It is also closed in Proj([F.sub.1][[[X.sub.i]].sub.i[member of]V]), and the corresponding closed subscheme is the projective space P "without the edge [mu]"; the coordinate ring is [F.sub.1][[[X.sub.i]].sub.i[member of]V]/[I.sub.[mu]] (where ([X.sub.j][X.sub.l]) =: [I.sub.[mu]]) and its scheme is the Proj-scheme defined by this ring.

In this presentation, an edge corresponds to a relation, and we construct a coordinate ring for [THETA]([GAMMA]) = S([GAMMA]) by deleting all relations of the ambient space P([GAMMA]) which are defined by edges in the complement of [GAMMA].

The [S.sub.n+1]-module we construct will be a subspace of the coordinate ring of the reflection representation of type [A.sub.n] and will inherit the polynomial grading of this coordinate ring.

In fact, the space [V.sub.n] sits inside the copy of the coordinate ring of the reflection representation of type [A.sub.n] sitting inside C[[x.sub.1], ..., [x.sub.n+1]] generated by [x.sub.i] - [x.sub.i+1] for 1 [less than or equal to] i [less than or equal to] n.

Let [Mathematical Expression Omitted], where the vector ([m.sub.j]) ranges in [N.sup.r], be the multi-graded

coordinate ring of G/B with respect to [L.sub.1],..., [L.sub.r].

(i) The

coordinate ring R of G/B with respect to ([L.sub.1], .

Previous approaches to these problems for constructing (quantum) cluster algebra structures on (quantum)

coordinate rings arising in Lie theory were done on a case-by-case basis, relying on the combinatorics of each concrete family, they say, and these findings will make that unnecessary.

1078 174184 Free resolutions of

coordinate rings of projective varieties and related topics (Kyoto 1998).

Among cluster algebras are

coordinate rings of many algebraic varieties that play a prominent rule in representation theory, invariant theory, the study of total positivity, and other areas, but the discussion here is limited to the relations of cluster algebra theory to Poisson geometry and the theory of integrable systems.