Conjectures on character degrees for the simple Thompson group.
Over the past 20 years or so, a whole collection of conjectures has been developed, relating the character degrees of finite groups to the character degrees of various \(p\)-local subgroups. The simplest version, due to McKay, is that the number of characters of \(G\) of degree prime to \(p\) is equal to the same number for the Sylow \(p\)-normalizer. Alperin extended this to a block version, relating character degrees of blocks with defect group \(D\) to the character degrees of \(N_G(D)\). Further refinements due to Dade extend this further by considering the numbers of characters in each \(p\)-block whose degrees are divisible by any given power of \(p\). The Dade conjectures are expressed in terms of an alternating sum of these numbers, summed over the normalizers of the so-called `radical \(p\)-chains'. In a different direction, Isaacs and Navarro refine the Alperin-McKay conjecture by considering the congruences modulo \(p\) of the \(p'\)-parts of the character degrees. In this paper, Uno proposes a marriage of the two, known by other authors as Uno's conjecture. Essentially, Dade's conjecture is split into a sum of \((p-1)/2\) conjectures, one for each pair \(\{\pm k\}\) of non-zero values modulo \(p\). The bulk of the present paper is devoted to a verification of this new conjecture for the sporadic simple Thompson group.
scientific article; zbMATH DE number 2068624