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

Mackey-Decomposition

: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

.

Mackey-decomposition

=⇒ 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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