Partition algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 9 April 2013

Partition algebras

For $k\in \frac{1}{2}{\mathbb{Z}}_{>0}$ and $n\in ℂ,$ the partition algebra $ℂA_k\left(n\right)$ is the associative algebra over $ℂ$ with basis $A_k,$

$$ℂA_k\left(n\right)=ℂ\text{-span}\{d\in A_k\},\text{and multiplication defined by}{d}_1{d}_2=n^{\ell }(d_1\circ d_2),$$

where, for $d_1,d_2\in A_k,$ $d_1\circ d_2$ is the product in the monoid $A_k$ and $\ell$ is the number of blocks removed from the middle row when constructing the composition $d_1\circ d_2\text{.}$ For example,

$$\begin{array}{cc}\begin{array}{c}\begin{array}{ccccc}\text{if}{d}_1=& & \text{and}{d}_2=& & \text{then}\end{array}\\ \begin{array}{ccccc}{d}_1{d}_2=& & =n^2& & ,\end{array}\end{array}& \text{(2.1)}\end{array}$$

since two blocks are removed from the middle row. There are inclusions of algebras given by

$$\begin{array}{cc}\begin{array}{ccc}ℂA_{k-\frac{1}{2}}& \hookrightarrow & ℂA_k\\ 1 k d & ⟼& 1 k d \end{array}\text{and}\begin{array}{ccc}ℂA_{k-1}& \hookrightarrow & ℂA_{k-\frac{1}{2}}\\ 1 k-1 d & ⟼& 1 k d \end{array}& \text{(2.2)}\end{array}$$

For $d_1,d_2\in A_k,$ define

$$d_1\le d_2,\text{if the set partition}\hspace{0.17em}{d}_2\hspace{0.17em}\text{is coarser than the set partition}\hspace{0.17em}{d}_1,$$

i.e., $i$ and $j$ in the same block of $d_1$ implies that $i$ and $j$ are in the same block of $d_2\text{.}$ Let $\{x_d\in ℂA_k\mid d\in A_k\}$ be the basis of $ℂA_k$ uniquely defined by the relation

$$\begin{array}{cc}d=\sum _{d\prime \le d}{x}_{d\prime },\text{for all}\hspace{0.17em}d\in A_k\text{.}& \text{(2.3)}\end{array}$$

Under any linear extension of the partial order $\le$ the transition matrix between the basis $\{d\mid d\in A_k\}$ of $ℂA_k\left(n\right)$ and the basis $\{x_d\mid d\in A_k\}$ of $ℂA_k\left(n\right)$ is upper triangular with 1s on the diagonal and so the $x_d$ are well defined.

The maps

$${\epsilon }_{\frac{1}{2}}:ℂA_k⟶ℂA_{k-\frac{1}{2}},{\epsilon }^{\frac{1}{2}}:ℂA_{k-\frac{1}{2}}⟶ℂA_{k-1}\text{and}{\text{tr}}_k:ℂA_k⟶ℂ\text{.}$$

Let $k\in {\mathbb{Z}}_{>0}\text{.}$ Define linear maps

$$\begin{array}{cccc}{\epsilon }_{\frac{1}{2}}:& ℂA_k& ⟶& ℂA_{k-\frac{1}{2}}\\ & 1 k d & ⟼& 1 k d \end{array}\text{and}\begin{array}{cccc}{\epsilon }^{\frac{1}{2}}:& ℂA_{k-\frac{1}{2}}& ⟶& ℂA_{k-1}\\ & 1 k d & ⟼& 1 k-1 d \end{array}$$

so that ${\epsilon }_{\frac{1}{2}}\left(d\right)$ is the same as $d$ except that the block containing $k$ and the block containing $k\prime$ are combined, and ${\epsilon }^{\frac{1}{2}}\left(d\right)$ has the same blocks as $d$ except with $k$ and $k\prime$ removed. There is a factor of $n$ in ${\epsilon }^{\frac{1}{2}}\left(d\right)$ if the removal of $k$ and $k\prime$ reduces the number of blocks by 1. For example,

$$\begin{array}{cc}\begin{array}{ccccc}{\epsilon }_{\frac{1}{2}}& (∑A ∑A) & =& & ,\end{array}& \begin{array}{ccccc}{\epsilon }^{\frac{1}{2}}& (∑A ∑A) & =& & ,\end{array}\end{array}$$

and

$$\begin{array}{cc}\begin{array}{ccccc}{\epsilon }_{\frac{1}{2}}& (∑A ∑A) & =& & ,\end{array}& \begin{array}{ccccc}{\epsilon }^{\frac{1}{2}}& (∑A ∑A) & =n& & \text{.}\end{array}\end{array}$$

The map ${\epsilon }^{\frac{1}{2}}$ is the composition $ℂA_{k-\frac{1}{2}}\hookrightarrow ℂA_k\stackrel{{\epsilon }_1}{\to }ℂA_{k-1}\text{.}$ The composition of ${\epsilon }_{\frac{1}{2}}$ and ${\epsilon }^{\frac{1}{2}}$ is the map

$$\begin{array}{cc}\begin{array}{cccc}{\epsilon }_1:& ℂA_k& ⟶& ℂA_{k-1}\\ & 1 k d & ⟼& 1 k-1 d & \text{.}\end{array}& \text{(2.4)}\end{array}$$

By drawing diagrams it is straightforward to check that, for $k\in {\mathbb{Z}}_{>0},$

$$\begin{array}{cc}\begin{array}{cccc}{\epsilon }_{\frac{1}{2}}\left({\alpha }_{1}b{a}_2\right)& =& a_1{\epsilon }_{\frac{1}{2}}\left(b\right)a_2,& \text{for}\hspace{0.17em}{a}_1,a_2\in A_{k-\frac{1}{2}},b\in A_k\\ {\epsilon }^{\frac{1}{2}}\left({\alpha }_{1}b{a}_2\right)& =& a_1{\epsilon }^{\frac{1}{2}}\left(b\right)a_2,& \text{for}\hspace{0.17em}{a}_1,a_2\in A_{k-1},b\in A_{k-\frac{1}{2}}\\ {\epsilon }_1\left({\alpha }_{1}b{a}_2\right)& =& a_1{\epsilon }_1\left(b\right)a_2,& \text{for}\hspace{0.17em}{a}_1,a_2\in A_{k-1},b\in A_k\end{array}& \text{(2.5)}\end{array}$$

and

$$\begin{array}{cc}\begin{array}{cccccc}{p}_{k+\frac{1}{2}}b{p}_{k+\frac{1}{2}}& =& {\epsilon }_{\frac{1}{2}}\left(b\right)p_{k+\frac{1}{2}}& =& p_{k+\frac{1}{2}}{\epsilon }_{\frac{1}{2}}\left(b\right),& \text{for}\hspace{0.17em}b\in A_k\\ p_{k}b{p}_k& =& {\epsilon }^{\frac{1}{2}}\left(b\right)p_k& =& p_k{\epsilon }^{\frac{1}{2}}\left(b\right),& \text{for}\hspace{0.17em}b\in A_{k-\frac{1}{2}}\\ e_{k}b{e}_k& =& {\epsilon }_1\left(b\right)e_k& =& e_k{\epsilon }_1\left(b\right),& \text{for}\hspace{0.17em}b\in A_k\text{.}\end{array}& \text{(2.6)}\end{array}$$

Define ${\text{tr}}_k:ℂA_k\to ℂ$ and ${\text{tr}}_{k-\frac{1}{2}}:ℂA_k\to ℂ$ by the equations

$$\begin{array}{cc}{\text{tr}}_k\left(b\right)={\text{tr}}_{k-\frac{1}{2}}\left({\epsilon }_{\frac{1}{2}}\left(b\right)\right),\text{for}\hspace{0.17em}b\in A_k,\text{and}{\text{tr}}_{k-\frac{1}{2}}\left(b\right)={\text{tr}}_{k-1}\left({\epsilon }^{\frac{1}{2}}\left(b\right)\right),\text{for}\hspace{0.17em}b\in A_{k-\frac{1}{2}},& \text{(2.7)}\end{array}$$

so that

$$\begin{array}{cc}{\text{tr}}_k\left(b\right)={\epsilon }_1^k\left(b\right),\text{for}\hspace{0.17em}b\in A_k,\text{and}{\text{tr}}_{k-\frac{1}{2}}\left(b\right)={\epsilon }_1^{k-1}{\epsilon }^{\frac{1}{2}}\left(b\right),\text{for}\hspace{0.17em}b\in A_{k-\frac{1}{2}}\text{.}& \text{(2.8)}\end{array}$$

Pictorially ${\text{tr}}_k\left(d\right)=n^c$ where $c$ is the number of connected components in the closure of the diagram $d,$

$$\begin{array}{cc}\begin{array}{ccc}{\text{tr}}_k\left(d\right)=& d & ,\text{for}\hspace{0.17em}d\in A_k\text{.}\end{array}& \text{(2.9)}\end{array}$$

The ideal $ℂI_k\left(n\right)$

For $k\in \frac{1}{2}{\mathbb{Z}}_{\ge 0}$ define

$$\begin{array}{cc}ℂI_k\left(n\right)=ℂ\text{-span}\{d\in I_k\}\text{.}& \text{(2.10)}\end{array}$$

By (1.3),

$$\begin{array}{cc}ℂI_k\left(n\right)\hspace{0.17em}\text{is an ideal of}\hspace{0.17em}ℂA_k\left(n\right)\text{and}ℂA_k\left(n\right)/ℂI_k\left(n\right)\cong ℂS_k,& \text{(2.11)}\end{array}$$

since the set partitions with propagating number $k$ are exactly the permutations in the symmetric group $S_k$ (by convention $S_{\ell +\frac{1}{2}}=S_{\ell }$ for $\ell \in {\mathbb{Z}}_{>0};$ see (2.2)).

View $ℂI_k\left(n\right)$ as an algebra (without identity). Since $ℂA_k\left(n\right)/ℂI_k\cong ℂS_k$ and $ℂS_k$ is semisimple, $\text{Rad}\left(ℂA_k\left(n\right)\right)\subseteq ℂI_k\left(n\right)\text{.}$ Since $ℂI_k\left(n\right)/\text{Rad}\left(ℂA_k\left(n\right)\right)$ is an ideal in $ℂA_k\left(n\right)/\text{Rad}\left(ℂA_k\left(n\right)\right)$ the quotient $ℂI_k\left(n\right)/\text{Rad}\left(ℂA_k\left(n\right)\right)$ is semisimple. Therefore $\text{Rad}\left(ℂI_k\left(n\right)\right)\subseteq \text{Rad}\left(ℂA_k\left(n\right)\right)\text{.}$ On the other hand, since $\text{Rad}\left(ℂA_k\left(n\right)\right)$ is an ideal of nilpotent elements in $ℂA_k\left(n\right),$ it is an ideal of nilpotent elements in $ℂI_k\left(n\right)$ and so $\text{Rad}\left(ℂA_k\left(n\right)\right)\supseteq \text{Rad}\left(ℂA_k\left(n\right)\right)\text{.}$ Thus

$$\begin{array}{cc}\text{Rad}\left(ℂA_k\left(n\right)\right)=\text{Rad}\left(ℂI_k\left(n\right)\right)\text{.}& \text{(2.12)}\end{array}$$

Let $k\in {\mathbb{Z}}_{\ge 0}\text{.}$ By (2.5) the maps

$${\epsilon }_{\frac{1}{2}}:ℂA_k⟶ℂA_{k-\frac{1}{2}}\text{and}{\epsilon }^{\frac{1}{2}}:ℂA_{k-\frac{1}{2}}⟶ℂA_{k-1}$$

are $(ℂA_{k-\frac{1}{2}},ℂA_{k-\frac{1}{2}})\text{-bimodule}$ and $(ℂA_{k-1},ℂA_{k-1})\text{-bimodule}$ homomorphisms, respectively. The corresponding basic constructions (see Section 4) are the algebras

$$\begin{array}{cc}ℂA_k\left(n\right){\otimes }_{ℂA_{k-\frac{1}{2}}\left(n\right)}ℂA_k\left(n\right)\text{and}ℂA_{k-\frac{1}{2}}\left(n\right){\otimes }_{ℂA_{k-1}\left(n\right)}ℂA_{k-\frac{1}{2}}\left(n\right)& \text{(2.13)}\end{array}$$

with products given by

$$\begin{array}{cc}(b_1\otimes b_2)(b_3\otimes b_4)=b_1\otimes {\epsilon }_{\frac{1}{2}}\left(b_2{b}_3\right)b_4,\text{and}(c_1\otimes c_2)(c_3\otimes c_4)=c_1\otimes {\epsilon }^{\frac{1}{2}}\left(c_2{c}_3\right)c_4,& \text{(2.14)}\end{array}$$

for $b_1,b_2,b_3,b_4\in ℂA_k\left(n\right),$ and for $c_1,c_2,c_3,c_4\in ℂA_{k-\frac{1}{2}}\left(n\right)\text{.}$

Let $k\in \frac{1}{2}{\mathbb{Z}}_{>0}\text{.}$ Then, by the relations in (2.6) and the fact that

$$\begin{array}{cc}\text{every}\hspace{0.17em}d\in I_k\text{can be written as}d=d_1{p}_k{d}_2,\text{with}\hspace{0.17em}{d}_1,d_2\in A_{k-\frac{1}{2}},& \text{(2.15)}\end{array}$$

the maps

$$\begin{array}{cc}\begin{array}{ccc}ℂA_{k-\frac{1}{2}}\left(n\right){\otimes }_{ℂA_{k-1}\left(n\right)}ℂA_{k-\frac{1}{2}}\left(n\right)& ⟶& ℂI_k\left(n\right)\\ b_1\otimes b_2& ⟼& b_1{p}_k{b}_2\end{array}& \text{(2.16)}\end{array}$$

are algebra isomorphisms. Thus the ideal $ℂI_k\left(n\right)$ is always isomorphic to a basic construction (in the sense of Section 4).

Representations of the symmetric group

A partition $\lambda$ is a collection of boxes in a corner. We shall conform to the conventions in [Mac1995] and assume that gravity goes up and to the left, i.e.,

$$$$

Numbering the rows and columns in the same way as for matrices, let

$$\begin{array}{cc}\begin{array}{ccc}{\lambda }_i& =& \text{the number of boxes in row}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda ,\\ {\lambda }_j^{\prime }& =& \text{the number of boxes in column}\hspace{0.17em}j\hspace{0.17em}\text{of}\hspace{0.17em}\lambda ,\text{and}\\ \mid \lambda \mid & =& \text{the total number of boxes in}\hspace{0.17em}\lambda \text{.}\end{array}& \text{(2.17)}\end{array}$$

Any partition $\lambda$ can be identified with the sequence $\lambda =({\lambda }_1\ge {\lambda }_2\ge \dots )$ and the conjugate partition to $\lambda$ is the partition $\lambda \prime =({\lambda }_1^{\prime },{\lambda }_2^{\prime },\dots )\text{.}$ The hook length of the box $b$ of $\lambda$ is

$$\begin{array}{cc}h\left(b\right)=({\lambda }_i-i)+({\lambda }_j^{\prime }-j)+1,\text{if}\hspace{0.17em}b\hspace{0.17em}\text{is in position}\hspace{0.17em}(i,j)\hspace{0.17em}of\hspace{0.17em}\lambda \text{.}& \text{(2.18)}\end{array}$$

Write $\lambda ⊢n$ if $\lambda$ is a partition with $n$ boxes. In the example above, $\lambda =\left(553311\right)$ and $\lambda ⊢18\text{.}$

See [Mac1995, Section I.7] for details on the representation theory of the symmetric group. The irreducible $ℂS_k\text{-modules}$ $S_k^{\lambda }$ are indexed by the elements of

$$\begin{array}{cc}{\hat{S}}_k=\{\lambda ⊢n\}\text{and}\text{dim}\left(S_k^{\lambda }\right)=\frac{k!}{\prod _{b\in \lambda }h\left(b\right)}\text{.}& \text{(2.19)}\end{array}$$

For $\lambda \in {\hat{S}}_k$ and $\mu \in {\hat{S}}_{k-1},$

$$\begin{array}{cc}{\text{Res}}_{S_{k-1}}^{S_k}\left(S_k^{\lambda }\right)\cong \underset{\lambda /\nu =\square }{⨁}{S}_{k-1}^{\nu }\text{and}{\text{Ind}}_{S_{k-1}}^{S_k}\left(S_{k-1}^{\mu }\right)\cong \underset{\nu /\mu =\square }{⨁}{S}_k^{\nu },& \text{(2.20)}\end{array}$$

where the first sum is over all partitions $\nu$ that are obtained from $\lambda$ by removing a box, and the second sum is over all partitions $\nu$ which are obtained from $\mu$ by adding a box (this result follows, for example, from [Mac1995, Section I.7 Example 22(d)]).

The Young lattice is the graph $\hat{S}$ given by setting

$$\begin{array}{cc}\begin{array}{c}\text{vertices on level}\hspace{0.17em}k:{\hat{S}}_k=\left\{\text{partitions}\hspace{0.17em}\lambda \hspace{0.17em}\text{with}\hspace{0.17em}k\hspace{0.17em}\text{boxes}\right\},\text{and}\\ \text{an edge}\hspace{0.17em}\lambda \to \mu ,\lambda \in {\hat{S}}_k,\mu \in {\hat{S}}_{k+1}\hspace{0.17em}\text{if}\hspace{0.17em}\mu \hspace{0.17em}\text{is obtained from}\hspace{0.17em}\lambda \hspace{0.17em}\text{by adding a box.}\end{array}& \text{(2.21)}\end{array}$$

It encodes the decompositions in (2.20). The first few levels of $\hat{S}$ are given by

$$∅ k=0: k=1: k=2: k=3: k=4:$$

For $\mu \in {\hat{S}}_k$ define

$${\hat{S}}_k^{\mu }=\{T=(T^{\left(0\right)},T^{\left(1\right)},\dots ,T^{\left(k\right)})\hspace{0.17em}|\hspace{0.17em}\begin{array}{c}{T}^{\left(0\right)}=\varnothing ,T^{\left(k\right)}=\mu ,\text{and},\text{for each}\hspace{0.17em}\ell ,\\ T^{\left(\ell \right)}\in {\hat{S}}_{\ell }\hspace{0.17em}\text{and}\hspace{0.17em}{T}^{\left(\ell \right)}\to T^{(\ell +1)}\hspace{0.17em}\text{is an edge in}\hspace{0.17em}\hat{S}\end{array}\}$$

so that ${\hat{S}}_k^{\mu }$ is the set of paths from $\varnothing \in {\hat{S}}_0$ to $\mu \in {\hat{S}}_k$ in the graph $\hat{S}\text{.}$ In terms of the Young lattice,

$$\begin{array}{cc}\text{dim}\left(S_k^{\mu }\right)=\text{Card}\left({\hat{S}}_k^{\mu }\right)\text{.}& \text{(2.22)}\end{array}$$

This is a translation of the classical statement (see [Mac1995, Section I.7.6(ii)]) that $\text{dim}\left(S_k^{\mu }\right)$ is the number of standard Young tableaux of shape $\lambda$ (the correspondence is obtained by putting the entry $\ell$ in the box of $\lambda$ which is added at the $\ell \text{th}$ step $T^{(\ell -1)}\to T^{\left(\ell \right)}$ of the path).

Structure of the algebra $ℂA_k\left(n\right)$

Build a graph $\hat{A}$ by setting

$$\begin{array}{cc}\begin{array}{c}\text{vertices on level}\hspace{0.17em}k:{\hat{A}}_k=\{\text{partitions}\hspace{0.17em}\mu \mid k-\mid \mu \mid \in {\mathbb{Z}}_{\ge 0}\},\\ \text{vertices on level}\hspace{0.17em}k+\frac{1}{2}:$ {\hat{A}}_{k+1}$={\hat{A}}_k=\{\text{partitions}\hspace{0.17em}\mu \mid k-\mid \mu \mid \in {\mathbb{Z}}_{\ge 0}\},\\ \text{an edge}\hspace{0.17em}\lambda \to \mu ,\lambda \in {\hat{A}}_k,\mu \in {\hat{A}}_{k+\frac{1}{2}}\hspace{0.17em}\text{if}\hspace{0.17em}\lambda =\mu \hspace{0.17em}\text{or if}\hspace{0.17em}\mu \hspace{0.17em}\text{is obtained from}\hspace{0.17em}\lambda \hspace{0.17em}\text{by removing a box,}\\ \text{an edge}\hspace{0.17em}\mu \to \lambda ,\mu \in {\hat{A}}_{k+\frac{1}{2}},\lambda \in {\hat{A}}_{k+1}\hspace{0.17em}\text{if}\hspace{0.17em}\lambda =\mu \hspace{0.17em}\text{or if}\hspace{0.17em}\lambda \hspace{0.17em}\text{is obtained from}\hspace{0.17em}\mu \hspace{0.17em}\text{by adding a box.}\end{array}& \text{(2.23)}\end{array}$$

The first few levels of $\hat{A}$ are given by

$$∅ ∅ ∅ ∅ ∅ ∅ ∅ k=0: k=0+12: k=1: k=1+12: k=2: k=2+12: k=3:$$

The following result is an immediate consequence of the Tits deformation theorem, Theorems 5.10 and 5.13 in this paper (see also [CRe1987, (68.17)]).

Theorem 2.24.

1. For all but a finite number of $n\in ℂ$ the algebra $ℂA_k\left(n\right)$ is semisimple.
2. If $ℂA_k\left(n\right)$ is semisimple then the irreducible $ℂA_k\left(n\right)\text{-modules,}$ $A_k^{\mu }$ are indexed by elements of the set ${\hat{A}}_k=\{\text{partitions}\hspace{0.17em}\mu \mid k-\mid \mu \mid \in {\mathbb{Z}}_{\ge 0}\},$ and $\text{dim}\left(A_k^{\mu }\right)=$ (number of paths from $\varnothing \in {\hat{A}}_0$ to $\mu \in {\hat{A}}_k$ in the graph $\hat{A}\text{).}$

Let

$${\hat{A}}_k^{\mu }=\{T=(T^{\left(0\right)},T^{\left(\frac{1}{2}\right)},\dots ,T^{(k-\frac{1}{2})},T^{\left(k\right)})\hspace{0.17em}|\hspace{0.17em}\begin{array}{c}{T}^{\left(0\right)}=\varnothing ,T^{\left(k\right)}=\mu ,\text{and},\text{for each}\hspace{0.17em}\ell ,\\ T^{\left(\ell \right)}\in {\hat{A}}_{\ell }\hspace{0.17em}\text{and}\hspace{0.17em}{T}^{\left(\ell \right)}\to T^{(\ell +\frac{1}{2})}\hspace{0.17em}\text{is an edge in}\hspace{0.17em}\hat{A}\end{array}\}$$

sothat ${\hat{A}}_k^{\mu }$ is the set of paths from $\varnothing \in {\hat{A}}_0$ to $\mu \in {\hat{A}}_k$ in the graph $\hat{A}\text{.}$ If $\mu \in {\hat{S}}_k$ then $\mu \in {\hat{A}}_k$ and $\mu \in {\hat{A}}_{k+\frac{1}{2}}$ and, for notational convenience in the following theorem,

$$\begin{array}{cccc}\text{identify}& P& =& (P^{\left(0\right)},P^{\left(1\right)},\dots ,P^{\left(k\right)})\in {\hat{S}}_k^{\mu }\hspace{0.17em}\text{with the corresponding}\\ & P& =& (P^{\left(0\right)},P^{\left(0\right)},P^{\left(1\right)},P^{\left(1\right)},\dots ,P^{(k-1)},P^{(k-1)},P^{\left(k\right)})\in {\hat{A}}_k^{\mu },\\ \text{and}& P& =& (P^{\left(0\right)},P^{\left(0\right)},P^{\left(1\right)},P^{\left(1\right)},\dots ,P^{(k-1)},P^{(k-1)},P^{\left(k\right)},P^{\left(k\right)})\in {\hat{A}}_{k+\frac{1}{2}}^{\mu }\text{.}\end{array}$$

For $\ell \in \frac{1}{2}{\mathbb{Z}}_{\ge 0}$ and $n\in ℂ$ such that $ℂA_{\ell }\left(n\right)$ is semisimple let ${\chi }_{A_{\ell }\left(n\right)}^{\mu },$ $\mu \in {\hat{A}}_{\ell },$ be the irreducible characters of $ℂA_{\ell }\left(n\right)\text{.}$ Let ${\text{tr}}_{\ell }:ℂA_{\ell }\left(n\right)\to ℂ$ be the traces on $ℂA_{\ell }\left(n\right)$ defined in (2.8) and define constants ${\text{tr}}_{\ell }^{\mu }\left(n\right),$ $\mu \in {\hat{A}}_{\ell },$ by

$$\begin{array}{cc}{\text{tr}}_{\ell }=\sum _{\mu \in {\hat{A}}_{\ell }}{\text{tr}}_{\ell }^{\mu }\left(n\right){\chi }_{A_{\ell }\left(n\right)}^{\mu }\text{.}& \text{(2.25)}\end{array}$$

Theorem 2.26.

1. Let $n\in ℂ$ and let $k\in \frac{1}{2}{\mathbb{Z}}_{\ge 0}\text{.}$ Assume that $${\text{tr}}_{\ell }^{\lambda }\left(n\right)\ne 0,\text{for all}\hspace{0.17em}\lambda \in {\hat{A}}_{\ell },\ell \in \frac{1}{2}{\mathbb{Z}}_{\ge 0},\ell <k\text{.}$$ Then the partition algebras $$\begin{array}{cc}ℂA_{\ell }\left(n\right)\hspace{0.17em}\text{are semisimple for all}\hspace{0.17em}\ell \in \frac{1}{2}{\mathbb{Z}}_{\ge 0},\ell \le k\text{.}& \text{(2.27)}\end{array}$$ For each $\ell \in \frac{1}{2}{\mathbb{Z}}_{\ge 0},$ $\ell \le k-\frac{1}{2},$ define $${\epsilon }_{\mu }^{\lambda }=\frac{{\text{tr}}_{\ell -\frac{1}{2}}^{\lambda }\left(n\right)}{{\text{tr}}_{\ell -1}^{\mu }\left(n\right)}\text{for each edge}\hspace{0.17em}\mu \to \lambda ,\mu \in {\hat{A}}_{\ell -1},\lambda \in {\hat{A}}_{\ell -\frac{1}{2}},\hspace{0.17em}\text{in the graph}\hspace{0.17em}\hat{A}\text{.}$$ Inductively define elements in $ℂA_{\ell }\left(n\right)$ by $$\begin{array}{cc}{e}_{PQ}^{\mu }=\frac{1}{\sqrt{{\epsilon }_{\mu }^{\tau }{\epsilon }_{\mu }^{\gamma }}}{e}_{P-T}^{\tau }{p}_{\ell }{e}_{T{Q}^{-}}^{\gamma },\text{for}\hspace{0.17em}\mu \in {\hat{A}}_{\ell },\mid \mu \mid \le \ell -1,P,Q\in {\hat{A}}_{\ell }^{\mu },& \text{(2.28)}\end{array}$$ where $\tau =P^{(\ell -\frac{1}{2})},$ $\gamma =Q^{(\ell -\frac{1}{2})},$ $R^{-}=(R^{\left(0\right)},\dots ,R^{(\ell -\frac{1}{2})})$ for $R=(R^{\left(0\right)},\dots ,R^{(\ell -\frac{1}{2})},R^{\left(\ell \right)})\in {\hat{A}}_{\ell }^{\mu }$ and $T$ is an element of ${\hat{A}}_{\ell -1}^{\mu }$ (the element $e_{PQ}^{\lambda }$ does not depend on the choice of $T\text{).}$ Then define $$\begin{array}{cc}{e}_{PQ}^{\lambda }=(1-z)s_{PQ}^{\lambda },\text{for}\lambda \in {\hat{S}}_{\ell },P,Q\in {\hat{S}}_{\ell }^{\lambda },\text{where}z=\sum _{\underset{\mid \mu \mid \le \ell -1}{\mu \in {\hat{A}}_{\ell }}}\sum _{P\in {\hat{A}}_{\ell }^{\mu }}{e}_{PP}^{\mu }& \text{(2.29)}\end{array}$$ and $\{s_{PQ}^{\lambda }\mid \lambda \in {\hat{S}}_{\ell },P,Q\in {\hat{S}}_{\ell }^{\lambda }\}$ is any set of matrix units for the group algebra of the symmetric group $ℂS_{\ell }\text{.}$ Together, the elements in (2.28) and (2.29) form a set of matrix units in $ℂA_{\ell }\left(n\right)\text{.}$
2. Let $n\in {\mathbb{Z}}_{\ge 0}$ and let $k\in \frac{1}{2}{\mathbb{Z}}_{>0}$ be minimal such that ${\text{tr}}_k^{\lambda }\left(n\right)=0$ for some $\lambda \in {\hat{A}}_k\text{.}$ Then $ℂA_{k+\frac{1}{2}}\left(n\right)$ is not semisimple.
3. Let $n\in {\mathbb{Z}}_{\ge 0}$ and $k\in \frac{1}{2}{\mathbb{Z}}_{>0}\text{.}$ If $ℂA_k\left(n\right)$ is not semisimple then $ℂA_{k+j}\left(n\right)$ is not semisimple for $j\in {\mathbb{Z}}_{>0}\text{.}$

		Proof.
	(a) Assume that $ℂA_{\ell -1}\left(n\right)$ and $ℂA_{\ell -\frac{1}{2}}\left(n\right)$ are both semisimple and that ${\text{tr}}_{\ell -1}^{\mu }\left(n\right)\ne 0$ for all $\mu \in {\hat{A}}_{\ell -1}\text{.}$ If $\lambda \in {\hat{A}}_{\ell -\frac{1}{2}}$ then ${\epsilon }_{\mu }^{\lambda }\ne 0$ if and only if ${\text{tr}}_{\ell -\frac{1}{2}}^{\lambda }\left(n\right)\ne 0,$ and, since the ideal $ℂI_{\ell }\left(n\right)$ is isomorphic to the basic construction $ℂA_{\ell -\frac{1}{2}}\left(n\right){\otimes }_{ℂA_{\ell -1}\left(n\right)}ℂA_{\ell -\frac{1}{2}}\left(n\right)$ (see (2.13)), it then follows from Theorem 4.28 that $ℂI_{\ell }\left(n\right)$ is semisimple if and only if ${\text{tr}}_{\ell -1}^{\mu }\left(n\right)\ne 0$ for all $\lambda \in {\hat{A}}_{\ell -\frac{1}{2}}\text{.}$ Thus, by(2.12), if $ℂA_{\ell -1}\left(n\right)$ and $ℂA_{\ell -\frac{1}{2}}\left(n\right)$ are both semisimple and ${\text{tr}}_{\ell -1}^{\mu }\left(n\right)\ne 0$ for all $\mu \in {\hat{A}}_{\ell -1}$ then $$\begin{array}{cc}ℂA_{\ell }\left(n\right)\text{is semisimple if and only if}{\text{tr}}_{\ell -\frac{1}{2}}^{\lambda }\left(n\right)\ne 0\hspace{0.17em}\text{for all}\hspace{0.17em}\lambda \in {\hat{A}}_{\ell -\frac{1}{2}}\text{.}& \text{(2.30)}\end{array}$$ By Theorem 4.28, when ${\text{tr}}_{\ell -\frac{1}{2}}^{\lambda }\left(n\right)\ne 0$ for all $\lambda \in {\hat{A}}_{\ell -\frac{1}{2}},$ the algebra $ℂI_{\ell }\left(n\right)$ has matrix units given by the formulas in (2.28). The element $z$ in (2.29) is the central idempotent in $ℂA_{\ell }\left(n\right)$ such that $ℂI_{\ell }\left(n\right)=zℂA_{\ell }\left(n\right)\text{.}$ Hence the complete set of elements in (2.28) and (2.29) form a set of matrix units for $ℂA_{\ell }\left(n\right)\text{.}$ This completes the proof of (a) and (b) follows from Theorem 4.28(b). (c) Part (g) of Theorem 4.28 shows that if $ℂA_{\ell -1}\left(n\right)$ is not semisimple then $ℂA_{\ell }\left(n\right)$ is not semisimple. $\square$

Specht modules

Let $A$ be an algebra. An idempotent is a nonzero element $p\in A$ such that $p^2=p\text{.}$ A minimal idempotent is an idempotent p which cannot be written as a sum $p=p_1+p_2$ with $p_1{p}_2=p_2{p}_1=0\text{.}$ If $p$ is an idempotent in $A$ and $pAp=ℂp$ then $p$ is a minimal idempotent of $A$ since, if $p=p_1+p_2$ with $p_1^2=p_1,$ $p_2^2=p_2,$ and $p_1{p}_2=p_2{p}_1=0,$ then $p{p}_{1}p=kp$ for some constant $p$ and so $k{p}_1=kp{p}_1=p{p}_{1}p{p}_1=p_1$ giving that either $p_1=0$ or $k=1,$ in which case $p_1=p{p}_{1}p=p\text{.}$

Let $p$ be an idempotent in $A\text{.}$ Then the map

$$\begin{array}{cc}\begin{array}{ccc}{\left(pAp\right)}^{\text{op}}& \stackrel{\sim }{⟶}& {\text{End}}_A\left(Ap\right),\\ pbp& ⟼& {\varphi }_{pbp}\end{array}\text{where}{\varphi }_{pbp}\left(ap\right)=\left(ap\right)\left(pbp\right)=apbp,\text{for}\hspace{0.17em}ap\in Ap,& \text{(2.31)}\end{array}$$

is a ring isomorphism.

If $p$ is a minimal idempotent of $A$ and $Ap$ is a semisimple $A\text{-module}$ then $Ap$ must be a simple $A\text{-module.}$ To see this suppose that $Ap$ is not simple so that there are $A\text{-submodules}$ $V_1$ and $V_2$ of $Ap$ such that $Ap=V_1\oplus V_2\text{.}$ Let ${\varphi }_1,{\varphi }_2\in {\text{End}}_A\left(Ap\right)$ be the $A\text{-invariant}$ projections on $V_1$ and $V_2\text{.}$ By (2.31) ${\varphi }_1$ and ${\varphi }_2$ are given by right multiplication by $p_1=p{\stackrel{\sim }{p}}_{1}p$ and $p_2=p{\stackrel{\sim }{p}}_{2}p,$ respectively, and it follows that $p=p_1+p_2,$ $V_1=A{p}_1,$ $V_2=A{p}_2,$ and $Ap=A{p}_1\oplus A{p}_2\text{.}$ Then $p_1^2={\varphi }_1\left(p_1\right)={\varphi }_1^2\left(p\right)=p_1$ and $p_1{p}_2={\varphi }_2\left(p_1\right)={\varphi }_2\left({\varphi }_1\left(p\right)\right)=0\text{.}$ Similarly $p_2^2=p_2$ and $p_2{p}_1=0\text{.}$ Thus $p$ is not a minimal idempotent.

If $p$ is an idempotent in $A$ and $Ap$ is a simple $A\text{-module}$ then

$$pAp={\text{End}}_A{\left(Ap\right)}^{\text{op}}=ℂ(p·1·p)=ℂp,$$

by (2.31) and Schur’s lemma (Theorem 5.3).

The group algebra of the symmetric group $S_k$ over the ring $\mathbb{Z}$ is

$$\begin{array}{cc}{S}_{k,\mathbb{Z}}=\mathbb{Z}{S}_k\text{and}ℂS_k=ℂ{\otimes }_{\mathbb{Z}}{S}_{k,\mathbb{Z}},& \text{(2.32)}\end{array}$$

where the tensor product is defined via the inclusion $\mathbb{Z}\hookrightarrow ℂ\text{.}$ Let $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{\ell })$ be a partition of $k\text{.}$ Define subgroups of $S_k$ by

$$\begin{array}{cc}{S}_{\lambda }=S_{{\lambda }_1}\times \dots \times S_{{\lambda }_{\ell }}\text{and}{S}_{\lambda \prime }=S_{{\lambda }_1^{\prime }}\times \dots \times S_{{\lambda }_r^{\prime }},& \text{(2.33)}\end{array}$$

where $\lambda \prime =({\lambda }_1^{\prime },{\lambda }_2^{\prime },\dots ,{\lambda }_r^{\prime })$ is the conjugate partition to $\lambda ,$ and let

$$\begin{array}{cc}{1}_{\lambda }=\sum _{w\in S_{\lambda }}w\text{and}{\epsilon }_{\lambda \prime }=\sum _{w\in S_{\lambda }^{\prime }}{(-1)}^{\ell \left(w\right)}w\text{.}& \text{(2.34)}\end{array}$$

Let $\tau$ be the permutation in $S_k$ that takes the row reading tableau of shape $\lambda$ to the column reading tableau of shape $\lambda$ For example for $\lambda =\left(553311\right),$

$$\begin{array}{c}\tau =(2,7,8,12,9,16,14,4,15,10,18,6)(3,11)(5,17),\text{since}\\ \begin{array}{ccccc}\tau ·& 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 & =& 1 7 11 15 17 2 8 12 16 18 3 9 13 4 10 14 5 6 & \text{.}\end{array}\end{array}$$

The Specht module for $S_k$ is the $\mathbb{Z}{S}_k\text{-module}$

$$\begin{array}{cc}{S}_{k,\mathbb{Z}}^{\lambda }=\text{im}\hspace{0.17em}{\Psi }_{S_k}=\left(\mathbb{Z}{S}_k\right)p_{\lambda },\text{where}\hspace{0.17em}{p}_{\lambda }=1_{\lambda }\tau {\epsilon }_{\lambda \prime }{\tau }^{-1},\text{and}& \text{(2.35)}\end{array}$$

where ${\Psi }_{S_k}$ is the $\mathbb{Z}{S}_k\text{-module}$ homomorphism given by

$$\begin{array}{cc}\begin{array}{cccccc}{\Psi }_{S_k}:& \left(\mathbb{Z}{S}_k\right)1_{\lambda }& \stackrel{\iota }{⟶}& \mathbb{Z}{S}_k& \stackrel{\pi }{⟶}& \left(\mathbb{Z}{S}_k\right)\tau {\epsilon }_{\lambda \prime }{\tau }^{-1}\\ & b{1}_{\lambda }& ⟼& b{1}_{\lambda }& ⟼& b{1}_{\lambda }\tau {\epsilon }_{\lambda \prime }{\tau }^{-1}\end{array}& \text{(2.36)}\end{array}$$

By induction and restriction rules for the representations of the symmetric groups, the $ℂS_k\text{-modules}$ $\left(ℂS_k\right)1_{\lambda }$ and $\left(ℂS_k\right)\tau {\epsilon }_{\lambda \prime }{\tau }^{-1}$ have only one irreducible component in common and it follows (see [Mac1995, Section I.7, Example 15]) that

$$\begin{array}{cc}{S}_k^{\lambda }=ℂ{\otimes }_{\mathbb{Z}}{S}_{k,\mathbb{Z}}^{\lambda }\text{is the irreducible}\hspace{0.17em}ℂS_k\text{-module indexed by}\hspace{0.17em}\lambda ,& \text{(2.37)}\end{array}$$

once one shows that ${\Psi }_{S_k}$ is not the zero map.

Let $k\in \frac{1}{2}{\mathbb{Z}}_{>0}\text{.}$ For an indeterminate $x,$ define the $\mathbb{Z}\left[x\right]\text{-algebra}$ by

$$\begin{array}{cc}{A}_{k,\mathbb{Z}}=\mathbb{Z}\left[x\right]\text{-span}\{d\in A_k\}& \text{(2.38)}\end{array}$$

with multiplication given by replacing $n$ with $x$ in (2.1). For each $n\in ℂ,$

$$\begin{array}{cc}ℂA_k\left(n\right)=ℂ{\otimes }_{\mathbb{Z}\left[x\right]}{A}_{k,\mathbb{Z}},\text{where the}\hspace{0.17em}\mathbb{Z}\text{-module homomorphism}\begin{array}{cccc}{\text{ev}}_n:& \mathbb{Z}\left[x\right]& ⟶& ℂ,\\ & x& ⟼& n\end{array}& \text{(2.39)}\end{array}$$

is used to define the tensor product. Let $\lambda$ be a partition with $\le k$ boxes. Let $b\otimes p_k^{\otimes (k-\mid \lambda \mid )}$ denote the image of $b\in A_{\mid \lambda \mid ,\mathbb{Z}}$ under the map given by

$$\begin{array}{ccc}{A}_{\mid \lambda \mid ,\mathbb{Z}}& ⟶& A_{k,\mathbb{Z}}\\ b & ⟼& b ⋯ ⋯ ...........................⏟k-|λ| & ,\text{if}\hspace{0.17em}k\hspace{0.17em}\text{is an integer, and}\\ \\ A_{\mid \lambda \mid +\frac{1}{2},\mathbb{Z}}& ⟶& A_{k,\mathbb{Z}}\\ b & ⟼& b ⋯ ⋯ ...........................⏟k-|λ|-12 & ,\text{if}\hspace{0.17em}k-\frac{1}{2}\hspace{0.17em}\text{is an integer.}\end{array}$$

For $k\in \frac{1}{2}{\mathbb{Z}}_{>0},$ define an $A_{k,\mathbb{Z}}\text{-module}$ homomorphism

$$\begin{array}{cc}\begin{array}{cccccc}{\Psi }_{A_k}:& A_{k,\mathbb{Z}}{t}_{\lambda }& \stackrel{{\psi }_1}{⟶}& A_{k,\mathbb{Z}}{s}_{\lambda \prime }& \stackrel{{\psi }_2}{⟶}& A_{k,\mathbb{Z}}/I_{\mid \lambda \mid ,\mathbb{Z}}\\ & b{t}_{\lambda }& ⟼& b{t}_{\lambda }{s}_{\lambda \prime }& ⟼& \overline{b{t}_{\lambda }{s}_{\lambda \prime }},\end{array}& \text{(2.40)}\end{array}$$

where $I_{\mid \lambda \mid ,\mathbb{Z}}$ is the ideal

$$I_{\mid \lambda \mid ,\mathbb{Z}}=\mathbb{Z}\left[x\right]\text{-span}\{d\in A_k\mid d\hspace{0.17em}\text{has propogating number}\hspace{0.17em}<\mid \lambda \mid \}$$

and $t_{\lambda },s_{\lambda \prime }\in A_{k,\mathbb{Z}}$ are defined by

$$\begin{array}{cc}{t}_{\lambda }=1_{\lambda }\otimes p_k^{\otimes (k-\mid \lambda \mid )}\text{and}{s}_{\lambda \prime }=\tau {\epsilon }_{\lambda \prime }{\tau }^{-1}\otimes p_k^{\otimes (k-\mid \lambda \mid )}\text{.}& \text{(2.41)}\end{array}$$

The Specht module for $ℂA_k\left(n\right)$ is the $A_{k,\mathbb{Z}}\text{-module}$

$$\begin{array}{cc}{A}_{k,\mathbb{Z}}^{\lambda }=\text{im}\hspace{0.17em}{\Psi }_{A_k}=(\text{image of}\hspace{0.17em}{A}_{k,\mathbb{Z}}{e}_{\lambda }\hspace{0.17em}\in \hspace{0.17em}{A}_{k,\mathbb{Z}}/I_{\mid \lambda \mid ,\mathbb{Z}}),\text{where}{e}_{\lambda }=p_{\lambda }\otimes p_k^{\otimes (k-\mid \lambda \mid )}\text{.}& \text{(2.42)}\end{array}$$

Proposition 2.43. Let $k\in \frac{1}{2}{\mathbb{Z}}_{>0},$ and let $\lambda$ be a partition with $\le k$ boxes. If $n\in ℂ$ such that $ℂA_k\left(n\right)$ is semisimple, then

$$A_k^{\lambda }\left(n\right)=ℂ{\otimes }_{\mathbb{Z}\left[x\right]}{A}_{k,\mathbb{Z}}^{\lambda }\text{is the irreducible}\hspace{0.17em}ℂA_k\left(n\right)\text{-module indexed by}\hspace{0.17em}\lambda ,$$

where the tensor product is defined via the $\mathbb{Z}\text{-module}$ homomorphism in (2.39).

		Proof.
	Let $r=\mid \lambda \mid \text{.}$ Since $$ℂA_r\left(n\right)/ℂI_r\left(n\right)\cong ℂS_r$$ and $p_{\lambda }$ is a minimal idempotent of $ℂS_r,$ it follows from (4.20) that $\overline{e_{\lambda }},$ the image of $e_{\lambda }$ in $\left(ℂA_k\left(n\right)\right)/\left(ℂI_r\left(n\right)\right),$ is a minimal idempotent in $\left(ℂA_k\left(n\right)\right)/\left(ℂI_r\left(n\right)\right)\text{.}$ Thus $$\left(\frac{ℂA_k\left(n\right)}{ℂI_r\left(n\right)}\right){\bar{e}}_{\lambda }\text{is a simple}\hspace{0.17em}\left(ℂA_k\left(n\right)\right)/\left(ℂI_r\left(n\right)\right)\text{-module.}$$ Since the projection $ℂA_k\left(n\right)\to \left(ℂA_k\left(n\right)\right)/\left(ℂI_r\left(n\right)\right)$ is surjective, any simple $\left(ℂA_k\left(n\right)\right)/\left(ℂI_r\left(n\right)\right)\text{-module}$ is a simple $ℂA_k\left(n\right)\text{-module.}$ $\square$

Notes and References

This is an excerpt of a paper entitled Partition algebras, written by Tom Halverson (Mathematics and Computer Science, Macalester College, Saint Paul, MN 55105, United States) and Arun Ram.

This research was supported in part by National Science Foundation Grant DMS-0100975. This research was also supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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