The quasi-partition algebra

In this paper, we develop the Robinson-Schensted correspondence for the G-vertex colored partition algebras, which gives the bijection between the set of G-vertex colored partition diagrams Pk(n, G) and the pairs of [Formula: see text]-vacillating tableaux of shape λ, [Formula: see text], [Formula: see text] where Γk= {µ ⊢ m|0 ≤ m ≤ k}, Ψq= {λ | λ = (q - j, 1j), j = 0, 1,…, q - 1} and G is a cyclic group of order q. We also derive the Knuth relations for the G-vertex colored partition algebra by using the Robinson-Schensted correspondence for the [Formula: see text]-vacillating tableau of shape [Formula: see text].

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On the modular representation theory of the partition algebra

Journal of Algebraic Combinatorics ◽

10.1007/s10801-014-0580-z ◽

2015 ◽

Vol 42 (1) ◽

pp. 245-282 ◽

Cited By ~ 2

Author(s):

Armin Shalile

Keyword(s):

Representation Theory ◽

Modular Representation ◽

Modular Representation Theory ◽

Partition Algebra

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On the decomposition matrix of the partition algebra in positive characteristic

Journal of Algebra ◽

10.1016/j.jalgebra.2015.11.038 ◽

2016 ◽

Vol 451 ◽

pp. 115-144 ◽

Cited By ~ 1

Author(s):

Oliver King

Keyword(s):

Positive Characteristic ◽

Partition Algebra ◽

Decomposition Matrix

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Jucys–Murphy elements of partition algebras for the rook monoid

International Journal of Algebra and Computation ◽

10.1142/s0218196721500399 ◽

2021 ◽

pp. 1-34

Author(s):

Ashish Mishra ◽

Shraddha Srivastava

Keyword(s):

Representation Theory ◽

Irreducible Representations ◽

Spectral Approach ◽

Inductive Approach ◽

Partition Algebra ◽

Rook Monoid ◽

Partition Algebras ◽

Weyl Duality ◽

Multiplicity Free

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra [Formula: see text] and the subalgebra [Formula: see text] of the partition algebra [Formula: see text] acting on [Formula: see text]. In this paper, we consider a subalgebra [Formula: see text] of [Formula: see text] such that there is Schur–Weyl duality between the actions of [Formula: see text] and [Formula: see text] on [Formula: see text]. This paper studies the representation theory of partition algebras [Formula: see text] and [Formula: see text] for rook monoids inductively by considering the multiplicity free tower [Formula: see text] Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of [Formula: see text] and [Formula: see text]. Also, we describe the Jucys–Murphy elements of [Formula: see text] which play a central role in the demonstration of the actions of Jucys–Murphy elements of [Formula: see text] and [Formula: see text].

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