8 Piecewise Linear Parametrization of Canonical Bases

In [L1] the author introduced the canonical basis for the plus part of a quantized enveloping algebra of type A,D or E. (The same method applies for nonsimplylaced types, see [L3, 12.1].) Another approach to the canonical basis was later found in [Ka]. In [L1] we have also found that the set parametrizing the canonical basis has a natural piecewise linear structure that is, a collection of bijections with N such that any two of these bijections differ by composition with a piecewise linear automorphism of N (an automorphism which can be expressed purely in terms of operations of the form a+ b, a− b,min(a, b)). This led to the first purely combinatorial formula (involving only counting) for the dimension of a weight space of an irreducible finite dimensional representation [L1] or the dimension of the space of coinvariants in a triple tensor product [L2, 6.5(f)]. (Later, different formulas in the same spirit were obtained by Littelman.) The construction of an analogous piecewise linear structure for the canonical basis in the nonsimplylaced case (based on a reduction to the simplylaced case) was only sketched in [L3] partly because it involved an assertion whose proof only appeared later (in [L4, 14.4.9]): as Berenstein and Zelevinsky write in [BZ, Proof of Theorem 5.2], ”Lusztig (implicitly) claims that the transition map R 2121 for B2 is obtained from the transition map R 132132 for type A3...”. In this paper we fill the gap in [L3] by making use of [L4, 14.4.9] which gives a relation between the canonical basis for a nonsimplylaced type and the canonical basis for a simplylaced type with a given (admissible) automorphism. At the same time we slightly extend [L4, 14.4.9] by allowing type A2n with its non-admissible involution. As an application we show that the canonical basis has a natural monoid structure and we define certain ”Frobenius” endomorphisms of this monoid.