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The partition algebra Pk(n) is an associative algebra with a basis of set partition diagrams and a multiplication given by diagram concatenation. It contains the group algebra of the symmetric group Sk as a subalgebra, and it is in Schur- Weyl duality with the symmetric group Sn on the k-fold tensor power of its n-dimensional permutation module. When n ≥ 2k, the partition algebra is semisimple, and its irreducible modules Pλk,n are indexed by integer partitions λ = [λ1,...,λi] of n with λ2 + ··· + λi ≤ k. The dimension of Pλk,n is given by the number of standard set-partition tableaux of shape λ. We give explicit descriptions of the action of Pk(n) on a basis indexed by set partition tableaux. This is a generalization of Young’s natural basis of the Specht module for Sk. As an application, we express the characters of Pk(n) as linear combinations of symmetric group characters. The coefficients in the expansion are non-negative integers which count fixed points in a quotient of the partition algebra. From these coefficient we obtain a matrix which maps the direct sum of character tables of S0, . . . , Sk to the character table of Pk(n).
Jacobson, Theo N., "Set-Partition Tableaux and Irreducible Representations of the Partition Algebra" (2018). Mathematics, Statistics, and Computer Science Honors Projects. 42.
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