Partition Algebras and the Invariant Theory of the Symmetric Group

Abstract

The symmetric group S n and the partition algebra P k (n) centralize one another in their actions on the k-fold tensor power M ⊗k n of the n-dimensional permutation module M n of S n . The duality afforded by the commuting actions determines an algebra homomorphism Φ k,n : P k (n) → End S n (M ⊗k n ) from the partition algebra to the centralizer algebra End S n (M ⊗k n ), which is a surjection for all k, n ∈ Z ≥1 , and an isomorphism when n ≥ 2k. We present results that can be derived from the duality between S n and P k (n); for example, (i) expressions for the multiplicities of the irreducible S n -summands of M ⊗k n , (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra End S n (M ⊗k n ), (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra P k (n). When 2k > n, the map Φ k,n has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of Φ k,n in terms of the orbit basis of P k (n) and explain how the surjection Φ k,n can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

An insertion algorithm on multiset partitions with applications to diagram algebras

“…In [HL06,MR98], the authors introduce RSK-type algorithms between partition algebra diagrams and pairs of paths in the Bratteli diagram of the partition algebras; in [HL06] these paths are called vacillating tableaux. In [BH17], the authors define a bijection between vacillating tableaux and standard multiset tableaux. In this section we provide a different bijection from partition algebra diagrams to standard multiset tableaux.…”

“…The map φ from Proposition 6.11 allows us to establish a bijection between standard multiset tableaux and vacillating tableaux. A vacillating tableau is a sequence partitions satisfying the condition λ (r) ⊢ n and λ (r+ 1 2 ) ⊢ n−1 with λ (r) ← λ (r+ 1 2 ) and λ (r+ 1 2 ) → λ (r+1) for 0 r < k [HL06,BH17]. A different bijection appears in [BH17].…”

“…A vacillating tableau is a sequence partitions satisfying the condition λ (r) ⊢ n and λ (r+ 1 2 ) ⊢ n−1 with λ (r) ← λ (r+ 1 2 ) and λ (r+ 1 2 ) → λ (r+1) for 0 r < k [HL06,BH17]. A different bijection appears in [BH17]. The bijection we provide here is compatible with the families of tableaux for each of the subalgebras and the Bratteli diagrams for those subalgebras.…”

“…Recently, a new combinatorial interpretation for the dimensions of the irreducible representations for the partition algebra has appeared in the literature [BH17,BHH17,OZ16,HJ18]. In particular, Benkart and Halverson [BH17] presented a bijection between vacillating tableaux and "set-partition tableaux" (tableaux whose entries are sets of positive integers). There are two main advantages to working with set-partition tableaux instead of vacillating tableaux.…”

“…We prove that the dimensions of the irreducible representations of the various algebras is equal to the number of our combinatorially-defined tableaux by establishing that the branching rules are encoded in the tableaux (see Section 6.3 and Corollary 6.16). Our insertion is different from combining the insertion of Halverson and Lewandowski [HL06] from partition diagrams to paths in the Bratteli diagram with the bijection of Halverson and Benkart [BH17] from paths in the Bratteli diagram to set-partition tableaux (see Section 6.3).…”