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 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).

Diagram representation of a partition of 14 elements

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.

Concatenation of two partitions of 22 elements

Generators and relations

The partition algebra is generated by elements of the type

Generators of the partition algebra

These generators obey relations that include[2]

Other elements that are useful for generating subalgebras include

Elements of the partition algebra that are useful for generating subalgebras

In terms of the original generators, these elements are

Properties

The partition algebra is an associative algebra. It has a multiplicative identity

Identity element of the partition algebra

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 Name Generators Dimension Example
Partition A partition
Planar partition A planar partition
Rook Brauer A rook Brauer partition
Motzkin A Motkzin partition
Brauer A Brauer partition
Temperley–Lieb A Temperley-Lieb partition
Rook A rook monoid partition
Planar rook A planar rook monoid partition
Symmetric group A permutation partition

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:

Inclusions from constraints on subset size:

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

We have the isomorphism:

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 Translation partition 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 :

Example of a state in a representation of the partition algebra

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

The dimension of this representation is[1]

where is a Stirling number of the second kind, is a binomial coefficient, and 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

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 Parameter Conditions Dimension

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]

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

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 on the tensor product space ,

Moreover, as a bimodule over , the tensor product space decomposes into irreducible representations as[1]

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 for an irreducible -dimensional representation of the first group or algebra:

Tensor product space Group or algebra Dual algebra or group Comments
The duality for the full partition algebra
Case of a partition algebra with a half-integer index[2]
The original Schur-Weyl duality
Duality between the orthogonal group and the Brauer algebra
Duality between the orthogonal group and the rook Brauer algebra[9]
Duality between the rook algebra and the totally propagating partition algebra[10][5]
Duality between a Lie superalgebra and the planar rook algebra[11]
Duality between the symmetric group and the quasi-partition algebra[7]
Duality involving the walled Brauer algebra.[12]

References

  1. Halverson, Tom; Jacobson, Theodore N. (2020). "Set-partition tableaux and representations of diagram algebras". Algebraic Combinatorics. 3 (2): 509–538. arXiv:1808.08118v2. doi:10.5802/alco.102. ISSN 2589-5486. S2CID 119167251.
  2. Halverson, Tom; Ram, Arun (2005). "Partition algebras". European Journal of Combinatorics. 26 (6): 869–921. arXiv:math/0401314v2. doi:10.1016/j.ejc.2004.06.005. S2CID 1168919.
  3. Colmenarejo, Laura; Orellana, Rosa; Saliola, Franco; Schilling, Anne; Zabrocki, Mike (2020). "An insertion algorithm on multiset partitions with applications to diagram algebras". Journal of Algebra. 557: 97–128. arXiv:1905.02071v2. doi:10.1016/j.jalgebra.2020.04.010. S2CID 146121089.
  4. Jacobsen, Jesper Lykke; Ribault, Sylvain; Saleur, Hubert (2022). "Spaces of states of the two-dimensional O(n) and Potts models". arXiv:2208.14298. {{cite journal}}: Cite journal requires |journal= (help)
  5. Mishra, Ashish; Srivastava, Shraddha (2021). "Jucys–Murphy elements of partition algebras for the rook monoid". International Journal of Algebra and Computation. 31 (5): 831–864. arXiv:1912.10737v3. doi:10.1142/S0218196721500399. ISSN 0218-1967. S2CID 209444954.
  6. Maltcev, Victor (2007-03-16). "On a new approach to the dual symmetric inverse monoid I*X". arXiv:math/0703478v1.
  7. Daugherty, Zajj; Orellana, Rosa (2014). "The quasi-partition algebra". Journal of Algebra. 404: 124–151. arXiv:1212.2596v1. doi:10.1016/j.jalgebra.2013.11.028. S2CID 117848394.
  8. Orellana, Rosa; Saliola, Franco; Schilling, Anne; Zabrocki, Mike (2021-12-27). "Plethysm and the algebra of uniform block permutations". arXiv:2112.13909v1 [math.CO].
  9. Halverson, Tom; delMas, Elise (2014-01-02). "Representations of the Rook-Brauer Algebra". Communications in Algebra. 42 (1): 423–443. arXiv:1206.4576v2. doi:10.1080/00927872.2012.716120. ISSN 0092-7872. S2CID 38469372.
  10. Kudryavtseva, Ganna; Mazorchuk, Volodymyr (2008). "Schur–Weyl dualities for symmetric inverse semigroups". Journal of Pure and Applied Algebra. 212 (8): 1987–1995. arXiv:math/0702864. doi:10.1016/j.jpaa.2007.12.004. S2CID 13564450.
  11. Benkart, Georgia; Moon, Dongho (2013-05-28). "Planar Rook Algebras and Tensor Representations of 𝔤𝔩(1 | 1)". Communications in Algebra. 41 (7): 2405–2416. arXiv:1201.2482v1. doi:10.1080/00927872.2012.658533. ISSN 0092-7872. S2CID 119125305.
  12. Cox, Anton; Visscher, De; Doty, Stephen; Martin, Paul (2007-09-06). "On the blocks of the walled Brauer algebra". arXiv:0709.0851v1 [math.RT].

Further reading

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