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 with simple roots
for
in a Euclidean vector space
with inner product
. For any
define
A weight lattice for
is a lattice spanning
such that
and
for all
and
. Fix a weight lattice
(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
. Elements of the weight lattice are called weights. The set of dominant weights is
Now let be a semisimple Lie algebra with Cartan subalgebra
(a Cartan subalgebra is a Euclidean vector space) and associated root system
. It is well-known from representation theory that if
is a finite dimensional representation, i.e. a finite dimensional vector space with a
action
respecting the Lie bracket, then
admits a decomposition
The are weight spaces:
That is, consists of those vectors in
that act like eigenvectors with respect to
for the Cartan subalgebra
. The amazing fact is that if
is a dominant weight, then there exists a unique irreducible representation
with highest weight
. That is
We would like to find a nice basis for to understand the representation theory of
. To this end, we would hope that the subbases for each
interact nicely. Kashiawara crystals our are answer.
Crystals
For each dominant weight there exists a crystal
of highest weight
. It is intimately connected to
. There exist other crystals
that are not intimately connected to highest weights, but we will not be concerned with these. In short, the elements of
are basis vectors for the
in the decomposition
and they interact nicely. The upshot is that the combinatorics of encodes much of the representation theory of
and is easier to understand than the representation theory directly. The crystal
is a nonempty set together with maps
where and
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
in another way. We view
as a connected directed graph whose vertices are basis elements of the representation
. In particular, there is one element of the crystal (i.e., vertex) for each basis element of each weight space
with
. Every
also has a weight
assigned to it. Notice that
really is a weight in the sense of a weight lattice since the target of
is
. The weight map
is not in general injective; elements of the crystal belonging to the same weight space
are prescribed the same weight
. We still need to describe the edges in the crystal
. The edges between vertices are indexed by the simple roots
, or equivalently, by the set
. The
and
maps describe how we are permitted to move between elements of the crystal (i.e, the vertices). The
are called raising operators and the
are called lowering operators. The reason we call them raising and lowering operators is because if
then
if and only if
. In this case,
Take a moment to stare at these identities. What they are saying is that raises
by
,
lowers
by
, and these operators send
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
might have more than one basis element). So why do the
and
possibly send an element of the crystal to
? Let’s illustrate this by first stating an albeit non-obvious fact:
Theorem: In, there exists a unique element
with weight
.
Essentially, all this is saying is that in the decomposition
the weight space is
-dimensional. Now consider
for any
. What element of the crystal should
be? Since the
raise the weight and
is the unique element of the crystal with highest weight, it is natural to have
in the sense that
maps
out of the crystal. Similarly, it is natural to imagine that if we apply successive
operators to any
we lower the weight enough for it to leave the crystal. So what are the
and
measuring? For a given
,
(respectively
) is the maximum integer
such that
(respectively
). That is,
is the number of times we can apply
to
before leaving the crystal (respectively the number of times we can apply
to
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
and
above to confirm this. So why then is
the target of the
and
and not simply
? For the crystals we will be concerned with (i.e., normal crystals) the
and
only map into
, but there exist crystals where they can take value
. So,
was used for completion
So what have we learned? The crystal as a set is a basis for
. But it does more than that. The crystal
comes equip with raising and lowering operators that describe how we move between elements of the basis. In other words,
contains information about how the root spaces
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 be a root system of type
with weight Lattice
, let
be our semisimple Lie algebra of type
with the usual Lie bracket, and
the set of trace zero diagonal matrices as the Cartan subalgebra. Dominant weights then correspond to partitions
. Consider the weight
and the corresponding unique irreducible representation
. 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 -dimensional weight space
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 . 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
and the solid arrows correspond to
.
Computing a Crystal Using Tableaux
Having defined crystals abstractly, and given an example of the crystal of highest weight
, it remains to explain how we systematically construct a crystal of highest weight. For type
root systems, and in general other standard types, many crystals can be constructed from semistandard Young tableaux. We will explain the construction for
. Given a dominant weight
(i.e. a partition), the elements of
are semistandard Young tableaux
of shape
in the alphabet
. The weight map is
where is the number of
‘s in
. The highest weight element is the Yamanouchi tableau. It is the tableaux
such that all entries in the
-th row are equal to
. 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
be the
th row of
. If
has
rows, then the row reading of
is
If we are applying or
to
, then put a
under each element
in the row reading and a
under each element
in the row reading. Then bracket the signs as follows
,
, etc. If we apply
, change the rightmost unbracketed
from
to
. If we apply
, change the leftmost unbracketed
from
to
. Then replaced the modified reading row back into
. This algorithm is often called the signature rule. Since
is connected, we can construct the entire crystal by taking the Yamanouchi tableau for
and successively applying lowering operators in all possible ways.
Let’s run through part of the construction for the and crystal
of highest weight
. The elements of
are semistandard Young tableaux of shape
in the alphabet
. The highest weight tableaux is the Yamanouchi tableau
of shape
with weight
. To construct the rest of the crystal we need to apply successive raising and lowering operators
. The row reading of
is
Applying the first lowering operator:
so the tableau is exacatly the same as
except the rightmost
in the first row is a
. Applying the second lowering operator:
so the tableau is exacatly the same as
except the rightmost
in the second row is a
. 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