Partition algebra

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation.^[1] 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.

Definition

Diagrams

A partition of ${\displaystyle 2k}$ elements labelled ${\displaystyle 1,{\bar {1}},2,{\bar {2}},\dots ,k,{\bar {k}}}$ is represented as a diagram, with lines connecting elements in the same subset. In the following example, the subset ${\displaystyle \{{\bar {1}},{\bar {4}},{\bar {5}},6\}}$ gives rise to the lines ${\displaystyle {\bar {1}}-{\bar {4}},{\bar {4}}-{\bar {5}},{\bar {5}}-6}$, and could equivalently be represented by the lines ${\displaystyle {\bar {1}}-6,{\bar {4}}-6,{\bar {5}}-6,{\bar {1}}-{\bar {5}}}$ (for instance).

For ${\displaystyle n\in \mathbb {C} }$ and ${\displaystyle k\in \mathbb {N} ^{*}}$, the partition algebra ${\displaystyle P_{k}(n)}$ is defined by a ${\displaystyle \mathbb {C} }$-basis made of partitions, and a multiplication given by diagram concatenation. The concatenated diagram comes with a factor ${\displaystyle n^{D}}$, where ${\displaystyle D}$ is the number of connected components that are disconnected from the top and bottom elements.

Generators and relations

The partition algebra ${\displaystyle P_{k}(n)}$ is generated by ${\displaystyle 3k-2}$ elements of the type

These generators obey relations that include^[2]

${\displaystyle s_{i}^{2}=1\quad ,\quad s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}\quad ,\quad p_{i}^{2}=np_{i}\quad ,\quad b_{i}^{2}=b_{i}\quad ,\quad p_{i}b_{i}p_{i}=p_{i}}$

Other elements that are useful for generating subalgebras include

In terms of the original generators, these elements are

${\displaystyle e_{i}=b_{i}p_{i}p_{i+1}b_{i}\quad ,\quad l_{i}=s_{i}p_{i}\quad ,\quad r_{i}=p_{i}s_{i}}$

Properties

The partition algebra ${\displaystyle P_{k}(n)}$ is an associative algebra. It has a multiplicative identity

The partition algebra ${\displaystyle P_{k}(n)}$ is semisimple for ${\displaystyle n\in \mathbb {C} -\{0,1,\dots ,2k-2\}}$. For any two ${\displaystyle n,n'}$ in this set, the algebras ${\displaystyle P_{k}(n)}$ and ${\displaystyle P_{k}(n')}$ are isomorphic.^[1]

The partition algebra is finite-dimensional, with ${\displaystyle \dim P_{k}(n)=B_{2k}}$ (a Bell number).

Subalgebras

Eight subalgebras

Subalgebras of the partition algebra can be defined by the following properties:^[3]

• Whether they are planar i.e. whether lines may cross in diagrams.
• Whether subsets are allowed to have any size ${\displaystyle 1,2,\dots ,2k}$, or size ${\displaystyle 1,2}$, or only size ${\displaystyle 2}$.
• Whether we allow top-top and bottom-bottom lines, or only top-bottom lines. In the latter case, the parameter ${\displaystyle n}$ is absent, or can be eliminated by ${\displaystyle p_{i}\to {\frac {1}{n}}p_{i}}$.

Combining these properties gives rise to 8 nontrivial subalgebras, in addition to the partition algebra itself:^[1][3]

Notation	Name	Generators	Dimension	Example
${\displaystyle P_{k}(n)}$	Partition	${\displaystyle s_{i},p_{i},b_{i}}$	${\displaystyle B_{2k}}$
${\displaystyle PP_{k}(n)}$	Planar partition	${\displaystyle p_{i},b_{i}}$	${\displaystyle {\frac {1}{2k+1}}{\binom {4k}{2k}}}$
${\displaystyle RB_{k}(n)}$	Rook Brauer	${\displaystyle s_{i},e_{i},p_{i}}$	${\displaystyle \sum _{\ell =0}^{k}{\binom {2k}{2\ell }}(2\ell -1)!!}$
${\displaystyle M_{k}(n)}$	Motzkin	${\displaystyle e_{i},l_{i},r_{i}}$	${\displaystyle \sum _{\ell =0}^{k}{\frac {1}{\ell +1}}{\binom {2\ell }{\ell }}{\binom {2k}{2\ell }}}$
${\displaystyle B_{k}(n)}$	Brauer	${\displaystyle s_{i},e_{i}}$	${\displaystyle (2k-1)!!}$
${\displaystyle TL_{k}(n)}$	Temperley–Lieb	${\displaystyle e_{i}}$	${\displaystyle {\frac {1}{k+1}}{\binom {2k}{k}}}$
${\displaystyle R_{k}}$	Rook	${\displaystyle s_{i},p_{i}}$	${\displaystyle \sum _{\ell =0}^{k}{\binom {k}{\ell }}^{2}\ell !}$
${\displaystyle PR_{k}}$	Planar rook	${\displaystyle l_{i},r_{i}}$	${\displaystyle {\binom {2k}{k}}}$
${\displaystyle \mathbb {C} S_{k}}$	Symmetric group	${\displaystyle s_{i}}$	${\displaystyle k!}$

The symmetric group algebra ${\displaystyle \mathbb {C} S_{k}}$ is the group ring of the symmetric group ${\displaystyle S_{k}}$ over ${\displaystyle \mathbb {C} }$. The Motzkin algebra is sometimes called the dilute Temperley–Lieb algebra in the physics literature.^[4]

Properties

The listed subalgebras are semisimple for ${\displaystyle n\in \mathbb {C} -\{0,1,\dots ,2k-2\}}$.

Inclusions of planar into non-planar algebras:

${\displaystyle PP_{k}(n)\subset P_{k}(n)\quad ,\quad M_{k}(n)\subset RB_{k}(n)\quad ,\quad TL_{k}(n)\subset B_{k}(n)\quad ,\quad PR_{k}\subset R_{k}}$

Inclusions from constraints on subset size:

${\displaystyle B_{k}(n)\subset RB_{k}(n)\subset P_{k}(n)\quad ,\quad TL_{k}(n)\subset M_{k}(n)\subset PP_{k}(n)\quad ,\quad \mathbb {C} S_{k}\subset R_{k}}$

Inclusions from allowing top-top and bottom-bottom lines:

${\displaystyle R_{k}\subset RB_{k}(n)\quad ,\quad PR_{k}\subset M_{k}(n)\quad ,\quad \mathbb {C} S_{k}\subset B_{k}(n)}$

We have the isomorphism:

${\displaystyle PP_{k}(n^{2})\cong TL_{2k}(n)\quad ,\quad \left\{{\begin{array}{l}p_{i}\mapsto ne_{2i-1}\\b_{i}\mapsto {\frac {1}{n}}e_{2i}\end{array}}\right.}$

More subalgebras

In addition to the eight subalgebras described above, other subalgebras have been defined:

• The totally propagating partition subalgebra ${\displaystyle {\text{prop}}P_{k}}$ is generated by diagrams whose blocks all propagate, i.e. partitions whose subsets all contain top and bottom elements.^[5] These diagrams from the dual symmetric inverse monoid, which is generated by ${\displaystyle s_{i},b_{i}p_{i+1}b_{i+1}}$.^[6]
• The quasi-partition algebra ${\displaystyle QP_{k}(n)}$ is generated by subsets of size at least two. Its generators are ${\displaystyle s_{i},b_{i},e_{i}}$ and its dimension is ${\displaystyle 1+\sum _{j=1}^{2k}(-1)^{j-1}B_{2k-j}}$.^[7]
• The uniform block permutation algebra ${\displaystyle U_{k}}$ is generated by subsets with as many top elements as bottom elements. It is generated by ${\displaystyle s_{i},b_{i}}$.^[8]

An algebra with a half-integer index ${\displaystyle k+{\frac {1}{2}}}$ is defined from partitions of ${\displaystyle 2k+2}$ elements by requiring that ${\displaystyle k+1}$ and ${\displaystyle {\overline {k+1}}}$ are in the same subset. For example, ${\displaystyle P_{k+{\frac {1}{2}}}}$ is generated by ${\displaystyle s_{i\leq k-1},b_{i\leq k},p_{i\leq k}}$ so that ${\displaystyle P_{k}\subset P_{k+{\frac {1}{2}}}\subset P_{k+1}}$, and ${\displaystyle \dim P_{k+{\frac {1}{2}}}=B_{2k+1}}$.^[2]

Periodic subalgebras are generated by diagrams that can be drawn on an annulus without line crossings. Such subalgebras include a translation element ${\displaystyle u=}$

such that ${\displaystyle u^{k}=1}$. The translation element and its powers are the only combinations of ${\displaystyle s_{i}}$ that belong to periodic subalgebras.

Representations

Structure

For an integer ${\displaystyle 0\leq \ell \leq k}$, let ${\displaystyle D_{\ell }}$ be the set of partitions of ${\displaystyle k+\ell }$ elements ${\displaystyle 1,2,\dots ,k}$ (bottom) and ${\displaystyle {\bar {1}},{\bar {2}},\dots ,{\bar {\ell }}}$ (top), such that no two top elements are in the same subset, and no top element is alone. Such partitions are represented by diagrams with no top-top lines, with at least one line for each top element. For example, in the case ${\displaystyle k=12,\ell =5}$:

Partition diagrams act on ${\displaystyle D_{\ell }}$ from the bottom, while the symmetric group ${\displaystyle S_{\ell }}$ acts from the top. For any Specht module ${\displaystyle V_{\lambda }}$ of ${\displaystyle S_{\ell }}$ (with therefore ${\displaystyle |\lambda |=\ell }$), we define the representation of ${\displaystyle P_{k}(n)}$

${\displaystyle {\mathcal {P}}_{\lambda }=\mathbb {C} D_{|\lambda |}\otimes _{\mathbb {C} S_{|\lambda |}}V_{\lambda }\ .}$

The dimension of this representation is^[1]

${\displaystyle \dim {\mathcal {P}}_{\lambda }=f_{\lambda }\sum _{\ell =|\lambda |}^{k}\left\{{k \atop \ell }\right\}{\binom {\ell }{|\lambda |}}\ ,}$

where ${\displaystyle \left\{{k \atop \ell }\right\}}$ is a Stirling number of the second kind, ${\displaystyle {\binom {\ell }{|\lambda |}}}$ is a binomial coefficient, and ${\displaystyle f_{\lambda }=\dim S_{\lambda }}$ is given by the hook length formula.

A basis of ${\displaystyle {\mathcal {P}}_{\lambda }}$ can be described combinatorially in terms of set-partition tableaux: Young tableaux whose boxes are filled with the blocks of a set partition.^[1]

Assuming that ${\displaystyle P_{k}(n)}$ is semisimple, the representation ${\displaystyle {\mathcal {P}}_{\lambda }}$ is irreducible, and the set of irreducible finite-dimensional representations of the partition algebra is

${\displaystyle {\text{Irrep}}\left(P_{k}(n)\right)=\left\{{\mathcal {P}}_{\lambda }\right\}_{0\leq |\lambda |\leq k}\ .}$

Representations of subalgebras

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 ${\displaystyle 0\leq \ell \leq k}$ with ${\displaystyle \ell \equiv k{\bmod {2}}}$, and a basis is simply given by a set of partitions.

The following table lists the irreducible representations of the partition algebra and eight subalgebras.^[3]

Algebra	Parameter	Conditions	Dimension
${\displaystyle P_{k}(n)}$	${\displaystyle \lambda }$	${\displaystyle 0\leq |\lambda |\leq k}$	${\displaystyle f_{\lambda }\sum _{\ell =|\lambda |}^{k}\left\{{k \atop \ell }\right\}{\binom {\ell }{|\lambda |}}}$
${\displaystyle PP_{k}(n)}$	${\displaystyle \ell }$	${\displaystyle 0\leq \ell \leq k}$	${\displaystyle {\binom {2k}{k+\ell }}-{\binom {2k}{k+\ell +1}}}$
${\displaystyle RB_{k}(n)}$	${\displaystyle \lambda }$	${\displaystyle 0\leq |\lambda |\leq k}$	${\displaystyle f_{\lambda }{\binom {k}{|\lambda |}}\sum _{m=0}^{\left\lfloor {\frac {k-|\lambda |}{2}}\right\rfloor }{\binom {k-|\lambda |}{2m}}(2m-1)!!}$
${\displaystyle M_{k}(n)}$	${\displaystyle \ell }$	${\displaystyle 0\leq \ell \leq k}$	${\displaystyle \sum _{m=0}^{\left\lfloor {\frac {k-\ell }{2}}\right\rfloor }{\binom {k}{\ell +2m}}\left\{{\binom {\ell +2m}{m}}-{\binom {\ell +2m}{m-1}}\right\}}$
${\displaystyle B_{k}(n)}$	${\displaystyle \lambda }$	${\displaystyle {\begin{array}{c}0\leq |\lambda |\leq k\\|\lambda |\equiv k{\bmod {2}}\end{array}}}$	${\displaystyle f_{\lambda }{\binom {k}{|\lambda |}}(k-|\lambda |-1)!!}$
${\displaystyle TL_{k}(n)}$	${\displaystyle \ell }$	${\displaystyle {\begin{array}{c}0\leq \ell \leq k\\\ell \equiv k{\bmod {2}}\end{array}}}$	${\displaystyle {\binom {k}{\frac {k+\ell }{2}}}-{\binom {k}{\frac {k+\ell +2}{2}}}}$
${\displaystyle R_{k}}$	${\displaystyle \lambda }$	${\displaystyle 0\leq |\lambda |\leq k}$	${\displaystyle f_{\lambda }{\binom {k}{|\lambda |}}}$
${\displaystyle PR_{k}}$	${\displaystyle \ell }$	${\displaystyle 0\leq \ell \leq k}$	${\displaystyle {\binom {k}{\ell }}}$
${\displaystyle \mathbb {C} S_{k}}$	${\displaystyle \lambda }$	${\displaystyle |\lambda |=k}$	${\displaystyle f_{\lambda }}$

The irreducible representations of ${\displaystyle {\text{prop}}P_{k}}$ are indexed by partitions such that ${\displaystyle 0<|\lambda |\leq k}$ and their dimensions are ${\displaystyle f_{\lambda }\left\{{k \atop |\lambda |}\right\}}$.^[5] The irreducible representations of ${\displaystyle QP_{k}}$ are indexed by partitions such that ${\displaystyle 0\leq |\lambda |\leq k}$.^[7] The irreducible representations of ${\displaystyle U_{k}}$ are indexed by sequences of partitions.^[8]

Schur-Weyl duality

Assume ${\displaystyle n\in \mathbb {N} ^{*}}$. For ${\displaystyle V}$ a ${\displaystyle n}$-dimensional vector space with basis ${\displaystyle v_{1},\dots ,v_{n}}$, there is a natural action of the partition algebra ${\displaystyle P_{k}(n)}$ on the vector space ${\displaystyle V^{\otimes k}}$. This action is defined by the matrix elements of a partition ${\displaystyle \{1,{\bar {1}},2,{\bar {2}},\dots ,k,{\bar {k}}\}=\sqcup _{h}E_{h}}$ in the basis ${\displaystyle (v_{j_{1}}\otimes \cdots \otimes v_{j_{k}})}$:^[2]

${\displaystyle \left(\sqcup _{h}E_{h}\right)_{j_{1},j_{2},\dots ,j_{k}}^{j_{\bar {1}},j_{\bar {2}},\dots ,j_{\bar {k}}}=\mathbf {1} _{r,s\in E_{h}\implies j_{r}=j_{s}}\ .}$

This matrix element is one if all indices corresponding to any given partition subset coincide, and zero otherwise. For example, the action of a Temperley–Lieb generator is

${\displaystyle e_{i}\left(v_{j_{1}}\otimes \cdots \otimes v_{j_{i}}\otimes v_{j_{i+1}}\otimes \cdots \otimes v_{j_{k}}\right)=\delta _{j_{i},j_{i+1}}\sum _{j=1}^{n}v_{j_{1}}\otimes \cdots \otimes v_{j}\otimes v_{j}\otimes \cdots \otimes v_{j_{k}}\ .}$

Duality between the partition algebra and the symmetric group

Let ${\displaystyle n\geq 2k}$ be integer. Let us take ${\displaystyle V}$ to be the natural permutation representation of the symmetric group ${\displaystyle S_{n}}$. This ${\displaystyle n}$-dimensional representation is a sum of two irreducible representations: the standard and trivial representations, ${\displaystyle V=[n-1,1]\oplus [n]}$.

Then the partition algebra ${\displaystyle P_{k}(n)}$ is the centralizer of the action of ${\displaystyle S_{n}}$ on the tensor product space ${\displaystyle V^{\otimes k}}$,

${\displaystyle P_{k}(n)\cong {\text{End}}_{S_{n}}\left(V^{\otimes k}\right)\ .}$

Moreover, as a bimodule over ${\displaystyle P_{k}(n)\times S_{n}}$, the tensor product space decomposes into irreducible representations as^[1]

${\displaystyle V^{\otimes k}=\bigoplus _{0\leq |\lambda |\leq k}{\mathcal {P}}_{\lambda }\otimes V_{[n-|\lambda |,\lambda ]}\ ,}$

where ${\displaystyle [n-|\lambda |,\lambda ]}$ is a Young diagram of size ${\displaystyle n}$ built by adding a first row to ${\displaystyle \lambda }$, and ${\displaystyle V_{[n-|\lambda |,\lambda ]}}$ is the corresponding Specht module of ${\displaystyle S_{n}}$.

Dualities involving subalgebras

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 ${\displaystyle V_{n}}$ for an irreducible ${\displaystyle n}$-dimensional representation of the first group or algebra:

Tensor product space	Group or algebra	Dual algebra or group	Comments
${\displaystyle \left(V_{n-1}\oplus V_{1}\right)^{\otimes k}}$	${\displaystyle S_{n}}$	${\displaystyle P_{k}(n)}$	The duality for the full partition algebra
${\displaystyle \left(V_{n-2}\oplus V_{1}\oplus V_{1}\right)^{\otimes k}}$	${\displaystyle S_{n-1}}$	${\displaystyle P_{k+{\frac {1}{2}}}(n)}$	Case of a partition algebra with a half-integer index[2]
${\displaystyle V_{n}^{\otimes k}}$	${\displaystyle GL_{n}(\mathbb {C} )}$	${\displaystyle S_{k}}$	The original Schur-Weyl duality
${\displaystyle V_{n}^{\otimes k}}$	${\displaystyle O(n)}$	${\displaystyle B_{k}(n)}$	Duality between the orthogonal group and the Brauer algebra
${\displaystyle \left(V_{n}\oplus V_{1}\right)^{\otimes k}}$	${\displaystyle O(n)}$	${\displaystyle RB_{k}(n+1)}$	Duality between the orthogonal group and the rook Brauer algebra[9]
${\displaystyle V_{n}^{\otimes k}}$	${\displaystyle R_{n}}$	${\displaystyle {\text{prop}}P_{k}}$	Duality between the rook algebra and the totally propagating partition algebra[10][5]
${\displaystyle V_{2}^{\otimes k}}$	${\displaystyle gl(1|1)}$	${\displaystyle PR_{k-1}}$	Duality between a Lie superalgebra and the planar rook algebra[11]
${\displaystyle V_{n-1}^{\otimes k}}$	${\displaystyle S_{n}}$	${\displaystyle QP_{k}(n)}$	Duality between the symmetric group and the quasi-partition algebra[7]
${\displaystyle V_{n}^{\otimes r}\otimes \left(V_{n}^{*}\right)^{\otimes s}}$	${\displaystyle GL_{n}(\mathbb {C} )}$	${\displaystyle B_{r,s}(n)}$	Duality involving the walled Brauer algebra.[12]

References

Further reading

• Kauffman, Louis H. (1991). Knots and Physics. World Scientific. ISBN 978-981-02-0343-6.
• Kauffman, Louis H. (1990). "An invariant of regular isotopy". Transactions of the American Mathematical Society. 318 (2): 417–471. doi:10.1090/S0002-9947-1990-0958895-7. ISSN 0002-9947.