partition algebra

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

Diagram representation of a partition of 14 elements

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

File:Partition concatenation.svg

The partition algebra $P\_k(n)$ is generated by $3k-2$ elements of the type

File:Partition generators.svg

These generators obey relations that include

:$$

s_i^2 = 1 \quad , \quad s_is_{i+1}s_i = s_{i+1}s_is_{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

File:Partition subgenerators.svg

In terms of the original generators, these elements are

:$e\_i\; =\; b\_ip\_ip\_\{i+1\}b\_i\; \backslash quad\; ,\; \backslash quad\; l\_i\; =\; s\_ip\_i\; \backslash quad\; ,\; \backslash quad\; r\_i=p\_is\_i$

The partition algebra $P\_k(n)$ is an associative algebra. It has a multiplicative identity

File:Partition identity.svg

The partition algebra $P\_k(n)$ is semisimple for $n\backslash in\backslash mathbb\{C\}\; -\; \backslash \{0,1,\backslash dots,\; 2k-2\backslash \}$. For any two $n,n\text{'}$ in this set, the algebras $P\_k(n)$ and $P\_k(n\text{'})$ are isomorphic.

The partition algebra is finite-dimensional, with $\backslash dim\; P\_k(n)\; =\; B\_\{2k\}$ (a Bell number).

Subalgebras of the partition algebra can be defined by the following properties:

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

Combining these properties gives rise to 8 nontrivial subalgebras, in addition to the partition algebra itself:

class="wikitable" ; text-align:center;"
Notation ! Name ! Generators ! Dimension ! Example
$P\_k(n)$ | Partition | $s\_i,p\_i,b\_i$ | $B\_\{2k\}$ | File:Partition ex P.svg
$PP\_k(n)$ | Planar partition | $p\_i,\; b\_i$ | $\backslash frac\{1\}\{2k+1\}\backslash binom\{4k\}\{2k\}$ | File:Partition ex PP.svg
$RB\_k(n)$ | Rook Brauer | $s\_i,\; e\_i,\; p\_i$ | $\backslash sum\_\{\backslash ell=0\}^k\; \backslash binom\{2k\}\{2\backslash ell\}\; (2\backslash ell-1)!!$ | File:Partition ex RB.svg
$M\_k(n)$ | Motzkin | $e\_i,\; l\_i,\; r\_i$ | $\backslash sum\_\{\backslash ell=0\}^k\; \backslash frac\{1\}\{\backslash ell+1\}\; \backslash binom\{2\backslash ell\}\{\backslash ell\}\; \backslash binom\{2k\}\{2\backslash ell\}$ | File:Partition ex M.svg
$B\_k(n)$ | Brauer | $s\_i,\; e\_i$ | $(2k-1)!!$ | File:Partition ex B.svg
$TL\_k(n)$ | Temperley–Lieb | $e\_i$ | $\backslash frac\{1\}\{k+1\}\; \backslash binom\{2k\}\{k\}$ | File:Partition ex TL.svg
$R\_k$ | Rook | $s\_i,\; p\_i$ | $\backslash sum\_\{\backslash ell=0\}^k\; \backslash binom\{k\}\{\backslash ell\}^2\; \backslash ell!$ | File:Partition ex R.svg
$PR\_k$ | Planar rook | $l\_i,\; r\_i$ | $\backslash binom\{2k\}\{k\}$ | File:Partition ex PR.svg
$\backslash mathbb\{C\}\; S\_k$ | Symmetric group | $s\_i$ | $k!$ | File:Partition ex S.svg

The symmetric group algebra $\backslash mathbb\{C\}\; S\_k$ is the group ring of the symmetric group $S\_k$ over $\backslash mathbb\{C\}$. The Motzkin algebra is sometimes called the dilute Temperley–Lieb algebra in the physics literature.

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

Inclusions of planar into non-planar algebras:

:$$

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:

:$$

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:

:$$

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:

:$$

PP_k(n^2) \cong TL_{2k}(n) \quad , \quad \left\{\begin{array}{l} p_i \mapsto n e_{2i-1} \\ b_i \mapsto \frac{1}{n} e_{2i} \end{array}\right.

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

• The totally propagating partition subalgebra $\backslash text\{prop\}P\_k$ is generated by diagrams whose blocks all propagate, i.e. partitions whose subsets all contain top and bottom elements. These diagrams from the dual symmetric inverse monoid, which is generated by $s\_i,\; b\_ip\_\{i+1\}b\_\{i+1\}$.
• The quasi-partition algebra $QP\_k(n)$ is generated by subsets of size at least two. Its generators are $s\_i,b\_i,e\_i$ and its dimension is $1+\backslash sum\_\{j=1\}^\{2k\}\; (-1)^\{j-1\}\; B\_\{2k-j\}$.
• The uniform block permutation algebra $U\_k$ is generated by subsets with as many top elements as bottom elements. It is generated by $s\_i,\; b\_i$.

An algebra with a half-integer index $k+\backslash frac12$ is defined from partitions of $2k+2$ elements by requiring that $k+1$ and $\backslash overline\{k+1\}$ are in the same subset. For example, $P\_\{k+\backslash frac12\}$ is generated by $s\_\{i\backslash leq\; k-1\},b\_\{i\backslash leq\; k\},p\_\{i\backslash leq\; k\}$ so that $P\_k\backslash subset\; P\_\{k+\backslash frac12\}\backslash subset\; P\_\{k+1\}$, and $\backslash dim\; P\_\{k+\backslash frac12\}\; =B\_\{2k+1\}$.

Periodic subalgebras are generated by diagrams that can be drawn on an annulus without line crossings. Such subalgebras include a translation element $u=$File:Partition translation.svg such that $u^k=1$. The translation element and its powers are the only combinations of $s\_i$ that belong to periodic subalgebras.

For an integer $0\backslash leq\; \backslash ell\; \backslash leq\; k$, let $D\_\backslash ell$ be the set of partitions of $k+\backslash ell$ elements $1,2,\backslash dots,\; k$ (bottom) and $\backslash bar\; 1,\backslash bar\; 2,\backslash dots,\backslash bar\; \backslash ell$ (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 $k=12,\; \backslash ell\; =\; 5$:

Example of a state in a representation of the partition algebra

Partition diagrams act on $D\_\backslash ell$ from the bottom, while the symmetric group $S\_\backslash ell$ acts from the top. For any Specht module $V\_\backslash lambda$ of $S\_\backslash ell$ (with therefore $|\backslash lambda|=\backslash ell$), we define the representation of $P\_k(n)$

:$$

\mathcal{P}_\lambda = \mathbb{C} D_

\lambda

\otimes_{\mathbb{C} S_

\lambda

} V_\lambda\ .

The dimension of this representation is

:$$

\dim\mathcal{P}_\lambda = f_\lambda \sum_{\ell = |\lambda|}^k \left\{ {k\atop \ell} \right\} \binom{\ell}

\lambda

\ ,

where $\backslash left\backslash \{\; \{k\backslash atop\; \backslash ell\}\; \backslash right\backslash \}$ is a Stirling number of the second kind, $\backslash binom\{\backslash ell\}$

\lambda

is a binomial coefficient, and $f\_\backslash lambda\; =\; \backslash dim\; V\_\backslash lambda$ is given by the hook length formula.

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

Assuming that $P\_k(n)$ is semisimple, the representation $\backslash mathcal\{P\}\_\backslash lambda$ is irreducible, and the

set of irreducible finite-dimensional representations of the partition algebra is

:$$

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

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 $0\backslash leq\; \backslash ell\backslash leq\; k$ with $\backslash ell\backslash equiv\; k\backslash bmod\; 2$, 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.

class="wikitable" ; text-align:center;"
Algebra ! Parameter ! Conditions ! Dimension
$P\_k(n)$ | $\backslash lambda$ | $0\backslash leq\; |\backslash lambda|\backslash leq\; k$ | $f\_\backslash lambda\; \backslash sum\_\{\backslash ell\; =\; |\backslash lambda$

^k \left\{ {k\atop \ell} \right\} \binom{\ell}

\lambda

|-

| $PP\_k(n)$

| $\backslash ell$

| $0\backslash leq\; \backslash ell\backslash leq\; k$

| $\backslash binom\{2k\}\{k+\backslash ell\}-\backslash binom\{2k\}\{k+\backslash ell+1\}$

|-

| $RB\_k(n)$

| $\backslash lambda$

| $0\backslash leq\; |\backslash lambda|\backslash leq\; k$

| $f\_\backslash lambda\; \backslash binom\{k\}$

\lambda

\sum_{m=0}^{\left\lfloor\frac{k-|\lambda|}{2}\right\rfloor} \binom{k-|\lambda|}{2m} (2m-1)!!

|-

| $M\_k(n)$

| $\backslash ell$

| $0\backslash leq\; \backslash ell\backslash leq\; k$

| $\backslash sum\_\{m=0\}^\{\backslash left\backslash lfloor\backslash frac\{k-\backslash ell\}\{2\}\backslash right\backslash rfloor\}\; \backslash binom\{k\}\{\backslash ell+2m\}\; \backslash left\backslash \{\backslash binom\{\backslash ell+2m\}\{m\}-\backslash binom\{\backslash ell+2m\}\{m-1\}\backslash right\backslash \}$

|-

| $B\_k(n)$

| $\backslash lambda$

| $\backslash begin\{array\}\{c\}\; 0\backslash leq\; |\backslash lambda|\backslash leq\; k\; \backslash \backslash \; |\backslash lambda|\backslash equiv\; k\backslash bmod\; 2\backslash end\{array\}$

| $f\_\backslash lambda\; \backslash binom\{k\}$

\lambda

(k-|\lambda|-1)!!

|-

| $TL\_k(n)$

| $\backslash ell$

| $\backslash begin\{array\}\{c\}\; 0\backslash leq\; \backslash ell\backslash leq\; k\; \backslash \backslash \; \backslash ell\backslash equiv\; k\backslash bmod\; 2\backslash end\{array\}$

| $\backslash binom\{k\}\{\backslash frac\{k+\backslash ell\}\{2\}\}\; -\backslash binom\{k\}\{\backslash frac\{k+\backslash ell+2\}\{2\}\}$

|-

| $R\_k$

| $\backslash lambda$

| $0\backslash leq\; |\backslash lambda|\backslash leq\; k$

| $f\_\backslash lambda\; \backslash binom\{k\}$

\lambda

|-

| $PR\_k$

| $\backslash ell$

| $0\backslash leq\; \backslash ell\backslash leq\; k$

| $\backslash binom\{k\}\{\backslash ell\}$

|-

| $\backslash mathbb\{C\}\; S\_k$

| $\backslash lambda$

| $|\backslash lambda|=k$

| $f\_\backslash lambda$

|}

The irreducible representations of $\backslash text\{prop\}P\_k$ are indexed by partitions such that and their dimensions are $f\_\backslash lambda\; \backslash left\backslash \{\; \{k\backslash atop\; |\backslash lambda|\}\; \backslash right\backslash \}$. The irreducible representations of $QP\_k$ are indexed by partitions such that $0\backslash leq|\backslash lambda|\backslash leq\; k$. The irreducible representations of $U\_k$ are indexed by sequences of partitions.

Assume $n\backslash in\; \backslash mathbb\{N\}^*$.

For $V$ a $n$-dimensional vector space with basis $v\_1,\backslash dots,\; v\_n$, there is a natural action of the partition algebra $P\_k(n)$ on the vector space $V^\{\backslash otimes\; k\}$. This action is defined by the matrix elements of a partition $\backslash \{1,\backslash bar\; 1,\; 2,\backslash bar\; 2,\backslash dots,\; k,\backslash bar\; k\backslash \}=\backslash sqcup\_h\; E\_h$ in the basis $(v\_\{j\_1\}\backslash otimes\; \backslash cdots\; \backslash otimes\; v\_\{j\_k\})$:

:$$

\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

:$$

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}\ .

Let $n\backslash geq\; 2k$ be integer.

Let us take $V$ to be the natural permutation representation of the symmetric group $S\_n$. This $n$-dimensional representation is a sum of two irreducible representations: the standard and trivial representations, $V=[n-1,1]\backslash oplus\; [n]$.

Then the partition algebra $P\_k(n)$ is the centralizer of the action of $S\_n$ on the tensor product space $V^\{\backslash otimes\; k\}$,

:$$

P_k(n) \cong \text{End}_{S_n}\left(V^{\otimes k}\right)\ .

Moreover, as a bimodule over $P\_k(n)\backslash times\; S\_n$, the tensor product space decomposes into irreducible representations as

:$$

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

where $[n-|\backslash lambda|,\backslash lambda]$ is a Young diagram of size $n$ built by adding a first row to $\backslash lambda$, and $V\_\{[n-|\backslash lambda|,\backslash lambda]\}$ is the corresponding Specht module of $S\_n$.

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 $V\_n$ for an irreducible $n$-dimensional representation of the first group or algebra:

class="wikitable" ; text-align:center;"
Tensor product space ! Group or algebra ! Dual algebra or group ! Comments
$\backslash left(V\_\{n-1\}\backslash oplus\; V\_1\backslash right)^\{\backslash otimes\; k\}$ | $S\_n$ | $P\_k(n)$ | The duality for the full partition algebra
$\backslash left(V\_\{n-2\}\backslash oplus\; V\_1\backslash oplus\; V\_1\backslash right)^\{\backslash otimes\; k\}$ | $S\_\{n-1\}$ | $P\_\{k+\backslash frac12\}(n)$ | Case of a partition algebra with a half-integer index
$V\_n^\{\backslash otimes\; k\}$ | $GL\_n(\backslash mathbb\{C\})$ | $S\_k$ | The original Schur-Weyl duality
$V\_n^\{\backslash otimes\; k\}$ | $O(n)$ | $B\_k(n)$ | Duality between the orthogonal group and the Brauer algebra
$\backslash left(V\_n\backslash oplus\; V\_1\backslash right)^\{\backslash otimes\; k\}$ | $O(n)$ | $RB\_k(n+1)$ | Duality between the orthogonal group and the rook Brauer algebra
$V\_n^\{\backslash otimes\; k\}$ | $R\_n$ | $\backslash text\{prop\}P\_k$ | Duality between the rook algebra and the totally propagating partition algebra
$V\_2^\{\backslash otimes\; k\}$ | $gl(1|1)$ | $PR\_\{k-1\}$ | Duality between a Lie superalgebra and the planar rook algebra
$V\_\{n-1\}^\{\backslash otimes\; k\}$ | $S\_n$ | $QP\_k(n)$ | Duality between the symmetric group and the quasi-partition algebra
$V\_n^\{\backslash otimes\; r\}\; \backslash otimes\; \backslash left(V\_n^*\backslash right)^\{\backslash otimes\; s\}$ | $GL\_n(\backslash mathbb\{C\})$ | $B\_\{r,s\}(n)$ | Duality involving the walled Brauer algebra.

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Category:Representation theory

Category:Diagram algebras