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no. 3
Super major index and Thrall’s problem
Armon, Sam1; Swanson, Joshua P.1
1 University of Southern California Dept. of Mathematics 3620 S. Vermont Ave KAP 104 Los Angeles CA 90089-2532 (USA)
Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 795-815.

Thrall’s problem asks for the Schur decomposition of the higher Lie modules $\mathcal{L}_\lambda $, which are defined using the free Lie algebra and decompose the tensor algebra as a general linear group module. Although special cases have been solved, Thrall’s problem remains open in general. We generalize Thrall’s problem to the free Lie superalgebra, and prove extensions of three known results in this setting: Brandt’s formula, Klyachko’s identification of the Schur–Weyl dual of $\mathcal{L}_n$, and Kráskiewicz–Weyman’s formula for the Schur decomposition of $\mathcal{L}_n$. The latter involves a new version of the major index on super tableaux, which we show corresponds to a $q,t$-hook formula of Macdonald.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.421
Classification: 05E10, 17B01
Keywords: free Lie superalgebras, Thrall’s problem, major index, super tableaux

Armon, Sam 1; Swanson, Joshua P. 1

1 University of Southern California Dept. of Mathematics 3620 S. Vermont Ave KAP 104 Los Angeles CA 90089-2532 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Armon, Sam; Swanson, Joshua P. Super major index and Thrall’s problem. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 795-815. doi : 10.5802/alco.421. https://alco.centre-mersenne.org/articles/10.5802/alco.421/

[1] Ahlbach, Connor; Swanson, Joshua P. Cyclic sieving, necklaces, and branching rules related to Thrall’s problem, Electron. J. Combin., Volume 25 (2018) no. 4, Paper no. 4.42, 38 pages | DOI | MR | Zbl

[2] Berele, A.; Regev, A. Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math., Volume 64 (1987) no. 2, pp. 118-175 | DOI | MR | Zbl

[3] Berele, A.; Remmel, J. B. Hook flag characters and their combinatorics, J. Pure Appl. Algebra, Volume 35 (1985) no. 3, pp. 225-245 | DOI | MR | Zbl

[4] Bergeron, F.; Bergeron, N.; Garsia, A. M. Idempotents for the free Lie algebra and q-enumeration, Invariant theory and tableaux (Minneapolis, MN, 1988) (IMA Vol. Math. Appl.), Volume 19, Springer, New York, 1990, pp. 166-190 | MR | Zbl

[5] Billey, Sara C.; Konvalinka, Matjaž; Swanson, Joshua P. Tableau posets and the fake degrees of coinvariant algebras, Adv. Math., Volume 371 (2020), Paper no. 107252, 46 pages | DOI | MR | Zbl

[6] Billey, Sara C.; Swanson, Joshua P. The metric space of limit laws for q-hook formulas, Comb. Theory, Volume 2 (2022) no. 2, Paper no. 5, 58 pages | MR | Zbl

[7] Brandt, Angeline The free Lie ring and Lie representations of the full linear group, Trans. Amer. Math. Soc., Volume 56 (1944), pp. 528-536 | DOI | MR | Zbl

[8] Fulton, William Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR | Zbl

[9] Garsia, Adriano M. Combinatorics of the free Lie algebra and the symmetric group, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 309-382 | DOI | Zbl

[10] Gessel, Ira M.; Reutenauer, Christophe Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A, Volume 64 (1993) no. 2, pp. 189-215 | DOI | MR | Zbl

[11] Haglund, J.; Haiman, M.; Loehr, N.; Remmel, J. B.; Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., Volume 126 (2005) no. 2, pp. 195-232 | DOI | MR | Zbl

[12] Klyachko, Aleksandr Anatolévich Lie elements in a tensor algebra, Sibirsk. Mat. Ž., Volume 15 (1974), p. 1296-1304, 1430 | MR | Zbl

[13] Kraśkiewicz, Witold; Weyman, Jerzy Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. (2001) no. 63, pp. 265-284 | MR | Zbl

[14] Krattenthaler, C. A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin, Electron. J. Combin., Volume 3 (1996) no. 2, Paper no. 14, 24 pages (The Foata Festschrift) | DOI | MR | Zbl

[15] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | DOI | MR | Zbl

[16] Petrogradsky, V. M. On Witt’s formula and invariants for free Lie superalgebras, Formal power series and algebraic combinatorics (Moscow, 2000), Springer, Berlin, 2000, pp. 543-551 | DOI | MR | Zbl

[17] Reiner, Victor Thrall’s problem and coarsenings, Banff workshop on positivity in algebraic combinatorics lecture slides, 2015 https://www-users.cse.umn.edu/...

[18] Reutenauer, Christophe Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) (Lecture Notes in Math.), Volume 1234, Springer, Berlin, 1986, pp. 267-284 | DOI | MR | Zbl

[19] Reutenauer, Christophe Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993, xviii+269 pages (Oxford Science Publications) | DOI | MR | Zbl

[20] Schocker, Manfred Multiplicities of higher Lie characters, J. Aust. Math. Soc., Volume 75 (2003) no. 1, pp. 9-21 | DOI | MR | Zbl

[21] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl

[22] Sundaram, Sheila The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math., Volume 104 (1994) no. 2, pp. 225-296 | DOI | MR | Zbl

[23] Swanson, Joshua P. On the existence of tableaux with given modular major index, Algebr. Comb., Volume 1 (2018) no. 1, pp. 3-21 | DOI

[24] Thrall, R. M. On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math., Volume 64 (1942), pp. 371-388 | DOI

[25] Witt, Ernst Treue Darstellung Liescher Ringe, J. Reine Angew. Math., Volume 177 (1937), pp. 152-160 | DOI

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