic construction"

$$A^: B^:$$ $$A^: B^: C^:$$ $$\sum _{ P ↓ μ ↓ ↓ λ γ }$$

From "Classification of Irreducible Representations for Rank 2"

$$\begin{array}{c} Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 β0 β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 β13 \\ Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 Hβ7 Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 Hβ7 β0 β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 β13 β14 β15 \\ \text{Figure 1. Hyperplanes and roots for}\hspace{0.17em}{I}_2\left(7\right)\hspace{0.17em}\text{and}\hspace{0.17em}{I}_2\left(8\right)\end{array}$$ $$\begin{array}{c} \\ \\ \text{Figure 2. Hyperplanes for}\hspace{0.17em}{I}_2\left(7\right)\hspace{0.17em}\text{and}\hspace{0.17em}{I}_2\left(n\right)\text{.}\end{array}$$ $$Hβ0 Hβk Hβn-k k chambers k chambers k chambers k chambers n-2k chambers n-2k chambers J=∅ J={βn-k} J={βk,βn-k}$$ $$Hβ0 Hβn/2 n/2 chambers n/2 chambers n/2 chambers n/2 chambers J=∅ J={βn/2}$$ $$Hβ0 Hβk Hβn-k Hβℓ Hβn-ℓ k chambers ℓ-k chambers n-2ℓ chambers ℓ-k chambers k chambers J=∅ J={βn-k} J={βn-k,βn-ℓ} J={βℓ,βn-ℓ,βn-k} J=P(γq)$$ $$Hβ0 Hβk-1 Hβn-k k chambers k-1 chambers n-2k+1 chambers n-2k+1 chambers k chambers k-1 chambers J=∅ J={βn-k} J={βk-1} J=P(γc,k)$$ $$Hβ0 n chambers n chambers J=∅ J={β0}$$

From "Hyperplane systems and Galleries"

$$\begin{array}{c} S T K d σ γ γ1 γ′ γ1′ d′ aS aT \\ \text{Figure 1.}\end{array}$$ $$\begin{array}{c} S0 S1 S2 aS0=γ(t0) aS1 aS2=γ(t3) γ(t1) γ(t2) γ1 γ2 γ1′ γ2′ \\ \text{Figure 2.}\end{array}$$ $$\begin{array}{c} d S T U aS aT aU γ γ′ γ′′ \\ \text{Figure 3.}\end{array}$$ $$\begin{array}{c} A=S0 S1 S2 S3 S4=B A=T0 T1 T2 T3 T4=B \\ \text{Figure 4.}\end{array}$$ $$\begin{array}{c} aA aB γ γ γ′ γj+1 (i,j) Ii,j \\ Figure\; 5.\end{array}$$ $$\begin{array}{c} G G G U F u f \\ \text{Figure 6.}\end{array}$$ $$\begin{array}{c} C0 C1 C2 C3 \\ \text{Figure. 7}\end{array}$$ $$\begin{array}{c} A -A \\ \text{Figure 8.}\end{array}$$ $$\begin{array}{c} A B M M′ a b x a′ \\ \text{Figure 9.}\end{array}$$ $$\begin{array}{c} 1 1 2 3 3 4 4 P C0=D0 C4=D4 U=C1 C2 C3 D1 D2 D3 \\ \text{Figure 10.}\end{array}$$ $$\begin{array}{cccc} 1 2 3 4 & 1 2 3 4 & 1 2 3 4 & 1 2 3 4 \\ \underset{\_}{1}\hspace{0.17em}2\hspace{0.17em}3\hspace{0.17em}4& \underset{\_}{1}\hspace{0.17em}2\hspace{0.17em}3\hspace{0.17em}4\hspace{0.17em}1\hspace{0.17em}\underset{\_}{1}& \underset{\_}{1}\hspace{0.17em}1\hspace{0.17em}4\hspace{0.17em}3\hspace{0.17em}2\hspace{0.17em}\underset{\_}{1}& 4\hspace{0.17em}3\hspace{0.17em}2\hspace{0.17em}\underset{\_}{1}\\ \multicolumn{4}{c}{Figure\; 11.}\end{array}$$ $$\begin{array}{cc} 1 2 3 & 1 2 3 \\ 1\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}\underset{\_}{1}& \underset{\_}{3}\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}3\\ \multicolumn{2}{c}{Figure\; 12.}\end{array}$$

From "Artin groups"

$$\begin{array}{c} 0 i-1 i ℓ \\ \text{Figure 13.}\end{array}$$ $$\begin{array}{cc} 0 i-1 i i+1 ℓ si si+1 si & 0 i-1 i i+1 ℓ si+1 si si+1 \\ \multicolumn{2}{c}{\text{Figure 14.}}\end{array}$$ $$\begin{array}{c} C C′ C′′ w(C) \\ \text{Figure 15.}\end{array}$$

From "Extended Artin groups"

$$\begin{array}{c} 12 0 1 12 \\ \text{Figure 16.}\end{array}$$ $$\begin{array}{cc} 12U(αi∨,n) αi & Mαi∨ C c \\ \text{real part of}\hspace{0.17em}{s}_i^{\left(n\right)}& \text{imaginary part of}\hspace{0.17em}{s}_i^{\left(n\right)}\\ \multicolumn{2}{c}{\text{Figure 17.}}\end{array}$$ $$\begin{array}{c} q+U(αi∨,n)=U(αi∨,n+αi∨(q)) } } } n αi∨(q) n+αi∨(q) 0 q \\ \text{Figure 18.}\end{array}$$ $$\begin{array}{c} 0 -1 n2+1 n2 n1 n1+1 \\ \text{Figure 19.}\end{array}$$ $$\begin{array}{c} 0 0 0 -1 -1 0 -1 -1 n4+1 n4 n1 n1+1 \\ \text{Figure 20.}\end{array}$$ $$\begin{array}{c} 0 0 0 0 0 0 0 -1 0 0 -1 0 0 -1 0 0 0 -1 0 -1 0 -1 -1 0 -1 0 0 -1 -1 0 -1 0 -1 0 -1 -1 -1 -1 0 -1 -1 -1 -1 0 -1 -1 -1 -1 n6+1 n6 n1 n1+1 \\ \text{Figure 21.}\end{array}$$ $$\begin{array}{cc} α0 α1 α2 σ0(c) c & c C σα0(c) \\ \text{Real part of}\hspace{0.17em}{\lambda }_1\gamma {\lambda }_2^{-1}& \text{Imaginary part of}\hspace{0.17em}{\lambda }_1\gamma {\lambda }_2^{-1}\\ \multicolumn{2}{c}{\text{Figure 22.}}\end{array}$$

From "The K(Pi,1)-Problem"

$$\begin{array}{c} y f F B r(y) St(F) \\ \text{Figure 23.}\end{array}$$ $$\begin{array}{c} 1 1 2 2 3 3 P1 P2 P3 C D \\ \text{Figure 24.}\end{array}$$ $$\begin{array}{c} G G1 G2 St(F1) St(F2) \\ \text{Figure 25.}\end{array}$$ $$\begin{array}{c} λ1 λ2 λ3 γ γ1 γ2 -1 -1c -1d a+-1c a+-1d Re-1(A) ?(F)=Im-1(St(F)) \\ \text{Figure 26.}\end{array}$$ $$\begin{array}{ccccc} 1 1 2 2 & 1 1 2 2 & 1 1 2 2 & 1 1 2 2 & 1 1 2 2 \\ 1\hspace{0.17em}1\hspace{0.17em}2\hspace{0.17em}2& 1\hspace{0.17em}2\hspace{0.17em}1\hspace{0.17em}2& 1\hspace{0.17em}2\hspace{0.17em}2\hspace{0.17em}1& 2\hspace{0.17em}1\hspace{0.17em}2\hspace{0.17em}1& 2\hspace{0.17em}2\hspace{0.17em}1\hspace{0.17em}1\\ \multicolumn{5}{c}{\text{Figure 27.}}\end{array}$$ $$\begin{array}{c} (a,1) (a,2) (a,3) (b,1) (b,2) (b,3) \\ \text{Figure 28.}\end{array}$$ $$\begin{array}{c} λ1 λ2 λ3 λ4 γ -1c -1d a+-1d w(q+-1c) w(q)+-1d Re-1(A) Im-1(D) \\ \text{Figure 29.}\end{array}$$ $$\begin{array}{c} 0 1 q ℝ Z \\ H=𝕍=ℝ\hspace{0.17em}\text{and}\hspace{0.17em}ℳ=\{0,1\}\\ \text{Figure 30.}\end{array}$$ $$\begin{array}{c} B C D D′ \\ \text{Figure 31.}\end{array}$$ $$\begin{array}{c} 1 2 3 4 5 5 A B C \\ \text{Figure 32.}\end{array}$$ $$\begin{array}{c} x b E d(x) A‾ \\ \text{Figure 33.}\end{array}$$

From "Representation Theory Lecture 12"

$$\begin{array}{c} ε1∨-ε2∨ ε2∨-ε3∨ ε3∨-ε4∨ ε4∨-ε5∨ ε4∨+ε5∨ \\ ε1∨-ε2∨ ε2∨-ε3∨ ε3∨-ε4∨ ε4∨-ε5∨ ε4∨+ε5∨ \end{array}$$ $$\begin{array}{cc}{SO}_{10}& \\ ⊆ & ⊆ \\ {SU}_5& \\ ⊆ & ⊆ \\ {SU}_3\times U_1\times {SU}_2& \end{array}$$ $$ε1+ε2 f∼2 ε1+ε3 f∼1 f∼3 ε2+ε3 ε1+ε4 f∼3 f∼1 f∼5 f∼4 ε2+ε4 ε1-ε5 ε1+ε5 f∼1 f∼1 f∼4 f∼2 f∼5 f∼1 f∼4 f∼5 ε2+ε5 ε3+ε4 ε2-ε5 ε1-ε4 f∼2 f∼4 f∼5 f∼2 f∼4 f∼1 f∼3 ε3+ε5 ε3-ε5 ε2-ε4 ε1-ε3 f∼3 f∼3 f∼4 f∼2 f∼3 f∼1 f∼2 ε4+ε5 ε4-ε5 ε3-ε4 ε2-ε3 ε1-ε2 f∼5 f∼4 f∼3 f∼2 f∼1 12(-ε4-ε5) 12(-ε4+ε5) 12(-ε3+ε4) 12(-ε2+ε3) 12(-ε1+ε2) +12(ε4+ε5) +12(ε4-ε5) +12(ε3-ε4) +12(ε2-ε3) +12(ε1-ε2) f∼5 f∼4 f∼3 f∼2 f∼1 -ε4-ε5 -ε4+ε5 -ε3+ε4 -ε2+ε3 -ε1+ε2 f∼3 f∼3 f∼4 f∼2 f∼3 f∼1 f∼2 -ε3-ε5 -ε3+ε5 -ε2+ε4 -ε1+ε3 f∼2 f∼4 f∼5 f∼2 f∼4 f∼1 f∼3 -ε2-ε5 -ε3-ε4 -ε2+ε5 -ε1+ε4 f∼1 f∼1 f∼4 f∼2 f∼5 f∼1 f∼4 f∼5 -ε2-ε4 -ε1+ε5 -ε1-ε5 f∼3 f∼1 f∼5 f∼4 -ε2-ε3 -ε1-ε4 f∼1 f∼3 -ε1-ε3 f∼2 -ε1-ε2$$

$$ε1-ε3 f∼1 f∼2 ε4-ε5 ε2-ε3 ε1-ε2 f∼4 f∼2 f∼1 12(-ε4-ε5) 12(-ε3+ε4) 12(-ε2+ε3) 12(-ε1+ε2) +12(ε4-ε5) +12(ε3-ε4) +12(ε2-ε3) +12(ε1-ε2) f∼4 f∼2 f∼1 -ε4+ε5 -ε2+ε3 -ε1+ε2 f∼1 f∼2 -ε1+ε3 3 bosons 1 photon 8 gluons of QCD w+,z,w-$$

From "The Fano plane as a flag variety"

$$𝔽23 ( 10 01 00 ) ( 10 00 01 ) ( 10 01 01 ) ( 00 10 01 ) ( 10 10 01 ) ( 01 10 01 ) ( 11 10 01 ) (100) (010) (110) (001) (101) (011) (111) (000)$$ $$(001) (011) (111) (101) (010) (110) (100)$$

From "Incidences and projective geometries"

$$p1 p2 p3 p6 ℓ(p1,p2) ℓ(p1,p3) ℓ(p2,p3) ℓ$$

From "Notes on Schubert Polynomials"

$$1 2 3 4 5 6 1 2 3 4 5 6$$ $$\begin{array}{ccc} & & \\ 1223& 2231& 22301\\ \\ & & \\ 1232& 2321& 23201\\ \\ & & \\ 1322& 3221& 32201\\ \\ & & \\ 3122& 302201& 302201\end{array}$$ $$\begin{array}{ccccc} & & & & \end{array}$$ $$\begin{array}{cc} A B C D r j & A B C D r j \\ G\left(w\right)& G\left(u\right)\end{array}$$ $$\begin{array}{ccc} A′ B′ C′ D′ i r j & i r j & i r j \\ G\left(w\right)& G\left(u\right)& G\left(v\right)\end{array}$$ $$\begin{array}{cc} 0 M 0 0 P N 0 i r j & i r j \\ G\left(w\right)& G\left(v\right)\end{array}$$ $$1 2 3 4 5 6 7 8 9 r r+1$$ $$\begin{array}{ccc} & & \end{array}$$ $$\begin{array}{cc} 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 & 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 \\ D\left(w\right)& D\left(v\right)\end{array}$$ $$D=\begin{array}{c} \end{array}$$ $$\begin{array}{ccc} & & \\ D_0& D_1& D_2\end{array}$$

From "Isomorphisms and automorphisms"

$$α β α+β α+2β L$$ $$α α+β 2α+3β α+2β α+3β β$$ $$\begin{array}{ccc}{A}_n(n\ge 2):& & {\sigma }^2=1\\ D_n(n\ge 4):& & {\sigma }^2=1\\ D_4:& & {\sigma }^3=1\\ E_6:& & {\sigma }^2=1\end{array}$$ $$\begin{array}{ccc}{C}_2& & p=2\\ F_4& & p=2\\ G_2& & p=3\end{array}$$ $$$$

From "Some twisted Groups"

$$\begin{array}{ccccc} & =& & ⟶& \end{array}$$ $$\begin{array}{ccccc} & =& & ⟶& \end{array}$$ $\begin{array}{c} α β γ δ \end{array}$ $\begin{array}{c} α β γ \end{array},$

From $\text{"}𝔭\text{-adic}$ groups"

$$s1s0s1 s1s0 s1 1 s0 s0s1 s0s1s0 Hα1,-1 Hα0,-1 Hα1,0 Hα0 Hα1,1 Hα0,0$$ $$s1 1 Hα1 c1$$ $$1 s0 H-α1+δ c1$$ $$s1s0 s1 1 Hα1+δ Hα1 c2 c1$$ $$1 s0 s0s1 H-α1+δ H-α1+2δ c1 c2$$ $$s1 1 Hα1 c1$$ $$s1s0 s1 1 Hα1+δ Hα1 c2 c1$$ $$1 s0 Hα1-δ c1-1$$ $$1 s0 H-α1+δ 0$$ $$s1 1 s0 s0s1 Hα1 Hα1-δ c2′ c1-1$$ $$1 s0 s0s1 H-α1+δ H-α1+2δ 0 c2-1$$ $$1 s0 s0s1 H-α1+δ H-α1+2δ 0 0$$

From "Representations of affine Hecke algebras"

$$\[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1} \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ + \] \[ - \] \[ s_0{s}_1 \] \[ s_1{s}_0 \] \[ s_0 \] \[ s_0{s}_2 \] \[ s_1 \] \[ 1 \] \[ s_1{s}_2 \] \[ s_2 \] \[ 0 \]$$ $$\begin{array}{c} i - + \end{array}=T_i\text{and}\begin{array}{c} i - + \end{array}=T_i^{-1}$$ $$X^{\lambda }=\begin{array}{c} \[ 0 \] \[ \lambda \]\end{array}\hspace{0.17em}\text{a minimal length walk.}$$ $$γ$$ $$$$ $$μ ρ ν$$ $$\begin{array}{c} \end{array}$$ $$γ0 γ1 γ C$$

From "Diagram algebras as tantalizers"

$$∅ 1 ∅ 1 1 1 3 1 2 3 ∅ 3 6 6 1 3 2 3 1 k=0: k=1: k=2: k=3: k=4:$$ $$\begin{array}{c} \end{array}\in S_7\text{and}\text{Card}\left(S_5\right)=5·4·3·2·1\text{.}$$ $$\begin{array}{c} \end{array}\in B_7\text{and},\text{Card}\left(B_5\right)=9·7·5·3·1\text{.}$$ $$\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\text{.}$$ $$t_{s_i}=\hspace{0.17em}\begin{array}{c} i i+1 ⋯ ⋯ \end{array}\text{and}{t}_{e_i}=\hspace{0.17em}\begin{array}{c} i i+1 ⋯ ⋯ \end{array},$$ $$\begin{array}{c} \end{array}\in {ℬ}_5\text{and}\begin{array}{c} \end{array}\in {\stackrel{\sim }{ℬ}}_5$$ $$\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})\begin{array}{c} \end{array}$$ $$\begin{array}{c}\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})\\ \begin{array}{c} \end{array}=z\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=z^{-1}\begin{array}{c} \end{array}\\ r\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}=Q_r\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=z^{-1}·\begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}=\frac{z-z^{-1}}{q-q^{-1}}+1\text{.}\end{array}$$ $$\begin{array}{c} i i+1 ⋯ ⋯ V ⊗ ⋯ ⊗ V ⊗ V ⊗ V ⊗ V ⊗ ⋯ ⊗ V V ⊗ ⋯ ⊗ V ⊗ V ⊗ V ⊗ V ⊗ ⋯ ⊗ V \end{array}\text{and}\begin{array}{c} i i+1 ⋯ ⋯ V ⊗ ⋯ ⊗ V ⊗ V ⊗ V ⊗ V ⊗ ⋯ ⊗ V V ⊗ ⋯ ⊗ V ⊗ V ⊗ V ⊗ V ⊗ ⋯ ⊗ V \end{array}$$ $$\begin{array}{c} V vi V vi \end{array},\begin{array}{c} V ⊗ V vi ⊗ vj V ⊗ V vj ⊗ vi \end{array},\begin{array}{c} V ⊗ V vi ⊗ vj V ⊗ V δij∑ℓvℓ⊗vℓ \end{array}$$ $$x_1+\cdots +x_i=\begin{array}{c} i i+1 ↭c↭ ⋯ V ⊗ ⋯ ⊗ V ⊗ V ⊗ V ⊗ ⋯ ⊗ V V ⊗ ⋯ ⊗ V ⊗ V ⊗ V ⊗ ⋯ ⊗ V \end{array}$$ $${Ř}_{MN}:M\otimes N⟶N\otimes M\begin{array}{c} M ⊗ N N ⊗ M \end{array}$$ $$\begin{array}{c} (M⊗N)⊗P P⊗(M⊗N) \end{array}=\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\begin{array}{c} M⊗(N⊗P) (N⊗P)⊗M \end{array}=\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}$$ $$\begin{array}{c} M ⊗ V ⊗ V ⊗ V ⊗ V ⊗ V M ⊗ V ⊗ V ⊗ V ⊗ V ⊗ V \end{array}$$ $$M\left(\lambda \right)\text{is}\begin{array}{c} vλ \end{array}\text{with}\hspace{0.17em}{K}_i{v}_{\lambda }=q^{{\lambda }_i}{v}_{\lambda }\text{.}$$ $$\begin{array}{c} vλ vμ vν \end{array}$$

From "p-compact groups"

$$1 2 7 10 3 8 11 12 4 6 9$$

From "The center of the affine and degenerate affine BMW algebras"

$$\begin{array}{cc}{E}_i=\begin{array}{c} \end{array}\text{.}& \text{(4.15)}\end{array}$$ $$d=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\text{and}{T}_d=\begin{array}{c} \end{array}$$

From "Representation theory Lecture Notes: Chapter 2"

$$\begin{array}{c}{t}_i^k{t}_j^{-k}(i,j),k\in (\mathbb{Z}/r\mathbb{Z}),1\le i<j\le n,\text{and}\\ t_i^k=t_{(0,\dots ,0,k,0,\dots ,0)}=\begin{array}{c} ξk ith ⋯ ⋯ \end{array},k\in (\mathbb{Z}/r\mathbb{Z}),1\le i\le n\text{.}\end{array}$$ $$\begin{array}{c}{t}_1=\begin{array}{c} ξ ⋯ \end{array},s_1=\begin{array}{c} ξ ξ-1 ⋯ \end{array},\\ s_i=(i-1,i)=\begin{array}{c} ⋯ ⋯ \end{array},2\le i\le n\text{.}\end{array}$$ $$\begin{array}{c} id s1 s2 s1s2 s2s1 s1s2s1 s2s1s2 s1s2s1s2 \end{array}$$ $$\begin{array}{c}\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\end{array}$$ $$w=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=s_1{s}_3{s}_5{s}_4{s}_2{s}_5{s}_3\text{.}$$ $$\begin{array}{cc}\lambda =\begin{array}{c} \end{array}& \text{(3.2)}\end{array}$$ $$T=\begin{array}{c} 1 2 3 4 5 3 4 7 12 15 8 9 14 11 16 18 13 17 \end{array}$$ $$\begin{array}{c} 0 1 2 3 4 -1 0 1 2 3 -2 -1 0 -3 -2 -1 -4 -5 \\ \text{Contents of boxes}\end{array}$$ $$\begin{array}{ccc}\begin{array}{c} \\ \text{Case (A)}\end{array}& \begin{array}{c} \\ \text{Case (B)}\end{array}& \begin{array}{c} \\ \text{Case (C)}\end{array}\end{array}$$ $$\begin{array}{cccc}\begin{array}{c} \\ \text{Case (1)}\end{array}& \begin{array}{c} \\ \text{Case (2)}\end{array}& \begin{array}{c} \\ \text{Case (3)}\end{array}& \begin{array}{c} \\ \text{Case (4)}\end{array}\end{array}$$ $$\begin{array}{c} a b c \end{array}$$ $$\begin{array}{ccc}\begin{array}{c} a b c \end{array}& & \begin{array}{c} a c b \end{array}\end{array}$$ $$\begin{array}{ccccc}\begin{array}{c} c a b \end{array}& & \begin{array}{c} b a c \end{array}& & \begin{array}{c} a b c \end{array}\end{array}$$ $$\begin{array}{ccccccccccc}\begin{array}{c} c b a \end{array}& & \begin{array}{c} c a b \end{array}& & \begin{array}{c} b c a \end{array}& & \begin{array}{c} a c b \end{array}& & \begin{array}{c} b a c \end{array}& & \begin{array}{c} a b c \end{array}\end{array}$$ $$T=\begin{array}{c} 1 2 5 10 3 4 9 11 6 7 8 13 12 14 15 \end{array}$$ $$\begin{array}{c} k k+1 \end{array}$$ $$\begin{array}{c}C=\begin{array}{c} 1 6 9 12 14 2 7 10 13 15 3 8 11 4 5 \end{array}\\ \text{Column reading tableau}\end{array}$$ $$\begin{array}{c} 6 9 11 13 15 4 7 10 12 14 3 5 8 2 1 \\ \text{Numbering of boxes}\end{array}$$ $$\begin{array}{ccccc}\begin{array}{c} w(i) w(j) > \\ \text{Case (A)}\end{array}& & \begin{array}{c} w(i) w(j) > \\ \text{Case (B)}\end{array}& & \begin{array}{c} w(i) w(j) < \\ \text{Case (C)}\end{array}\end{array}$$ $$\begin{array}{ccccc}\begin{array}{c} -1 -1 \end{array}=\begin{array}{c} \end{array}& & \begin{array}{c} -1 \end{array}=\begin{array}{c} -1 \end{array}=\begin{array}{c} -1 \end{array}& & \begin{array}{c} -1 -1 \end{array}=\begin{array}{c} -1 -1 \end{array}=\begin{array}{c} -1 -1 \end{array}\end{array}$$ $$t_i=(i,-i)=\begin{array}{c} -1 ith ⋯ ⋯ \end{array}=s_i{s}_{i-1}\cdots s_3{s}_2{s}_1{s}_2{s}_3\cdots s_{i-1}{s}_i\text{.}$$ $$w=\begin{array}{c} -1 -1 -1 \end{array}=\begin{array}{c} -1 -1 -1 \end{array}=\pi t_1{t}_3{t}_6$$ $$T=\begin{array}{c} 1 5 7 4 10 12 8 13 2 3 9 6 11 α β \end{array}$$ $$\begin{array}{cc}\begin{array}{c} 0 1 2 -1 0 1 -2 -1 0 1 2 -1 0 \\ \text{Contents of boxes}\end{array}& \begin{array}{c} 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 \\ \text{Signs of boxes}\end{array}\end{array}$$ $$\begin{array}{ccc}\begin{array}{cc}{A}_n& \begin{array}{c} 2 3 n-1 n \end{array}\end{array}& & \begin{array}{cc}{B}_n& \begin{array}{c} 1 2 3 n-1 n \end{array}\end{array}\\ \begin{array}{cc}{D}_n& \begin{array}{c} 1 2 3 4 n-1 n \end{array}\end{array}& & \begin{array}{cc}{E}_6& \begin{array}{c} 2 3 4 5 6 1 \end{array}\end{array}\\ \begin{array}{cc}{E}_7& \begin{array}{c} 2 3 4 5 6 7 1 \end{array}\end{array}& & \begin{array}{cc}{E}_8& \begin{array}{c} 2 3 4 5 6 7 8 1 \end{array}\end{array}\\ \begin{array}{cc}{F}_4& \begin{array}{c} 1 2 3 4 \end{array}\end{array}& & \begin{array}{cc}{H}_3& \begin{array}{c} 1 2 3 5 \end{array}\end{array}\\ \begin{array}{cc}{H}_4& \begin{array}{c} 1 2 3 4 5 \end{array}\end{array}& & \begin{array}{cc}{I}_2\left(m\right)& \begin{array}{c} 1 2 m \end{array}\end{array}\end{array}$$ $$t_{\lambda }=t_{({\lambda }_1,\dots ,{\lambda }_n)}=\begin{array}{c} λ1 λ2 ⋯ λn \end{array}$$ $$\begin{array}{c}\stackrel{\sim }{w}=t_{\lambda }w,\text{with}\hspace{0.17em}\lambda \in {\mathbb{Z}}^n\hspace{0.17em}\text{and}\hspace{0.17em}w\in S_n\text{.}\\ w=\begin{array}{c} 5 32 0 -7 -61 1 \end{array}=\begin{array}{c} 5 32 0 -7 -61 1 \end{array}=t_{(5,32,0,-7,-61,1)}\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 6& 2& 4& 1& 5\end{array}\right)\text{.}\end{array}$$ $$\begin{array}{c}{s}_0=\begin{array}{c} -1 0 0 0 0 1 ⋯ ⋯ \end{array},\omega =\begin{array}{c} -1 0 0 0 0 0 ⋯ ⋯ \end{array},\\ s_i=\begin{array}{c} 0 0 0 0 0 0 0 ⋯ ⋯ \end{array},1\le i\le n-1,\end{array}$$

From "Affine Braid Group Representations and the Functors $F_{\lambda }$"

$$\begin{array}{c} \\ \\ \end{array}$$ $${\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}={\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2\text{.}$$

From "Markov Traces"

$$\begin{array}{cc}\begin{array}{ccc}{\stackrel{\sim }{ℬ}}_k& \hookrightarrow & {\stackrel{\sim }{ℬ}}_{k+1}\\ \begin{array}{c} b 1 ⋯ k \end{array}& ⟼& \begin{array}{c} b 1 ⋯ k k+1 \end{array}\end{array}& \text{(5.1)}\end{array}$$ $${\stackrel{\sim }{X}}^{{\epsilon }_{k+1}}=T_k{T}_{k-1}\cdots T_2{X}^{{\epsilon }_1}{T}_2^{-1}\cdots T_{k-1}^{-1}{T}_k^{-1}=\begin{array}{c} 1 2 ⋯ k+1 \end{array}$$ $${\epsilon }_k\left(\begin{array}{c} z 1 ⋯ k \end{array}\right)=\begin{array}{c} z \end{array}=\begin{array}{c} εk(z) 1 ⋯ k-1 \end{array}\text{.}$$ $$r\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c}\\ \end{array}\begin{array}{c} \end{array}={\epsilon }_1\left({\left(X^{{\epsilon }_1}\right)}^r\right)=\xi ·{\text{id}}_M,\text{for some}\hspace{0.17em}\xi \in ℂ\text{.}$$ $$\begin{array}{ccc}{\text{mt}}_{k+1}\left(\begin{array}{c} b (X∼εk+1)r 1 ⋯ k \end{array}\right)& =& {\text{mt}}_{k+1}\left(\begin{array}{c} b (X∼εk+1)r 1 ⋯ k \end{array}\right)\\ & =& {\text{mt}}_k\left(\begin{array}{c} b (X∼εk+1)r 1 ⋯ k \end{array}\right)\\ & =& \xi ·{\text{mt}}_k\left(\begin{array}{c} b 1 ⋯ k \end{array}\right)\\ & =& \xi ·{\text{mt}}_k\left(\begin{array}{c} b 1 ⋯ k \end{array}\right)\text{.}\end{array}$$

From "Examples"

$${\text{mt}}_k\left(b\right)={\text{mt}}_1\left(\begin{array}{c} b \end{array}\right)$$ $$\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})$$ $$\begin{array}{c}{r}_1\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}\\ ⋮\\ r_k\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}\end{array}=\frac{Q_{r_1}\cdots Q_{r_k}}{{\text{dim}}_q{\left(V\right)}^k}·\text{mt}\left(\begin{array}{c} \end{array}\right)=\frac{Q_{r_1}\cdots Q_{r_k}}{{\text{dim}}_q{\left(V\right)}^k}\text{.}$$

From section 2 of "Classification of graded Hecke algebras for complex reflection groups"

$$\begin{array}{c}\begin{array}{cc}{A}_{n-1}\begin{array}{c} 1 2 n-2 n-1 \end{array}& B_n\begin{array}{c} 1 2 3 n-1 n \end{array}\\ D_n\begin{array}{c} 1 2 3 4 n-1 n \end{array}& E_6\begin{array}{c} 2 3 1 4 5 6 \end{array}\\ E_7\begin{array}{c} 2 3 1 4 5 6 7 \end{array}& E_8\begin{array}{c} 2 3 1 4 5 6 7 8 \end{array}\\ F_4\begin{array}{c} 1 2 3 4 \end{array}& H_3\begin{array}{c} 1 2 3 5 \end{array}\\ H_4\begin{array}{c} 1 2 3 4 5 \end{array}& I_2\left(m\right)\begin{array}{c} 1 2 m \end{array}\end{array}\\ \text{Figure 1.}\hspace{0.17em}\text{Coxeter-Dynkin diagrams for real reflection groups.}\end{array}$$

Notes and References

See the SVG primer for notes on drawing SVG pictures.

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