RSK Insertion for Set Partitions and Diagram Algebras

Halverson, Tom; Lewandowski, Tim

doi:10.37236/1881

Cited by 26 publications

(45 citation statements)

References 18 publications

(16 reference statements)

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“…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.…”

Section: Application: Diagram Algebrasmentioning

confidence: 99%

“…Paths in the Brauer algebra Bratteli diagram are often called updown tableaux or oscillating tableaux in the literature; see [HL06] and the references therein. Proposition 6.11.…”

Section: A Kmentioning

confidence: 99%

“…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].…”

Section: A Kmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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An insertion algorithm on multiset partitions with applications to diagram algebras

et al. 2020

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We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson [HJ18].

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Section: Application: Diagram Algebrasmentioning

confidence: 99%

“…Paths in the Brauer algebra Bratteli diagram are often called updown tableaux or oscillating tableaux in the literature; see [HL06] and the references therein. Proposition 6.11.…”

Section: A Kmentioning

confidence: 99%

Section: A Kmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An insertion algorithm on multiset partitions with applications to diagram algebras

et al. 2020

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“…We also give the Knuth relations and the determinantal formula for the signed Brauer algebra. Since the Brauer algebra is the subalgebra of the signed Brauer algebra, our correspondence restricted to the Brauer algebra is the same as in [DS,HL,Ro1,Ro2,Su]. As a biproduct, we give the Knuth relations and the determinantal formula for the Brauer algebra.…”

Section: Introductionmentioning

confidence: 99%

Robinson-Schensted Correspondence for the Signed Brauer Algebras

Parvathi¹,

Tamilselvi²

2007

Electron. J. Combin.

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In this paper, we develop the Robinson-Schensted correspondence for the signed Brauer algebra. The Robinson-Schensted correspondence gives the bijection between the set of signed Brauer diagrams $d$ and the pairs of standard bi-dominotableaux of shape $\lambda=(\lambda_1,\lambda_2)$ with $\lambda_1=(2^{2f}),\lambda_2 \in \overline{\Gamma}_{f,r}$ where $\overline{\Gamma}_{f,r}=\{ \lambda | \lambda\vdash 2(n-2f)+|\delta_r| {\rm \ whose } \ 2{\rm-core \ is \ \delta_r, \ } \delta_r=(r,r-1,\ldots,1,0)\}$, for fixed $r\geq 0$ and $0\leq f \leq \left[{n\over 2}\right]$. We also give the Robinson-Schensted for the signed Brauer algebra using the vacillating tableau which gives the bijection between the set of signed Brauer diagrams ${\overline{V}_n}$ and the pairs of $d$-vacillating tableaux of shape $\lambda \in \overline{\Gamma}_{f,r}$ and $0\leq f \leq \left[{n\over 2}\right]$. We derive the Knuth relations and the determinantal formula for the signed Brauer algebra by using the Robinson-Schensted correspondence for the standard bi-dominotableau whose core is $\delta_{r}$, $r \geq n-1$.

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Partition Algebras and the Invariant Theory of the Symmetric Group

Benkart

Halverson

2019

Association for Women in Mathematics Series

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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.

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RSK Insertion for Set Partitions and Diagram Algebras

Cited by 26 publications

References 18 publications

An insertion algorithm on multiset partitions with applications to diagram algebras

An insertion algorithm on multiset partitions with applications to diagram algebras

Robinson-Schensted Correspondence for the Signed Brauer Algebras

Partition Algebras and the Invariant Theory of the Symmetric Group

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