Schur-Positivity proof using Kashiwara crystals.

Question

It is well known that the polynomial $$(x_1+\cdots+x_n)^d$$ is Schur positive with coefficients $c_\lambda$ in the Schur expansion equal to the dimension of the irreducible representation $\lambda$ of $S_d$. I know that there is a way to proof this using Kashiwara crystals but I don't understand how to implement that technique, any hint of how to do it?

Answer 1

This is not particularly difficult if one is willing to accept a lot of results about crystals. We can stick to crystals of $\operatorname{GL}_n$. Here is a rough sketch of how things unfold:

1. Let $B$ be the basic crystal of $\operatorname{GL}_n$. This crystal has $n$ vertices, and looks like a path $$ 1 \xrightarrow{1} 2 \xrightarrow{2} 3 \xrightarrow{4} \cdots \xrightarrow{n-1} n,$$ where the vertex labelled $i$ is in weight $x_i$. The character of this crystal is $x_1 + \cdots + x_n$.
2. Form the $d$-fold tensor product crystal $B^{\otimes d}$, which is the crystal of words. More specifically, its vertices are in bijection with words of length $d$ on the alphabet $\{1, \ldots, n\}$. The character of this crystal is $(x_1 + \cdots + x_n)^d$.
3. Figure out how the tensor product crystal decomposes into irreducible components. Each irreducible component $B(\lambda)$ is labelled by some partition $\lambda$ of $d$, and the character of each $B(\lambda)$ is the Schur polynomial $s_\lambda(x_1, \ldots, x_n)$. Let $c_\lambda$ be the number of times that $B(\lambda)$ occurs in $B^{\otimes d}$: we are looking for how to compute $c_\lambda$ (but by this point, we already know that each $c_\lambda$ is a nonnegative integer).
4. A much more careful examination of the internals of $B^{\otimes d}$ shows that each connected component can be labelled by a pair $(P, \lambda)$ where $\lambda$ is a partition of $d$, and $P$ is a standard tableau of shape $\lambda$. For a given $\lambda$, the number of standard tableaux of shape $\lambda$ is equal to the dimension of the irreducible representation $S^\lambda$ of the symmetric group $S_d$.

What is nice is that you get Schur-positivity as a result of the fact that (a) any $\operatorname{GL}_n$ crystal decomposes into a disjoint union of various $B(\lambda)$, and (b) the tensor product of two crystals is again a crystal.

There is a fantastic account of this in some notes called Crystals for Dummies by Mark Shimozono, which you can find online.