Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

Benkart, Georgia; Halverson, Tom; Harman, Nate

doi:10.1007/s10801-017-0748-4

Cited by 20 publications

(22 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Our interest is in one specific choice of

λ

, namely, the two‐part partition

[n - k, k] ⊢ n

. When

n ⩾ 2 k

, the corresponding irreducible

{sans-serifP}_{k} false(n false)

‐module

P_{k, n}^{false[n - k, k false]}

is one‐dimensional (see [, Corollary 5.14]). The diagram basis elements

d_{π}

with

pn (π) < k

act as 0 on

P_{k, n}^{false[n - k, k false]}

, and the permutation diagrams act as the identity element.…”

Section: Connections With Primitive Central Idempotentsmentioning

confidence: 99%

“…When

n = 2 k

, the kernel of the surjection

{sans-serifP}_{k + 1 / 2} false(2 k false) \to {sans-serifEnd}_{S_{2 k - 1}} false(M_{2 k}^{\otimes k} false)

is generated by

, as in Theorem (b). In fact, we know from [, Theorem 5.5] that the kernel is one‐dimensional, since the dimension of

{sans-serifEnd}_{S_{2 k - 1}} false(M_{2 k}^{\otimes k} false)

B false(2 k + 1 false) - 1 = \dim {sans-serifP}_{k + 1 / 2} false(2 k false) - 1

, so the kernel is the

double-struckF

‐span of

(see Theorem (b)). By Theorem (b),

{normalΞ}_{k + 1 / 2, 2 k} = false({false(- 1 false)}^{k} / k! false) {sans-serife}_{k + 1 / 2, 2 k}

, when

n = 2 k

.…”

Section: Connections With Primitive Central Idempotentsmentioning

confidence: 99%

“…With this identification, Ψ k,n (ε λ ) for any λ n can be expressed in the diagram basis {d π | π ∈ Π 2k } or in the orbit basis {x π | π ∈ Π 2k } of P k (n). Our interest is in one specific choice of λ, namely, the two-part partition [n − k, k] n. When n 2k, the corresponding irreducible P k (n)-module P [n−k,k] k,n is one-dimensional (see [2,Corollary 5.14]). The diagram basis elements d π with pn(π) < k act as 0 on P [n−k,k] k,n , and the permutation diagrams act as the identity element.…”

Section: The Irreducible Module Labeled By the Partition [N − K K]mentioning

confidence: 99%

See 2 more Smart Citations

Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

Benkart

Halverson

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

Assume Mn is the n‐dimensional permutation module for the symmetric group Sn, and let Mn⊗k be its k‐fold tensor power. The partition algebra sans-serifPkfalse(nfalse) maps surjectively onto the centralizer algebra sans-serifEndSnfalse(Mn⊗kfalse) for all k,n∈double-struckZ⩾1 and isomorphically when n⩾2k. We describe the image of the surjection normalΦk,n:sans-serifPkfalse(nfalse)→sans-serifEndSnfalse(Mn⊗kfalse) explicitly in terms of the orbit basis of sans-serifPkfalse(nfalse) and show that when 2k>n the kernel of Φk,n is generated by a single essential idempotent ek,n, which is an orbit basis element. We obtain a presentation for sans-serifEndSnfalse(Mn⊗kfalse) by imposing one additional relation, sans-serifek,n=0, to a presentation of the partition algebra sans-serifPkfalse(nfalse) when 2k>n. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group Sn. We show under the natural embedding of the partition algebra sans-serifPnfalse(nfalse) into sans-serifPkfalse(nfalse) for k⩾n that the essential idempotent en,n generates the kernel of Φk,n. Therefore, the relation sans-serifen,n=0 can replace sans-serifek,n=0 when k⩾n.

show abstract

“…Our interest is in one specific choice of

λ

, namely, the two‐part partition

[n - k, k] ⊢ n

. When

n ⩾ 2 k

, the corresponding irreducible

{sans-serifP}_{k} false(n false)

‐module

P_{k, n}^{false[n - k, k false]}

is one‐dimensional (see [, Corollary 5.14]). The diagram basis elements

d_{π}

with

pn (π) < k

act as 0 on

P_{k, n}^{false[n - k, k false]}

, and the permutation diagrams act as the identity element.…”

Section: Connections With Primitive Central Idempotentsmentioning

confidence: 99%

“…When

n = 2 k

, the kernel of the surjection

{sans-serifP}_{k + 1 / 2} false(2 k false) \to {sans-serifEnd}_{S_{2 k - 1}} false(M_{2 k}^{\otimes k} false)

is generated by

, as in Theorem (b). In fact, we know from [, Theorem 5.5] that the kernel is one‐dimensional, since the dimension of

{sans-serifEnd}_{S_{2 k - 1}} false(M_{2 k}^{\otimes k} false)

B false(2 k + 1 false) - 1 = \dim {sans-serifP}_{k + 1 / 2} false(2 k false) - 1

, so the kernel is the

double-struckF

‐span of

(see Theorem (b)). By Theorem (b),

{normalΞ}_{k + 1 / 2, 2 k} = false({false(- 1 false)}^{k} / k! false) {sans-serife}_{k + 1 / 2, 2 k}

, when

n = 2 k

.…”

Section: Connections With Primitive Central Idempotentsmentioning

confidence: 99%

Section: The Irreducible Module Labeled By the Partition [N − K K]mentioning

confidence: 99%

See 1 more Smart Citation

Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

Benkart

Halverson

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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).…”

Section: Introductionmentioning

confidence: 99%

An insertion algorithm on multiset partitions with applications to diagram algebras

et al. 2020

View full text Add to dashboard Cite

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].

show abstract

“…The Schur-Weyl duality afforded by Φ k,n between the partition algebra P k (n) and the symmetric group S n in their commuting actions on M ⊗k n enables information to flow back and forth between S n and P k (n). Indeed, the symmetric group S n has been used to • develop the combinatorial representation theory of the partition algebras P k (n) as k varies [M3,MR,H,FaH,HL,HR,BH,BHH],…”

Section: Introductionmentioning

confidence: 99%

Partition Algebras and the Invariant Theory of the Symmetric Group

Benkart

Halverson

2019

Association for Women in Mathematics Series

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

Cited by 20 publications

References 21 publications

Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

An insertion algorithm on multiset partitions with applications to diagram algebras

Partition Algebras and the Invariant Theory of the Symmetric Group

Contact Info

Product

Resources

About