ALGEBRAIC COMBINATORICS

no. 6
The poset of Specht ideals for hyperoctahedral groups
Debus, Sebastian1; Moustrou, Philippe2; Riener, Cordian3; Verdure, Hugues3
1 Technische Universität Chemnitz Fakultät für Mathematik 09107 Chemnitz (Germany)
2 Université Toulouse Jean Jaures Institut de Mathématiques de Toulouse UMR 5219, UT2J, 31058 Toulouse (France)
3 UiT - the Arctic University of Norway Department of Mathematics and Statistics 9037 Tromsø  (Norway)
Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1593-1619.

Specht polynomials classically realize the irreducible representations of the symmetric group. The ideals defined by these polynomials provide a strong connection with the combinatorics of Young tableaux and have been intensively studied by several authors. We initiate similar investigations for the ideals defined by the Specht polynomials associated to the hyperoctahedral group B n . We introduce a bidominance order on bipartitions which describes the poset of inclusions of these ideals and study algebraic consequences on general B n -invariant ideals and varieties, which can lead to computational simplifications.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.316
Classification: 10X99, 14A12, 11L05
Keywords: bipartitions, Specht polynomials, hyperoctahedral group, invariant ideals
Debus, Sebastian 1; Moustrou, Philippe 2; Riener, Cordian 3; Verdure, Hugues 3

1 Technische Universität Chemnitz Fakultät für Mathematik 09107 Chemnitz (Germany)
2 Université Toulouse Jean Jaures Institut de Mathématiques de Toulouse UMR 5219, UT2J, 31058 Toulouse (France)
3 UiT - the Arctic University of Norway Department of Mathematics and Statistics 9037 Tromsø  (Norway)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2023__6_6_1593_0,
     author = {Debus, Sebastian and Moustrou, Philippe and Riener, Cordian and Verdure, Hugues},
     title = {The poset of {Specht} ideals for hyperoctahedral groups},
     journal = {Algebraic Combinatorics},
     pages = {1593--1619},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {6},
     year = {2023},
     doi = {10.5802/alco.316},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.316/}
}
TY  - JOUR
AU  - Debus, Sebastian
AU  - Moustrou, Philippe
AU  - Riener, Cordian
AU  - Verdure, Hugues
TI  - The poset of Specht ideals for hyperoctahedral groups
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 1593
EP  - 1619
VL  - 6
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.316/
DO  - 10.5802/alco.316
LA  - en
ID  - ALCO_2023__6_6_1593_0
ER  - 
%0 Journal Article
%A Debus, Sebastian
%A Moustrou, Philippe
%A Riener, Cordian
%A Verdure, Hugues
%T The poset of Specht ideals for hyperoctahedral groups
%J Algebraic Combinatorics
%D 2023
%P 1593-1619
%V 6
%N 6
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.316/
%R 10.5802/alco.316
%G en
%F ALCO_2023__6_6_1593_0
Debus, Sebastian; Moustrou, Philippe; Riener, Cordian; Verdure, Hugues. The poset of Specht ideals for hyperoctahedral groups. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1593-1619. doi : 10.5802/alco.316. https://alco.centre-mersenne.org/articles/10.5802/alco.316/

[1] Abe, Takuro; Horiguchi, Tatsuya; Masuda, Mikiya; Murai, Satoshi; Sato, Takashi Hessenberg varieties and hyperplane arrangements, J. Reine Angew. Math., Volume 2020 (2020) no. 764, pp. 241-286 | MR | Zbl

[2] Aguiar, Marcelo; Mahajan, Swapneel Topics in hyperplane arrangements, Math. surv. monogr., 226, AMS, Providence, RI, 2017, xxiv+611 pages | DOI | MR

[3] Al-Aamily, Eyad; Morris, Alun O.; Peel, Michael H. The representations of the Weyl groups of type B n , J. Algebra, Volume 68 (1981) no. 2, pp. 298-305 | DOI | MR | Zbl

[4] Ariki, Susumu On the classification of simple modules for cyclotomic Hecke algebras of type G(m,1,n) and Kleshchev multipartitions, Osaka J. Math., Volume 38 (2001) no. 4, pp. 827-837 | MR | Zbl

[5] Basu, Saugata; Riener, Cordian Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets, Found. Comput. Math. (2021), pp. 1-68

[6] Blekherman, Grigoriy; Riener, Cordian Symmetric non-negative forms and sums of squares, Discrete Comput. Geom., Volume 65 (2021) no. 3, pp. 764-799 | DOI | MR | Zbl

[7] Brylawski, Thomas The lattice of integer partitions, Discrete Math., Volume 6 (1973) no. 3, pp. 201-219 | DOI | MR | Zbl

[8] Chen, William Y.C. Induced cycle structures of the hyperoctahedral group, SIAM J. Discrete Math., Volume 6 (1993) no. 3, pp. 353-362 | DOI | MR | Zbl

[9] Chen, William Y.C.; Stanley, Richard P. Derangements on the n-cube, Discrete Math., Volume 115 (1993) no. 1-3, pp. 65-75 | DOI | MR | Zbl

[10] Chevalley, Claude Invariants of finite groups generated by reflections, Am. J. Math., Volume 77 (1955) no. 4, pp. 778-782 | DOI | MR | Zbl

[11] Choi, Man-Duen; Lam, Tsit-Yuen; Reznick, Bruce Even symmetric sextics, Math. Z., Volume 195 (1987) no. 4, pp. 559-580 | DOI | MR | Zbl

[12] Dipper, Richard; James, Gordon; Murphy, Eugene Hecke algebras of type B n at roots of unity, Proc. Lond. Math. Soc., Volume 3 (1995) no. 3, pp. 505-528 | DOI | MR | Zbl

[13] Dostert, Maria; Guzmán, Cristóbal; de Oliveira Filho, Fernando Mário; Vallentin, Frank New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry, Discrete Comput. Geom., Volume 58 (2017) no. 2, pp. 449-481 | DOI | MR | Zbl

[14] Faugère, Jean-Charles; Labahn, George; El Din, Mohab Safey; Schost, Éric; Vu, Thi Xuan Computing critical points for invariant algebraic systems, J. Symb. Comput., Volume 116 (2023), pp. 365-399 | DOI | MR | Zbl

[15] Geissinger, Ladnor; Kinch, Dennis Representations of the hyperoctahedral groups, J. Algebra, Volume 53 (1978) no. 1, pp. 1-20 | DOI | MR | Zbl

[16] Hall, Philip The algebra of partitions, Proceedings of the Fourth Canadian Mathematical Congress, Banff, 1957, University of Toronto Press, Toronto, Ont., Banff (1959), pp. 147-159 | Zbl

[17] Harris, William R. Real even symmetric ternary forms, J. Algebra, Volume 222 (1999) no. 1, pp. 204-245 | DOI | MR | Zbl

[18] Heaton, Alexander; Shankar, Isabelle An SOS counterexample to an inequality of symmetric functions, J. Pure Appl. Algebra, Volume 225 (2021) no. 8, Paper no. 106656, 10 pages | DOI | MR | Zbl

[19] Kurpisz, Adam; Leppänen, Samuli; Mastrolilli, Monaldo Sum-of-squares hierarchy lower bounds for symmetric formulations, Math. Program., Volume 182 (2020) no. 1, pp. 369-397 | DOI | MR | Zbl

[20] Li, Shuo-Yen Robert; Li, Wen-Ch’ing Winnie Independence numbers of graphs and generators of ideals, Combinatorica, Volume 1 (1981) no. 1, pp. 55-61 | MR | Zbl

[21] Lien, Arne Symmetric ideals, Masters thesis, UiT Norges arktiske universitet (2021)

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

[23] Moustrou, Philippe; Naumann, Helen; Riener, Cordian; Theobald, Thorsten; Verdure, Hugues Symmetry reduction in AM/GM-based optimization, SIAM J. Optim, Volume 32 (2022) no. 2, pp. 765-785 | DOI | MR | Zbl

[24] Moustrou, Philippe; Riener, Cordian; Verdure, Hugues Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations, J. Symb. Comput., Volume 107 (2021), pp. 106-121 | DOI | MR | Zbl

[25] Murai, Satoshi; Ohsugi, Hidefumi; Yanagawa, Kohji A note on the reducedness and Gröbner bases of Specht ideals, Commun. Algebra, Volume 50 (2022) no. 12, pp. 5430-5434 | DOI | Zbl

[26] Musili, Chitikila Representations of the Hyperoctahedral Group B n , Representations of Finite Groups, Springer, 1993, pp. 197-220 | DOI

[27] Riener, Cordian Specht Ideals, their varieties and connections to symmetric ideals, MFO-RIMS tandem workshop: Symmetries on polynomial ideals and varieties, Volume Workshops 2021, 2021

[28] Riener, Cordian; Safey el Din, Mohab Real root finding for equivariant semi-algebraic systems, Proc. Int. Symp. Symb. Algebr. Comput. ISSAC (2018), pp. 335-342 | Zbl

[29] Riener, Cordian; Theobald, Thorsten; Andrén, Lina Jansson; Lasserre, Jean B Exploiting symmetries in SDP-relaxations for polynomial optimization, Math. Oper. Res., Volume 38 (2013) no. 1, pp. 122-141 | DOI | MR | Zbl

[30] Shephard, Geoffrey Colin; Todd, John Arthur Finite Unitary Reflection Groups, Can. J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl

[31] Specht, Wilhelm Darstellungstheorie der Hyperoktaedergruppe, Math. Z., Volume 42 (1937) no. 1, pp. 629-640 | DOI | MR | Zbl

[32] Specht, Wilhelm Zur Darstellungstheorie der symmetrischen Gruppe, Math. Z., Volume 42 (1937) no. 1, pp. 774-779 | DOI | MR | Zbl

[33] Woo, Alexander Ideals of the polynomial ring generated by irreducible symmetric group representations and Ellingsrud-Strømme cells on the Hilbert scheme, Ph. D. Thesis, University of California, Berkeley (2005)

[34] Xia, Jianqiao A partial order on bipartitions from the generalized Springer correspondence, J. Comb. Algebra, Volume 2 (2018) no. 3, pp. 301-309 | MR | Zbl

[35] Yanagawa, Kohji When is a Specht ideal Cohen–Macaulay?, J. Commut. Algebra, Volume 13 (2021) no. 4, pp. 589-608 | MR | Zbl

Cited by Sources: