Vyjayanthi Chari
University of California, Riverside
August 27, 2013
Posets, Tensor Products and Schur positivity: Let $\mathfrak g$ be a simple Lie algebra and suppose that we are given four irreducible representations $V_j, \ j=1, \ldots, 4$ of $\mathfrak g$. In this talk, we shall be interested in the problem of giving sufficient conditions for the the following to hold: $$ \dim \mathrm{Hom}_\mathfrak g (V, V_1 \otimes V_2) \leq \dim \mathrm{Hom}_\mathfrak g (V, V_3 \otimes V_4), $$ Where $V$ is an arbitrary irreducible representation of $\mathfrak g$. If we take $\mathfrak g$ 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 $\mathfrak g$ 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 the inequality discussed above holds along the partial order. We shall discuss the cases when the conjecture is known to be true. Observe that if the conjecture is true, there is obviously a non-canonical surjection $V_3 \otimes V_4 \rightarrow V_1 \otimes V_2 \rightarrow 0$ of $\mathfrak g$-modules. We shall use some joint work with R. Venkatesh to show that by passing to representations of certain infinite-dimensional Lie algebras $\mathfrak g[t]$, one could expect to define a canonical map of these modules as representations of $\mathfrak g[t]$.
University of California, Riverside
August 27, 2013
Posets, Tensor Products and Schur positivity: Let $\mathfrak g$ be a simple Lie algebra and suppose that we are given four irreducible representations $V_j, \ j=1, \ldots, 4$ of $\mathfrak g$. In this talk, we shall be interested in the problem of giving sufficient conditions for the the following to hold: $$ \dim \mathrm{Hom}_\mathfrak g (V, V_1 \otimes V_2) \leq \dim \mathrm{Hom}_\mathfrak g (V, V_3 \otimes V_4), $$ Where $V$ is an arbitrary irreducible representation of $\mathfrak g$. If we take $\mathfrak g$ 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 $\mathfrak g$ 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 the inequality discussed above holds along the partial order. We shall discuss the cases when the conjecture is known to be true. Observe that if the conjecture is true, there is obviously a non-canonical surjection $V_3 \otimes V_4 \rightarrow V_1 \otimes V_2 \rightarrow 0$ of $\mathfrak g$-modules. We shall use some joint work with R. Venkatesh to show that by passing to representations of certain infinite-dimensional Lie algebras $\mathfrak g[t]$, one could expect to define a canonical map of these modules as representations of $\mathfrak g[t]$.