## 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.