## Partition algebras

Last update: 9 April 2013

## Partition algebras

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

$ℂAk(n)=ℂ-span {d∈Ak}, and multiplication defined byd1d2= nℓ(d1∘d2),$

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,

$ifd1= andd2= then d1d2= =n2 , (2.1)$

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

$ℂAk-12 ↪ ℂAk 1 k d ⟼ 1 k d and ℂAk-1 ↪ ℂAk-12 1 k-1 d ⟼ 1 k d (2.2)$

For ${d}_{1},{d}_{2}\in {A}_{k},$ define

$d1≤d2, if the set partition d2 is coarser than the set partition d1,$

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 $\left\{{x}_{d}\in ℂ{A}_{k}\mid d\in {A}_{k}\right\}$ be the basis of $ℂ{A}_{k}$ uniquely defined by the relation

$d=∑d′≤d xd′, for all d∈Ak. (2.3)$

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

The maps

$ε12:ℂAk⟶ℂ Ak-12, ε12:ℂAk-12 ⟶ℂAk-1and trk:ℂAk⟶ℂ.$

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

$ε12: ℂAk ⟶ ℂAk-12 1 k d ⟼ 1 k d and ε12: ℂAk-12 ⟶ ℂAk-1 1 k d ⟼ 1 k-1 d$

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,

$ε12 (∑A ∑A) = , ε12 (∑A ∑A) = ,$

and

$ε12 (∑A ∑A) = , ε12 (∑A ∑A) =n .$

The map ${\epsilon }^{\frac{1}{2}}$ is the composition $ℂ{A}_{k-\frac{1}{2}}↪ℂ{A}_{k}\stackrel{{\epsilon }_{1}}{\to }ℂ{A}_{k-1}\text{.}$ The composition of ${\epsilon }_{\frac{1}{2}}$ and ${\epsilon }^{\frac{1}{2}}$ is the map

$ε1: ℂAk ⟶ ℂAk-1 1 k d ⟼ 1 k-1 d . (2.4)$

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

$ε12 (α1ba2) = a1ε12 (b)a2, for a1,a2∈ Ak-12,b∈ Ak ε12 (α1ba2) = a1ε12 (b)a2, for a1,a2∈ Ak-1,b∈ Ak-12 ε1 (α1ba2) = a1ε1 (b)a2, for a1,a2∈ Ak-1,b∈ Ak (2.5)$

and

$pk+12b pk+12 = ε12(b) pk+12 = pk+12 ε12(b), for b∈Ak pkb pk = ε12(b) pk = pk ε12(b), for b∈Ak-12 ekbek = ε1(b)ek = ekε1(b), for b∈Ak. (2.6)$

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

$trk(b)= trk-12 (ε12(b)), for b∈Ak, andtrk-12 (b)=trk-1 (ε12(b)), for b∈Ak-12 , (2.7)$

so that

$trk(b)= ε1k(b), for b∈Ak,and trk-12(b) =ε1k-1ε12 (b),for b∈ Ak-12. (2.8)$

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

$trk(d)= d ,for d∈Ak. (2.9)$

The ideal $ℂ{I}_{k}\left(n\right)$

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

$ℂIk(n)=ℂ -span{d∈Ik}. (2.10)$

By (1.3),

$ℂIk(n) is an ideal of ℂAk(n) andℂAk(n) /ℂIk(n)≅ ℂSk, (2.11)$

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 $\ell \in {ℤ}_{>0};$ see (2.2)).

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

$Rad(ℂAk(n))= Rad(ℂIk(n)). (2.12)$

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

$ε12:ℂAk⟶ ℂAk-12and ε12:ℂ Ak-12⟶ℂ Ak-1$

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

$ℂAk(n) ⊗ℂAk-12(n) ℂAk(n)and ℂAk-12(n) ⊗ℂAk-1(n) ℂAk-12(n) (2.13)$

with products given by

$(b1⊗b2) (b3⊗b4)= b1⊗ε12 (b2b3)b4 ,and (c1⊗c2) (c3⊗c4)= c1⊗ε12 (c2c3)c4, (2.14)$

for ${b}_{1},{b}_{2},{b}_{3},{b}_{4}\in ℂ{A}_{k}\left(n\right),$ and for ${c}_{1},{c}_{2},{c}_{3},{c}_{4}\in ℂ{A}_{k-\frac{1}{2}}\left(n\right)\text{.}$

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

$every d∈Ik can be written asd=d1 pkd2,with d1,d2∈ Ak-12, (2.15)$

the maps

$ℂAk-12(n) ⊗ℂAk-1(n) ℂAk-12(n) ⟶ ℂIk(n) b1⊗b2 ⟼ b1pkb2 (2.16)$

are algebra isomorphisms. Thus the ideal $ℂ{I}_{k}\left(n\right)$ 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

$λi = the number of boxes in row i of λ, λj′ = the number of boxes in column j of λ,and ∣λ∣ = the total number of boxes in λ. (2.17)$

Any partition $\lambda$ can be identified with the sequence $\lambda =\left({\lambda }_{1}\ge {\lambda }_{2}\ge \dots \right)$ and the conjugate partition to $\lambda$ is the partition $\lambda \prime =\left({\lambda }_{1}^{\prime },{\lambda }_{2}^{\prime },\dots \right)\text{.}$ The hook length of the box $b$ of $\lambda$ is

$h(b)= (λi-i)+ (λj′-j) +1,if b is in position (i,j) of λ. (2.18)$

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 $ℂ{S}_{k}\text{-modules}$ ${S}_{k}^{\lambda }$ are indexed by the elements of

$S^k= {λ⊢n}and dim(Skλ)= k!∏b∈λh(b) . (2.19)$

For $\lambda \in {\stackrel{^}{S}}_{k}$ and $\mu \in {\stackrel{^}{S}}_{k-1},$

$ResSk-1Sk (Skλ)≅ ⨁λ/ν=□ Sk-1νand IndSk-1Sk (Sk-1μ)≅ ⨁ν/μ=□ Skν, (2.20)$

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 $\stackrel{^}{S}$ given by setting

$vertices on level k:S^k ={partitions λ with k boxes} ,and an edge λ→μ,λ∈ S^k,μ∈ S^k+1 if μ is obtained from λ by adding a box. (2.21)$

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

$∅ k=0: k=1: k=2: k=3: k=4:$

For $\mu \in {\stackrel{^}{S}}_{k}$ define

$S^kμ= { T= ( T(0), T(1),…, T(k) ) | T(0)=∅, T(k)=μ,and ,for each ℓ, T(ℓ)∈ S^ℓ and T(ℓ)→ T(ℓ+1) is an edge in S^ }$

so that ${\stackrel{^}{S}}_{k}^{\mu }$ is the set of paths from $\varnothing \in {\stackrel{^}{S}}_{0}$ to $\mu \in {\stackrel{^}{S}}_{k}$ in the graph $\stackrel{^}{S}\text{.}$ In terms of the Young lattice,

$dim(Skμ)= Card(S^kμ) . (2.22)$

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 ${T}^{\left(\ell -1\right)}\to {T}^{\left(\ell \right)}$ of the path).

Structure of the algebra $ℂ{A}_{k}\left(n\right)$

Build a graph $\stackrel{^}{A}$ by setting

$vertices on level k:A^k ={partitions μ∣k-∣μ∣∈ℤ≥0} , vertices on level k+12: {\stackrel{^}{A}}_{k+1}= A^k ={partitions μ∣k-∣μ∣∈ℤ≥0} , an edge λ→μ,λ∈ A^k,μ∈ A^k+12 if λ=μ or if μ is obtained from λ by removing a box, an edge μ→λ,μ∈ A^k+12,λ∈ A^k+1 if λ=μ or if λ is obtained from μ by adding a box. (2.23)$

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

$∅ ∅ ∅ ∅ ∅ ∅ ∅ k=0: k=0+\frac{1}{2}: k=1: k=1+\frac{1}{2}: k=2: k=2+\frac{1}{2}: k=3:$

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 $ℂ{A}_{k}\left(n\right)$ is semisimple.
2. If $ℂ{A}_{k}\left(n\right)$ is semisimple then the irreducible $ℂ{A}_{k}\left(n\right)\text{-modules,}$ ${A}_{k}^{\mu }$ are indexed by elements of the set ${\stackrel{^}{A}}_{k}=\left\{\text{partitions} \mu \mid k-\mid \mu \mid \in {ℤ}_{\ge 0}\right\},$ and $\text{dim}\left({A}_{k}^{\mu }\right)=$ (number of paths from $\varnothing \in {\stackrel{^}{A}}_{0}$ to $\mu \in {\stackrel{^}{A}}_{k}$ in the graph $\stackrel{^}{A}\text{).}$

Let

$A^kμ= { T= ( T(0), T(12),…, T(k-12), T(k) ) | T(0)=∅, T(k)=μ, and,for each ℓ, T(ℓ)∈ A^ℓ and T(ℓ)→ T(ℓ+12) is an edge in A^ }$

sothat ${\stackrel{^}{A}}_{k}^{\mu }$ is the set of paths from $\varnothing \in {\stackrel{^}{A}}_{0}$ to $\mu \in {\stackrel{^}{A}}_{k}$ in the graph $\stackrel{^}{A}\text{.}$ If $\mu \in {\stackrel{^}{S}}_{k}$ then $\mu \in {\stackrel{^}{A}}_{k}$ and $\mu \in {\stackrel{^}{A}}_{k+\frac{1}{2}}$ and, for notational convenience in the following theorem,

$identify P= ( P(0), P(1),…, P(k) ) ∈S^kμ with the corresponding P= ( P(0), P(0), P(1), P(1),…, P(k-1), P(k-1), P(k) ) ∈A^kμ, andP= ( P(0), P(0), P(1), P(1),…, P(k-1), P(k-1), P(k), P(k) ) ∈A^k+12μ.$

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

$trℓ= ∑μ∈A^ℓ trℓμ(n) χAℓ(n)μ. (2.25)$

Theorem 2.26.

1. Let $n\in ℂ$ and let $k\in \frac{1}{2}{ℤ}_{\ge 0}\text{.}$ Assume that $trℓλ(n)≠0, for all λ∈ A^ℓ,ℓ∈12 ℤ≥0,ℓ Then the partition algebras $ℂAℓ(n) are semisimple for all ℓ∈ 12ℤ≥0,ℓ ≤k. (2.27)$ For each $\ell \in \frac{1}{2}{ℤ}_{\ge 0},$ $\ell \le k-\frac{1}{2},$ define $εμλ= trℓ-12λ(n) trℓ-1μ(n) for each edge μ→λ, μ∈A^ℓ-1, λ∈A^ℓ-12 , in the graph A^.$ Inductively define elements in $ℂ{A}_{\ell }\left(n\right)$ by $ePQμ= 1εμτεμγ eP-Tτpℓ eTQ-γ, for μ∈A^ℓ ,∣μ∣≤ℓ-1, P,Q∈A^ℓμ, (2.28)$ where $\tau ={P}^{\left(\ell -\frac{1}{2}\right)},$ $\gamma ={Q}^{\left(\ell -\frac{1}{2}\right)},$ ${R}^{-}=\left({R}^{\left(0\right)},\dots ,{R}^{\left(\ell -\frac{1}{2}\right)}\right)$ for $R=\left({R}^{\left(0\right)},\dots ,{R}^{\left(\ell -\frac{1}{2}\right)},{R}^{\left(\ell \right)}\right)\in {\stackrel{^}{A}}_{\ell }^{\mu }$ and $T$ is an element of ${\stackrel{^}{A}}_{\ell -1}^{\mu }$ (the element ${e}_{PQ}^{\lambda }$ does not depend on the choice of $T\text{).}$ Then define $ePQλ= (1-z)sPQλ ,forλ∈ S^ℓ,P,Q∈ S^ℓλ, wherez= ∑μ∈A^ℓ∣μ∣≤ℓ-1 ∑P∈A^ℓμ ePPμ (2.29)$ and $\left\{{s}_{PQ}^{\lambda }\mid \lambda \in {\stackrel{^}{S}}_{\ell },P,Q\in {\stackrel{^}{S}}_{\ell }^{\lambda }\right\}$ is any set of matrix units for the group algebra of the symmetric group $ℂ{S}_{\ell }\text{.}$ Together, the elements in (2.28) and (2.29) form a set of matrix units in $ℂ{A}_{\ell }\left(n\right)\text{.}$
2. Let $n\in {ℤ}_{\ge 0}$ and let $k\in \frac{1}{2}{ℤ}_{>0}$ be minimal such that ${\text{tr}}_{k}^{\lambda }\left(n\right)=0$ for some $\lambda \in {\stackrel{^}{A}}_{k}\text{.}$ Then $ℂ{A}_{k+\frac{1}{2}}\left(n\right)$ is not semisimple.
3. Let $n\in {ℤ}_{\ge 0}$ and $k\in \frac{1}{2}{ℤ}_{>0}\text{.}$ If $ℂ{A}_{k}\left(n\right)$ is not semisimple then $ℂ{A}_{k+j}\left(n\right)$ is not semisimple for $j\in {ℤ}_{>0}\text{.}$

 Proof. (a) Assume that $ℂ{A}_{\ell -1}\left(n\right)$ and $ℂ{A}_{\ell -\frac{1}{2}}\left(n\right)$ are both semisimple and that ${\text{tr}}_{\ell -1}^{\mu }\left(n\right)\ne 0$ for all $\mu \in {\stackrel{^}{A}}_{\ell -1}\text{.}$ If $\lambda \in {\stackrel{^}{A}}_{\ell -\frac{1}{2}}$ 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 $ℂ{I}_{\ell }\left(n\right)$ is isomorphic to the basic construction $ℂ{A}_{\ell -\frac{1}{2}}\left(n\right){\otimes }_{ℂ{A}_{\ell -1}\left(n\right)}ℂ{A}_{\ell -\frac{1}{2}}\left(n\right)$ (see (2.13)), it then follows from Theorem 4.28 that $ℂ{I}_{\ell }\left(n\right)$ is semisimple if and only if ${\text{tr}}_{\ell -1}^{\mu }\left(n\right)\ne 0$ for all $\lambda \in {\stackrel{^}{A}}_{\ell -\frac{1}{2}}\text{.}$ Thus, by(2.12), if $ℂ{A}_{\ell -1}\left(n\right)$ and $ℂ{A}_{\ell -\frac{1}{2}}\left(n\right)$ are both semisimple and ${\text{tr}}_{\ell -1}^{\mu }\left(n\right)\ne 0$ for all $\mu \in {\stackrel{^}{A}}_{\ell -1}$ then $ℂAℓ(n) is semisimple if and only if trℓ-12λ (n)≠0 for all λ∈A^ℓ-12 . (2.30)$ By Theorem 4.28, when ${\text{tr}}_{\ell -\frac{1}{2}}^{\lambda }\left(n\right)\ne 0$ for all $\lambda \in {\stackrel{^}{A}}_{\ell -\frac{1}{2}},$ the algebra $ℂ{I}_{\ell }\left(n\right)$ has matrix units given by the formulas in (2.28). The element $z$ in (2.29) is the central idempotent in $ℂ{A}_{\ell }\left(n\right)$ such that $ℂ{I}_{\ell }\left(n\right)=zℂ{A}_{\ell }\left(n\right)\text{.}$ Hence the complete set of elements in (2.28) and (2.29) form a set of matrix units for $ℂ{A}_{\ell }\left(n\right)\text{.}$ This completes the proof of (a) and (b) follows from Theorem 4.28(b). (c) Part (g) of Theorem 4.28 shows that if $ℂ{A}_{\ell -1}\left(n\right)$ is not semisimple then $ℂ{A}_{\ell }\left(n\right)$ 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

$(pAp)op ⟶∼ EndA(Ap), pbp ⟼ ϕpbp whereϕpbp (ap)=(ap) (pbp)=apbp, for ap∈Ap, (2.31)$

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

$pAp=EndA (Ap)op=ℂ (p·1·p)=ℂp,$

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

The group algebra of the symmetric group ${S}_{k}$ over the ring $ℤ$ is

$Sk,ℤ=ℤSk andℂSk= ℂ⊗ℤSk,ℤ, (2.32)$

where the tensor product is defined via the inclusion $ℤ↪ℂ\text{.}$ Let $\lambda =\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{\ell }\right)$ be a partition of $k\text{.}$ Define subgroups of ${S}_{k}$ by

$Sλ=Sλ1×… ×Sλℓand Sλ′= Sλ1′×…× Sλr′, (2.33)$

where $\lambda \prime =\left({\lambda }_{1}^{\prime },{\lambda }_{2}^{\prime },\dots ,{\lambda }_{r}^{\prime }\right)$ is the conjugate partition to $\lambda ,$ and let

$1λ=∑w∈Sλ wandελ′ =∑w∈Sλ′ (-1)ℓ(w)w. (2.34)$

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 $\lambda =\left(553311\right),$

$τ= ( 2,7,8,12,9,16,14, 4,15,10,18,6 ) (3,11)(5,17), since τ· 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 = 1 7 11 15 17 2 8 12 16 18 3 9 13 4 10 14 5 6 .$

The Specht module for ${S}_{k}$ is the $ℤ{S}_{k}\text{-module}$

$Sk,ℤλ=im ΨSk= (ℤSk)pλ, where pλ=1λ τελ′ τ-1,and (2.35)$

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

$ΨSk: (ℤSk)1λ ⟶ι ℤSk ⟶π (ℤSk)τελ′τ-1 b1λ ⟼ b1λ ⟼ b1λτελ′τ-1 (2.36)$

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

$Skλ=ℂ⊗ℤ Sk,ℤλ is the irreducible ℂSk-module indexed by λ, (2.37)$

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

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

$Ak,ℤ=ℤ[x] -span{d∈Ak} (2.38)$

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

$ℂAk(n)=ℂ ⊗ℤ[x] Ak,ℤ, where the ℤ-module homomorphism evn: ℤ[x] ⟶ ℂ, x ⟼ n (2.39)$

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

$A∣λ∣,ℤ ⟶ Ak,ℤ b ⟼ b ⋯ ⋯ ...........................⏟k-|λ| ,if k is an integer, and A∣λ∣+12,ℤ ⟶ Ak,ℤ b ⟼ b ⋯ ⋯ ...........................⏟k-|λ|-12 ,if k-12 is an integer.$

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

$ΨAk: Ak,ℤtλ ⟶ψ1 Ak,ℤsλ′ ⟶ψ2 Ak,ℤ/I∣λ∣,ℤ btλ ⟼ btλsλ′ ⟼ btλsλ′‾, (2.40)$

where ${I}_{\mid \lambda \mid ,ℤ}$ is the ideal

$I∣λ∣,ℤ=ℤ [x]-span {d∈Ak∣d has propogating number <∣λ∣}$

and ${t}_{\lambda },{s}_{\lambda \prime }\in {A}_{k,ℤ}$ are defined by

$tλ=1λ⊗ pk⊗(k-∣λ∣) andsλ′= τελ′τ-1 ⊗pk⊗(k-∣λ∣). (2.41)$

The Specht module for $ℂ{A}_{k}\left(n\right)$ is the ${A}_{k,ℤ}\text{-module}$

$Ak,ℤλ=im ΨAk= ( image of Ak,ℤeλ ∈ Ak,ℤ/ I∣λ∣,ℤ ) ,whereeλ= pλ⊗ pk⊗(k-∣λ∣). (2.42)$

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

$Akλ(n)=ℂ ⊗ℤ[x] Ak,ℤλ is the irreducible ℂAk (n)-module indexed by λ,$

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

 Proof. Let $r=\mid \lambda \mid \text{.}$ Since $ℂAr(n)/ℂ Ir(n)≅ℂSr$ and ${p}_{\lambda }$ is a minimal idempotent of $ℂ{S}_{r},$ it follows from (4.20) that $\stackrel{‾}{{e}_{\lambda }},$ the image of ${e}_{\lambda }$ in $\left(ℂ{A}_{k}\left(n\right)\right)/\left(ℂ{I}_{r}\left(n\right)\right),$ is a minimal idempotent in $\left(ℂ{A}_{k}\left(n\right)\right)/\left(ℂ{I}_{r}\left(n\right)\right)\text{.}$ Thus $( ℂAk(n) ℂIr(n) ) e‾λ is a simple (ℂAk(n))/ (ℂIr(n)) -module.$ Since the projection $ℂ{A}_{k}\left(n\right)\to \left(ℂ{A}_{k}\left(n\right)\right)/\left(ℂ{I}_{r}\left(n\right)\right)$ is surjective, any simple $\left(ℂ{A}_{k}\left(n\right)\right)/\left(ℂ{I}_{r}\left(n\right)\right)\text{-module}$ is a simple $ℂ{A}_{k}\left(n\right)\text{-module.}$ $\square$

## 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).