Newell-Littlewood numbers

Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander

doi:10.1090/tran/8375

Newell-Littlewood numbers
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by Shiliang Gao, Gidon Orelowitz and Alexander Yong PDF

Trans. Amer. Math. Soc. 374 (2021), 6331-6366 Request permission

Abstract:

The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Koike-I. Terada basis of the ring of symmetric functions. Recent work of H. Hahn studies them, motivated by R. Langlands’ beyond endoscopy proposal; we address her work with a simple characterization of detection of Weyl modules. This motivates further study of the combinatorics of the numbers. We consider analogues of ideas of J. De Loera-T. McAllister, H. Derksen-J. Weyman, S. Fomin–W. Fulton-C.-K. Li–Y.-T. Poon, W. Fulton, R. King-C. Tollu-F. Toumazet, M. Kleber, A. Klyachko, A. Knutson-T. Tao, T. Lam-A. Postnikov-P. Pylyavskyy, K. Mulmuley-H. Narayanan-M. Sohoni, H. Narayanan, A. Okounkov, J. Stembridge, and H. Weyl.

References

David Anderson, Edward Richmond, and Alexander Yong, Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians, Compos. Math. 149 (2013), no. 9, 1569–1582. MR 3109734, DOI 10.1112/S0010437X13007343
Hélène Barcelo and Arun Ram, Combinatorial representation theory, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 23–90. MR 1731814
Allan Berele, A Schensted-type correspondence for the symplectic group, J. Combin. Theory Ser. A 43 (1986), no. 2, 320–328. MR 867655, DOI 10.1016/0097-3165(86)90070-1
Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, DOI 10.1007/s002220000102
Rajendra Bhatia, Linear algebra to quantum cohomology: the story of Alfred Horn’s inequalities, Amer. Math. Monthly 108 (2001), no. 4, 289–318. MR 1836940, DOI 10.2307/2695237
Jesús A. De Loera and Tyrrell B. McAllister, On the computation of Clebsch-Gordan coefficients and the dilation effect, Experiment. Math. 15 (2006), no. 1, 7–19. MR 2229381
Harm Derksen and Jerzy Weyman, On the Littlewood-Richardson polynomials, J. Algebra 255 (2002), no. 2, 247–257. MR 1935497, DOI 10.1016/S0021-8693(02)00125-4
Sergey Fomin, William Fulton, Chi-Kwong Li, and Yiu-Tung Poon, Eigenvalues, singular values, and Littlewood-Richardson coefficients, Amer. J. Math. 127 (2005), no. 1, 101–127. MR 2115662
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Martin Grötschel, László Lovász, and Alexander Schrijver, Geometric algorithms and combinatorial optimization, 2nd ed., Algorithms and Combinatorics, vol. 2, Springer-Verlag, Berlin, 1993. MR 1261419, DOI 10.1007/978-3-642-78240-4
Heekyoung Hahn, On tensor third $L$-functions of automorphic representations of $\textrm {GL}_n(\Bbb {A}_F)$, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5061–5069. MR 3556252, DOI 10.1090/proc/13134
Heekyoung Hahn, On classical groups detected by the triple tensor product and the Littlewood-Richardson semigroup, Res. Number Theory 2 (2016), Paper No. 19, 12. MR 3553350, DOI 10.1007/s40993-016-0049-3
Alfred Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225–241. MR 140521
Michael Kapovich, Shrawan Kumar, and John J. Millson, The eigencone and saturation for Spin(8), Pure Appl. Math. Q. 5 (2009), no. 2, Special Issue: In honor of Friedrich Hirzebruch., 755–780. MR 2508902, DOI 10.4310/PAMQ.2009.v5.n2.a7
J. Kiers, On the saturation conjecture for ${\mathsf {Spin}}(2n)$, Exp. Math., 2019, to appear. arXiv:1804.09229
R. C. King, Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Mathematical Phys. 12 (1971), 1588–1598. MR 287816, DOI 10.1063/1.1665778
R. C. King, Branching rules for classical Lie groups using tensor and spinor methods, J. Phys. A 8 (1975), 429–449. MR 411400
R. C. King, C. Tollu, and F. Toumazet, Stretched Littlewood-Richardson and Kostka coefficients, Symmetry in physics, CRM Proc. Lecture Notes, vol. 34, Amer. Math. Soc., Providence, RI, 2004, pp. 99–112. MR 2056979, DOI 10.1090/crmp/034/10
Michael Kleber, Linearly independent products of rectangularly complementary Schur functions, Electron. J. Combin. 9 (2002), no. 1, Research Paper 39, 8. MR 1928791
Alexander A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419–445. MR 1654578, DOI 10.1007/s000290050037
Allen Knutson and Terence Tao, The honeycomb model of $\textrm {GL}_n(\textbf {C})$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055–1090. MR 1671451, DOI 10.1090/S0894-0347-99-00299-4
Allen Knutson, Terence Tao, and Christopher Woodward, The honeycomb model of $\textrm {GL}_n(\Bbb C)$ tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc. 17 (2004), no. 1, 19–48. MR 2015329, DOI 10.1090/S0894-0347-03-00441-7
Kazuhiko Koike and Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B_n,\;C_n,\;D_n$, J. Algebra 107 (1987), no. 2, 466–511. MR 885807, DOI 10.1016/0021-8693(87)90099-8
Shrawan Kumar, Tensor product decomposition, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1226–1261. MR 2827839
Shrawan Kumar, A survey of the additive eigenvalue problem, Transform. Groups 19 (2014), no. 4, 1051–1148. With an appendix by M. Kapovich. MR 3278861, DOI 10.1007/s00031-014-9287-4
Jae-Hoon Kwon, Combinatorial extension of stable branching rules for classical groups, Trans. Amer. Math. Soc. 370 (2018), no. 9, 6125–6152. MR 3814326, DOI 10.1090/tran/7104
Thomas Lam, Alexander Postnikov, and Pavlo Pylyavskyy, Schur positivity and Schur log-concavity, Amer. J. Math. 129 (2007), no. 6, 1611–1622. MR 2369890, DOI 10.1353/ajm.2007.0045
R. P. Langlands, Letter to André Weil (1967), http:publications.ias.edu/rpl/section/21.
R. P. Langlands, Beyond endoscopy, Contributions to automorphic forms, geometry, and number theory, 611–697, Johns Hopkins Univ. Press, Baltimore, MD, 2004.
D. E. Littlewood, Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Canadian J. Math. 10 (1958), 17–32. MR 95209, DOI 10.4153/CJM-1958-002-7
Dudley E. Littlewood, The theory of group characters and matrix representations of groups, AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the second (1950) edition. MR 2213154, DOI 10.1090/chel/357
Ketan D. Mulmuley, Hariharan Narayanan, and Milind Sohoni, Geometric complexity theory III: on deciding nonvanishing of a Littlewood-Richardson coefficient, J. Algebraic Combin. 36 (2012), no. 1, 103–110. MR 2927658, DOI 10.1007/s10801-011-0325-1
Hariharan Narayanan, On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients, J. Algebraic Combin. 24 (2006), no. 3, 347–354. MR 2260022, DOI 10.1007/s10801-006-0008-5
M. J. Newell, Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A 54 (1951), 153–163. MR 0043093
Soichi Okada, Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux, Electron. J. Combin. 23 (2016), no. 4, Paper 4.43, 27. MR 3604801
Andrei Okounkov, Log-concavity of multiplicities with application to characters of $\textrm {U}(\infty )$, Adv. Math. 127 (1997), no. 2, 258–282. MR 1448715, DOI 10.1006/aima.1997.1622
N. Ressayre, Horn inequalities for nonzero Kronecker coefficients, Adv. Math. 356 (2019), 106809, 21. MR 4011023, DOI 10.1016/j.aim.2019.106809
Brendon Rhoades and Mark Skandera, Temperley-Lieb immanants, Ann. Comb. 9 (2005), no. 4, 451–494. MR 2205034, DOI 10.1007/s00026-005-0268-0
Brendon Rhoades and Mark Skandera, Kazhdan-Lusztig immanants and products of matrix minors, J. Algebra 304 (2006), no. 2, 793–811. MR 2264279, DOI 10.1016/j.jalgebra.2005.07.017
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
John R. Stembridge, Multiplicity-free products of Schur functions, Ann. Comb. 5 (2001), no. 2, 113–121. MR 1904379, DOI 10.1007/s00026-001-8008-6
John R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404–439. MR 2017064, DOI 10.1090/S1088-4165-03-00150-X
Sheila Sundaram, ON THE COMBINATORICS OF REPRESENTATIONS OF THE SYMPLECTIC GROUP, ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2941115
Éva Tardos, A strongly polynomial algorithm to solve combinatorial linear programs, Oper. Res. 34 (1986), no. 2, 250–256. MR 861043, DOI 10.1287/opre.34.2.250
Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479 (German). MR 1511670, DOI 10.1007/BF01456804
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
Andrei Zelevinsky, Littlewood-Richardson semigroups, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 337–345. MR 1731821