Stretched Newell–Littlewood coefficients
Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1227-1256.

Newell–Littlewood coefficients n μ,ν λ are the multiplicities occurring in the decomposition of products of universal characters of the orthogonal and symplectic groups. They may also be expressed, or even defined directly in terms of Littlewood–Richardson coefficients, c μ,ν λ . Both sets of coefficients have stretched forms c tμ,tν tλ and n tμ,tν tλ , where tκ is the partition obtained by multiplying each part of the partition κ by the integer t. It is known that c tμ,tν tλ is a polynomial in t and here it is shown that n tμ,tν tλ is an Ehrhart quasi-polynomial in t with minimum quasi-period at most 2. The evaluation of n tμ,tν tλ is effected both by deriving its generating function and by establishing a hive model analogous to that used for the calculation of c tμ,tν tλ . These two approaches lead to a whole battery of conjectures about the nature of the quasi-polynomials n tμ,tν tλ . These include both positivity, stability and saturation conjectures that are supported by a significant amount of data from a range of examples.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.186
Classification: 05E10, 20C15
Keywords: Orthogonal and symplectic algebras, universal characters, multiplicities, generating functions, hive model, Ehrhart quasi-polynomials.

King, Ronald C. 1

1 Mathematical Sciences University of Southampton Southampton SO17 1BJ UK
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2022__5_6_1227_0,
     author = {King, Ronald C.},
     title = {Stretched {Newell{\textendash}Littlewood} coefficients},
     journal = {Algebraic Combinatorics},
     pages = {1227--1256},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {6},
     year = {2022},
     doi = {10.5802/alco.186},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.186/}
}
TY  - JOUR
AU  - King, Ronald C.
TI  - Stretched Newell–Littlewood coefficients
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 1227
EP  - 1256
VL  - 5
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.186/
DO  - 10.5802/alco.186
LA  - en
ID  - ALCO_2022__5_6_1227_0
ER  - 
%0 Journal Article
%A King, Ronald C.
%T Stretched Newell–Littlewood coefficients
%J Algebraic Combinatorics
%D 2022
%P 1227-1256
%V 5
%N 6
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.186/
%R 10.5802/alco.186
%G en
%F ALCO_2022__5_6_1227_0
King, Ronald C. Stretched Newell–Littlewood coefficients. Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1227-1256. doi : 10.5802/alco.186. https://alco.centre-mersenne.org/articles/10.5802/alco.186/

[1] Beck, Matthias; Sottile, Frank Irrational proofs for three theorems of Stanley, European J. Combin., Volume 28 (2007) no. 1, pp. 403-409 | DOI | MR | Zbl

[2] Belkale, Prakash; Kumar, Shrawan Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups, J. Algebraic Geom., Volume 19 (2010) no. 2, pp. 199-242 | DOI | MR | Zbl

[3] Berenstein, Arkady; Zelevinsky, Andrei Triple multiplicities for sl(r+1) and the spectrum of the exterior algebra of the adjoint representation, J. Algebraic Combin., Volume 1 (1992) no. 1, pp. 7-22 | DOI | MR | Zbl

[4] Berenstein, Arkady; Zelevinsky, Andrei Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., Volume 143 (2001) no. 1, pp. 77-128 | DOI | MR | Zbl

[5] Black, G. R. E.; King, Ronald C.; Wybourne, Brian G. Kronecker products for compact semisimple Lie groups, J. Phys. A, Volume 16 (1983) no. 8, pp. 1555-1589 | DOI | MR | Zbl

[6] Buch, Anders Skovsted The saturation conjecture (after A. Knutson and T. Tao). With an appendix by William Fulton, Enseign. Math. (2), Volume 46 (2000) no. 1-2, pp. 43-60 | MR | Zbl

[7] De Loera, Jesús A.; McAllister, Tyrrell B. On the computation of Clebsch–Gordan coefficients and the dilation effect, Experiment. Math., Volume 15 (2006) no. 1, pp. 7-19 | DOI | MR | Zbl

[8] Ehrhart, Eugène Polynômes arithmétiques et méthode des polyèdres en combinatoire, International Series of Numerical Mathematics, 35, Birkhäuser Verlag, Basel-Stuttgart, 1977, 165 pages | MR | Zbl

[9] Fauser, Bertfried; Jarvis, Peter D.; King, Ronald C. The Hopf algebra structure of the character rings of classical groups, J. Phys. A, Volume 46 (2013) no. 3, Paper no. 035205, 30 pages | DOI | MR | Zbl

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

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

[12] Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander Newell–Littlewood numbers II: extended Horn inequalities, Algebr. Comb., Volume 5 (2022) no. 6, pp. 1287-1297 | DOI | MR | Zbl

[13] 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

[14] Kapovich, Michael; Millson, John J. A path model for geodesics in Euclidean buildings and its applications to representation theory, Groups Geom. Dyn., Volume 2 (2008) no. 3, pp. 405-480 | DOI | MR | Zbl

[15] King, Ronald C. Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Mathematical Phys., Volume 12 (1971), pp. 1588-1598 | DOI | MR | Zbl

[16] King, Ronald C. S-functions and characters of Lie algebras and superalgebras, Invariant theory and tableaux (Minneapolis, MN, 1988) (IMA Vol. Math. Appl.), Volume 19, Springer, New York, 1990, pp. 226-261 | MR | Zbl

[17] King, Ronald C. Some remarks on characters of symmetric groups, Schur functions, Littlewood–Richardson and Kronecker coefficients, 2009 (Mathematical Foundations of Quantum Information Workshop. Sevilla, 23-27 de noviembre de 2009, http://congreso.us.es/enredo2009/Workshop_files/Sevilla_King.pdf)

[18] King, Ronald C.; Tollu, Christophe; Toumazet, Frédéric The hive model and the polynomial nature of stretched Littlewood–Richardson coefficients, Sém. Lothar. Combin., Volume 54A (2005/07), Paper no. B54Ad, 19 pages | MR | Zbl

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

[20] Knutson, Allen; Tao, Terence; Woodward, Christopher The honeycomb model of GL n () 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

[21] Koike, Kazuhiko; Terada, Itaru Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank, Adv. Math., Volume 79 (1990) no. 1, pp. 104-135 | DOI | MR | Zbl

[22] Littlewood, Dudley E. The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940, viii+292 pages | MR | Zbl

[23] 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

[24] Littlewood, Dudley E.; Richardson, Archibald R. Group characters and algebra, Phil. Trans. Roy. Soc. London A, Volume 233 (1934), pp. 99-141 | Zbl

[25] Murnaghan, F. D. The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group, Amer. J. Math., Volume 60 (1938) no. 3, pp. 761-784 | DOI | MR | Zbl

[26] 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

[27] Newell, Martin J. On the representations of the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A, Volume 54 (1951), pp. 143-152 | MR | Zbl

[28] Rassart, Etienne A polynomiality property for Littlewood-Richardson coefficients, J. Combin. Theory Ser. A, Volume 107 (2004) no. 2, pp. 161-179 | DOI | MR | Zbl

[29] Sam, Steven V. Symmetric quivers, invariant theory, and saturation theorems for the classical groups, Adv. Math., Volume 229 (2012) no. 2, pp. 1104-1135 | DOI | MR | Zbl

[30] SCHUR6.09 Interactive Program For Calculating Properties Of Symmetric Functions, 2014 (http://sourceforge.net/projects/schur)

[31] Stanley, Richard P. Decompositions of rational convex polytopes, Ann. Discrete Math., Volume 6 (1980), pp. 333-342 | DOI | MR | Zbl

[32] Stanley, Richard P. Enumerative Combinatorics. Vol. 1. With a foreword by Gian-Carlo Rota, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 1997, xii+325 pages | DOI | MR

[33] Terada, Itaru; King, Ronald C.; Azenhas, OOlga The symmetry of Littlewood-Richardson coefficients: a new hive model involutory bijection, SIAM J. Discrete Math., Volume 32 (2018) no. 4, pp. 2850-2899 | DOI | MR | Zbl

[34] Xin, Guoce A fast algorithm for MacMahon’s partition analysis, Electron. J. Combin., Volume 11 (2004), Paper no. 58, 20 pages | MR | Zbl

Cited by Sources: