Séminaire Lotharingien de Combinatoire, 95B.65 (2026), 12 pp.
Álvaro Gutiérrez, Álvaro L. Martínez,
Michał Szwej and Mark Wildon
Plethystic Lifts of q-Binomial Identities
Abstract.
We develop a framework to study plethysms of
SL2-representations over fields of arbitrary
characteristic. Each representation of the type
SymnSymmF2 is identified with a space of symmetric polynomials.
A certain map on polynomials (defined by substitution) gives
morphisms between these representations. We study the map
combinatorially, and use it to categorify numerous q-binomial
identities including the q-analogues of the team-and-leader
identity
(n + 1)
= 
(n + d + 1)
and its generalisations. We conclude with a
potential application to the generalised Foulkes Conjecture
for SL2(C), and
some further conjectures and directions raised by this work.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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