Crystal Bases are combinatorial analogs of finite-dimensional representations. This course will be based on the forthcoming book Crystal Bases: Representations and Combinatorics by Daniel Bump and Anne Schilling. It will focus on two aspects: giving purely combinatorial foundations for the subject, and explaning many analogies of the field with topics in representation theory of Lie groups.
In Lie group representations, Schur-Weyl duality and $GL(n)\times GL(m)$ duality are powerful tools. They have exact combinatorial analogs in the Robinson-Schensted-Knuth (RSK) correspondence. In combinatorics, tableaux are at the heart of symmetric function theory. Lascoux and Schützenberger defined a monoidal structure on tableaux, called the plactic monoid. In retrospect, crystal bases give a natural context for understanding and also generalizing both RSK and the plactic monoid. We will explain the analogies with facts from Lie group representations in the context of these ideas.
Regarding the foundations, in the past foundations of the subject have depended on either the theory of quantum groups or the theory of Littelmann paths. Instead, we will give combinatorial proofs based on the Stembridge axioms supplemented by the theory of virtual crystals. Just as in Weyl's theory there is a unique Lie group representation with a given dominant weight as highest weight, we will show that there is a unique normal crystal with a given highest weight. Using the theory of Demazure crystals we will then show the behavior of crystals exactly mirrors the behavior of representations under tensor product and Levi branching.
The following exercises will be in the next posted beta version of the book. I hope this will be released by the end of this week (January 15).
To gain familiarity with crystals it would be good to learn Sage's capabilities. These are explained in the Thematic Tutorial You should be able to do the following exercises by hand, but you may find Sage useful for checking your work.
Exercise 1. Let $\mathcal{B}$ be the $\text{GL}(3)$ crystal from Example 2.18. Compute $f_i(x\otimes y)$ by hand for every $x$ and $y$ in $\mathcal{B}$ and thus compute by hand the tensor product $\mathcal{B}\otimes\mathcal{B}$. Check that it decomposes as a disjoint union of two connected crystals, one with three elements and one with six.
In addition to the standard crystal found in Example 2.18, there is a dual standard crystal that looks like this: \[\newcommand\boxt[1]{\begin{array}{|l|}\hline \,#1\,\\\hline\end{array}} \begin{array}{cccccc} &\scriptstyle r&&\scriptstyle r-1&&\scriptstyle 2&&\scriptstyle 1\\ \boxt{\overline{r+1}}&\longrightarrow&\boxt{\overline{r}}&\longrightarrow&\cdots&\longrightarrow&\boxt{\overline{2}}&\longrightarrow&\boxt{\overline{1}} \end{array} \] Here the weight of $\boxt{\overline{i}}$ is $-\mathbf{e}_i$, where $\mathbf{e}_i$ is the $i$-th standard basis vector of the weight lattice~$\Lambda$.Exercise 2. Show that one of the two connected subcrystals of $\mathcal{B}\otimes\mathcal{B}$ that you found in Exercise 1 is a twist of the dual standard crystal defined above. Explain why it is necessary to twist.
Exercise 3. Generalize Exercise 2 by showing that the dual standard crystal for of type $\text{GL}(n)$ is a twist of a subcrystal of $\bigoplus^{n-1}\mathcal{B}$.
Hint The $n$ elements of this subcrystal have the shape \[\boxt{n}\otimes\cdots\otimes\widehat{\boxt{j}} \otimes\cdots\otimes{\boxt{1}}\] with the entries in decreasing order, where the notation means that the factor ${\boxt{j}}$ is omitted.
Exercise 4. Let $\mathcal{B}$ be the $\text{GL}(3)$ crystal from Example 2.18 and let $\widehat{\mathcal{B}}$ be the dual standard crystal defined above. Compute the tensor product $\mathcal{B}\otimes\widehat{\mathcal{B}}$. You should find that this crystal has two components, one of degree one and one of degree 8.
Exercise 5. Let $\Lambda$ be the $C_2$ weight lattice, and let $\mathcal{B}$ be the standard crystal of degree $4$. Show that $\mathcal{B}\otimes\mathcal{B}$ has three connected components, of degrees $1$, $5$ and $10$. What are their highest weight elements?
Exercise 6. To generalize the last exercise, let $\mathcal{B}$ be the $C_r$ standard crystal. In order to look for connected components of $\mathcal{B}\otimes\mathcal{B}$, a shortcut is to look for highest weight elements. That is, assume that \[\varepsilon_i(\boxt{x}\otimes\boxt{y})=0\] for all $i$ and try to deduce what the possibilities are for $x$ and $y$ using (2.7). You should find that there are exactly three highest weight vectors. Can you conjecture the degrees of the crystals containing them?
Exercise 7. Construct a connected $\text{GL}(4)$ crystal $\mathcal{C}$ with highest weight $(1,1,0,0)$ having six elements. Compute its Levi branching to a $\text{GL}(2)\times\text{GL}(2)$ crystal.
Exercise 8. Every one of these Exercises is the analog of some computation involving irreducible representations of Lie groups. Describe these.