J an 2 00 2 Dual Standard Monomial Theoretic Basis and Canonical Basis for Type A

Let U − q (g) be the negative part of the quantized universal enveloping algebra constructed from a Cartan matrix associated to a complex semisimple Lie algebra g. Let λ be a dominant integral weight and V (λ) the irreducible U − q (g)-module with highest weight λ. There is on the one hand the canonical basis for U − q (g) [4, 12, 13] and on the other the standard monomial theoretic basis for the dual of V (λ) [7, 8, 11]. It is natural to ask if there is any relationship between these two bases. To quote Littelmann [11, page 552], ". .. the properties of the path basis suggest that the transformation matrix should be upper triangular. .. ". It is the purpose of this note to prove that such is indeed the case when the Cartan matrix is of type A. As to other types, we have nothing to say. Let us indicate a little more precisely what is proved here. We show first of all that the duals of the standard monomial theoretic bases for various V (λ) patch together to give what can be called a dual standard monomial theoretic basis for U − q (g). This basis lives in the crystal lattice and the image modulo q of a basis element is—as is perhaps to be expected—the corresponding standard tableau thought of as a crystal. The main result is that the transformation matrix between this basis and the canonical basis * Both authors were partially supported by DST under grant # MS/I–73/97. New address for P. 1 is unipotent upper triangular with respect to a natural partial order on the set of standard tableaux. And, finally, all this holds over the integral forms. These results are stated and proved in §5. The key to the results is the proposition proved in §4. In §2 we give a procedure to associate monomials to tableaux on which everything else is based. The combinatorial properties of this procedure are stated in the lemma of §3. These properties are crucial for the proofs. Finally, in §6, we compute explicitly the dual standard monomial theoretic basis for U − q (sl 3). We assume throughout that g = sl n (C). We set ℓ := n − 1 and denote by α i the simple root ǫ i − ǫ i+1. The terminology and notation of [3] are …