CORRIGENDA: “ON EXTENSIONS OF SIMPLE MODULES OVER SYMMETRIC AND ALGEBRAIC GROUPS”

A.S. KLESHCHEV AND J. SHETH

We correct some errors and an inconsistency appearing in [1].

1. We are grateful to Kai Meng Tan for pointing out that remark (ii) on page 706 does not make sense, as the partition λ = (1^{p}) is not p-regular. In fact, the remark should be applied to Theorem 2.9, not 2.10.

2. We are grateful to Karin Erdmann and Anne Henke for pointing out that calculations of cohomology in Theorem 3.5 and Corollary 3.6 contain errors. Below is a corrected version of these results. We note that Erdmann and Henke have recently obtained the same results for symmetric and general linear groups using different methods [2].

Theorem 3.5 (i) Assume G = GL_{n}(F), λ = (2^{u}, 1^{v−u}), μ = (2^{r}, 1^{s−r}), 0 ≤ u ≤ v ≤ n, 0 ≤ r ≤ s ≤ n, u ≥ r, and let v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) = 0, unless s−v = u−r = (p−a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p−1 or min(u, n − v) < p^{i+1}. In this case the Ext is 1-dimensional.

(ii) Assume G = SL_{n}(F), λ = ω_{u} + ω_{v}, μ = ω_{r} + ω_{s}, 0 ≤ u ≤ v ≤ n, 0 ≤ r ≤ s ≤ n, u ≥ r, (ω_{0}, ω_{n} are interpreted as 0), and let v −u+1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) = 0, unless s − v = u − r = (p − a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p − 1 or min(u, n − v) < p^{i+1}. In this case the Ext is 1-dimensional.

(iii) Assume G = Sp_{2n}(F), λ = ω_{u}, μ = ω_{r}, 0 ≤ r ≤ u ≤ n, (ω_{0} is interpreted as 0) and let n − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) = 0, unless u − r = 2(p − a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p − 1 or u < 2p^{i+1}. In this case the Ext is 1-dimensional.

(iv) Assume p > 2, G = Σ_{n}, and λ = (v, u), μ = (s, r) be partitions of n with 0 ≤ u ≤ v, 0 ≤ r ≤ s, and u ≥ r. Let v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then Ext_{G}^{1}(D^{λ}, D^{μ}) ≅ Ext_{G}^{1}(D^{μ}, D^{λ}) = 0, unless u − r = (p − a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p − 1 or u < p^{i+1}. In this case the Ext is 1-dimensional.

Proof. (i) By Proposition 3.1 and Theorem 3.3, Ext_{G}^{1}(L(λ), L(μ)) = 0 unless μ = ν_{I}(λ) for a minimal non-empty I in A_{λ}, in which case the Ext is 1-dimensional. To describe the minimal non-empty sets I in A_{λ}, recall that any I ∈ A_{λ} is a union of the form [i_{1}, i_{2}) ∪ [i_{3}, i_{4}) ∪ ⋯, where [i_{1}, i_{2}) clearly belongs to A_{λ}. So we may assume that I = [i, j), with a_{i} ≠ 0 and a_{j} ≠ p − 1. Now, if j = i + 1, then I is clearly minimal. Let j > i + 1. If there exists k ∈ (i, j) with a_{k} < p − 1, then δ_{[i,k)} ≤ δ_{I}, whence [i, k) ∈ A_{λ}, and I is not minimal. So we may assume that a_{k} = p − 1 for all k ∈ (i, j). In this case δ_{[i+1,j)} = p^{i+1}, and it follows that I is minimal if and only if p^{i+1} > min(u, n − v). This proves (i).

The proofs of (ii) and (iii) are similar, as is the proof of (iv), except that in (iv) we use Proposition 3.2 in place of Proposition 3.1 and Corollary 3.4 in place of Theorem 3.3. □

Corollary 3.6. (i) Let G = SL_{n}(F), λ = ω_{u}+ω_{v} for 0 ≤ u ≤ v ≤ n, and let v−u+1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then H^{1}(G, L(λ)) = 0, unless n − v = u = (p − a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p − 1 or u < p^{i+1}. In this case H^{1}(G, L(λ)) ≅ F.

(ii) Let G = Sp_{2n}(F), λ = ω_{u} for 0 ≤ u ≤ n, and let n−u+1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then H^{1}(G, L(λ)) = 0, unless u = 2(p − a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p − 1 or u < 2p^{i+1}. In this case H^{1}(G, L(λ)) ≅ F.

(iii) Let p > 2, G = Σ_{n}, λ = (v, u) be a partition of n, and v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then H^{1}(G, D^{λ}) = 0, unless u = (p − a_{i})p^{i} for some i such that a_{i} > 0 and either a_{i+1} < p − 1 or u < p^{i+1}. In this case H^{1}(G, D^{λ}) ≅ F.

References

1. A.S. Kleshchev and J. Sheth, On Extensions of simple modules over symmetric and algebraic groups, J. Algebra 221 (1999), 705–722.

2. K. Erdmann and A. Henke, preprint, Oxford University, 2000.

Department of Mathematics, University of Oregon, Eugene OR 97403, USA

   E-mail address: klesh@math.uoregon.edu,  sheth@slu.edu