Restriction of Odd Degree Characters and Natural Correspondences

Abstract

Abstract. Let q be an odd prime power, n > 1, and let P denote a maximal parabolic subgroup of GLn(q) with Levi subgroup GLn−1(q) × GL1(q). We restrict the odd-degree irreducible characters of GLn(q) to P to discover a natural correspondence of characters, both for GLn(q) and SLn(q). A similar result is established for certain finite groups with self-normalizing Sylow p-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of Sn and those of M , where M is any maximal subgroup of Sn of odd index; as well as between the odd-degree irreducible characters of G = GLn(q) or GUn(q) with q odd and those of NG(P ), where P is a Sylow 2-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that the fields of values of character correspondents are the same. We use this to answer some questions of R. Gow.

Characters of odd degree of symmetric groups

“…In [4] it is shown that the process described above defines a bijection α n between Irr 2 (S n ) and H := H(2 n1 ) × · · · × H(2 nt ). The description of the Sylow 2-subgroup structure of S n given in Section 2.3, together with Theorem 1.1 implies the existence of a bijection β n between H and Irr 2 (P n ).…”

“…Let n = 6 and let P 6 be a Sylow 2-subgroup of S 6 . Then (4), and K 6 = {(6), (5, 1), (4, 2), (3 2 ), (2 3 ), (2 2 , 1 2 ), (2, 1 4 ), (1 6 )} ⊆ P (6).…”

“…For n = 6 the bijection γ 6 constructed in [4] and described above is as follows: 4), (2)) , χ (5,1) −→ ψ ((4),(1 2 )) , χ (4,2) −→ ψ ((3,1),(1 2 )) ,…”

“…We also remark that, building on the bijection described in Theorem for the basic case , it is possible to construct an explicit canonical bijection between and , where is any maximal subgroup of of odd index. This is done in [, Theorem D]. Similarly, the result obtained in Theorem is fundamental to determine a natural correspondence between and , where , is an odd prime power and is a Sylow 2‐subgroup of .…”

“…Similarly, the result obtained in Theorem is fundamental to determine a natural correspondence between and , where , is an odd prime power and is a Sylow 2‐subgroup of . This is done in [, Theorem E].…”