Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients

Álvaro Gutiérrez; Mercedes H. Rosas

doi:10.5802/crmath.468

Combinatoire, Théorie des représentations

Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients
[Conditions nécessaires de positivité pour les coefficients de Littlewood–Richardson et du pléthysme]

Álvaro Gutiérrez ¹ ; Mercedes H. Rosas ¹

¹ Universidad de Sevilla, Spain

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1163-1173.

Résumé
Abstract

Nous donnons des conditions nécessaires de positivité pour les coefficients de Littlewood–Richardson et pour les coefficients SXP. Nous en déduisons la condition nécessaire de positivité suivante pour les coefficients du pléthysme : si $S^{λ} (V)$ apparaît dans la décomposition en irréductibles de $S^{μ} (S^{ν} (V))$ , alors le diagramme de $ν$ est contenu dans celui de $λ$ .

We give necessary conditions for the positivity of Littlewood–Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if $S^{λ} (V)$ appears as a summand in the decomposition into irreducibles of $S^{μ} (S^{ν} (V))$ , then $ν$ ’s diagram is contained in $λ$ ’s diagram.

Reçu le : 2021-09-28
Révisé le : 2022-12-30
Accepté le : 2023-01-26
Publié le : 2023-10-24

DOI : 10.5802/crmath.468

Classification : 05E05, 05E18, 05A17

Affiliations des auteurs :

Álvaro Gutiérrez ¹ ; Mercedes H. Rosas ¹

¹ Universidad de Sevilla, Spain

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G7_1163_0,
     author = {\'Alvaro Guti\'errez and Mercedes H. Rosas},
     title = {Necessary conditions for the positivity of {Littlewood{\textendash}Richardson} and plethystic coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1163--1173},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.468},
     language = {en},
}

TY  - JOUR
AU  - Álvaro Gutiérrez
AU  - Mercedes H. Rosas
TI  - Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1163
EP  - 1173
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.468
LA  - en
ID  - CRMATH_2023__361_G7_1163_0
ER  -

%0 Journal Article
%A Álvaro Gutiérrez
%A Mercedes H. Rosas
%T Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients
%J Comptes Rendus. Mathématique
%D 2023
%P 1163-1173
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.468
%G en
%F CRMATH_2023__361_G7_1163_0

Álvaro Gutiérrez; Mercedes H. Rosas. Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1163-1173. doi : 10.5802/crmath.468. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.468/

Bibliographie
Cité par

[1] Emmanuel Briand; Rosa Orellana; Mercedes Rosas Reduced Kronecker coefficients and counter-examples to Mulmuley’s strong saturation conjecture SH, Comput. Complexity, Volume 18 (2009) no. 4, pp. 577-600 (with an appendix by Ketan Mulmuley) | DOI | MR | Zbl

[2] Emmanuel Briand; Rosa Orellana; Mercedes Rosas The stability of the Kronecker product of Schur functions, J. Algebra, Volume 331 (2011), pp. 11-27 | DOI | MR | Zbl

[3] Peter Bürgisser; Christian Ikenmeyer Deciding positivity of Littlewood–Richardson coefficients, SIAM J. Discrete Math., Volume 27 (2013) no. 4, pp. 1639-1681 | DOI | MR | Zbl

[4] Y. M. Chen; Adriano M. Garsia; Jeffrey B. Remmel Algorithms for plethysm, Combinatorics and algebra (Boulder, Colo., 1983) (Contemporary Mathematics), Volume 34, American Mathematical Society, 1984, pp. 109-153 | DOI | MR | Zbl

[5] Laura Colmenarejo; Rosa Orellana; Franco Saliola; Anne Schilling; Mike Zabrocki The mystery of plethysm coefficients (2022) | arXiv

[6] Yoav Dvir On the Kronecker product of $S_{n}$ characters, J. Algebra, Volume 154 (1993) no. 1, pp. 125-140 | DOI | MR | Zbl

[7] Nick Fischer; Christian Ikenmeyer The computational complexity of plethysm coefficients, Comput. Complexity, Volume 29 (2020) no. 2, 8, 43 pages | DOI | MR | Zbl

[8] William Fulton Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997, x+260 pages | MR

[9] William Fulton; Joe Harris Representation theory. A first course, Readings in Mathematics, Graduate Texts in Mathematics, 129, Springer, 1991, xvi+551 pages | DOI | MR | Zbl

[10] Christian Ikenmeyer; Ketan D. Mulmuley; Michael Walter On vanishing of Kronecker coefficients, Comput. Complexity, Volume 26 (2017) no. 4, pp. 949-992 | DOI | MR | Zbl

[11] T. M. Langley; Jeffrey B. Remmel The plethysm $s_{λ} [s_{μ}]$ at hook and near-hook shapes, Electron. J. Comb., Volume 11 (2004) no. 1, 11, 26 pages | MR | Zbl

[12] Dudley E. Littlewood Modular representations of symmetric groups, Proc. R. Soc. Lond., Ser. A, Volume 209 (1951), pp. 333-353 | DOI | MR | Zbl

[13] Ian G. Macdonald Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, Clarendon Press, 2015, xii+475 pages (with contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition) | MR

[14] Marni Mishna; Mercedes Rosas; Sheila Sundaram Vector partition functions and Kronecker coefficients, J. Phys. A. Math. Gen., Volume 54 (2021) no. 20, 205204, 29 pages | DOI | MR | Zbl

[15] Ketan D. Mulmuley; Milind Sohoni Geometric complexity theory. I. An approach to the P vs. NP and related problems, SIAM J. Comput., Volume 31 (2001) no. 2, pp. 496-526 | DOI | MR | Zbl

[16] Igor Pak; Greta Panova On the complexity of computing Kronecker coefficients, Comput. Complexity, Volume 26 (2017) no. 1, pp. 1-36 | DOI | MR | Zbl

[17] Igor Pak; Greta Panova Breaking down the reduced Kronecker coefficients, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 4, pp. 463-468 | DOI | Numdam | MR | Zbl

[18] Jeffrey B. Remmel The combinatorics of $(k, l)$ -hook Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) (Contemporary Mathematics), Volume 34, American Mathematical Society, 1984, pp. 253-287 | DOI | MR | Zbl

[19] Mercedes Rosas The Kronecker product of Schur functions indexed by two-row shapes or hook shapes, J. Algebr. Comb., Volume 14 (2001) no. 2, pp. 153-173 | DOI | MR | Zbl

[20] Richard P. Stanley Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999, xii+581 pages (with a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin) | DOI | MR

[21] Mark Wildon A generalized SXP rule proved by bijections and involutions, Ann. Comb., Volume 22 (2018) no. 4, pp. 885-905 | DOI | MR | Zbl

[22] Mei Yang The first term in the expansion of plethysm of Schur functions, Discrete Math., Volume 246 (2002) no. 1-3, pp. 331-341 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique