Let $\mathfrak{g}$ be a semisimple Lie algebra. One example to keep in mind is $\mathfrak{sl}_n$, the Lie algebra of trace-zero $n\times n$ matrices. Let $\Lambda$ be the *weight lattice* of $\mathfrak{g}$, the lattice of generalized eigenvalues of the Cartan subalgebra of $\mathfrak{g}$. Let $I$ be the index set of the simple roots $\{\alpha_i\}_{i\in I}$ of $\mathfrak{g}$. For instance, in the case of $\mathfrak{sl}_n$ (referred to as `type A’), we have that $\Lambda$ is the quotient of $\mathbb{Z}^n$ by the subspace generated by the vector $(1,1,\ldots,1)$, and $I=\{1,\ldots, n-1\}$ with simple roots $$\alpha_i=(0,0,\ldots,0,1,-1,0\ldots,0)=v_i-v_{i+1}$$ where $v_i=(0,0,\ldots,0,1,0,0,\ldots,0)$ is the standard basis vector with a $1$ in the $i$-th position.

In the simplest nontrivial case, $\mathfrak{sl}_2$, the Lie algebra is generated by the matrices $$e_1=\left(\begin{array}{cc} 0 & 1 \\ 0 & 0\end{array}\right),\hspace{0.5cm} f_1=\left(\begin{array}{cc} 0 & 0 \\ 1 & 0\end{array}\right), \hspace{0.5cm}h_1=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right).$$ In general there are generators $e_i,f_i,h_i$ for $i\in I$ that generate $\mathfrak{g}$, and hence $U_q(\mathfrak{g})$. The $e_i$ and $f_i$ elements, and their actions on weight spaces in $U_q(\mathfrak{g})$-representations, give rise to the corresponding crystal operators $E_i$ and $F_i$ in Kashiwara’s construction.

We now come to the main definition: a **combinatorial crystal** (sometimes referred to as a **Kashiwara crystal**) is a nonempty set $\mathcal{B}$ along with:

- Raising and lowering operators $$e_i,f_i:\mathcal{B}\to \mathcal{B}\sqcup \{\varnothing\},$$
- `String length’ operators $$\varepsilon_i,\varphi_i:\mathcal{B}\to \mathbb{Z}\sqcup \{-\infty\},$$
- A weight function $$\mathrm{wt}:\mathcal{B}\to \Lambda,$$

satisfying the following properties:

- The operators $e_i$ and $f_i$ are partial inverses: if $x,y\in \mathcal{B}$ then $$e_i(x)=y\text{ iff }f_i(y)=x.$$

Moreover, in this case we have:

$$\begin{eqnarray*}

\mathrm{wt}(y)&=&\mathrm{wt}(x)+\alpha_i \\

\varepsilon_i(y)&=&\varepsilon_i(x)-1 \\

\varphi_i(y)&=&\varphi_i(x)+1.

\end{eqnarray*}

$$ - For any $x\in \mathcal{B}$ and $i\in I$, we have $$\varphi_i(x)-\varepsilon_i(x)=\langle \mathrm{wt}(x),\alpha_i^\vee \rangle$$ where $\alpha_i^\vee = 2\alpha_i/\langle \alpha_i,\alpha_i\rangle$ is the corresponding simple coroot. In particular if $\varphi_i(x)=-\infty$, then $\epsilon_i=-\infty$ as well, and in this case $e_i(x)=f_i(x)=0$.

The string lengths $\varphi_i(x)$ and $\varepsilon_i(x)$ often measure actual lengths to the end of the $i$-string through $x$ in the crystal graph, and such a crystal is called **seminormal**. More precisely, a crystal is **seminormal** if $\varphi_i(x)$ is the largest $k$ for which $f_i^k(x)\neq \varnothing$, and $\varepsilon_i(x)$ is the largest $k$ for which $e_i^k(x)\neq \varnothing$, for all $x$ and $i$.

One can now define the category of Kashiwara crystals by defining a **crystal morphism** from $\mathcal{B}$ to $\mathcal{C}$ to be a map $\psi:\mathcal{B}\to \mathcal{C}\cup \{\varnothing\}$ that preserves the string length, weight, and raising and lowering operators. In fact, it is a tensor category under the following tensor product. If $\mathcal{B}$ and $\mathcal{C}$ are two crystals of the same type, their **tensor product** $\mathcal{B}\otimes \mathcal{C}$ is the crystal having:

- Ground set $\mathcal{B}\times \mathcal{C}$, the Cartesian product of the sets.
- Weight function $\mathrm{wt}(x\otimes y)=\mathrm{wt}(x)+\mathrm{wt}(y)$.
- Raising operators $$e_i(x\otimes y)=\begin{cases} e_i(x)\otimes y & \varphi_i(y) \lt \varepsilon_i(x) \\ x\otimes e_i(y) & \varphi_i(y)\ge \varepsilon_i(x) \end{cases}$$ and lowering operators $$f_i(x\otimes y)=\begin{cases} f_i(x)\otimes y & \varphi_i(y) \le \varepsilon_i(x) \\ x\otimes f_i(y) & \varphi_i(y)\gt \varepsilon_i(x) \end{cases}.$$
- String lengths $$\begin{eqnarray*}\varphi_i(x\otimes y)&=&\mathrm{max}(\varphi_i(x),\varphi_i(y)+\langle\mathrm{wt}(x),\alpha_i^\vee\rangle) \\ \varepsilon_i(x\otimes y)&=&\mathrm{max}(\varepsilon_i(y),\varepsilon_i(x)+\langle\mathrm{wt}(y),\alpha_i^\vee\rangle)

\end{eqnarray*}$$

While this abstract category has many nice properties, it is in some sense too large. In particular, not every crystal in this category comes from a representation of $U_q(\mathfrak{g})$. In this paper, Stembridge introduced a set of local axioms that identify which objects in the category of crystals correspond to actual representations, for any simply-laced Lie type. (He restricts to the case of integrable representations, meaning that the operators $e_\alpha$ and $f_\alpha$ that generate $\mathfrak{g}$ act locally nilpotently). Such crystals, called **Stembridge crystals**, have unique highest weight elements, and the highest weight element uniquely determines its Stembridge crystal.

It’s not hard to check that the tableau crystals, and their tensor products, satisfy these axioms for the type A root system, using string lengths for the operators $\varphi_i$ and $\varepsilon_i$. They satisfy Stembridge’s axioms as well, and there is one tableau crystal for every possible highest weight, so in fact:

The tableau crystals are precisely the Stembridge crystals in type A, corresponding to the integrable representations of $U_q(\mathfrak{sl}_n)$.

In future posts, we’ll discuss some new developments in crystal base theory and their applications to symmetric function theory. An excellent reference for more crystal essentials is the book Crystal Bases by Dan Bump and Anne Schilling.