The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
Relatives of the Symmetric Group Algebra.
Götz Pfeiffer
goetz.pfeiffer@nuigalway.ie
Department of Mathematics
NUI,Galway
Groups in Galway,19.05.2006
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
Outline
1
The Descent Algebra.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
2
The Iwahori–Hecke Algebra.
The Iwahori–Hecke Algebra.
Properties.
Other
q
analogues.
3
Relations.
Wallach’s Formula.
Lusztig’s
q
analogue.
Other
q
analogues?
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
ZigZag.
Associate to the permutation
α = (1,3,8,7)(4,9,5) =
1
2
3
4
5
6
7
8
9
3
2
8
9
4
6
1
7
5
∈ Sym(9)
the pattern
Let
I = {1,...,n−1}
.The
descent set
of
α ∈ Sym(n)
is
the set
D(α) = {i ∈ I:α(i) > α(i +1)}
.
E.g.,
D((1,3,8,7)(4,9,5)) = {1,4,6,8} ⊆ {1,...8}
.
The symmetric group
Sym(n)
is partitioned into
2
n−1
descent classes
Y
K
= {α ∈ Sym(n):D(α) = I\K}
,
K ⊆ I
.
E.g.,
Y
I
= {id}
and
Y
∅
= {
1
2
...
n−1
n
n
n−1
...
2
1
}
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
Descent Algebra.
Denote
W = Sym(n)
.
For
J ⊆ I
deﬁne
X
J
=
K⊇J
Y
K
and
x
J
=
X
−1
J
∈ Q[W]
.
Theorem (Solomon,1976)
The elements
x
J
,
J ⊆ I
,form a basis of a subalgebra of
dimension
2
I
of the group algebra
Q[W]
.More precisely,
x
J
x
K
=
L⊆I
a
JKL
x
L
,for all
J,K ⊆ I
,and certain
a
JKL
∈ Z
.
The algebra
Σ(W)
spanned by the
x
J
,
J ⊆ I
,is called the
descent algebra
of
W
.
This theorem holds for all
ﬁnite Coxeter groups
W
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
Coxeter Groups.
A
Coxeter group
(W,S)
is a group
W
with
generators
S = {s,t,...}
and
relations
(st)
m(s,t)
= 1
,for certain
m(s,t) = m(t,s) ∈ {2,3,...,∞}
,if
s = t
,and
m(s,s) = 1
.
Example (Typical Finite Coxeter Groups.)
Sym(n) = (i,i +1):i ∈ I
.
D
2m
= s,t  s
2
= t
2
= (st)
m
= 1
.
Length function
:
l(ws) = l(w) ±1
.
Descent set
:
D(w) = {s ∈ S:l(sw) < l(w)}
.
Descent class
:
Y
K
= {w ∈ W:D(w) = S\K}
,
K ⊆ S
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
Distinguished Coset Representatives.
For
J ⊆ S
,the
standard parabolic subgroup
W
J
= J
of
W
is a Coxeter group
(W
J
,J)
.
The set
X
J
=
K⊇J
Y
K
= {w ∈ W:D(w) ∩J = ∅}
is a
right
transversal
of
W
J
in
W = W
J
∙ X
J
consisting of the unique
elements of minimal length
in each coset.
X
−1
J
is a minimal length
left transversal
of
W
J
in
W
.
Inductively
,if
J ⊆ K ⊆ S
,then
X
K
J
:= X
J
∩W
K
is a minimal
length right transversal of
W
J
in
W
K
.
X
JK
= X
J
∩X
−1
K
is a set of minimal length
double coset
representatives
of
W
J
and
W
K
in
W =
d∈X
JK
W
J
dW
K
.
If
d ∈ X
JK
then
W
d
J
∩W
K
= W
L
,where
L = J
d
∩K
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
The Coset Graph.
Represent
X
J
as a
directed colored graph
with
vertices
:
x ∈ X
J
and
edges
:
x
s
−
→xs
if
l(xs) > l(x)
,
s ∈ S
.
Example (
Sym(6) =
s
1
,
s
2
,
s
3
,
s
4
,
s
5
≥ Sym(5)
)
Example (
Sym(6) ≥ Sym(2) ×Sym(4)
)
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
Properties.
1
Transitivity
:Let
J ⊆ K ⊆ S
.Then
X
J
= X
K
∙ X
K
J
.
(
W
J
∙ X
J
= W = W
K
∙ X
K
= W
J
∙ X
K
J
∙ X
K
.)
Example (
Sym(5) ≥ Sym(3)
)
2
Shift:
Let
J,K ⊆ S
.Then
dX
J
d
∩K
= X
J∩
d
K
for all
d ∈ X
JK
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
More Properties.
3
MackeyDecomposition
:Let
J,K ⊆ S
.Then
X
J
=
d∈X
JK
d ∙ X
J
J
d
∩K
.
Example (
Sym(6) ≥ Sym(2) ×Sym(4)
)
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
Proof of Solomon’s Theorem.
For
J ⊆ S
,let
x
J
=
X
−1
J
.Then,for
J,K ⊆ S
,
x
J
x
K
(3)
= x
J
d∈X
JK
x
J
J∩
d
K
d
(1)
=
d∈X
JK
x
J∩
d
K
d
(2)
=
d∈X
JK
x
J
d
∩K
.
Example
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
ZigZag.
Distinguished Coset Representatives in Coxeter Groups.
Positive Elements are Diagonalizable.
Positive Elements are Diagonalizable.
Let
θ:x
J
→1
W
W
J
be the map from
Σ(W)
into the character
ring of
W
,which assigns to
x
J
the
permutation character
of
W
on the cosets of
W
J
.
θ
is an algebra homomorphism with
ker θ = RadΣ(W)
.
Let
Σ
+
(W)
be the set of
nonnegative linear
combinations
of the
x
J
,
J ⊆ S
.
a,b ∈ Σ
+
(W) =⇒ a +b ∈ Σ
+
(W)
,clearly.
a,b ∈ Σ
+
(W) =⇒ ab ∈ Σ
+
(W)
,by Solomon’s Theorem.
Theorem (Bonnafé–Pf.2005)
Let
a ∈ Σ
+
(W)
.Then
λ∈{θ(a)(w):w∈W}
(a −λ) = 0.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
The Iwahori–Hecke Algebra.
Properties.
Other
q
analogues.
The Iwahori–Hecke Algebra.
Deﬁnition
Let
q
be an indeterminate.Let
H
be the
Z[q]
algebra with
generators
T
1
,...,T
n−1
and
relations
:
(T
i
−q)(T
i
+1) = 0
for
i = 1,...,n−1
,
T
i
T
i+1
T
i
= T
i+1
T
i
T
i+1
for
i = 1,...,n−2
,
T
i
T
j
= T
j
T
i
for
i −j = 1
.
H
is a
q
analogue
of the group algebra
Z[W]
for
W = Sym(n)
:if
q 1
then
H Z[W]
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
The Iwahori–Hecke Algebra.
Properties.
Other
q
analogues.
Properties.
Every ﬁnite Coxeter group
W
has an Iwahori–Hecke
algebra
H
.
H
has a basis
{T
w
:w ∈ W}
.
Matsumoto’s theorem
:
T
v
T
w
= T
vw
if
l(vw) = l(v) +l(w)
.
Irr(H) ↔Irr(W):χ
q
χ
.
Choose,for each conjugacy class
C
of
W
,a representative
w
C
of
minimal length
.
Then
(χ
q
(T
w
C
))
χ,C
is the
character table
of
H
,a
q
analogue of the character table
(χ(w
C
))
χ,C
of
W
.
(Starkey,1975)
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
The Iwahori–Hecke Algebra.
Properties.
Other
q
analogues.
Other
q
analogues.
The
q
bracket
[k]
q
= 1 +q +∙ ∙ ∙ +q
k−1
∈ Z[q]
.
The
q
factorial
[n]
q
!= [1]
q
[2]
q
∙ ∙ ∙ [n]
q
.
The
q
binomial coefﬁcient
n
k
q
=
[n]
q
!
[k]
q
![n −k]
q
!
counts
the number of
k
dimensional subspaces of
F
n
q
.
P
W
=
w∈W
q
l(w)
is the
Poincaré polynomial
of
W
.
...
Question
Is there a
q
analogue of
Σ(W)
???
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
Wallach’s Formula.
Lusztig’s
q
analogue.
Other
q
analogues?
Wallach’s Formula.
Let
s
i
= (i,i +1) ∈ Sym(n)
.
Let
X = {1,s
n−1
,s
n−2
s
n−1
,...,s
2
s
3
∙ ∙ ∙ s
n−1
,s
1
s
2
∙ ∙ ∙ s
n−1
}
.
Let
t =
X
.
Then (Wallach,1988)
(t −n)
n−2
k=0
(t −k) = 0.
Note
:
X
is the distinguished left transversal of
Sym(n −1)
in
Sym(n)
.
Thus
t ∈ Σ(Sym(n))
and the formula is a special case of
our theorem.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
Wallach’s Formula.
Lusztig’s
q
analogue.
Other
q
analogues?
Lusztig’s
q
analogue.
Let
H
be the Iwahori–Hecke algebra of
Sym(n)
(with
parameter
q
).
Let
t =
x∈X
T
x
∈ H
.
Then (Lusztig,2004)
(t −[n]
q
)
n−2
k=0
(t −[k]
q
) = 0.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
Wallach’s Formula.
Lusztig’s
q
analogue.
Other
q
analogues?
Proof of Lusztig’s
q
analogue.
Exploit the properties of
X
J
and Matsumoto’s Theorem...
Let
S
k
= {s
1
,...,s
k
}
generate
Sym(k +1)
,
k < n
.
For
0 ≤ i ≤ k
let
X
(k)
i
= X
S
k
S
i
.
Let
t
(k)
i
=
{T
x
:x ∈ X
(k)
i
}
.Then
t = t
(n−1)
n−2
.
Transitivity
=⇒ t
(k)
i
= t
(j)
i
t
(k)
j
,for
i ≤ j ≤ k
.
Mackeydecomposition
=⇒ t
(k)
k−i
t
(k)
k−1
= q
i
t
(k)
k−i−1
+[i]
q
t
(k)
k−i
.
Conclude that
l−1
i=0
(t
(k)
k−1
−[i]
q
)
t
(k)
k
=1
= q
(
l
2
)
t
(k)
k−l
.
Set
l = k = n −1
and use
t t
(n−1)
0
= [n]
q
t
(n−1)
0
.
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
The Descent Algebra.
The Iwahori–Hecke Algebra.
Relations.
Wallach’s Formula.
Lusztig’s
q
analogue.
Other
q
analogues?
Other
q
analogues?
Other identities of this type with
q
analogues in
H
are:
t −1 = 0
for
t = 1 ∈ H
(if
J = S
then
x
J
= 1
and
θ(x
J
)
is the
trivial character
of
W
with constant value
1
).
t(t −P
W
) = 0
for
t =
w∈W
T
w
∈ H
(if
J = ∅
then
x
J
=
W
and
θ(x
J
)
is the
regular character
of
W
with values
W
and
0
).
Except for these
extreme cases
,Lusztig’s
q
analogue
seems to be the
only valid identity
of this kind in
H
!?!
Götz Pfeiffer
Relatives of the Symmetric Group Algebra.
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