Kashiwara Crystals and Bases of Representations

The theorem of highest weight tells us that for each dominant weight there exists a unique irreducible finite dimensional representation with the dominant weight as the highest weight. We may further decompose these representations into a sum of weight spaces. It turns out that bases of these representations admit beautiful combinatorial structure with respect to the weight space decomposition. In the following we describe this structure via Kashiwara crystals.

Motivation for Kashiwara Crystals

Let us fix an irreducible root system \Phi with simple roots \alpha_{i} for i \in I in a Euclidean vector space \mathfrak{h} \cong \mathfrak{h}^{\ast} with inner product \langle \cdot,\cdot \rangle. For any \alpha \in \mathfrak{h} define

\displaystyle \alpha^{\vee} = \frac{2\alpha}{\langle \alpha,\alpha \rangle}.

A weight lattice \Lambda for \Phi is a lattice spanning \mathfrak{h} such that \Phi \subset \Lambda and \langle \lambda,\alpha^{\vee} \rangle \in \mathbb{Z} for all \lambda \in \Lambda and \alpha \in \Phi. Fix a weight lattice \Lambda (in general there are more than one but if you’d like you can take the maximal one). The weight lattice comes equip with a natural partial ordering \le. Elements of the weight lattice are called weights. The set of dominant weights is

\Lambda^{+} = \{\lambda \in \Lambda \mid \langle \lambda,\alpha_{i}^{\vee} \rangle \ge 0 \,\, \forall \,\, i \in I\}.

Now let \mathfrak{g} be a semisimple Lie algebra with Cartan subalgebra \mathfrak{h} (a Cartan subalgebra is a Euclidean vector space) and associated root system \Phi. It is well-known from representation theory that if V is a finite dimensional representation, i.e. a finite dimensional vector space with a \mathfrak{g} action X \cdot v respecting the Lie bracket, then V admits a decomposition

\displaystyle V = \bigoplus_{\lambda \in \Lambda}V_{\lambda}.

The V_{\lambda} are weight spaces:

V_{\lambda} = \{v \in V \mid H \in \mathfrak{h}, H \cdot v = \lambda(H)v\}.

That is, V_{\lambda} consists of those vectors in V that act like eigenvectors with respect to \lambda for the Cartan subalgebra \mathfrak{h}. The amazing fact is that if \lambda is a dominant weight, then there exists a unique irreducible representation V(\lambda) with highest weight \lambda. That is

\displaystyle V(\lambda) = \bigoplus_{\mu \le \lambda}V_{\mu}.

We would like to find a nice basis for V(\lambda) to understand the representation theory of V(\lambda). To this end, we would hope that the subbases for each V_{\mu} interact nicely. Kashiawara crystals our are answer.

Crystals

For each dominant weight \lambda there exists a crystal \mathcal{B}(\lambda) of highest weight \lambda. It is intimately connected to V(\lambda). There exist other crystals \mathcal{B} that are not intimately connected to highest weights, but we will not be concerned with these. In short, the elements of \mathcal{B}(\lambda) are basis vectors for the V_{\mu} in the decomposition

\displaystyle V(\lambda) = \bigoplus_{\mu \le \lambda}V_{\mu}

and they interact nicely. The upshot is that the combinatorics of \mathcal{B}(\lambda) encodes much of the representation theory of V(\lambda) and is easier to understand than the representation theory directly. The crystal \mathcal{B}(\lambda) is a nonempty set together with maps

\begin{aligned} e_{i},f_{i}:\mathcal{B}(\lambda) &\to \mathcal{B}(\lambda) \sqcup \{0\}, \\ \epsilon_{i},\varphi_{i}:\mathcal{B}(\lambda) &\to \mathbb{Z} \sqcup \{-\infty\}, \\ \mathrm{wt}:\mathcal{B}(\lambda) &\to \Lambda, \end{aligned}

where i \in I and 0 \notin \mathcal{B}(\lambda) is an auxiliary element. These maps also satisfy some other axioms but they are not necessary at the moment. Let’s unpack what these maps mean by thinking about \mathcal{B}(\lambda) in another way. We view \mathcal{B}(\lambda) as a connected directed graph whose vertices are basis elements of the representation V(\lambda). In particular, there is one element of the crystal (i.e., vertex) for each basis element of each weight space V_{\mu} with \mu \le \lambda. Every v \in \mathcal{B}(\lambda) also has a weight \mathrm{wt}(v) assigned to it. Notice that \mathrm{wt}(v) really is a weight in the sense of a weight lattice since the target of \mathrm{wt} is \Lambda. The weight map \mathrm{wt} is not in general injective; elements of the crystal belonging to the same weight space V_{\mu} are prescribed the same weight \mu. We still need to describe the edges in the crystal \mathcal{B}(\lambda). The edges between vertices are indexed by the simple roots \alpha_{i}, or equivalently, by the set I. The e_{i} and f_{i} maps describe how we are permitted to move between elements of the crystal (i.e, the vertices). The e_{i} are called raising operators and the f_{i} are called lowering operators. The reason we call them raising and lowering operators is because if v,w \in \mathcal{B}(\lambda) then e_{i}(v) = w if and only if f_{i}(w) = v. In this case,

\mathrm{wt}(w) = \mathrm{wt}(v)+\alpha_{i}, \qquad \epsilon_{i}(w) = \epsilon_{i}(v)-1, \qquad \varphi_{i}(w) = \varphi_{i}(v)+1.

Take a moment to stare at these identities. What they are saying is that e_{i}(v) raises \mathrm{wt}(v) by \alpha_{i}, f_{i}(v) lowers \mathrm{wt}(v) by \alpha_{i}, and these operators send v to an element of the crystal with that prescribed weight. The subtly is that there might be multiple elements of the crystal with the same weight (since V_{\mu} might have more than one basis element). So why do the e_{i} and f_{i} possibly send an element of the crystal to 0? Let’s illustrate this by first stating an albeit non-obvious fact:

Theorem: In \mathcal{B}(\lambda) , there exists a unique element v  with weight \lambda .

Essentially, all this is saying is that in the decomposition

\displaystyle V(\lambda) = \bigoplus_{\mu \le \lambda}V_{\mu}

the weight space V_{\lambda} is 1-dimensional. Now consider e_{i}(v) for any i \in I. What element of the crystal should e_{i}(v) be? Since the e_{i} raise the weight and v is the unique element of the crystal with highest weight, it is natural to have e_{i}(v) = 0 in the sense that e_{i}(v) maps v out of the crystal. Similarly, it is natural to imagine that if we apply successive f_{i} operators to any v \in \mathcal{B}(\lambda) we lower the weight enough for it to leave the crystal. So what are the \epsilon_{i} and \varphi_{i} measuring? For a given v \in \mathcal{B}(\lambda), \epsilon_{i}(v) (respectively \varphi_{i}(v)) is the maximum integer k such that e_{i}^{k}(v) \neq 0 (respectively f_{i}^{k}(v) \neq 0). That is, \epsilon_{i}(v) is the number of times we can apply e_{i} to v before leaving the crystal (respectively the number of times we can apply f_{i} to v before leaving the crystal). Technically, this is true for normal crystals but the only crystals we care about will be normal. Look at the formulas for \epsilon_{i} and \varphi_{i} above to confirm this. So why then is \mathbb{Z} \sqcup \{-\infty\} the target of the \epsilon_{i} and \varphi_{i} and not simply \mathbb{Z}? For the crystals we will be concerned with (i.e., normal crystals) the \epsilon_{i} and \varphi_{i} only map into \mathbb{Z}, but there exist crystals where they can take value -\infty. So, \mathbb{Z} \sqcup \{-\infty\} was used for completion

So what have we learned? The crystal \mathcal{B}(\lambda) as a set is a basis for V(\lambda). But it does more than that. The crystal \mathcal{B}(\lambda) comes equip with raising and lowering operators that describe how we move between elements of the basis. In other words, \mathcal{B}(\lambda) contains information about how the root spaces V_{\mu} interact with respect to the weight lattice!

An Example of Weight Spaces and Crystals

Let’s run through an example of what the weight space decomposition of a representation looks like with its associated crystal. Let \Phi be a root system of type A_{2} with weight Lattice \mathbb{Z}^{3}, let \mathfrak{sl}{3}(\mathbb{C}) be our semisimple Lie algebra of type A_{2} with the usual Lie bracket, and \mathfrak{h} the set of trace zero diagonal matrices as the Cartan subalgebra. Dominant weights then correspond to partitions (\lambda_{1},\lambda_{2},\lambda_{3}). Consider the weight \lambda = (5,2,0) and the corresponding unique irreducible representation V(\lambda). It decomposes into weight spaces as illustrated below:

In the image each circle represents a weight space and the encircled number is its dimension. Notice that there exists a 1-dimensional weight space V_{\lambda} in the top right as expected. The corresponding crystal should then have one vertex for each dimension of each weight space. Indeed, the diagram below

is the corresponding crystal of highest weight \lambda. Notice that the crystal has been displayed such that vertices close together correspond to basis elements in the same weight space. The dashes arrows correspond to f_{1} and the solid arrows correspond to f_{2}.

Computing a Crystal Using Tableaux

Having defined crystals abstractly, and given an example of the A_{2} crystal of highest weight \lambda = (5,2,0), it remains to explain how we systematically construct a crystal of highest weight. For type A_{n} root systems, and in general other standard types, many crystals can be constructed from semistandard Young tableaux. We will explain the construction for A_{n}. Given a dominant weight \lambda (i.e. a partition), the elements of \mathcal{B}(\lambda) are semistandard Young tableaux T of shape \lambda in the alphabet [n+1]. The weight map is

\mathrm{wt}:\mathcal{B}(\lambda) \to \mathbb{Z}^{n+1} \qquad \mathrm{wt}(T) = (\mu_{1},\mu_{2},\ldots,\mu_{n+1})

where \mu_{i} is the number of i‘s in T. The highest weight element is the Yamanouchi tableau. It is the tableaux T such that all entries in the i-th row are equal to i. What are the raising and lowering operators? There are several, equivalent, ways to define them but we will use row reading and the signature rule. Let T_{i} be the ith row of T. If T has k rows, then the row reading of T is

(T_{k},T_{k-1},\ldots,T_{1}).

If we are applying e_{i} or f_{i} to T, then put a + under each element i+1 in the row reading and a - under each element i in the row reading. Then bracket the signs as follows (+-), (+(+-)-), etc. If we apply f_{i}, change the rightmost unbracketed - from i to i+1. If we apply e_{i}, change the leftmost unbracketed + from i+1 to i. Then replaced the modified reading row back into T. This algorithm is often called the signature rule. Since \mathcal{B}(\lambda) is connected, we can construct the entire crystal by taking the Yamanouchi tableau for \lambda and successively applying lowering operators in all possible ways.

Let’s run through part of the construction for the A_{2} and crystal \mathcal{B}(\lambda) of highest weight \l = (5,2,0). The elements of \mathcal{B}(\lambda) are semistandard Young tableaux of shape (5,2,0) in the alphabet [3]. The highest weight tableaux is the Yamanouchi tableau T of shape (5,2,0) with weight (5,2,0). To construct the rest of the crystal we need to apply successive raising and lowering operators f_{i}. The row reading of T is

(2,2,1,1,1,1,1).

Applying the first lowering operator:

f_{1}(\underset{+}{2},\underset{+}{2},\underset{-}{1},\underset{-}{1},\underset{-}{1},\underset{-}{1},\underset{-}{1}) = (2,2,1,1,1,1,2)

so the tableau f_{1}(T) is exacatly the same as T except the rightmost 1 in the first row is a 2. Applying the second lowering operator:

f_{2}(\underset{-}{2},\underset{-}{2},1,1,1,1,1) = (2,3,1,1,1,1,1)

so the tableau f_{2}(T) is exacatly the same as T except the rightmost 2 in the second row is a 3. By repeatedly applying these lowering operators, the entire crystal above can be reconstructed as a crystal of tableaux.


References

Crystal Bases: Representations and Combinatorics – Daniel Bump, Anne Schilling

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