Q
ep
p
r
∆0
er
q
R
P
eq
ϕ ep
ϕ∆
0
ϕ er
ϕQ
Kaleidoscopic Tilings on Surfaces, Group
Algebras and Separability
S. Allen Broughton
Rose-Hulman Institute of Technology
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Credits
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Some of this work has been done and will
be done with numerous undergraduates
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Talk 2 Outline
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Groups again and group actions
Group algebras
Tilings and tiling groups again
Chains, boundaries, and homology
Separability
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Groups and group action
Definition for these talks
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A group G is set of transformations of a set or space X,
such that G is closed under composition and taking of
inverses.
If g,h ε G then gh =g◦h ε G and g-1 ε G
For x ε X and g ε G define the action of g on x by
gx=g(x).
Note that (gh)x =g◦h(x) =g(hx).
Typically a group will preserve some property of the space
– e.g., distance preserving or rigid motion.
Subgroups are subsets of G closed under compositions
and taking inverses.
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Groups and group actions
Examples
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X = {1,2,3…,n} and G is a group of permutations
of X.
X is a regular n-gon and G = Dn is the dihedral
group of symmetries of X.
X is tiled surface and G and G* are the tiling
groups acting on X as geometric transformations
G acts on subsets of X, e.g. k-subsets, vertices,
edges, tiles, faces, lines, or circles.
The tiling groups act as permutation groups on
vertices edges and tiles.
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Group algebras
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Let k be a field and G a finite group.
Group algebra k[G] is the set of formal k-sums of
element of G.
k[G] = {ΣgεG agg}
Let α=ΣgεG agg , β=ΣgεG bgg
Sums: α + β = ΣgεG (ag+ bg)g
Products:
α β= (ΣgεG agg) (ΣhεG bhh)=ΣgεG ΣhεG (agbhgh)
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Group algebra - sample calculations
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G = {1, g, g2} = cyclic group of order 3.
α =1+g+g2, β =1-g.
α β =1+g+g2 –1g – gg - g2g
= 1+g+g2-g-g2-1 = 0
Set ε = (1+g+g2)/3, α= a1+bg+cg2
Then α ε = ε α = (a+b+c) ε
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Group algebra – action modules
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X be a set on which G acts
k[X] = {ΣxεX axx} is a k-vector space
k[X] is a k[G] module
Let α=ΣgεG agg , v= ΣxεX bxx
The α v = (ΣgεG agg) (ΣxεX bxx )
=ΣgεG ΣxεX (ag bxgx)
(gh)x = g(hx) implies (αβ)v = α (βv)
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(2,3,5) – tiling – soccer ball
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(2,4,4) -tiling of the torus
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Separability
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A reflection in a tiling is separating if the
mirror of the reflection separates the tiling
into two pieces.
Problem: determine from the tiling group
whether a reflection is separating
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The tiling group - 1
Full Tiling Group for triangle (a finite group)
G =< p , q , r >
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Group Relations
p
2
= q
2
= r
( pq ) = ( qr )
l
2
m
= 1.
= ( rp )
n
= 1.
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The tiling group - 2
Observe/define:
a = pq,b = qr ,c = rp
Tiling Group:
G = < p, q , r >
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Orientation Preserving Tiling Group:
G = < a ,b, c >
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Chain modules for a tiling - 1
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Let Xn be the n dimensional components of
the tiling of S.
• n=0, vertices,
• n=1, edges,
• n=2, faces.
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Cn = k[Xn]
∂ : Cn → Cn-1 is the boundary map
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Chain modules for a tiling - 2
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∂∆0 = ep+ eq+ er
∂(q∆0)= - a-1ep - eq - ber
See picture on next slide
Extend by G-linearity
∂ζ∆0 + ∂ηq∆0 = ζ ∂∆0 + η∂(q∆0)
Don’t need to worry about n=1 or n=0.
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The master tile
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ep
p
r
∆0
er
q
R
P
eq
ϕ ep
ϕ∆
ϕQ
0
ϕ er
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Separability criterion
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Let k = {0,1}, A = k[G].
Let Mφ = αep+ βeq+ γer sum of all the edges
in the mirror of the reflection φ.
Then q is separating if and only if there are
ζ and η satisfying
∂ζ∆0 + ∂ηq∆0 = αep+ βeq+ γer
or η(1-a-1)= β- α, η(1-b)= β- γ
Note: Mφ can be easily computed.
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Solvability of equations in the group
algebra
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Let A = k[G].
Then ηα1 = β1, ηα2= β2 if and only if
α1 γ1 + α2 γ2= 0 implies β1 γ1 + β2 γ2= 0
for all γ1,γ2
The point is that we might only need to
check a small number of equations.
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