Séminaire Lotharingien de Combinatoire, 91B.22 (2024), 12 pp.
Seamus Albion
Character Factorisations, z-Asymmetric Partitions, and Plethysm
Abstract.
An old theorem of D. E. Littlewood asserts that the Schur function
with variables "twisted" by a primitive t-th root of unity vanishes
unless the t-core of the indexing partition is empty, in which case it
factors as a product of Schur functions indexed by the t-quotient.
Recently, Ayyer and Kumari generalised Littlewood's result to characters of
the classical groups O(2n,C), Sp(2n,C)
and SO(2n+1,C). We show that Ayyer and Kumari's results
may be lifted to the universal characters of the associated groups, and in
doing so give a uniform extension involving a determinant of Bressoud and Wei which
was later generalised by Hamel and King.
What facilitates this extension is a new property of the Littlewood
decomposition, extending results of Garvan, Kim and Stanton.
We also explain the connection between Littlewood's original result and
an instance of plethysm.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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