# ALGEBRAIC COMBINATORICS

Newell–Littlewood numbers II: extended Horn inequalities
Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1287-1297.

The Newell–Littlewood numbers ${N}_{\mu ,\nu ,\lambda }$ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions $\left(\mu ,\nu ,\lambda \right)$ does ${N}_{\mu ,\nu ,\lambda }>0$ hold? The Littlewood–Richardson coefficient case is solved by the Horn inequalities (in work of A. Klyachko and A. Knutson-T. Tao). We extend these celebrated linear inequalities to a much larger family, suggesting a general solution.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.217
Classification: 05E10,  22E46,  15A18
Keywords: Newell–Littlewood numbers, Weyl modules, Horn inequalities
Gao, Shiliang 1; Orelowitz, Gidon 1; Yong, Alexander 1

1 University of Illinois at Urbana-Champaign Dept. of Mathematics 1409 W. Green Street Urbana IL 61801, USA
@article{ALCO_2022__5_6_1287_0,
author = {Gao, Shiliang and Orelowitz, Gidon and Yong, Alexander},
title = {Newell{\textendash}Littlewood numbers {II:} extended {Horn} inequalities},
journal = {Algebraic Combinatorics},
pages = {1287--1297},
publisher = {The Combinatorics Consortium},
volume = {5},
number = {6},
year = {2022},
doi = {10.5802/alco.217},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.217/}
}
TY  - JOUR
AU  - Gao, Shiliang
AU  - Orelowitz, Gidon
AU  - Yong, Alexander
TI  - Newell–Littlewood numbers II: extended Horn inequalities
JO  - Algebraic Combinatorics
PY  - 2022
DA  - 2022///
SP  - 1287
EP  - 1297
VL  - 5
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.217/
UR  - https://doi.org/10.5802/alco.217
DO  - 10.5802/alco.217
LA  - en
ID  - ALCO_2022__5_6_1287_0
ER  - 
%0 Journal Article
%A Gao, Shiliang
%A Orelowitz, Gidon
%A Yong, Alexander
%T Newell–Littlewood numbers II: extended Horn inequalities
%J Algebraic Combinatorics
%D 2022
%P 1287-1297
%V 5
%N 6
%I The Combinatorics Consortium
%U https://doi.org/10.5802/alco.217
%R 10.5802/alco.217
%G en
%F ALCO_2022__5_6_1287_0
Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander. Newell–Littlewood numbers II: extended Horn inequalities. Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1287-1297. doi : 10.5802/alco.217. https://alco.centre-mersenne.org/articles/10.5802/alco.217/

[1] Belkale, Prakash Local systems on ${ℙ}^{1}-S$ for $S$ a finite set, Compositio Math., Volume 129 (2001) no. 1, pp. 67-86 | DOI | MR | Zbl

[2] Belkale, Prakash; Kumar, Shrawan Eigenvalue problem and a new product in cohomology of flag varieties, Invent. Math., Volume 166 (2006) no. 1, pp. 185-228 | DOI | MR | Zbl

[3] Bhatia, Rajendra Linear algebra to quantum cohomology: the story of Alfred Horn’s inequalities, Amer. Math. Monthly, Volume 108 (2001) no. 4, pp. 289-318 | DOI | MR | Zbl

[4] Briand, Emmanuel; Orellana, Rosa; Rosas, Mercedes Rectangular symmetries for coefficients of symmetric functions, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. 3.15, 18 pages | DOI | MR | Zbl

[5] Fulton, William Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.), Volume 37 (2000) no. 3, pp. 209-249 | DOI | MR | Zbl

[6] Gao, Shiliang; Orelowitz, Gidon; Ressayre, Nicolas; Yong, Alexander Newell–Littlewood numbers III: eigencones and GIT-semigroups (2021) (https://arxiv.org/abs/2107.03152)

[7] Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander Newell–Littlewood numbers, Trans. Amer. Math. Soc., Volume 374 (2021) no. 9, pp. 6331-6366 | DOI | MR | Zbl

[8] Hahn, Heekyoung On classical groups detected by the triple tensor product and the Littlewood–Richardson semigroup, Res. Number Theory, Volume 2 (2016), Paper no. 19, 12 pages | DOI | MR | Zbl

[9] Horn, Alfred Eigenvalues of sums of Hermitian matrices, Pacific J. Math., Volume 12 (1962), pp. 225-241 | DOI | MR | Zbl

[10] Klyachko, Alexander A. Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), Volume 4 (1998) no. 3, pp. 419-445 | DOI | MR | Zbl

[11] Knutson, Allen; Tao, Terence The honeycomb model of ${\mathrm{GL}}_{n}\left(ℂ\right)$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., Volume 12 (1999) no. 4, pp. 1055-1090 | DOI | MR | Zbl

[12] Knutson, Allen; Tao, Terence; Woodward, Christopher The honeycomb model of ${\mathrm{GL}}_{n}\left(ℂ\right)$ tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone, J. Amer. Math. Soc., Volume 17 (2004) no. 1, pp. 19-48 | DOI | MR | Zbl

[13] Kumar, Shrawan A survey of the additive eigenvalue problem. With an appendix by M. Kapovich, Transform. Groups, Volume 19 (2014) no. 4, pp. 1051-1148 | DOI | MR | Zbl

[14] Littlewood, Dudley E. Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Canadian J. Math., Volume 10 (1958), pp. 17-32 | DOI | MR | Zbl

[15] Newell, Martin J. Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A, Volume 54 (1951), pp. 153-163 | MR | Zbl

[16] Ressayre, Nicolas Horn inequalities for nonzero Kronecker coefficients, Adv. Math., Volume 356 (2019), Paper no. 106809, 21 pages | DOI | MR | Zbl

[17] Zelevinsky, Andrei Littlewood–Richardson semigroups, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) (Math. Sci. Res. Inst. Publ.), Volume 38, Cambridge Univ. Press, Cambridge, 1999, pp. 337-345 | MR | Zbl

Cited by Sources: