Abstract

Adapted algebras and standard monomials

Peter Littelmann (Wuppertal)

Let G be a semisimple complex algebraic group. For a dominant weight λ let V (λ) be the corresponding module, let Q be the associated parabolic subgroup and consider the embedding G/Q ↪ ℙ(V (λ)). A “canonical” basis of V (λ)^{∗} is B^{∗}(λ), the dual of the canonical basis B(λ) of V (λ). The aim of standard monomial theory is to give a presentation of the homogenous coordinate ring R of the variety G/Q ↪ ℙ(V (λ)), which is compatible with natural subvarieties like Schubert varieties, opposite Schubert varieties, intersections of these etc. B^{∗}(λ) looks like a perfect candidate as a starting basis for a standard monomial theory, but the multiplicative structure of the dual canonical basis is not at all understood. On the other hand, the construction in [2] provides a basis for R, and it is a natural question to ask for the relationship between these two bases.

An adapted algebra [1] is a maximal subalgebra of ℂ_{q}[U^{−}] which is spanned as a vector space by a subset S of the dual canonical basis B^{∗}, and all elements of S are multiplicative, i.e., their product is again an element of the dual canonical basis. The main result of [3] can be formulated as follows: In the (specialization of the) q-commutative parts of B^{∗} given by an adapted algebra, the dual canonical basis and the standard monomial basis [2] coincide.

References

[1] P. Caldero: Adapted algebras for the Berenstein–Zelevinsky conjecture, preprint arXiv:math.RT/0104165.

[2] P. Littelmann: Contracting modules and standard monomial theory. JAMS 11, 551–567 (1998)

[3] P. Caldero, P. Littelmann: Adapted algebras and standard monomials, preprint math.RT/0201026 (2002).