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.