Combinatorial Construction of the Weak Order of a Coxeter Group
Combinatorial Construction of the Weak Order
of a Coxeter Group
Přemysl Jedlička
Department of Mathematics
Faculty of Engineering (former Technical Faculty)
Czech University of Life Sciences (former Czech University of Agriculture), Prague
April 2008 – Denver, Colorado
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Combinatorial Construction of the Weak Order of a Coxeter Group
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Semidirect Products
Semidirect Product of Groups
G=K nH
K
H
1
(k, h) ∗ (k0 , h0 ) = (k · k0 , h · ϕk (h0 ))
Combinatorial Construction of the Weak Order of a Coxeter Group
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Semidirect Products
Semidirect Product of Groups
G=K nH
K
H
1
(k, h) ∗ (k0 , h0 ) = (k · k0 , h · ϕk (h0 ))
Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Products
A Lattice That Is a Semidirect Product
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Combinatorial Construction of the Weak Order of a Coxeter Group
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Semidirect Product of Lattices
Definition of the Mappings ϕ and ψ
(k∨k0 , ϕk,k0 (h))
(k∨k0 , 1H )
(k, 1H )
(k∨k0 , 0H )
(k, h)
(k, 0H )
(k0 , 0H ) ∨ (k, h) = (k ∨ k0 , ϕk,k0 (h)) (k00 , 1H )
(k00 , ϕ
00 (h))
k,k
(k, 1H ) ∧ (k0 , h)
= (k ∧ k0 , ψk0 ,k (h))
(k0 , 1H )
(k00 , 0H )
(k0 , ϕk,k0 (h))
Combinatorial Construction of the Weak Order of a Coxeter Group
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Semidirect Product of Lattices
Definition of the Mappings ϕ and ψ
(k∨k0 , ϕk,k0 (h))
(k∨k0 , 1H )
(k, 1H )
(k∨k0 , 0H )
(k, h)
(k, 0H )
(k0 , 0H ) ∨ (k, h) = (k ∨ k0 , ϕk,k0 (h)) (k00 , 1H )
(k00 , ϕ
00 (h))
k,k
(k, 1H ) ∧ (k0 , h)
= (k ∧ k0 , ψk0 ,k (h))
(k0 , 1H )
(k00 , 0H )
(k0 , ϕk,k0 (h))
Combinatorial Construction of the Weak Order of a Coxeter Group
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Semidirect Product of Lattices
(k, h)
(k,of
0Hthe
)
Properties
Mapping ϕ
(k00 , 1H )
(k00 , ϕk,k00 (h))
(k0 , 1H )
00
(k , 0H )
(k0 , ϕk,k0 (h))
(k, 1H )
(k0 , 0H )
(k, 0H )
(k, h)
ϕk,k00 = ϕk0 ,k00 ◦ ϕk,k0
for k 6 k0 6 k00
ϕk,k0 (h ∨ h0 ) = ϕk,k0 (h) ∨ ϕk,k0 (h0 )
(k, ϕk0 ,k (h))
(k, 1H )
Combinatorial Construction of the Weak Order of a Coxeter Group
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Semidirect Product of Lattices
(k, h)
(k,of
0Hthe
)
Properties
Mapping ϕ
(k00 , 1H )
(k00 , ϕk,k00 (h))
(k0 , 1H )
00
(k , 0H )
(k0 , ϕk,k0 (h))
(k, 1H )
(k0 , 0H )
(k, 0H )
(k, h)
ϕk,k00 = ϕk0 ,k00 ◦ ϕk,k0
for k 6 k0 6 k00
ϕk,k0 (h ∨ h0 ) = ϕk,k0 (h) ∨ ϕk,k0 (h0 )
(k, ϕk0 ,k (h))
(k, 1H )
H
Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Product of Lattices
(k0 , 0H )Between ϕ and ψ
Connection
(k, 0H )
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(k0 , ϕk,k0 (h))
(k, 1H )
(k, h)
(k, 1H )
(k, ϕk0 ,k (h))
(k0 , 1H )
(k, 0H )
(k0 , h)
(k0 , ψk,k0 ◦ϕk0 ,k (h))
0
(k , 0H )
ψk,k0 ◦ ϕk0 ,k (h) > h
Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Product of Lattices
Semidirect Product of Lattices
Theorem (P.J.)
Let K, H be two lattices and let ϕ, ψ : K 2 → H H be two mappings
satisfying eight specific conditions. Then the set K × H with the
operations ∨, ∧, defined by
(k, h) ∨ (k0 , h0 ) = (k ∨ k0 , ϕk,k0 (h) ∨ ϕk0 ,k (h0 )),
(k, h) ∧ (k0 , h0 ) = (k ∧ k0 , ψk,k0 (h) ∧ ψk0 ,k0 (h0 )),
forms a lattice, called the semidirect product of K and H and
ϕ
denoted by K nψ H .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Product of Lattices
Example of a Semidirect Product 1
Let K, H be two arbitrary lattices and let ϕk,k0 = ψk,k0 = idH , for all
ϕ
k, k0 in K . Then the semidirect product K nψ H is the carthesian
product K × H .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Product of Lattices
Example of a Semidirect Product 2
Let K, H be two arbitrary lattices and let, for each k 6 k0 and h
in H , be ϕk,k0 (h) = 0H , ψk0 ,k (h) = 1H . Then we
have (k, h) 6 (k0 , h0 ) if and only if k < k0 in K or k = k0 and h 6 h0
in H . Therefore, the semidirect product of K and H consists of |K|
copies of the lattice H arranged in the form of the lattice K .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Product of Lattices
Example of a Semidirect Product 3
Let K , H be arbitrary lattices and, for all k 6 k0 , let ϕk,k0 (h) = 1H ,
for h > 0H , and ψk0 ,k (h) = 0H , for h < 1H . In this case if K has at
ϕ
least 2 elements and H has at least 3 elements, the lattice K nψ H
is not modular.
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Combinatorial Construction of the Weak Order of a Coxeter Group
Semidirect Product of Lattices
Semidirect Product of Semilattices
Theorem (P.J.)
Let K and H be two meet-semilattices and let ψ : K 2 → End(H)
be a mapping satisfying, for all k, k0 and k00 from K ,
ψk,k = idH ;
ψk,k0 ∧k00 = ψk∧k0 ,k00 ◦ ψk,k0 .
Then the set K × H together with the operation ∧ defined by
(k, h) ∧ (k0 , h0 ) = (k ∧ k0 , ψk,k0 (h) ∧ ψk0 ,k (h0 ))
forms a semilattice. This semilattice is denoted K nψ H .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Coxeter Groups
Definition
A Coxeter system is a pair (W, S), where W is a group and S is a
subset of W such that W has a presentation in form
D
E
W = S; s2 = 1, (st)mst = 1; for all s, t ∈ S
where mst ∈ {2, 3, 4, . . . , ∞}.
Such a group W is called a Coxeter group.
Examples
Weyl groups
Dihedral groups
Groups of symmetries of a Euclidean space
Groups of symmetries of an affine space
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Coxeter Groups
Definition
A Coxeter system is a pair (W, S), where W is a group and S is a
subset of W such that W has a presentation in form
D
E
W = S; s2 = 1, (st)mst = 1; for all s, t ∈ S
where mst ∈ {2, 3, 4, . . . , ∞}.
Such a group W is called a Coxeter group.
Examples
Weyl groups
Dihedral groups
Groups of symmetries of a Euclidean space
Groups of symmetries of an affine space
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Weak Order
Definition
Let (W, S) be a Coxeter system and take w ∈ W . A reduced
expression of w is an expression w = s1 s2 · · · sk , for si ∈ S,
where k is minimal possible. The lenght of w, denoted by `(w), is
the lenght of this reduced expression.
Definition
Let (W, S) be a Coxeter system. We write w 4 w0 , for elements w
and w0 in W , if `(w0 ) = `(w) + `(w−1 w0 ). This relation is called the
weak order of W or sometimes the weak Bruhat order of W .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Weak Order
Definition
Let (W, S) be a Coxeter system and take w ∈ W . A reduced
expression of w is an expression w = s1 s2 · · · sk , for si ∈ S,
where k is minimal possible. The lenght of w, denoted by `(w), is
the lenght of this reduced expression.
Definition
Let (W, S) be a Coxeter system. We write w 4 w0 , for elements w
and w0 in W , if `(w0 ) = `(w) + `(w−1 w0 ). This relation is called the
weak order of W or sometimes the weak Bruhat order of W .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Properties of the Weak Order
Theorem (A. Björner)
The weak order on a Coxeter group W forms a meet-semilattice
with 1 as the smallest element. The order forms a lattice if and
only if W is finite.
Observation
Let (W, S) be a Coxeter system. The unoriented Hasse diagram of
the weak order on W and the unlabelled Cayley graph of the
presentation given by S are the same graphs.
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Properties of the Weak Order
Theorem (A. Björner)
The weak order on a Coxeter group W forms a meet-semilattice
with 1 as the smallest element. The order forms a lattice if and
only if W is finite.
Observation
Let (W, S) be a Coxeter system. The unoriented Hasse diagram of
the weak order on W and the unlabelled Cayley graph of the
presentation given by S are the same graphs.
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Weak Order of the Symmetric Group S4
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Standard Parabolic Subgroups
Definition
Let (W, S) be a Coxeter system and let X be a subset of S. The
subgroup of W generated by X is called a standard parabolic
subgroup and it is denoted by WX .
Fact (well known)
Let (W, S) be a Coxeter system and let X be a subset of S. Then
the pair (WX , X) is a Coxeter system. For each element in WX , the
length in WX and in W are the same.
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Standard Parabolic Subgroups
Definition
Let (W, S) be a Coxeter system and let X be a subset of S. The
subgroup of W generated by X is called a standard parabolic
subgroup and it is denoted by WX .
Fact (well known)
Let (W, S) be a Coxeter system and let X be a subset of S. Then
the pair (WX , X) is a Coxeter system. For each element in WX , the
length in WX and in W are the same.
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Reduced Elements
Definition
Let (W, S) be a Coxeter system and let X be a subset of S. An
element w in W is called X -reduced if x $ w, for all x ∈ X . The set
of all X -reduced elements is denoted W X
Proposition (V. Deodhar)
For each element w in W , there exists a unique decomposition
w = wX wX with wX ∈ WX and wX ∈ W X .
Proposition (P.J.)
Let θ be this equivalence: (w, w0 ) ∈ θ if and only if wX = w0X . Then
θ is a congruence of the semilattice (W, 4) with W/θ WX and
each of the congruence classes is isomorphic to W X .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Reduced Elements
Definition
Let (W, S) be a Coxeter system and let X be a subset of S. An
element w in W is called X -reduced if x $ w, for all x ∈ X . The set
of all X -reduced elements is denoted W X
Proposition (V. Deodhar)
For each element w in W , there exists a unique decomposition
w = wX wX with wX ∈ WX and wX ∈ W X .
Proposition (P.J.)
Let θ be this equivalence: (w, w0 ) ∈ θ if and only if wX = w0X . Then
θ is a congruence of the semilattice (W, 4) with W/θ WX and
each of the congruence classes is isomorphic to W X .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Coxeter Groups
Reduced Elements
Definition
Let (W, S) be a Coxeter system and let X be a subset of S. An
element w in W is called X -reduced if x $ w, for all x ∈ X . The set
of all X -reduced elements is denoted W X
Proposition (V. Deodhar)
For each element w in W , there exists a unique decomposition
w = wX wX with wX ∈ WX and wX ∈ W X .
Proposition (P.J.)
Let θ be this equivalence: (w, w0 ) ∈ θ if and only if wX = w0X . Then
θ is a congruence of the semilattice (W, 4) with W/θ WX and
each of the congruence classes is isomorphic to W X .
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Combinatorial Construction of the Weak Order of a Coxeter Group
Constructions of the Weak Orders
Construction for the Groups of Type A
Fact (well known)
The group of type An is the symmetric group on n + 1 elements,
with si = (i, i + 1). Let X = {s1 , . . . , sn−1 }. Then
WX = {1, sn , sn sn−1 , sn sn−1 sn−2 , . . . , sn sn−1 · · · s2 s1 }.
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Combinatorial Construction of the Weak Order of a Coxeter Group
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Constructions of the Weak Orders
Construction for the Groups of Type A
Fact (well known)
The group of type An is the symmetric group on n + 1 elements,
with si = (i, i + 1). Let X = {s1 , . . . , sn−1 }. Then
WX = {1, sn , sn sn−1 , sn sn−1 sn−2 , . . . , sn sn−1 · · · s2 s1 }.
si+2
sn
si
si
si
···
si
si+2
sn
si+1
sn−1
sn−1
si
si+1
si+1
si
si−1
si
si+1
s2
s1
si+1
si+1
s2
s1
si−1
···
si+1
Combinatorial Construction of the Weak Order of a Coxeter Group
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Constructions of the Weak Orders
Cayley Graph / Weak Order of the Group of Type A3
Combinatorial Construction of the Weak Order of a Coxeter Group
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Constructions of the Weak Orders
Cayley Graph / Weak Order of the Group of Type B3
Combinatorial Construction of the Weak Order of a Coxeter Group
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Constructions of the Weak Orders
Cayley Graph / Weak Order of the Group of Type D3
Combinatorial Construction of the Weak Order of a Coxeter Group
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Constructions of the Weak Orders
Cayley Graph / Weak Order of the Group of Type H3
Combinatorial Construction of the Weak Order of a Coxeter Group
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Constructions of the Weak Orders
Cayley Graph / Weak Order of the Group of Type Ã2
Combinatorial Construction of the Weak Order of a Coxeter Group
Bibliography
Bibliography
A. Björner: Orderings of Coxeter Groups
Cont. Math. 34, AMS: Providence, R.I., 1984, 175–195
V. Deodhar:
A splitting criterion for the Bruhat orderings on Coxeter groups
Comm. in Algebra 15 (9), 1987, 1889–1894
P. Jedlička: Combinatorial Construction of the Weak Order of a
Coxeter Group
Comm. in Algebra 33 (5), 2005, 1447–1460
P. Jedlička: Semidirect Product of Lattices
to appear in Algebra Universalis (2007)
C. Le Conte de Poly-Barbut:
Treillis de Cayley des groupes de Coxeter finis. Construction
par récurrence et décompositions sur des quotiens.
Math. Inf. Sci. Hum. 140, 1997, 11–33
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