Séminaire Lotharingien de Combinatoire, 80B.30 (2018), 12 pp.
Christian Korff and David Palazzo
Cylindric Reverse Plane Partitions and 2D TQFT
Abstract.
The ring of symmetric functions carries the structure of a Hopf
algebra. When computing the coproduct of complete symmetric functions
hλ one arrives at weighted sums over reverse plane partitions
(RPP) involving binomial coefficients. Employing the action of the
extended affine symmetric group at fixed level n we generalise these
weighted sums to cylindric RPP and define cylindric complete symmetric
functions. The latter are shown to be h-positive, that is, their
expansions coefficients in the basis of complete symmetric functions are
non-negative integers. We state an explicit formula in terms of tensor
multiplicities for irreducible representations of the generalised
symmetric group. Moreover, we relate the complete symmetric functions to
a 2D topological quantum field theory (TQFT) that is a generalisation of
the celebrated sl~n-Verlinde algebra or
Wess-Zumino-Witten fusion ring, which plays a prominent role in the
context of vertex operator algebras and algebraic geometry.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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