The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley–Lieb algebra, or the group algebra of the symmetric group. Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group.
A partition of
2k
1,\bar1,2,\bar2,...,k,\bark
\{\bar1,\bar4,\bar5,6\}
\bar1-\bar4,\bar4-\bar5,\bar5-6
\bar1-6,\bar4-6,\bar5-6,\bar1-\bar5
For
n\inC
k\inN*
Pk(n)
C
nD
D
The partition algebra
Pk(n)
3k-2
These generators obey relations that include
2 | |
s | |
i |
=1 , sisi+1si=si+1sisi+1 ,
2 | |
p | |
i |
=npi ,
2= | |
b | |
i |
bi , pibipi=pi
In terms of the original generators, these elements are
ei=bipipi+1bi , li=sipi , ri=pisi
The partition algebra
Pk(n)
The partition algebra
Pk(n)
n\inC-\{0,1,...,2k-2\}
n,n'
Pk(n)
Pk(n')
The partition algebra is finite-dimensional, with
\dimPk(n)=B2k
Subalgebras of the partition algebra can be defined by the following properties:
1,2,...,2k
1,2
2
n
pi\to
1 | |
n |
pi
Combining these properties gives rise to 8 nontrivial subalgebras, in addition to the partition algebra itself:
Notation | Name | Generators | Dimension | Example | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Pk(n) | Partition | si,pi,bi | B2k | ||||||||||
PPk(n) | Planar partition | pi,bi |
\binom{4k}{2k} | ||||||||||
RBk(n) | Rook Brauer | si,ei,pi |
\binom{2k}{2\ell}(2\ell-1)! | ||||||||||
Mk(n) | Motzkin | ei,li,ri |
\binom{2\ell}{\ell}\binom{2k}{2\ell} | ||||||||||
Bk(n) | si,ei | (2k-1)! | |||||||||||
TLk(n) | Temperley–Lieb | ei |
\binom{2k}{k} | ||||||||||
Rk | Rook | si,pi |
\binom{k}{\ell}2\ell! | ||||||||||
PRk | Planar rook | li,ri | \binom{2k}{k} | ||||||||||
CSk | Symmetric group | si | k! |
CSk
Sk
C
The listed subalgebras are semisimple for
n\inC-\{0,1,...,2k-2\}
Inclusions of planar into non-planar algebras:
PPk(n)\subsetPk(n) , Mk(n)\subsetRBk(n) , TLk(n)\subsetBk(n) , PRk\subsetRk
Bk(n)\subsetRBk(n)\subsetPk(n) , TLk(n)\subsetMk(n)\subsetPPk(n) , CSk\subsetRk
Rk\subsetRBk(n) , PRk\subsetMk(n) , CSk\subsetBk(n)
2) | |
PP | |
k(n |
\congTL2k(n) , \left\{\begin{array}{l}pi\mapstone2i-1\ bi\mapsto
1 | |
n |
e2i\end{array}\right.
In addition to the eight subalgebras described above, other subalgebras have been defined:
propPk
si,bipi+1bi+1
QPk(n)
si,bi,ei
2k | |
1+\sum | |
j=1 |
(-1)j-1B2k-j
Uk
si,bi
An algebra with a half-integer index
k+ | 12 |
2k+2
k+1
\overline{k+1}
P | ||||
|
si\leq,bi\leq,pi\leq
Pk\subset
P | ||||
|
\subsetPk+1
\dim
P | ||||
|
=B2k+1
Periodic subalgebras are generated by diagrams that can be drawn on an annulus without line crossings. Such subalgebras include a translation element
u=
uk=1
si
For an integer
0\leq\ell\leqk
D\ell
k+\ell
1,2,...,k
\bar1,\bar2,...,\bar\ell
k=12,\ell=5
Partition diagrams act on
D\ell
S\ell
Vλ
S\ell
|λ|=\ell
Pk(n)
l{P}λ=CD|λ|
⊗ | |
CS|λ| |
Vλ .
\diml{P}λ=fλ
k | |
\sum | |
\ell=|λ| |
\left\{{k\atop\ell}\right\}\binom{\ell}{|λ|} ,
\left\{{k\atop\ell}\right\}
\binom{\ell}{|λ|}
fλ=\dimSλ
A basis of
l{P}λ
Assuming that
Pk(n)
l{P}λ
Irrep\left(Pk(n)\right)=\left\{l{P}λ\right\}0\leq .
Representations of non-planar subalgebras have similar structures as representations of the partition algebra. For example, the Brauer-Specht modules of the Brauer algebra are built from Specht modules, and certain sets of partitions.
In the case of the planar subalgebras, planarity prevents nontrivial permutations, and Specht modules do not appear. For example, a standard module of the Temperley–Lieb algebra is parametrized by an integer
0\leq\ell\leqk
\ell\equivk\bmod2
The following table lists the irreducible representations of the partition algebra and eight subalgebras.
Algebra | Parameter | Conditions | Dimension | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Pk(n) | λ | 0\leq | \lambda | \leq k | fλ
\left\{{k\atop\ell}\right\}\binom{\ell}{|λ|} | |||||||||||
PPk(n) | \ell | 0\leq\ell\leqk | \binom{2k}{k+\ell}-\binom{2k}{k+\ell+1} | |||||||||||||
RBk(n) | λ | 0\leq | \lambda | \leq k | fλ\binom{k}{|λ|}
\binom{k-|λ|}{2m}(2m-1)! | |||||||||||
Mk(n) | \ell | 0\leq\ell\leqk |
\binom{k}{\ell+2m}\left\{\binom{\ell+2m}{m}-\binom{\ell+2m}{m-1}\right\} | |||||||||||||
Bk(n) | λ | \begin{array}{c}0\leq | \lambda | \leq k \\ | \lambda | \equiv k\bmod 2\end | fλ\binom{k}{|λ|}(k- | \lambda | -1)! | |||||||
TLk(n) | \ell | \begin{array}{c}0\leq\ell\leqk\ \ell\equivk\bmod2\end{array} |
| |||||||||||||
Rk | λ | 0\leq | \lambda | \leq k | fλ\binom{k}{|λ|} | |||||||||||
PRk | \ell | 0\leq\ell\leqk | \binom{k}{\ell} | |||||||||||||
CSk | λ | \lambda | =k | fλ |
propPk
0<|λ|\leqk
fλ\left\{{k\atop|λ|}\right\}
QPk
0\leq|λ|\leqk
Uk
Assume
n\inN*
V
n
v1,...,vn
Pk(n)
V ⊗
\{1,\bar1,2,\bar2,...,k,\bark\}=\sqcuphEh
(v | |
j1 |
⊗ … ⊗
v | |
jk |
)
\left(\sqcuphEh\right)
j\bar,j\bar,...,j\bar | |
j1,j2,...,jk |
=
1 | |
r,s\inEh\impliesjr=js |
.
ei
\left(v | |
j1 |
⊗ … ⊗
v | |
ji |
⊗
v | |
ji+1 |
⊗ … ⊗
v | |
jk |
\right) =
\delta | |
ji,ji+1 |
n | |
\sum | |
j=1 |
v | |
j1 |
⊗ … ⊗ vj ⊗ vj ⊗ … ⊗
v | |
jk |
.
Let
n\geq2k
V
Sn
n
V=[n-1,1] ⊕ [n]
Then the partition algebra
Pk(n)
Sn
V ⊗
Pk(n)\cong
End | |
Sn |
\left(V ⊗ \right) .
Pk(n) x Sn
V ⊗ =oplus0\leql{P}λ ⊗ V[n-|λ|,λ] ,
[n-|λ|,λ]
n
λ
V[n-|λ|,λ]
Sn
The duality between the symmetric group and the partition algebra generalizes the original Schur-Weyl duality between the general linear group and the symmetric group. There are other generalizations. In the relevant tensor product spaces, we write
Vn
n
Tensor product space | Group or algebra | Dual algebra or group | Comments | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\left(Vn-1 ⊕
| Sn | Pk(n) | The duality for the full partition algebra | |||||||||||||
\left(Vn-2 ⊕ V1 ⊕
| Sn-1 |
(n) | Case of a partition algebra with a half-integer index | |||||||||||||
| GLn(C) | Sk | The original Schur-Weyl duality | |||||||||||||
| O(n) | Bk(n) | Duality between the orthogonal group and the Brauer algebra | |||||||||||||
\left(Vn ⊕
| O(n) | RBk(n+1) | Duality between the orthogonal group and the rook Brauer algebra | |||||||||||||
| Rn | propPk | Duality between the rook algebra and the totally propagating partition algebra | |||||||||||||
| gl(1 | 1) | PRk-1 | Duality between a Lie superalgebra and the planar rook algebra | ||||||||||||
| Sn | QPk(n) | Duality between the symmetric group and the quasi-partition algebra | |||||||||||||
⊗
⊗ | GLn(C) | Br,s(n) | Duality involving the walled Brauer algebra. |