<i>G</i>-Vertex Colored Partition Algebras as Centralizer Algebras of Direct Products

Parvathi, M.; Kennedy, A. Joseph

doi:10.1081/agb-200034152

“…As noted by Grood [33], Solomon's discovery [72] of a Schur-Weyl duality for rook monoid algebras (see also [50]) led to the investigation of a number of other "rook diagram algebras", such the rook Brauer algebras [35,60], Motzkin algebras [5] and more. Such studies, and other considerations often to do with representation theory and/or statistical mechanics, have led to the discovery and investigation of a great many other families of diagram algebras [6,10,11,45,61,62,67,69].…”

Section: Introductionmentioning

confidence: 99%

Presentations for rook partition monoids and algebras and their singular ideals

East

¹

2019

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…The main motivation for studying the partition algebra is in generalizing the Temperly-Lieb algebras and the Potts model in statistical mechanics. In [PK1] Parvathi and Kennedy obtained a new class of algebras P k (x, G), the G-vertex colored partition algebras, where G is a finite group. These algebras, when x = n ≥ 2k, were shown to be the centralizer algebra of the direct product group S n × G acting on the tensor product space V ⊗k by the restricted action as a subgroup of the wreath product G S n as in [B].…”

Section: Introductionmentioning

confidence: 99%

R-S Correspondence for $({\Bbb Z}_2 \times {\Bbb Z}_2) \wr S_n$ and Klein-$4$ Diagram Algebras

Parvathi¹,

Sivakumar²

2008

Electron. J. Combin.

Self Cite

0

View full text Add to dashboard Cite

In [PS] a new family of subalgebras of the extended ${\Bbb Z}_2$-vertex colored algebras, called Klein-$4$ diagram algebras, are studied. These algebras are the centralizer algebras of $G_n:=({\Bbb Z}_2 \times {\Bbb Z}_2) \wr S_n$ when it acts on $V^{\otimes k},$ where $V$ is the signed permutation module for $G_n.$ In this paper we give the Robinson-Schensted correspondence for $G_n$ on $4$-partitions of $n,$ which gives a bijective proof of the identity $\sum_{[\lambda] \vdash n } (f^{[\lambda]})^2 = 4^n n!,$ where $f^{[\lambda]}$ is the degree of the corresponding representation indexed by $[\lambda]$ for $G_n.$ We give proof of the identity $2^kn^k = \sum_{[\lambda] \in \Gamma_{n,k}^G} f^{[\lambda]} m_{k}^{[\lambda]}$ where the sum is over $4$-partitions which index the irreducible $G_n$-modules appearing in the decomposition of $V^{\otimes k} $ and $m_{k}^{[\lambda]}$ is the multiplicity of the irreducible $G_n$-module indexed by $[\lambda ].$ Also, we develop an R-S correspondence for the Klein-$4$ diagram algebras by giving a bijection between the diagrams in the basis and pairs of vacillating tableau of same shape.

show abstract

“…In [11], we defined an equivalence relation ∼ on G-diagrams and a multiplication on G-diagrams, which is associative and well-defined up to equivalence of such diagrams, as follows:…”

Section: The Colored Partition Algebrasmentioning

confidence: 99%

“…The G-vertex colored partition algebra P k (n, G) has been introduced in [11] and has been realized as the centralizer algebra of the subgroup G × S n of G S n . The extended vertex colored partition algebras P k (n, G), which is the centralizer algebra of the subgroup S n of G × S n , and the representations of these algebras have been studied in [12] and [13].…”

mentioning

confidence: 99%

“…The G-vertex colored partition algebra P k (n, G) has been introduced in [11] and has been realized as the centralizer algebra of the subgroup G × S n of G S n . The extended vertex colored partition algebras P k (n, G), which is the centralizer algebra of the subgroup S n of G × S n , and the representations of these algebras have been studied in [12] and [13]. In the case |G| = m, the natural inclusion of groups S n ⊆ G × S n ⊆ G S n ⊆ S m S n ⊆ S mn induces the natural revers inclusion of the corresponding centralizer algebras as P k (mn) ⊆ P k (m, n) ⊆ − → P k (n, G) ⊆ P k (n, G) ⊆ P k (n, G), which have been studied explicitly in [6].…”

mentioning

confidence: 99%

See 1 more Smart Citation