## On the combinatorics of sl(n)-fusion algebra

Geanina Tudose
( York University)

The fusion algebra also known as the Verlinde algebra plays a
central role in the 2 dimensional Wess-Zumino-Witten
models of conformal field theory. The study of the multiplicative
structure of this algebra has received a lot of attention in the past
decade due to the fact that it appears in an increasing number of
mathematical contexts such as quantum cohomology, representations
of quantum groups and Hecke algebras, knot invariants, vertex
operator algebras, and others.

The $sl(n)$-fusion  algebra can be viewed as a quotient of  the ring of
symmetric functions in $n$ variables by the ideal generated by Schur functions $S_\lambda$ indexed by
partitions of length at most $n$ such that $\lambda_1-\lambda_n \leq k$
and $S_{1^n}-1$.

From representation theoretic arguments it is known that
its structure constants N_{\lambda \mu}^{\nu}, called fusion coefficients,
are non-negative integers. We will give a  combinatorial description for
these numbers for $\mu$ two column and hook partitions and a larger
family of partitions obtained via fusion invariants.
In addition, we present a number of applications for these cases
including the proof of the conjecture that the fusion coefficients are
increasing with respect to the level.