Partition Algebra
Ak and Sn
Partition Algebras
Erica Shannon
April 10, 2012
Representations of Sn
Partition Algebra
Ak and Sn
Elements of the Partition Monoid
Let k be a positive integer.
Definition
Ak = {set partitions of {1, 2, . . . , k, 10 , 20 , . . . , k 0 }}.
Representations of Sn
Partition Algebra
Ak and Sn
Elements of the Partition Monoid
Let k be a positive integer.
Definition
Ak = {set partitions of {1, 2, . . . , k, 10 , 20 , . . . , k 0 }}.
Example
Some elements of A4 are
• {{1, 2, 3, 4, 10 , 20 , 30 , 40 }}
• {{1, 2, 3, 4}, {10 , 20 , 30 , 40 }}
• {{1, 2, 10 , 20 }, {3}, {4}, {30 }, {40 }}
• {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Elements of the Partition Monoid
Let k be a positive integer.
Definition
Ak = {set partitions of {1, 2, . . . , k, 10 , 20 , . . . , k 0 }}.
Example
Some elements of A4 are
• {{1, 2, 3, 4, 10 , 20 , 30 , 40 }}
• {{1, 2, 3, 4}, {10 , 20 , 30 , 40 }}
• {{1, 2, 10 , 20 }, {3}, {4}, {30 }, {40 }}
• {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
We refer to each individual subset of {1, 2, . . . , k, 10 , 20 , . . . , k 0 } as
a block.
Partition Algebra
Ak and Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Example
{{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }} corresponds to the graph
1
2
3
4
u
u
@
u
u
u
u
@u
u
1’
2’
3’
4’
@
Partition Algebra
Ak and Sn
Representations of Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Example
{{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }} corresponds to the graph
1
2
3
4
u
u
@
u
u
u
u
@u
u
1’
2’
3’
4’
@
The graph is only defined up to its connected components.
Partition Algebra
Ak and Sn
Representations of Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Example
{{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }} corresponds to the graph
1
2
3
4
u
u
@
u
@
u
u
u
1’
2’
@
@u
3’
@
@u
4’
The graph is only defined up to its connected components.
Partition Algebra
Ak and Sn
Representations of Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Example
{{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }} corresponds to the graph
1
2
3
4
u
u
@
u
@
u
u
u
1’
2’
@
@u
3’
@
@u
4’
The graph is only defined up to its connected components.
Partition Algebra
Ak and Sn
Representations of Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Example
{{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }} corresponds to the graph
1
2
3
4
u
u
@
u
@
u
u
u
1’
2’
@
@u
3’
@
@u
4’
The graph is only defined up to its connected components.
Partition Algebra
Ak and Sn
Representations of Sn
Graphs
We represent a set partition d ∈ Ak by a graph.
Example
{{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }} corresponds to the graph
1
2
3
4
u
u
u
@
u
u
u
u
1’
2’
3’
@
@u
4’
The graph is only defined up to its connected components.
Partition Algebra
Ak and Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
@
u
u
@u
u
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
@u
u
u
@
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
@u
u
u
@
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
@u
u
u
@
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
@u
u
u
@
u
u
u
u
u
u
u
u
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
@u
u
u
@
u
u
u
u
u
u
u
u
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
u
u
u
u
u
u
u
u
@u
u
u
@
Partition Algebra
Ak and Sn
Representations of Sn
Composition in Ak
We define a composition operation on elements of Ak .
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
To compute d1 ◦ d2 , we draw a graph for d1 above a graph for d2 :
u
u
@
u
u
u
@
u
@
u
u
@u
u
u
u
u
u
u
u
u
u
u
u
u
@u
u
u
@
So d1 ◦ d2 = {{1, 2, 10 }, {20 , 30 , 3, 4}, {40 }}.
Partition Algebra
Ak and Sn
The Partition Monoid
The operation ◦ makes Ak into an associative monoid.
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
The Partition Monoid
The operation ◦ makes Ak into an associative monoid.
The identity is given by
1
2
3
4
u
u
u
u
u
u
u
u
u
1’
2’
3’
4’
k0
k
u
···
Partition Algebra
Ak and Sn
Representations of Sn
Propagating Number
Definition
For a partition d ∈ Ak , we define the propagating number of d:
pn(d) = number of blocks of d containing an element of
{1, 2, . . . , k} and an element of {10 , 20 , . . . , k 0 }.
Partition Algebra
Ak and Sn
Representations of Sn
Propagating Number
Definition
For a partition d ∈ Ak , we define the propagating number of d:
pn(d) = number of blocks of d containing an element of
{1, 2, . . . , k} and an element of {10 , 20 , . . . , k 0 }.
pn(d1 ◦ d2 ) ≤ min(pn(d1 ), pn(d2 ))
Partition Algebra
Ak and Sn
Representations of Sn
Propagating Number
Definition
For a partition d ∈ Ak , we define the propagating number of d:
pn(d) = number of blocks of d containing an element of
{1, 2, . . . , k} and an element of {10 , 20 , . . . , k 0 }.
pn(d1 ◦ d2 ) ≤ min(pn(d1 ), pn(d2 ))
Example
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
pn(d1 ) = 2
Partition Algebra
Ak and Sn
Representations of Sn
Propagating Number
Definition
For a partition d ∈ Ak , we define the propagating number of d:
pn(d) = number of blocks of d containing an element of
{1, 2, . . . , k} and an element of {10 , 20 , . . . , k 0 }.
pn(d1 ◦ d2 ) ≤ min(pn(d1 ), pn(d2 ))
Example
d1 = {{10 }, {1, 2, 20 , 30 }, {3, 4, 40 }}
d2 = {{1, 20 , 30 , 4}, {10 , 2}, {3}, {40 }}
pn(d1 ) = 2
pn(d2 ) = 2
Partition Algebra
Ak and Sn
Submonoids: Sk
• Sk = {d ∈ Ak |pn(d) = k}
(the symmetric group)
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Sk
• Sk = {d ∈ Ak |pn(d) = k}
1
2
(the symmetric group)
3
4
u
@
u
u u
@ @
@@
@
u @u @u @u
1’
2’
3’
4’
∈ S4
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Sk
Define
s
si =
s
s
···
s
i
i +1
s
s
@
s @s
i 0 (i + 1)0
s
s
s
···
s
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Sk
Define
s
si =
s
s
···
s
i
i +1
s
s
@
s @s
s
s
s
···
s
i 0 (i + 1)0
Theorem
The group Sk is presented by generators s1 , . . . , sk−1 and relations
si2 = 1,
si si+1 si = si+1 si si+1 ,
and
si sj = sj si ,
for |i − j| > 1.
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Sk
Define
s
si =
···
s
i +1
i
s
s
s
@
s @s
s
s
s
···
s
s
i 0 (i + 1)0
Theorem
The group Sk is presented by generators s1 , . . . , sk−1 and relations
si2 = 1,
si si+1 si = si+1 si si+1 ,
s
s
@
s @s
s1 s3 =
s
s
s
s
s
s
s
s
s
s
@
s @s
and
=
for |i − j| > 1.
si sj = sj si ,
s
s
s
s
s
s
@
s @s
s
s
@
s @s
s
s
s
s
= s3 s1
Partition Algebra
Ak and Sn
Submonoids: Ik
• Ik = {d ∈ Ak |pn(d) < k}
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Ik
• Ik = {d ∈ Ak |pn(d) < k}
u
u
u u
@
@ @
@@
@
u @u @u @u
∈ I4
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Ik
• Ik = {d ∈ Ak |pn(d) < k}
u
u
u u
@
@ @
@@
@
u @u @u @u
∈ I4
u
u
u u
@
@ @
@
@
@
u @u @u @u
∈
/ I4
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Ak+ 12
• Ak+ 1 = {d ∈ Ak+1 |(k +1) and (k +1)0 are in the same block}
2
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Ak+ 12
• Ak+ 1 = {d ∈ Ak+1 |(k +1) and (k +1)0 are in the same block}
2
u
u
u u
@
@ @@
u @ u @u
u
∈ A3+ 1
2
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Ak+ 12
• Ak+ 1 = {d ∈ Ak+1 |(k +1) and (k +1)0 are in the same block}
2
u
u
u u
@
@ @@
u @ u @u
u
∈ A3+ 1
u
u
u u
@
@ @
@
@@
u @u @u @u
∈
/ A3+ 1
2
2
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Pk
• Pk = {d ∈ Ak |d is planar}
d ∈ Ak is planar if it has some planar graph with all edges
inside the rectangle spanned by the vertices.
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Pk
• Pk = {d ∈ Ak |d is planar}
d ∈ Ak is planar if it has some planar graph with all edges
inside the rectangle spanned by the vertices.
u
u
u
@
@
@
@
u @ u @u
u
planar
u
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Pk
• Pk = {d ∈ Ak |d is planar}
d ∈ Ak is planar if it has some planar graph with all edges
inside the rectangle spanned by the vertices.
u
u
u
@
@
@
@
u @ u @u
u
@
u
planar
u
u
u u
@ @
@
u
@u @u
u
not planar
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Pk
• Pk = {d ∈ Ak |d is planar}
d ∈ Ak is planar if it has some planar graph with all edges
inside the rectangle spanned by the vertices.
u
u
u
@
@
@
@
u @ u @u
u
u
@
u
u
@
@
@u @u
u
u
u
@
planar
u
not planar
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Pk
s
Let
pi+ 1 =
2
···
s
s
pj =
s
s
s
s
s
s
···
i +1
i
s
s
s
s
s
i 0 (i + 1)0
j
s
s
j0
s
s
s
···
s
s
···
s
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Pk
s
Let
pi+ 1 =
2
···
s
s
pj =
s
s
s
s
s
s
···
i +1
i
s
s
s
s
s
i 0 (i + 1)0
j
s
s
s
s
s
···
s
s
···
s
j0
Theorem
The monoid Pk is presented by generators p 1 , p1 , p 3 , . . . , pk and
2
2
relations
pi2 = pi ,
pi pi±1 pi = pi ,
and
pi pj = pj pi ,
for |i − j| > 1/2.
Partition Algebra
Ak and Sn
Submonoids: Bk
• Bk = {d ∈ Ak |all blocks have size 2}
(gives the Brauer algebra)
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Bk
• Bk = {d ∈ Ak |all blocks have size 2}
(gives the Brauer algebra)
u
u
u
u
u
@
@
u
u @u
∈ B4
Partition Algebra
Ak and Sn
Submonoids: Tk
• Tk = Pk ∩ Bk
(gives the Temperley-Lieb algebra)
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Tk
• Tk = Pk ∩ Bk
(gives the Temperley-Lieb algebra)
u
u
u
u
u u
u
u
∈ T4
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Tk
s
Let
ei =
s
s
···
s
i
s
s
i +1
s
s
i 0 (i + 1)0
s
s
s
···
s
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Tk
s
Let
ei =
s
s
···
s
i
s
s
i +1
s
s
s
s
s
···
s
i 0 (i + 1)0
Theorem
The monoid Tk is presented by generators e1 , . . . , ek−1 and
relations
ei2 = ei ,
ei ei±1 ei = ei ,
and
ei ej = ej ei ,
for |i − j| > 1.
Partition Algebra
Ak and Sn
Submonoids: Tk
Example
The relation ei ei±1 ei = ei means that in T4 , we have
e1 e2 e1 =
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
=
Representations of Sn
Partition Algebra
Ak and Sn
Submonoids: Tk
Example
The relation ei ei±1 ei = ei means that in T4 , we have
e1 e2 e1 =
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
=
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Tk
Example
The relation ei ei±1 ei = ei means that in T4 , we have
e1 e2 e1 =
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
=
s
s
s
s
s
s
s
s
Partition Algebra
Ak and Sn
Representations of Sn
Submonoids: Tk
Example
The relation ei ei±1 ei = ei means that in T4 , we have
e1 e2 e1 =
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
=
s
s
s
s
s
s
s
s
= e1
Partition Algebra
Ak and Sn
Representations of Sn
Sizes of Submonoids
For ` ∈ Z>0 , let
(2k)!! = (2k − 1)(2k − 3) · · · 5 · 3 · 1,
B(`) = the number of set partitions of {1, 2, . . . , `},
1
2`
C (`) =
`+1 `
and
Partition Algebra
Ak and Sn
Representations of Sn
Sizes of Submonoids
For ` ∈ Z>0 , let
(2k)!! = (2k − 1)(2k − 3) · · · 5 · 3 · 1,
B(`) = the number of set partitions of {1, 2, . . . , `},
1
2`
C (`) =
`+1 `
The submonoids described have the following sizes:
Submonoid
Ak
Sk
Ik
Ak+ 1
2
Pk
Bk
Tk
Size
B(2k)
k!
B(2k) − k!
B(2k + 1)
C (2k)
(2k)!!
C (k)
and
Partition Algebra
Ak and Sn
Representations of Sn
Presentation for Ak
Theorem
Halverson & Ram
The monoid Ak is presented by generators s1 , . . . , sk−1 and
p 1 , p1 , p 3 , . . . , pk and relations
2
2
pi2 = pi ,
si2 = 1,
pi pi±1 pi = pi ,
and
si si+1 si = si+1 si si+1 ,
pi pj = pj pi ,
and
si pi pi+1 = pi pi+1 si = pi pi+1 ,
si pi si = pi+1 ,
si pj = pj si ,
for |i − j| > 1/2;
for |i − j| > 1;
si sj = sj si ,
si pi+ 1 = pi+ 1 si = pi+ 1 ,
2
2
si si+1 pi+ 1 si+1 si = pi+ 3 ,
2
2
and
2
1
3
for j 6= i − 1, i, i + , i + 1, i + .
2
2
Partition Algebra
Ak and Sn
Presentation for Ak
Example
The relation si pi+ 1 = pi+ 1 means that in A4 , we have
2
s
s2 p2+ 1 =
2
2
s
s
s
s
@
s @s
s
s
s
s
s
s
s
s
s
=
Representations of Sn
Partition Algebra
Ak and Sn
Presentation for Ak
Example
The relation si pi+ 1 = pi+ 1 means that in A4 , we have
2
s
s2 p2+ 1 =
2
2
s
s
s
s
@
s @s
s
s
s
s
s
s
s
s
s
=
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Presentation for Ak
Example
The relation si pi+ 1 = pi+ 1 means that in A4 , we have
2
s
s2 p2+ 1 =
2
s
2
s
s
@
s @s
s
s
s
s
s
s
s
s
s
s
=
s
s
s
s
s
s
s
s
Partition Algebra
Ak and Sn
Representations of Sn
Presentation for Ak
Example
The relation si pi+ 1 = pi+ 1 means that in A4 , we have
2
s
s2 p2+ 1 =
2
s
2
s
s
@
s @s
s
s
s
s
s
s
s
s
s
s
=
s
s
s
s
s
s
s
s
= p2+ 1
2
Partition Algebra
Ak and Sn
Representations of Sn
The Partition Algebra
Fix k ∈ 12 Z>0 and n ∈ C.
Definition
CAk (n) = C-span{d ∈ Ak }
with multiplication d1 d2 = n` (d1 ◦ d2 ) where ` is the number of
blocks removed from the center row during the composition d1 ◦ d2 .
Partition Algebra
Ak and Sn
Multiplication in CAk (n)
Example
Representations of Sn
Partition Algebra
Ak and Sn
Multiplication in CAk (n)
Example
d1 = s
s
s
s
s
s
s
s
s
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
d1 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d1 = s
d1 ◦ d2 =
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d1 = s
d1 ◦ d2 =
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d1 = s
d1 ◦ d2 =
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d1 = s
d1 ◦ d2 =
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d1 = s
d1 ◦ d2 =
= d3
Partition Algebra
Ak and Sn
Representations of Sn
Multiplication in CAk (n)
Example
s
s
s
s
s
s
s
s
s
s
s
s
s
d2 = s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d1 = s
d1 ◦ d2 =
So d1 d2 = n2 d3 in CAk (n).
= d3
Partition Algebra
Ak and Sn
Representations of Sn
CIk
Recall that Ik = {d ∈ Ak |pn(d) < k} is a submonoid of Ak .
Definition
Define CIk (n) = C-span{d ∈ Ik }.
Partition Algebra
Ak and Sn
Representations of Sn
CIk
Recall that Ik = {d ∈ Ak |pn(d) < k} is a submonoid of Ak .
Definition
Define CIk (n) = C-span{d ∈ Ik }.
• CIk is an ideal of CAk (n).
Partition Algebra
Ak and Sn
Representations of Sn
CIk
Recall that Ik = {d ∈ Ak |pn(d) < k} is a submonoid of Ak .
Definition
Define CIk (n) = C-span{d ∈ Ik }.
• CIk is an ideal of CAk (n).
• CAk (n)/CIk (n) ∼
= CSk .
Partition Algebra
Ak and Sn
Representations of Sn
CIk
Recall that Ik = {d ∈ Ak |pn(d) < k} is a submonoid of Ak .
Definition
Define CIk (n) = C-span{d ∈ Ik }.
• CIk is an ideal of CAk (n).
• CAk (n)/CIk (n) ∼
= CSk .
• CIk is always isomorphic to a basic construction.
Partition Algebra
Ak and Sn
Representations of Sn
When is CAk (n) semisimple?
Theorem
Halverson & Ram
Let n ∈ Z≥2 and k ∈ 12 Z≥0 . Then
CAk (n) is semisimple if and only if k ≤
n+1
.
2
Partition Algebra
Ak and Sn
Representations of Sn
When is CAk (n) semisimple?
Theorem
Halverson & Ram
Let n ∈ Z≥2 and k ∈ 12 Z≥0 . Then
CAk (n) is semisimple if and only if k ≤
n+1
.
2
To learn more about CAk (n), we’ll examine how it acts on a
familiar vector space. . .
Partition Algebra
Ak and Sn
Representations of Sn
V ⊗k
Let V be an n-dimensional C-vector space with basis {v1 , . . . , vn }.
For k a positive integer,
V ⊗k = V ⊗ V ⊗ · · · ⊗ V
(k times)
has basis
{vi1 ⊗ · · · ⊗ vik }i1 ,...,ik ∈{1,...,n} .
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Given d ∈ Ak , and integers i1 , . . . , ik , i10 , . . . , ik0 ∈ {1, . . . , n}, we
define

 1 if whenever r and s are in the same block of d,
k
we have ir = is
(d)ii10,...,i
=
1 ,...,ik 0

0 otherwise
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Given d ∈ Ak , and integers i1 , . . . , ik , i10 , . . . , ik0 ∈ {1, . . . , n}, we
define

 1 if whenever r and s are in the same block of d,
k
we have ir = is
(d)ii10,...,i
=
1 ,...,ik 0

0 otherwise
We define an action of Ak on V ⊗k : if d ∈ Ak , then
X
k
d(vi1 ⊗ · · · ⊗ vik ) =
(d)ii10,...,i
,...,i 0 vi10 ⊗ · · · ⊗ vik 0
1
i10 ,...,ik 0
k
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Given d ∈ Ak , and integers i1 , . . . , ik , i10 , . . . , ik0 ∈ {1, . . . , n}, we
define

 1 if whenever r and s are in the same block of d,
k
we have ir = is
(d)ii10,...,i
=
1 ,...,ik 0

0 otherwise
We define an action of Ak on V ⊗k : if d ∈ Ak , then
X
k
d(vi1 ⊗ · · · ⊗ vik ) =
(d)ii10,...,i
,...,i 0 vi10 ⊗ · · · ⊗ vik 0
1
k
i10 ,...,ik 0
This action extends to a homomorphism Φk : CAk → End(V ⊗k ).
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Example
We’ll look at how an element of A2 acts on V ⊗2 when V = C2
with basis {v1 , v2 }.
Let
d = {{1, 20 }, {2, 10 }}
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Example
We’ll look at how an element of A2 acts on V ⊗2 when V = C2
with basis {v1 , v2 }.
Let
d = {{1, 20 }, {2, 10 }}
(d)1,1
1,1 = 1
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Example
We’ll look at how an element of A2 acts on V ⊗2 when V = C2
with basis {v1 , v2 }.
Let
d = {{1, 20 }, {2, 10 }}
(d)1,1
1,1 = 1
(d)1,1
1,2 = 0
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Example
We’ll look at how an element of A2 acts on V ⊗2 when V = C2
with basis {v1 , v2 }.
Let
d = {{1, 20 }, {2, 10 }}
(d)1,1
1,1
(d)1,1
1,2
1,1
(d)2,1
(d)1,1
2,2
=1
=0
=0
=0
(d)1,2
1,1
(d)1,2
1,2
(d)1,2
2,1
(d)1,2
2,2
=0
=0
=1
=0
(d)2,1
1,1
(d)2,1
1,2
(d)2,1
2,1
(d)2,1
2,2
=0
=1
=0
=0
(d)2,2
1,1
(d)2,2
1,2
(d)2,2
2,1
(d)2,2
2,2
=0
=0
=0
=1
Partition Algebra
Ak and Sn
Representations of Sn
CAk acts on V ⊗k
Example
We’ll look at how an element of A2 acts on V ⊗2 when V = C2
with basis {v1 , v2 }.
Let
d = {{1, 20 }, {2, 10 }}
(d)1,1
1,1
(d)1,1
1,2
1,1
(d)2,1
(d)1,1
2,2
=1
=0
=0
=0
(d)1,2
1,1
(d)1,2
1,2
(d)1,2
2,1
(d)1,2
2,2
=0
=0
=1
=0
(d)2,1
1,1
(d)2,1
1,2
(d)2,1
2,1
(d)2,1
2,2
=0
=1
=0
=0
(d)2,2
1,1
(d)2,2
1,2
(d)2,2
2,1
(d)2,2
2,2
=0
=0
=0
=1
So
d(v1 ⊗ v1 ) = v1 ⊗ v1
d(v1 ⊗ v2 ) = v2 ⊗ v1
etc.
Partition Algebra
Ak and Sn
Representations of Sn
GLn (C) acts on V ⊗k
GLn (C) acts on V by linear transformations, and on V ⊗k by acting
on each coordinate from V .
Partition Algebra
Ak and Sn
Representations of Sn
GLn (C) acts on V ⊗k
GLn (C) acts on V by linear transformations, and on V ⊗k by acting
on each coordinate from V .
View Sn as the subgroup of permutation matrices in GLn (C).
Partition Algebra
Ak and Sn
Representations of Sn
GLn (C) acts on V ⊗k
GLn (C) acts on V by linear transformations, and on V ⊗k by acting
on each coordinate from V .
View Sn as the subgroup of permutation matrices in GLn (C).
Definition
The centralizer algebra of Sn is
EndSn (V ⊗k ) = {b ∈ End(V ⊗k )|bσv = σbv for all σ ∈ Sn , v ∈ V ⊗k }
Partition Algebra
Ak and Sn
The Centralizer Algebra
Theorem
Halverson & Ram
Let n ∈ Z>0 and Φk : CAk → End(V ⊗k ) as before. Then
imΦk = EndSn (V ⊗k ).
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
The Centralizer Algebra
Theorem
Halverson & Ram
Let n ∈ Z>0 and Φk : CAk → End(V ⊗k ) as before. Then
imΦk = EndSn (V ⊗k ).
Also, ker Φk is the C-span of some special basis elements in Ak .
Partition Algebra
Ak and Sn
Representations of Sn
Decomposition of V ⊗k
Now we can view V ⊗k as a (CSn , CAk (n))-bimodule.
Let λ = {λ1 , λ2 , . . . , } be a partition of n and let |λ>1 | = n − λ1 .
Partition Algebra
Ak and Sn
Representations of Sn
Decomposition of V ⊗k
Now we can view V ⊗k as a (CSn , CAk (n))-bimodule.
Let λ = {λ1 , λ2 , . . . , } be a partition of n and let |λ>1 | = n − λ1 .
Theorem
Halverson & Ram
As a bimodule,
V ⊗k '
M
Snλ ⊗ Aλk (n),
λ∈Âk (n)
where
Âk (n) = {partitions λ of n : k − |λ>1 | ≥ 0},
and Aλk (n) are irreducible CAk (n)-modules.
Partition Algebra
Ak and Sn
Representations of Sn
What are the irreducible
representations of Sn over C?
These correspond to irreducible CSn -modules.
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Representations of Sn
What are the irreducible
representations of Sn over C?
These correspond to irreducible CSn -modules.
The irreducible CSn -modules are indexed by
partitions λ of n. They are called Specht
modules, and written Snλ .
Wilhelm Specht
Partition Algebra
Ak and Sn
Representations of Sn
Representations of Sn
What are the irreducible
representations of Sn over C?
These correspond to irreducible CSn -modules.
The irreducible CSn -modules are indexed by
partitions λ of n. They are called Specht
modules, and written Snλ .
In order to understand Specht modules, we
need to know a little bit about tableaux. . .
Wilhelm Specht
Partition Algebra
Ak and Sn
Representations of Sn
Young diagrams
To any partition λ of n, we associate a Young diagram with n
boxes. This Young diagram
corresponds to the partition λ = {5, 3, 3, 1} of 12. We say that our
diagram has size 12 and shape λ.
Partition Algebra
Ak and Sn
Tableaux and Sn
A tableau is a Young diagram of size n with the numbers
{1, . . . , n} assigned to its boxes.
1
7
4
2
8
6
5
11
9
12
3
10
Representations of Sn
Partition Algebra
Ak and Sn
Tableaux and Sn
A tableau is a Young diagram of size n with the numbers
{1, . . . , n} assigned to its boxes.
1
7
4
2
8
6
5
11
9
3
10
12
Sn acts on tableaux of size n by relabeling.
Representations of Sn
Partition Algebra
Ak and Sn
Tableaux and Sn
A tableau is a Young diagram of size n with the numbers
{1, . . . , n} assigned to its boxes.
1
7
4
2
8
6
5
11
9
12
3
10
σ = (3 7)(6 9 11 8)
Sn acts on tableaux of size n by relabeling.
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Tableaux and Sn
A tableau is a Young diagram of size n with the numbers
{1, . . . , n} assigned to its boxes.
1
7
4
2
8
5
11
12
3
10
1
3
4
6
2
6
9
9
5
8
11
σ = (3 7)(6 9 11 8)
12
Sn acts on tableaux of size n by relabeling.
7
10
Partition Algebra
Ak and Sn
Representations of Sn
Tableaux and Sn
A tableau is a Young diagram of size n with the numbers
{1, . . . , n} assigned to its boxes.
1
7
4
2
8
5
11
12
3
10
1
3
4
6
2
6
9
9
5
8
11
σ = (3 7)(6 9 11 8)
7
10
12
Sn acts on tableaux of size n by relabeling.
A standard Young tableau is a Young diagram with the numbers
{1, . . . , n} in its boxes, increasing down columns and left to right.
Partition Algebra
Ak and Sn
Tabloids
Definition
Given a tableau T of size n, the row-stabilizer of T is
RT = {σ ∈ Sn : σ fixes the rows of T setwise}
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Tabloids
Definition
Given a tableau T of size n, the row-stabilizer of T is
RT = {σ ∈ Sn : σ fixes the rows of T setwise}
and the the column-stabilizer of T is
QT = {σ ∈ Sn : σ fixes the columns of T setwise}.
Partition Algebra
Ak and Sn
Representations of Sn
Tabloids
Definition
Given a tableau T of size n, the row-stabilizer of T is
RT = {σ ∈ Sn : σ fixes the rows of T setwise}
and the the column-stabilizer of T is
QT = {σ ∈ Sn : σ fixes the columns of T setwise}.
Definition
A tabloid is an equivalence class of tableaux of the same shape,
where T ∼ S provided σT = S for some σ ∈ RT .
Partition Algebra
Ak and Sn
Representations of Sn
Tabloids
Definition
Given a tableau T of size n, the row-stabilizer of T is
RT = {σ ∈ Sn : σ fixes the rows of T setwise}
and the the column-stabilizer of T is
QT = {σ ∈ Sn : σ fixes the columns of T setwise}.
Definition
A tabloid is an equivalence class of tableaux of the same shape,
where T ∼ S provided σT = S for some σ ∈ RT .
Sn acts on the set of tabloids of a given shape.
Partition Algebra
Ak and Sn
Specht Modules
• Start with a partition λ = {λ1 , . . . , λk }
Representations of Sn
Partition Algebra
Ak and Sn
Specht Modules
• Start with a partition λ = {λ1 , . . . , λk }
• Associate to λ the C-vector space
V λ = C-span{vT |T is a tabloid of shape λ}.
Representations of Sn
Partition Algebra
Ak and Sn
Specht Modules
• Start with a partition λ = {λ1 , . . . , λk }
• Associate to λ the C-vector space
V λ = C-span{vT |T is a tabloid of shape λ}.
• The dimension of V λ is
n!
.
λ1 !λ2 ! · · · λk !
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Specht Modules
• Let T be a standard Young tableau of shape λ. Associate to
T the vector in V λ given by
X
ET =
sgn(σ)vσ(T )
σ∈QT
This ET is called a standard polytabloid.
Partition Algebra
Ak and Sn
Representations of Sn
Specht Modules
• Let T be a standard Young tableau of shape λ. Associate to
T the vector in V λ given by
X
ET =
sgn(σ)vσ(T )
σ∈QT
This ET is called a standard polytabloid.
• The subspace of V λ spanned by all ET (corresponding to all
standard tableaux T of shape λ) is a CSn -module, the
Specht module Snλ .
Partition Algebra
Ak and Sn
Representations of Sn
Specht Modules
• Let T be a standard Young tableau of shape λ. Associate to
T the vector in V λ given by
X
ET =
sgn(σ)vσ(T )
σ∈QT
This ET is called a standard polytabloid.
• The subspace of V λ spanned by all ET (corresponding to all
standard tableaux T of shape λ) is a CSn -module, the
Specht module Snλ .
• These modules are irreducible and distinct, and make up all of
the irreducible CSn -modules.
Partition Algebra
Ak and Sn
Recap
As a (CSn , CAk (n))-bimodule,
M
V ⊗k '
Snλ ⊗ Aλk (n).
λ∈Âk (n)
Representations of Sn
Partition Algebra
Ak and Sn
Representations of Sn
Recap
As a (CSn , CAk (n))-bimodule,
M
V ⊗k '
Snλ ⊗ Aλk (n).
λ∈Âk (n)
We know
dimSnλ = Q
k!
,
h(b)
b∈λ
where h(b) is the hook length of box b in the Young diagram for λ.
Partition Algebra
Ak and Sn
Representations of Sn
Recap
As a (CSn , CAk (n))-bimodule,
M
V ⊗k '
Snλ ⊗ Aλk (n).
λ∈Âk (n)
We know
dimSnλ = Q
k!
,
h(b)
b∈λ
where h(b) is the hook length of box b in the Young diagram for λ.
We also have a formula for dimAλk (n).
Partition Algebra
Ak and Sn
Representations of Sn
Recap
As a (CSn , CAk (n))-bimodule,
M
V ⊗k '
Snλ ⊗ Aλk (n).
λ∈Âk (n)
We know
dimSnλ = Q
k!
,
h(b)
b∈λ
where h(b) is the hook length of box b in the Young diagram for λ.
We also have a formula for dimAλk (n).
This information allows us to break down V ⊗k as a CSn -module or
as a CAk (n)-module.
Partition Algebra
Ak and Sn
Representations of Sn
Recap
CAk (n)
V ⊗k
Sn
Partition Algebra
Ak and Sn
Representations of Sn
Recap
V ⊗k
Sn
⊂
CAk (n)
CSk
Partition Algebra
Ak and Sn
Representations of Sn
Recap
Sn
CAk (n)
V ⊗k
⊂
⊂
GLn (C)
CSk
Partition Algebra
Ak and Sn
Representations of Sn
Recap
GLn (C)
V ⊗k
CBk (n)
⊂
⊂
⊂
CAk (n)
Sn
CSk
Partition Algebra
Ak and Sn
Representations of Sn
Recap
GLn (C)
CBk (n)
⊂
V ⊗k
⊂
On (C)
⊂
⊂
CAk (n)
Sn
CSk
Partition Algebra
Ak and Sn
Representations of Sn
Recap
GLn (C)
CBk (n)
⊂
V ⊗k
⊂
On (C)
⊂
⊂
⊂
CAk (n)
Sn
CSk
CTk , CPk
Partition Algebra
Ak and Sn
Thank you!
Representations of Sn