Minimum Polynomials and Lower Bounds for Eigenvalue Multiplicities of Prime-Power Order Elements in Representations of Classical Groups

Recognition of simple groups U 3(Q) by element orders

“…(i) The fact that 2 t+2 ∈ ω(V : U ) follows from [12,13]; but we can also verify it directly. Suppose g ∈ V : U has order 2 t+2 .…”

On Eigenvalues of Group Elements in Representations of Simple Algebraic Groups and Finite Chevalley Groups

“…This is shown to be generically true with few exceptions; the exceptional pairs (g, G) have been determined in [53] and a full list of the exceptional triples (g, G, φ) has been provided in [39] (see also [54] and [46]). This work has been continued for elements g of p-power order in [55] for SL(n, q) and for other classical groups in [7]. In fact, the results have been obtained for representations over fields of any characteristic different from p, and the classification of all cases has been completed in [16].…”

“…This program outlined first in [47] is not yet completed, but the results available show that main difficulties are over. The case where G is classical and g belongs to a parabolic subgroup has been settled in [6]. In [50] and [52] I consider the case where g is of prime order.…”

“…In [50] and [52] I consider the case where g is of prime order. Further results are available in [7] and [42] where g is assumed to be of prime power order in a classical group G.…”

“…An important case where G is classical and g belongs to a parabolic subgroup has been settled in [6] for semisimple elements and in [7] for unipotent elements. If g does not belong to a parabolic subgroup, one cannot get help from the representation theory of groups with a cyclic Sylow p-subgroup unless p = .…”