**This class will meet 9 AM on Tuesdays and Thursdays in room 380Y.**

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.

- Lecture notes in the form of a beta version of the forthcoming book are available. I provided a link in class. Let me know if you didn't get this.
- Sage has excellent support for crystals. For details, see the Thematic Tutorial on Lie Methods and Related Combinatorics in Sage.

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.

**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.