**Vyjayanthi Chari**

University of California, Riverside

August 27, 2013

**Posets, Tensor Products and Schur positivity**:
Let $\gg$ be a simple Lie algebra and suppose that we are given four
irreducible representations $V_j, \ j=1, \ldots, 4$ of $\gg$. In this talk, we shall be interested in the problem of giving sufficient
conditions for the the following to hold:
$$
\dim \Hom_\gg (V, V_1 \otimes V_2) \leq \dim \Hom_\gg (V, V_3 \otimes V_4),
$$
Where $V$ is an arbitrary irreducible representation of $\gg$. If we take $\gg$ to be type $A_n$ for instance, this amounts to asking when the difference of the characters of $V_3 \otimes V_4$ and $V_1 \otimes V_2$ can be written as a non-negative integer linear combination of Schur functions. We propose a partial order on pairs (more generally $k$-tuples) of dominant weights for arbitrary $\gg$
which add up to a particular dominant weight. We shall see that the
maximal element in this partial order in the case of $A_n$ coincides with the row shuffle of partitions definedned by Fomin, Fulton, Li and
Poon. In joint work with Fourier and Sagaki, we conjecture that t