The partition algebra is an
. Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group.
Definition
Diagrams
A partition of
elements labelled
is represented as a diagram, with lines connecting elements in the same subset. In the following example, the subset
gives rise to the lines
, and could equivalently be represented by the lines
(for instance).
For
and
, the partition algebra
is defined by a
-basis made of partitions, and a multiplication given by diagram concatenation. The concatenated diagram comes with a factor
, where
is the number of connected components that are disconnected from the top and bottom elements.
Generators and relations
The partition algebra
is generated by
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a86530d6746d08293ae80110796ba5998ebcd85c)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45b72efa67afc32a7a24901909d51ac898591e3f)
Properties
The partition algebra
is an associative algebra. It has a multiplicative identity
The partition algebra
is semisimple for
. For any two
in this set, the algebras
and
are isomorphic.[1]
The partition algebra is finite-dimensional, with
(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
, or size
, or only size
.
- Whether we allow top-top and bottom-bottom lines, or only top-bottom lines. In the latter case, the parameter
is absent, or can be eliminated by
.
Combining these properties gives rise to 8 nontrivial subalgebras, in addition to the partition algebra itself:[1][3]
Notation
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Name
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Generators
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Dimension
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Example
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Partition
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Planar partition
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Rook Brauer
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Motzkin
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Brauer
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Temperley–Lieb
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Rook
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Planar rook
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Symmetric group
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The symmetric group algebra
is the group ring of the symmetric group
over
. The Motzkin algebra is sometimes called the dilute Temperley–Lieb algebra in the physics literature.[4]
Properties
The listed subalgebras are semisimple for
.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4377ca6255ac42388ddf212e3e3bcd2b8a592f)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd19442dd63c7126aad4811e45517478cf937523)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc291bc222a7146aee91573305518ac7d1647da3)
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e51e46accff83e0a6faed312fb53ae720ac121)
More subalgebras
In addition to the eight subalgebras described above, other subalgebras have been defined:
- The totally propagating partition subalgebra
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
.[6]
- The quasi-partition algebra
is generated by subsets of size at least two. Its generators are
and its dimension is
.[7]
- The uniform block permutation algebra
is generated by subsets with as many top elements as bottom elements. It is generated by
.[8]
An algebra with a half-integer index
is defined from partitions of
elements by requiring that
and
are in the same subset. For example,
is generated by
so that
, and
.[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=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea08ed932b6883d805e392918b1df37de2a891e)
such that
. The translation element and its powers are the only combinations of
that belong to periodic subalgebras.
Representations
Structure
For an integer
, let
be the set of partitions of
elements
(bottom) and
(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
:
Partition diagrams act on
from the bottom, while the symmetric group
acts from the top. For any Specht module
of
(with therefore
), we define the representation of
![{\displaystyle {\mathcal {P}}_{\lambda }=\mathbb {C} D_{|\lambda |}\otimes _{\mathbb {C} S_{|\lambda |}}V_{\lambda }\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ddf7bc29a8aa8c86f65419bc60059b23a4ea63)
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 |}}\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc00ea82b6ee2d07c7316848917b160ba7a62a94)
where
is a
Stirling number of the second kind
,
![{\displaystyle {\binom {\ell }{|\lambda |}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89b5c4bf166df548f740a0e9c0bd3171b91336cd)
is a
binomial coefficient, and
![{\displaystyle f_{\lambda }=\dim S_{\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf98a949dd10a1b4b6fb9aa2f75d246348c7190)
is given by the
hook length formula.
A basis of
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
is semisimple, the representation
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}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24fae4966d71642e852a180ad1e260404768ce12)
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
with
, 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
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Parameter
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Conditions
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Dimension
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The irreducible representations of
are indexed by partitions such that
and their dimensions are
.[5] The irreducible representations of
are indexed by partitions such that
.[7] The irreducible representations of
are indexed by sequences of partitions.[8]
Schur-Weyl duality
Assume
.
For
a
-dimensional vector space with basis
, there is a natural action of the partition algebra
on the vector space
. This action is defined by the matrix elements of a partition
in the basis
:[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}}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/588dbdc4d14639e044d3356e6dc20508e9e51641)
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}}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ee3a820c0888bfd252ecddb1c45370b4e0688)
Duality between the partition algebra and the symmetric group
Let
be integer.
Let us take
to be the natural permutation representation of the symmetric group
. This
-dimensional representation is a sum of two irreducible representations: the standard and trivial representations,
.
Then the partition algebra
is the
centralizer
of the action of
![{\displaystyle S_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4)
on the tensor product space
![{\displaystyle V^{\otimes k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4b3a680c9ed12084f31bd4ee6dabe0d93ce040)
,
![{\displaystyle P_{k}(n)\cong {\text{End}}_{S_{n}}\left(V^{\otimes k}\right)\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb318b9b2c653b95d4229336c22344d8efde28d)
Moreover, as a bimodule over
, 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 ]}\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8286ebe5ed8cd7d243b074a73df3e998032a7bf0)
where
is a Young diagram of size
built by adding a first row to
, and
is the corresponding Specht module of
.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767)
for an irreducible
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-dimensional representation of the first group or algebra:
Tensor product space
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Group or algebra
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Dual algebra or group
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Comments
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The duality for the full partition algebra
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Case of a partition algebra with a half-integer index[2]
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The original Schur-Weyl duality
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Duality between the orthogonal group and the Brauer algebra
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Duality between the orthogonal group and the rook Brauer algebra[9]
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Duality between the rook algebra and the totally propagating partition algebra[10][5]
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Duality between a Lie superalgebra and the planar rook algebra[11]
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Duality between the symmetric group and the quasi-partition algebra[7]
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Duality involving the walled Brauer algebra.[12]
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References
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Further reading