Representations of SL 2 and the distribution of points in ℙ n

Sagbi bases of Cox–Nagata rings

“…The formulas in Examples 1.1 and 1.2 are well-known. They can be found, for instance, in the article on our problem by Kuttler and Wallach [20]. The next example illustrates the new type of formulas to be derived in this paper.…”

“…This induces an isomorphism of the blow-up X G which replaces the divisor class r • L + n j=1 (r − u j )E j by the divisor class r ′ • L + n j=1 (r ′ − u ′ j )E j . An alternative proof using representation theory of SL 2 (K) was given by Kuttler and Wallach [20]. The reader may find it instructive to verify (8) in Examples 1.1, 1.2 and 1.3.…”

“…Let us first consider the case i = j = 0. A monomial in the algebra generated by (20) has degree (k, u) = (k, 1, . .…”

“…We now illustrate formula (20) The Hilbert function of Z G is the following specialization of ψ : N 7 → N:…”

“…(20) This is seen either from the matrix (17), whose diagonal entries multiply to (20), or from the expansion (19). Let T be the caterpillar tree whose splits are splits(T ) = {{A, B} : max(A) < min(B)}.…”