A classical result of Littlewood gives a factorisation for the Schur function at a set of variables “twisted” by a primitive $t$-th root of unity, characterised by the core and quotient of the indexing partition. While somewhat neglected, it has proved to be an important tool in the character theory of the symmetric group, the cyclic sieving phenomenon, plethysms of symmetric functions and more. Recently, similar factorisations for the characters of the groups $\mathrm{O}(2n,\u2102)$, $\mathrm{Sp}(2n,\u2102)$ and $\mathrm{SO}(2n+1,\u2102)$ were obtained by Ayyer and Kumari. We lift these results to the level of universal characters, which has the benefit of making the proofs simpler and the structure of the factorisations more transparent. Our approach also allows for universal character extensions of some factorisations of a different nature originally discovered by Ciucu and Krattenthaler, and generalised by Ayyer and Behrend.

Accepted:

Published online:

Keywords: Schur functions, symplectic characters, orthogonal characters, universal characters, $t$-core, $t$-quotient

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@article{ALCO_2023__6_6_1653_0, author = {Albion, Seamus P.}, title = {Universal characters twisted by roots of unity}, journal = {Algebraic Combinatorics}, pages = {1653--1676}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {6}, year = {2023}, doi = {10.5802/alco.320}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.320/} }

TY - JOUR AU - Albion, Seamus P. TI - Universal characters twisted by roots of unity JO - Algebraic Combinatorics PY - 2023 SP - 1653 EP - 1676 VL - 6 IS - 6 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.320/ DO - 10.5802/alco.320 LA - en ID - ALCO_2023__6_6_1653_0 ER -

Albion, Seamus P. Universal characters twisted by roots of unity. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1653-1676. doi : 10.5802/alco.320. https://alco.centre-mersenne.org/articles/10.5802/alco.320/

[1] Skew symplectic and orthogonal characters through lattice paths, 2023 | arXiv

[2] Skew characters and cyclic sieving, Forum Math. Sigma, Volume 9 (2021), Paper no. e41, 31 pages | MR | Zbl

[3] Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 78-105 | DOI | MR | Zbl

[4] Bijective proofs of skew Schur polynomial factorizations, J. Combin. Theory Ser. A, Volume 174 (2020), Paper no. 105241, 40 pages | MR | Zbl

[5] Factorization of classical characters twisted by roots of unity, J. Algebra, Volume 609 (2022), pp. 437-483 | DOI | MR | Zbl

[6] Algebraic aspects of increasing subsequences, Duke Math. J., Volume 109 (2001), pp. 1-65 | MR | Zbl

[7] Algorithms for plethysm, Combinatorics and algebra (Boulder, Colo., 1983) (Contemp. Math.), Volume 34, Amer. Math. Soc., Providence, RI, 1984, pp. 109-153 | DOI | MR | Zbl

[8] A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings, Advances in combinatorial mathematics, Springer, Berlin, 2009, pp. 39-59 | DOI | Zbl

[9] Character deflations and a generalization of the Murnaghan–Nakayama rule, J. Group Theory, Volume 17 (2006), pp. 1035-1070 | DOI | MR | Zbl

[10] On $p$-quotients and star diagrams of the symmetric group, Proc. Cambridge Phil. Soc., Volume 49 (1953), pp. 157-160 | DOI | MR | Zbl

[11] On the representations of the symmetric group, Proc. London Math. Soc., Volume 4 (1954), pp. 303-316 | DOI | Zbl

[12] On Schur functions, Proc. London Math. Soc., Volume 8 (1958), pp. 621-630 | DOI | Zbl

[13] Cranks and $t$-cores, Invent. Math., Volume 101 (1990), pp. 1-17 | DOI | MR | Zbl

[14] The $q$,$t$-Catalan numbers and the space of diagonal harmonics, University Lecture Series, 41, American Mathematical Society, Providence, RI, 2008, viii+167 pages | MR

[15] Combining hook length formulas and BG-ranks for partitions via the Littlewood decomposition, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 1041-1060 | MR | Zbl

[16] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981

[17] Skew-type symplectic/orthogonal Schur functions, 2022 | arXiv

[18] On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math., Volume 74 (1989), pp. 57-86 | DOI | MR | Zbl

[19] Representations of spinor groups and the difference characters of $SO\left(2n\right)$, Adv. Math., Volume 128 (1997), pp. 40-81 | DOI | MR | Zbl

[20] Young-diagrammatic methods for the representation theory of the classical groups of type ${B}_{n}$, ${C}_{n}$, ${D}_{n}$, J. Algebra, Volume 107 (1987), pp. 466-511 | DOI | Zbl

[21] Factorization of classical characters twisted by roots of unity: II, 2022 | arXiv

[22] Skew hook Schur functions and the cyclic sieving phenomenon, 2022 | arXiv

[23] Symmetric functions, 2001 https://www.emis.de/journals/SLC/wpapers/s68vortrag/ALCoursSf2.pdf (unpublished notes)

[24] Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, 99, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003, xii+268 pages | DOI

[25] An elliptic hypergeometric function approach to branching rules, SIGMA, Volume 16 (2020), Paper no. 142, 52 pages | MR | Zbl

[26] Edge sequences, ribbon tableaux, and an action of affine permutations, European J. Combin., Volume 20 (1999), pp. 179-195 | DOI | MR | Zbl

[27] The theory of group characters and matrix representations of groups, Oxford University Press, New York, 1940, viii+292 pages | MR

[28] On invariant theory under restricted groups, Philos. Trans. Roy. Soc. London Ser. A, Volume 239 (1944), pp. 387-417 | MR | Zbl

[29] Modular representations of symmetric groups, Proc. Roy. Soc. London Ser. A, Volume 209 (1951), pp. 333-353 | MR | Zbl

[30] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | DOI

[31] Factorization of Schur’s $Q$-functions and plethysm, Ann. Comb., Volume 6 (2002), pp. 87-101 | DOI | MR | Zbl

[32] On some modular properties of irreducible representations of symmetric groups. II, Jpn. J. Math., Volume 17 (1941), pp. 411-423 | DOI | MR

[33] Note on blocks of symmetric groups, Nagoya Math. J., Volume 2 (1951), pp. 111-117 | DOI | MR | Zbl

[34] Ribbon tile invariants, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 5525-5561 | MR | Zbl

[35] A refinement of the Murnaghan–Nakayama rule by descents for border strip tableaux, Combin. Theory, Volume 2 (2022), Paper no. 16, 16 pages | MR | Zbl

[36] A character relationship on ${\mathrm{GL}}_{n}\left(\u2102\right)$, Israel J. Math., Volume 211 (2016), pp. 257-270 | DOI | MR | Zbl

[37] The cyclic sieving phenomenon, J. Combin. Theory Ser. A, Volume 108 (2004), pp. 17-50 | DOI | MR

[38] On the representations of the symmetric group. III, Amer. J. Math., Volume 70 (1948), pp. 277-294 | DOI | MR | Zbl

[39] Star diagrams and the symmetric group, Can. J. Math., Volume 2 (1950), pp. 79-92 | DOI | MR | Zbl

[40] Rational tableaux and the tensor algebra of $g{l}_{n}$, J. Combin. Theory Ser. A, Volume 46 (1987), pp. 79-120 | DOI | MR | Zbl

[41] Modular Nekrasov–Okounkov formulas, Sém Lothar. Combin., Volume 81 (2020), Paper no. B81c, 28 pages | MR | Zbl

[42] The classical groups: Their invariants and representations, Princeton University Press, Princeton, N.J., 1939

[43] Review of the article “A character relationship between symmetric group and hyperoctahedral group” by F. Lübeck and D. Prasad, Mathematical Reviews, Volume MR4196561 (2021)

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