Equivariant Embeddings of Algebraic Groups

k sets. (The first formula is obviously true and the second is valid because there are f(n − 1, k − 1)
partitions with k members where π consists of {n − 1} and k − 1 other subsets while all others have
the form of a partition of {1, 2, . . . , n−2} into k sets with n−1 then placed into any of these (which is
why we multiply f(n−1, k) by k).) Clearly, f(n, 0) = 0, f(n, 1) = 1, f(n, n−1) = 1 and f(n, k) = 0
whenever k ≥ n for all n. A little work shows f(2) = 1, f(3) = 2, f(4) = 5, f(5) = 15, f(6) = 52
and f(7) = 203.
We must observe that these Y π are not the only compactifications of B ⊂ SL n . For instance,
in the SL 2 case, P 2 is the only Y π . However, the blow-up of P 2 at the origin is another B-
compactification, which produces a distinct SL 2 -compactification, SL 2 × B Bl 0 P 2 .
Lemma 3. Let B be the Borel subgroup of upper triangular matrices in SL n . Let U be the unipotent
radical of B and let T be the maximal torus in B consisting of the diagonal matrices of SL n . Let
π be a partition of the set {2, 3, . . . , n} and let Y π be the associated B-embedding. Then the closure
of T in Y π , and thus in SL n × B Y π , is T ∼ = ∏ P ∈π P#P , where #P denotes the cardinality of the
set P .
Therefore, the closure of T in any SL n × B Y π only depends on the cardinalities of the elements
of π, and not specifically on the partition. As a result, only when n = 2 or 3 will the closure T
of T in SL n differentiate between non-isomorphic embeddings. When n = 4, all three embeddings
SL 4 × B Y {{2,3},{4}} , SL 4 × B Y {{2,4},{3}} and SL 4 × B Y {{2},{3,4}} have T ∼ = P 1 × P 2 . {{2, 5, 6}, {3, 4}}
have T = P 2 × P 3 and
Proposition 4. For all n ≥ 4, there are distinct partitions π 1 , π 2 of {2, . . . , n} whose associated
SL n -embeddings are non-isomorphic while their associated toric varieties T are isomorphic.
Proof. Let T be the maximal torus of SL n consisting of diagonal matrices in SL n .
Consider
the partitions π 1 = {{2, 3}, {4}, {5, . . . , n}} and π 2 = {{2}, {3, 4}, {5, . . . , n}} of {2, . . . , n}. The
corresponding B-embeddings are Y π1 = P 5 × P 4 × P 1 2 n(n+1)−10 and Y π2 = P 2 × P 7 × P 1 2 n(n+1)−10 ,
which are non-isomorphic. Observe that when n = 4, 1 n(n + 1) − 10 = 0, as there are only two
2
terms in Y π1 = P 5 × P 4 and in Y π2 = P 2 × P 7 . Hence the associated SL n -embeddings, SL n × B Y π1
and SL n × B Y π2 , are non-isomorphic as varieties and so also as SL n -embeddings.
In SL n × B Y π1 , if we identify SL n with its open orbit using the base point [I n , ([0 : 1 :
20