The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Weyl group of given type. The $\mu$-coefficient of the Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is the coefficient of $q^{\frac{l(w)-l(u)-1}{2}}$ in $P_{u,w}(q)$.

The largest coefficient in a Kazhdan Lusztig polynomial of the Weyl group of given type.

The minimal size of a base for the Weyl group of the Cartan type. A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter. Any base has at least $\log |G|/n$ elements, where $n$ is the degree of the group, i.e., the size of its domain.

The rank of the Cartan type. The rank of a Cartan type $X_n$ is equal to the rank of the corresponding Cartan matrix.

The maximal dimension of an irreducible representation of the Weyl group of a finite Cartan type.

The number of distinct dimensions of the irreducible representations of the Weyl group of a finite Cartan type.

The maximum down-degree of the Hasse diagram of the strong Bruhat order in the Weyl group of the Cartan type.

The number of commutative positive roots in the root system of the given finite Cartan type. An upper ideal $I$ in the root poset $\Phi^+$ is called '''abelian''' if $\alpha,\beta \in I$ implies that $\alpha+\beta \notin \Phi^+$. A positive root is called '''commutative''' if the upper ideal it generates is abelian. The numbers are then given in [1, Theorem 4.4].

The exponent of the Weyl group of given type. This is the least common multiple of the orders of the elements of the group. In a comment to [4], see also [5], it is asked whether this is the same as the least common multiple of the degrees of the Weyl group.

The number of maximal elements of a poset.