On representation theory of partition algebras for complex reflection groups
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 389-432.

This paper defines the partition algebra, denoted by 𝒯 k (r,p,n), for complex reflection group G(r,p,n) acting on k-fold tensor product ( n ) k , where n is the reflection representation of G(r,p,n). A basis of the centralizer algebra of this action of G(r,p,n) was given by Tanabe and for p=1, the corresponding partition algebra was studied by Orellana. We also define a subalgebra 𝒯 k+1 2 (r,p,n) such that 𝒯 k (r,p,n)𝒯 k+1 2 (r,p,n)𝒯 k+1 (r,p,n) and establish this subalgebra as partition algebra of a subgroup of G(r,p,n) acting on ( n ) k . We call the algebras 𝒯 k (r,p,n) and 𝒯 k+1 2 (r,p,n) Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras

𝒯0(r,p,n)𝒯12(r,p,n)𝒯1(r,p,n)𝒯32(r,p,n)𝒯n2(r,p,n).

We conclude the paper by giving Jucys–Murphy elements of Tanabe algebras and their actions on the Gelfand–Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.

Received: 2018-12-22
Revised: 2019-08-11
Accepted: 2019-09-18
Published online: 2020-04-01
DOI: https://doi.org/10.5802/alco.97
Classification: 05E10,  20F55,  20C15
Keywords: Complex reflection groups, Tanabe algebras, Schur–Weyl duality, Jucys–Murphy elements
@article{ALCO_2020__3_2_389_0,
     author = {Mishra, Ashish and Srivastava, Shraddha},
     title = {On representation theory of partition algebras for complex reflection groups},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     pages = {389-432},
     doi = {10.5802/alco.97},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_2_389_0/}
}
Mishra, Ashish; Srivastava, Shraddha. On representation theory of partition algebras for complex reflection groups. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 389-432. doi : 10.5802/alco.97. https://alco.centre-mersenne.org/item/ALCO_2020__3_2_389_0/

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