The essential dimension of the normalizer of a maximal torus in the projective linear group

Abstract

Let p be a prime, k be a field of characteristic = p containing a primitive pth root of unity and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension ed k (N ; p) of N at p for every n ≥ 1.

Essential p-dimension of the normalizer of a maximal torus

“…This work extends the results of [MR09], where ed(N ; p) was computed for G = P GL n , which appears in Table I as the cases 1 A +1 p r −1 and 1 A +1 p r s−1 (using notation of 1.2).…”

“…This is the same choice as in [MR09,6 and 7]. For all other cases, by Corollory 2.15 we have |Λ| ≥ p 2r−1 + p k .…”

“…In this case, we know from [MR09,Lemma 3.3] 2 that Λ satisfies (K H ) iff V Λ is generically free. By (ii), the general case now follows.…”

“…is given by an iterated wreath product of cyclic p-groups, as in [MR09]. The H r -action on R p r naturally partitions the p r coordinates into blocks of size p i , for 0 ≤ i ≤ r. We will call these p i -blocks, and we may consider them as elements of R p i ; for each i the number of p i -blocks is p r−i .…”

“…Then one can ask what is the essential p-dimension of the normalizer of a split maximal torus in G, ed(N ; p)? This question was answered in [MR09] for G = P GL n . Their main motivation came from the upper bound ed(G; p) ≤ ed(N ; p).…”