Partition algebras

By Tom Halverson, Arun Ram. European journal of combinatorics (Print), 2004

300 citations recorded by the source corpus.

The partition algebra CAk(n) is the centralizer algebra of Sn acting on the k-fold tensor product V×k of its n-dimensional permutation representation V. The partition algebra CAk + ½ (n) is the centralizer algebra of the restriction of V⊗k to Sn-1 ⊆ Sn. We apply the theory of the basic construction (generalized matrix algebras) to the tower of partition algebras CA0(n) ⊆ CA ½ (n) CA1 (n) ⊆ CA1½ (n) ⊆ .... Our main results are: (a) a presentation on generators and relations for CAk(n); (b) a derivation of "Specht modules" from the basic construction; (c) a proof that CAk(n) is semisimple if and only if k ≤ (n + 1)/2 (except for a few special cases); (d) Murphy elements for CAk(n); and (e) an exposition on the theory of the basic construction and semisimple algebras.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/009235c2666206796e10a37b6b4758d58469124f

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In Discrete Mathematics and Combinatorics

1. Elise delMas
2. Steven V. Sam
3. Andrew Snowden
4. Vaughan F. R. Jones
5. Richard P. Stanley
6. Tom Halverson
7. William Y. C. Chen
8. Persi Diaconis

In Geometry and Topology

1. Steven V. Sam
2. Andrew Snowden
3. Elise delMas
4. Vaughan F. R. Jones
5. Richard P. Stanley
6. Tom Halverson
7. William Y. C. Chen
8. Jan Draisma

In Mathematical Physics

1. Steven V. Sam
2. Andrew Snowden
3. Elise delMas
4. Vaughan F. R. Jones
5. Richard P. Stanley
6. Tom Halverson
7. William Y. C. Chen
8. Zhengwei Liu

Citations in the corpus

• Quadratic Tangles in Planar Algebras (2010, background): “Note the identity between the principal graphs of the Haagerup subfactor and the partition planar algebra for distance up to five from *. The partition algebra has been studied by several people ([17],[9]) with generic values of the parameter k. One might be tempted to think that the Haagerup subfact”
• STABILITY PATTERNS IN REPRESENTATION THEORY (2013, methodology): “th αin place of t; of course, An(α) is just the quotient of An by the two-sided ideal generated by t−α. The algebra An is called the partition algebra, and was introduced in [Mar] and [Jon] (see also [HR]). (6.3.3) The An(d)-module Td n . We now give Td n the structure of a An(d)-module. We use nota”
• The partition algebra and the Kronecker coefficients (2012, unknown): “r), and when n ≥ 2r, Pr(n) ∼= EndS n(V ⊗r n ). (b) Sn generates EndSn(Vn⊗r). We will denote Er(n) = EndSn(Vn⊗r). THE PARTITION ALGEBRA AND THE KRONECKER COEFFICIENTS 7 Theorem 3.1.2 ([Mar96] see also [HR05]). We have a decomposition of V⊗r nas a (S ,Pr(n))- bimodule V⊗r n = M S(λ[n]) ⊗Lr(λ) where th”
• Representations of the Rook-Brauer Algebra (2012, methodology): “O n(C)(V ⊗k) be the centralizer of O n(C) acting on V⊗k, so that Ck = n φ ∈End(V⊗k) φ(yw) = yφ(w) for all y ∈On(C), w ∈V⊗k o . (3.7) 6 Then by the classical double-centralizer theory (see for example [HR] or [CR, Secs. 3B and 68]), we know the following. Let mk,λ = multV⊗k(Vλ) denote the multiplicit”
• Crossings and nestings of matchings and partitions (2005, background): “ra P n is a certain semisimple algebra, say over C, whose dimension is the Bell number B(n) (the number of partitions of [n]). (The algebra P n depends on a parameter x which is irrelevant here.) See [15, 16] for a survey of this topic. Vacillating tableaux are related to irreducible representations”