On representation theory of partition algebras for complex reflection groups

Mishra, Ashish; Srivastava, Shraddha

doi:10.5802/alco.97

Mishra, Ashish ¹; Srivastava, Shraddha ²

¹ Instituto de Ciências Exatas e Naturais Universidade Federal do Pará Belém Pará Brazil
² The Institute of Mathematical Sciences Chennai India

Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 389-432.

Abstract

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 ${(ℂ^{n})}^{\otimes k}$ , where $ℂ^{n}$ 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 $𝒯_{k + \frac{1}{2}} (r, p, n)$ such that $𝒯_{k} (r, p, n) \subseteq 𝒯_{k + \frac{1}{2}} (r, p, n) \subseteq 𝒯_{k + 1} (r, p, n)$ and establish this subalgebra as partition algebra of a subgroup of $G (r, p, n)$ acting on ${(ℂ^{n})}^{\otimes k}$ . We call the algebras $𝒯_{k} (r, p, n)$ and $𝒯_{k + \frac{1}{2}} (r, p, n)$ 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) \subseteq 𝒯_{\frac{1}{2}} (r, p, n) \subseteq 𝒯_{1} (r, p, n) \subseteq 𝒯_{\frac{3}{2}} (r, p, n) \subseteq \dots \subseteq 𝒯_{⌊ \frac{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: 2018-12-22
Revised: 2019-08-11
Accepted: 2019-09-18
Published online: 2020-04-01

DOI: 10.5802/alco.97

Classification: 05E10, 20F55, 20C15
Keywords: Complex reflection groups, Tanabe algebras, Schur–Weyl duality, Jucys–Murphy elements

Author's affiliations:

Mishra, Ashish ¹; Srivastava, Shraddha ²

¹ Instituto de Ciências Exatas e Naturais Universidade Federal do Pará Belém Pará Brazil
² The Institute of Mathematical Sciences Chennai India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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/}
}

TY  - JOUR
AU  - Mishra, Ashish
AU  - Srivastava, Shraddha
TI  - On representation theory of partition algebras for complex reflection groups
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 389
EP  - 432
VL  - 3
IS  - 2
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.97/
DO  - 10.5802/alco.97
LA  - en
ID  - ALCO_2020__3_2_389_0
ER  -

%0 Journal Article
%A Mishra, Ashish
%A Srivastava, Shraddha
%T On representation theory of partition algebras for complex reflection groups
%J Algebraic Combinatorics
%D 2020
%P 389-432
%V 3
%N 2
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.97/
%R 10.5802/alco.97
%G en
%F 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/articles/10.5802/alco.97/

References
Cited by

[1] Bagno, Eli; Biagioli, Riccardo Colored-descent representations of complex reflection groups $G (r, p, n)$ , Israel J. Math., Volume 160 (2007), pp. 317-347 | DOI | MR | Zbl

[2] Benkart, Georgia; Halverson, Tom Partition algebras $P_{k} (n)$ with $2 k > n$ and the fundamental theorems of invariant theory for the symmetric group $S_{n}$ , J. Lond. Math. Soc. (2), Volume 99 (2019) no. 1, pp. 194-224 | DOI | MR | Zbl

[3] Bloss, Matthew Partition algebras and permutation representations of wreath products, Ph. D. Thesis, University of Wisconsin – Madison (USA) (2002) (http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3049298)

[4] Cohen, Arjeh M. Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4), Volume 9 (1976) no. 3, pp. 379-436 | DOI | Numdam | MR | Zbl

[5] Enyang, John A seminormal form for partition algebras, J. Combin. Theory Ser. A, Volume 120 (2013) no. 7, pp. 1737-1785 | DOI | MR | Zbl

[6] Green, James A. Polynomial representations of $G L_{n}$ , Lecture Notes in Mathematics, 830, Springer, Berlin, 2007, x+161 pages (With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, Green and M. Schocker) | MR

[7] Halverson, Tom; Ram, Arun Murnaghan–Nakayama rules for characters of Iwahori–Hecke algebras of the complex reflection groups $G (r, p, n)$ , Canad. J. Math., Volume 50 (1998) no. 1, pp. 167-192 | DOI | MR | Zbl

[8] Halverson, Tom; Ram, Arun Partition algebras, European J. Combin., Volume 26 (2005) no. 6, pp. 869-921 | DOI | MR | Zbl

[9] Jones, Vaughan F. R. The Potts model and the symmetric group, Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 259-267 | MR | Zbl

[10] Martin, Paul P. Temperley–Lieb algebras for nonplanar statistical mechanics – the partition algebra construction, J. Knot Theory Ramifications, Volume 3 (1994) no. 1, pp. 51-82 | DOI | MR | Zbl

[11] Martin, Paul P. The partition algebra and the Potts model transfer matrix spectrum in high dimensions, J. Phys. A, Volume 33 (2000) no. 19, pp. 3669-3695 | DOI | MR | Zbl

[12] Martin, Paul P.; Rollet, Geneviève The Potts model representation and a Robinson–Schensted correspondence for the partition algebra, Compositio Math., Volume 112 (1998) no. 2, pp. 237-254 | DOI | MR | Zbl

[13] Martin, Paul P.; Saleur, Hubert On an algebraic approach to higher-dimensional statistical mechanics, Comm. Math. Phys., Volume 158 (1993) no. 1, pp. 155-190 | DOI | MR | Zbl

[14] Martin, Paul P.; Saleur, Hubert Algebras in higher-dimensional statistical mechanics – the exceptional partition (mean field) algebras, Lett. Math. Phys., Volume 30 (1994) no. 3, pp. 179-185 | DOI | MR | Zbl

[15] Mendes, Anthony; Remmel, Jeffrey; Wagner, Jennifer A $λ$ -ring Frobenius characteristic for $G ≀ S_{n}$ , Electron. J. Combin., Volume 11 (2004) no. 1, Paper no. paper number 56, 33 pages | MR | Zbl

[16] Mishra, Ashish; Srinivasan, Murali K. The Okounkov–Vershik approach to the representation theory of $G \sim S_{n}$ , J. Algebraic Combin., Volume 44 (2016) no. 3, pp. 519-560 | DOI | MR | Zbl

[17] Morita, Hiroshi; Yamada, Hiro-Fumi Higher Specht polynomials for the complex reflection group $G (r, p, n)$ , Hokkaido Math. J., Volume 27 (1998) no. 3, pp. 505-515 | DOI | MR

[18] Okounkov, Andrei; Vershik, Anatoly A new approach to representation theory of symmetric groups, Selecta Math. (N.S.), Volume 2 (1996) no. 4, pp. 581-605 | DOI | MR | Zbl

[19] Orellana, Rosa C. On partition algebras for complex reflection groups, J. Algebra, Volume 313 (2007) no. 2, pp. 590-616 | DOI | MR | Zbl

[20] Pushkarev, I. A. On the theory of representations of the wreath products of finite groups and symmetric groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 240 (1997) no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, p. 229-244, 294–295 translation in J. Math. Sci. (New York) 96 (1999), no. 5, 3590–3599 | DOI | MR | Zbl

[21] Ram, Arun Notes (http://soimeme.org/~arunram/notes.html)

[22] Ram, Arun Notes titled Murphy elements $ℂ A_{k} (r, p, n)$ (2010) http://soimeme.org/~arunram/Notes/murphyelementsContent.xhtml (accessed 16th April 2018)

[23] Ram, Arun Notes titled Schur–Weyl dualities (2010) http://soimeme.org/~arunram/Notes/schurweyldualitiesContent.xhtml (accessed 28th December 2017)

[24] Read, E. W. On the finite imprimitive unitary reflection groups, J. Algebra, Volume 45 (1977) no. 2, pp. 439-452 | DOI | MR | Zbl

[25] Shephard, Geoffrey C.; Todd, J. A. Finite unitary reflection groups, Canadian J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl

[26] Stembridge, John R. On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math., Volume 140 (1989) no. 2, pp. 353-396 | DOI | MR | Zbl

[27] Tanabe, Kenichiro On the centralizer algebra of the unitary reflection group $G (m, p, n)$ , Nagoya Math. J., Volume 148 (1997), pp. 113-126 | DOI | MR | Zbl

[28] Vershik, Anatoly; Okounkov, Andrei A new approach to representation theory of symmetric groups. II, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 307 (2004) no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, p. 57-98, 281 translation in J. Math. Sci. (N. Y.) 131 (2005), no. 2, 5471–5494 | DOI | MR

[29] Wigner, Eugene P. Condition that the irreducible representations of a group, considered as representations of a subgroup, do not contain any representation of the subgroup more than once, The collected works of Eugene Paul Wigner. Part A. The scientific papers. Vol. I, Springer-Verlag, Berlin, 1993 Published in: Spectroscopic and Group Theoretic Methods in Physics, (F. Bloch et al., eds.). North-Holland, Amsterdam 1968, pp.131–136 | MR

Cited by Sources:

ALGEBRAIC COMBINATORICS

Editorial Team

Author Guidelines

About the Journal

List-server

Submit a paper