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no. 4
Shuffle Tableaux, Littlewood–Richardson Coefficients, and Schur Log-Concavity
Nguyen, Chau1; Nguyen, Son2; Woodruff, Dora2
1University of Minnesota, Minneapolis, MN, USA
2Massachusetts Institute of Technology, Cambridge, MA, USA
Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1119-1137

We give a new formula for the Littlewood–Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel–Whitney rule. This gives an efficient way to compute generalized Littlewood–Richardson coefficients for Temperley–Lieb immanants of Jacobi–Trudi matrices. We will also show that our rule behaves well with Bender–Knuth involutions, recovering the symmetry of Littlewood–Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam–Postnikov–Pylyavskyy.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.502
Classification: 05E05, 05E10, 14M15
Keywords: shuffle tableaux, Littlewood–Richardson coefficients, Schur log-concavity

Nguyen, Chau  1 ; Nguyen, Son  2 ; Woodruff, Dora  2

1 University of Minnesota, Minneapolis, MN, USA
2 Massachusetts Institute of Technology, Cambridge, MA, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Nguyen, Chau; Nguyen, Son; Woodruff, Dora. Shuffle Tableaux, Littlewood–Richardson Coefficients, and Schur Log-Concavity. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1119-1137. doi: 10.5802/alco.502
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[1] Beĭlinson, Alexandre; Bernstein, Joseph Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math., Volume 292 (1981) no. 1, pp. 15-18 | MR | Zbl

[2] Bender, Edward A.; Knuth, Donald E. Enumeration of plane partitions, J. Combinatorial Theory Ser. A, Volume 13 (1972), pp. 40-54 | DOI | MR | Zbl

[3] Berenshteĭn, A. D.; Zelevinskiĭ, A. V. When is the multiplicity of a weight equal to $1$?, Funktsional. Anal. i Prilozhen., Volume 24 (1990) no. 4, p. 1-13, 96 | DOI | MR | Zbl

[4] Berenstein, A. D.; Kirillov, A. N. Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, Algebra i Analiz, Volume 7 (1995) no. 1, pp. 92-152 | MR | Zbl

[5] Berenstein, A. D.; Zelevinsky, A. V. Triple multiplicities for ${\rm sl}(r+1)$ 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

[6] Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei Parametrizations of canonical bases and totally positive matrices, Adv. Math., Volume 122 (1996) no. 1, pp. 49-149 | DOI | MR | Zbl

[7] 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

[8] Bergeron, François; McNamara, Peter Some positive differences of products of Schur functions, 2004 | arXiv | Zbl

[9] Brylinski, J.-L.; Kashiwara, M. Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math., Volume 64 (1981) no. 3, pp. 387-410 | DOI | MR | Zbl

[10] Chiang, Judy Hsin-Hui; Hoang, Anh Trong Nam; Kendall, Matthew; Lynch, Ryan; Nguyen, Son; Przybocki, Benjamin; Xia, Janabel Bender-Knuth involutions on linear extensions of posets, Discrete Math., Volume 347 (2024) no. 9, Paper no. 114068, 20 pages | DOI | MR | Zbl

[11] Chmutov, Michael; Glick, Max; Pylyavskyy, Pavlo The Berenstein-Kirillov group and cactus groups, J. Comb. Algebra, Volume 4 (2020) no. 2, pp. 111-140 | DOI | MR | Zbl

[12] Dobrovolska, Galyna; Pylyavskyy, Pavlo On products of $\mathfrak{sl}_{\mathfrak{n}}$ characters and support containment, J. Algebra, Volume 316 (2007) no. 2, pp. 706-714 | DOI | MR | Zbl

[13] Du, Jie Canonical bases for irreducible representations of quantum ${\rm {GL}}_n$, Bull. London Math. Soc., Volume 24 (1992) no. 4, pp. 325-334 | DOI | MR | Zbl

[14] Du, Jie Canonical bases for irreducible representations of quantum ${\rm {GL}}_n$. II, J. London Math. Soc. (2), Volume 51 (1995) no. 3, pp. 461-470 | DOI | MR | Zbl

[15] Fomin, Sergey; Fulton, William; Li, Chi-Kwong; Poon, Yiu-Tung Eigenvalues, singular values, and Littlewood-Richardson coefficients, Amer. J. Math., Volume 127 (2005) no. 1, pp. 101-127 | MR | DOI | Zbl

[16] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl

[17] Haiman, Mark Hecke algebra characters and immanant conjectures, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 569-595 | DOI | MR | Zbl

[18] Kashiwara, Masaki On crystal bases of the ${Q}$-analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR | Zbl

[19] Kashiwara, Masaki Global crystal bases of quantum groups, Duke Math. J., Volume 69 (1993) no. 2, pp. 455-485 | DOI | MR | Zbl

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

[21] Lam, Thomas; Postnikov, Alexander Alcoved polytopes. I, Discrete Comput. Geom., Volume 38 (2007) no. 3, pp. 453-478 | DOI | MR | Zbl

[22] Lam, Thomas; Postnikov, Alexander Alcoved polytopes II, Lie groups, geometry, and representation theory (Progr. Math.), Volume 326, Birkhäuser/Springer, Cham, 2018, pp. 253-272 | DOI | MR | Zbl

[23] Lam, Thomas; Postnikov, Alexander; Pylyavskyy, Pavlo Schur positivity and Schur log-concavity, Amer. J. Math., Volume 129 (2007) no. 6, pp. 1611-1622 | DOI | MR | Zbl

[24] Lascoux, Alain; Leclerc, Bernard; Thibon, Jean-Yves Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys., Volume 38 (1997) no. 2, pp. 1041-1068 | DOI | MR | Zbl

[25] Littlewood, Dudley Ernest; Richardson, Archibald Read Group characters and algebra, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Volume 233 (1934) no. 721-730, pp. 99-141 | DOI | Zbl

[26] Lusztig, George Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., Volume 3 (1990) no. 2, pp. 447-498 | DOI | MR | Zbl

[27] Lusztig, George Total positivity in reductive groups, Lie theory and geometry (Progr. Math.), Volume 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531-568 | DOI | MR | Zbl

[28] Lusztig, George Total positivity and canonical bases, Algebraic groups and Lie groups (Austral. Math. Soc. Lect. Ser.), Volume 9, Cambridge Univ. Press, Cambridge, 1997, pp. 281-295 | MR | Zbl

[29] Nguyen, Son The Cactus Group Property for Ordinal Sums of Disjoint Unions of Chains, 2023 | arXiv | Zbl

[30] Nguyen, Son; Pylyavskyy, Pavlo Temperley-Lieb crystals, Int. Math. Res. Not. IMRN (2025) no. 7, Paper no. rnaf080, 28 pages | DOI | MR | Zbl

[31] Okounkov, Andrei Log-concavity of multiplicities with application to characters of ${\rm {U}}(\infty )$, Adv. Math., Volume 127 (1997) no. 2, pp. 258-282 | DOI | MR | Zbl

[32] Pylyavskyy, Pavlo Comparing products of Schur functions and quasisymmetric functions, Ph. D. Thesis, Massachusetts Institute of Technology (2007)

[33] Reiner, Victor; Shimozono, Mark Percentage-avoiding, northwest shapes and peelable tableaux, J. Combin. Theory Ser. A, Volume 82 (1998) no. 1, pp. 1-73 | DOI | MR | Zbl

[34] Remmel, J. B.; Whitney, R. Multiplying Schur functions, J. Algorithms, Volume 5 (1984) no. 4, pp. 471-487 | DOI | MR | Zbl

[35] Rhoades, Brendon; Skandera, Mark Temperley-Lieb immanants, Ann. Comb., Volume 9 (2005) no. 4, pp. 451-494 | DOI | MR | Zbl

[36] Rhoades, Brendon; Skandera, Mark Kazhdan-Lusztig immanants and products of matrix minors, J. Algebra, Volume 304 (2006) no. 2, pp. 793-811 | DOI | MR | Zbl

[37] Schützenberger, M. P. Promotion des morphismes d’ensembles ordonnés, Discrete Math., Volume 2 (1972), pp. 73-94 | DOI | MR | Zbl

[38] Skandera, Mark On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations, J. Pure Appl. Algebra, Volume 212 (2008) no. 5, pp. 1086-1104 | DOI | MR | Zbl

[39] Zelevinsky, A. V. A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence, J. Algebra, Volume 69 (1981) no. 1, pp. 82-94 | DOI | MR

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