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

𝒯 0 (r,p,n)𝒯 1 2 (r,p,n)𝒯 1 (r,p,n)𝒯 3 2 (r,p,n)𝒯 n 2 (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:
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Accepted:
Published online:
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},
     pages = {389--432},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     doi = {10.5802/alco.97},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.97/}
}
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/articles/10.5802/alco.97/

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