Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

A Combinatorial Model for Crystals of Kac-Moody Algebras

General Background

In this class we dealt with various crystals models, like the tableau models. However, many of these models only work for specific classes of Kac-Moody Lie algebras (eg finite dimensional), and often must be changed to fit with any particular Lie algebra. One of the ideas behind the model presented in this paper is that it can be applied to any complex, symmetrizable Kac-Moody algebra. We will just briefly go over the definitions, and important facts without proof.

\( \lambda \) Chains (of roots)

First we define a pair of counting functions to shorten our notation. Let \( \lbrace \beta_i \rbrace_{\lbrace i \in I\rbrace} \) be any set, indexed by a completed ordered set \( I \). Then:

\( N(\alpha)= \) number of times \( \alpha \) appears in the sequence.

\( N_i( \alpha )=  \)number of times \( \alpha \) appears in the sequence strictly before \( i \).

Definition: Let \( \lambda \in P_+\) be a dominant weight. A \( \lambda-chain\) is a sequence of positive roots, \( \lbrace \beta_i \rbrace_{\lbrace i \in I\rbrace}\), such that the following two conditions hold:

(1) \( N(\alpha) = \langle \lambda , \alpha^\vee \rangle\) for all positive roots \( \alpha \in \Phi^+\).

(2) If \( \gamma^\vee = \alpha^\vee + k \beta^\vee\), and \( \beta_i=\beta\), then: \( N_i(\gamma)=N_i(\alpha)+kN_i(\beta)\)

 

The idea is to pick any ambient \( \lambda \) chain, and that this will give us an equivalent crystal later on. However, in order to make sure that \( \lambda \)-chains exist, we will construct one.

Fix an ordering on the simple roots: \( \alpha_1,...\alpha_r \). We will order the set \( \lbrace ( \alpha,k)  :  \alpha \in \Phi^+,0 \leq k < \langle \lambda , \alpha^\vee \rangle \rbrace \). This will provide a \( \lambda \)-chain as long as the second condition holds.  Let \( \alpha^\vee = \sum\limits_p a_p \alpha_p^\vee \) ,  \( \beta^\vee = \sum\limits_p b_p \alpha_p^\vee \) . Then we let:

\( (\alpha , k) < (\beta , l) \) if and only if

\( \frac{1}{\langle \lambda , \alpha^\vee \rangle} (k,a_1,...,a_r) < \frac{1}{\langle \lambda , \beta^\vee \rangle} (l,b_1,...,b_r)  \)

in lexicographical order.

The proof that this gives a \( \lambda \)-chain, isn't particularly enlightening, but can be found in the paper.

Folding Chains (of roots)

Next we build a set of foldings recursively by first creating a basis element using the \( \lambda \)-chain, and then defining folding operators which generate the rest of the foldings.

\( \Gamma ( \varnothing ) = ( \lbrace (\beta_i , 1) \rbrace_{i \in I} , \rho ) \)

Where \( \rho \) is the dominant weight that pairs to one with all simple roots. More generally, a folding will take the following form:

\( \Gamma = ( \lbrace ( \gamma_i, \epsilon_i ) \rbrace_{i \in I} , \gamma_\infty ) \)

Where \(  \gamma_i \) are (possibly negative) roots, \( \epsilon_i \in \lbrace \pm 1 \rbrace \), and \(  \gamma_\infty \)  is just some weight (in the orbit of \( \rho \) under the Weyl group action) .

Our folding operators are as follows:

\( \phi_i ( \Gamma ) = ( \lbrace \delta_j , \zeta_j ) \rbrace_{j \in I} , t_i(\gamma_\infty )) \)

Where: \( t_i = s_{\gamma_i} \), the reflection about the hyperplane perpendicular to \( \gamma_i \) , and

\( \delta_j = \gamma_j \) if  \( j \leq i \), \( \delta_j = t_i(\gamma_j) \) if  \( j > i \),

\( \zeta_j = \epsilon_j \) if \( j \neq i \) , \( \zeta_j = - \epsilon_j \) if \( j = i \) .

Proposition: \( \phi_i , \phi_j \) commute for all \( i \) , \( j \)

and so in particular, each folding is uniquely determined by which \( \epsilon_i \) are negative, and so is in bijection with finite subsets of \( I \) . For our arbitrary \( \Gamma \) this subset will be \( J = \lbrace j_1, ... , j_k \rbrace \). We will often use  \( \Gamma \) and \( J \) interchangeably.

Since this is supposed to give us a crystal, we want to define the parts that make it a crystal. First we will define the weight, but to do this we need to define the level sequence.

Definition: The levels of a folding are: \( l_i = -\delta_{\gamma < 0} + \sum\limits_{j<i, \epsilon_j=1,\gamma_j = \pm \gamma_i} sgn(\gamma_j) \). We call the level sequence of folding is \( (l_i)_{i \in I} \) . In particular, for we write \( (l_i^\varnothing)_{i \in I} \) for the level sequence of \( \Gamma(\varnothing) \),

Definition:  \( \mu(\Gamma(J)) = \hat{r_{j_1}} \cdot \cdot \cdot \hat{r_{j_k}} \) which we call the Weight of \( \Gamma \). Where \( r_i = s_{\beta_i} \) , \( \hat{r_i} \) is the reflection about \( \langle \lambda , \beta_i \rangle = l^\varnothing_i \) . We also define \( \kappa(\Gamma) = r_{j_1} \cdot \cdot \cdot r_{j_k} \).

Finally we define some particular subsets of interest, of both \( I \), and the level sequences. 

Definition: \( I_\alpha = \lbrace i \in I : \gamma_i = \pm \alpha \rbrace \subseteq I \) , \( \hat{I_\alpha} = I_\alpha \cup \lbrace \infty \rbrace \), 

Definition: \( L_\alpha = \lbrace l_i \in I : i \in I_\alpha \rbrace \) , \( \hat{I_\alpha} =L_\alpha \cup \lbrace l_\alpha^\infty = \langle \mu(\gamma), \alpha^\vee \rangle \rbrace \), 

Admissible Foldings

We want to restrict to a subset of all possible foldings, which will give us our crystal. 

Definition: If  \( (1) < r_{j_1} < r_{j_1}r_{j_2}<...<r_r_{j_1}r_{j_2} \cdot \cdot \cdot r_{j_k} = \kappa(J) \) are in fact coverings in the Bruhat order, we call \( \Gamma(J) \) (the folding associated the subset \( J \) ), an admissible folding.

Now we will define the rest of the crystal.

Definition: \( \epsilon_p = M= \) maximum value in \( \hat{L_\alpha} \) . ( \( \phi_p = \langle \mu(J) , \alpha_p^\vee \rangle - M \) )

Definition: \(m= \) minimum index that gives \(l_m = M \) . Let k be the predecessor of m in \( \hat{I_\alpha} \) .

Proposition: \( l_k=l_m-1 \)

Proposition: Such a k always exists.

Definition: \( f_p(J) = \phi_k \phi_m (J) \). We define \( e_p \) similarly.

Proposition: \( \mu(F_p(J)) = \mu(J) -\alpha_p \)

Proposition: \( e_p\), \(f_p \) send admissible folds to admissible folds.

Finally, here is the main theorem:

Theorem: The collection of admissible subsets along with the above defined root operators form a (semi-perfect) crystal.

Works Cited

All information presented on this page is paraphrased or directly taken from the following paper:

A Combinatorial Model for Crystals of Kac-Moody Algebras​, Trans. of AMS, pg  4349-4381, August 2008

Contributors

  • Colin Hagemeyer (UC Davis)