Let N (d, n) be the variety of all d-tuples of commuting nilpotent n × n matrices. It is well-known that N (d, n) is irreducible if d = 2, if n ≤ 3 or if d = 3 and n = 4. On the other hand N (3, n) is known to be reducible for n ≥ 13. We study in this paper the reducibility of N (d, n) for various values of d and n. In particular, we prove that N (d, n) is reducible for all d, n ≥ 4. In the case d = 3, we show that it is irreducible for n ≤ 6.
Abstract. Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p and let g be the Lie algebra of G. It is well known that for p large enough the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of elements in the nilpotent cone of g [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, 10 (1997), 693-728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties Cr(gl 2 ), Cr(sl2) and Cr(N ) where N is the nilpotent cone of sl2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of r-tuples. Furthermore, in the case when g = sl2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of sl3, we are able to verify the aforementioned properties for Cr(u). Finally, applying our calculations on the commuting variety Cr(O sub ) where O sub is the closure of the subregular orbit in sl3, we prove that the nilpotent commuting variety Cr(N ) has singularities of codimension ≥ 2.
Abstract. Let SL 2 be the rank one simple algebraic group defined over an algebraically closed field k of characteristic p > 0. The paper presents a new method for computing the dimension of the cohomology spaces H n (SL 2 , V (m)) for Weyl SL 2 -modules V (m). We provide a closed formula for dim H n (SL 2 , V (m)) when n ≤ 2p − 3 and show that this dimension is bounded by the (n + 1)-th Fibonacci number. This formula is then used to compute dim H n (SL 2 , V (m)) for n = 1, 2, or 3. For n > 2p − 3, an exponential bound, only depending on n, is obtained for dim H n (SL 2 , V (m)). Analogous results are also established for the extension spaces Ext n SL2 (V (m 2 ), V (m 1 )) between Weyl modules V (m 1 ) and V (m 2 ). In particular, we determine the degree three extensions for all Weyl modules of SL 2 . As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of SL 2 , and the finite group of Lie type SL 2 (p s ).
The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent r-tuples in a classical Lie algebra g defined over an algebraically closed field k. As applications, we obtain some new results on the structure of the (even) cohomology ring of Frobenius kernels Gr for each r ≥ 1, where G is the simply connected, simple algebraic group such that Lie(G) = g. Explicit calculations for rank two groups are also presented.
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