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

Albion, Seamus P. ^{1}

@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/

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