×

A new approach to Glauberman’s correspondence. (English) Zbl 1037.20004

This interesting paper generalizes the Glauberman character correspondence of finite group characters to twisted group algebras. Special cases of this result include the Glauberman correspondence and correspondences similar to those obtained by other authors.
Let \(E\) be a finite group with a normal subgroup \(G\) such that \(F=E/G\) is cyclic. Let \(\mathcal A\) be a twisted group algebra of \(E\) over a field \(\mathcal F\) of characteristic zero that splits \({\mathcal A}[H]\) for each \(H\leq E\). For any such \(H\leq E\), the set \(\text{Ch}({\mathcal A}[H])\) of all virtual \(\mathcal F\)-characters of \({\mathcal A}[H]\) is a free \(\mathbb{Z}\)-module with basis \(\text{Irr}({\mathcal A}[H])\). The dual \(\widehat F\) of all linear \(\mathcal F\)-characters of \(F=E/G\) permutes \(\text{Irr}({\mathcal A}[H])\) and so \(\text{Ch}({\mathcal A}[H])\) becomes a \(\mathbb{Z} F\)-module and \(\widehat F\) preserves the natural inner product \((\phi,\psi)_{{\mathcal A}[H]}\in\mathbb{Z}\) of characters \(\phi,\psi\in\text{Ch}({\mathcal A}[H])\). A \(\mathbb{Z}\widehat F\)-submodule \(\text{Ch}({\mathcal A}[E]| E_0)\) of \(\text{Ch}({\mathcal A}[E])\) is defined, let \(\mu=\min\{(\theta,\theta)_{{\mathcal A}[E]}\mid\theta\in\text{Ch}({\mathcal A}[E]| E_0)\}\) and set \(\min\text{Ch}({\mathcal A}[E]| E_0)=\{\theta\in\text{Ch}({\mathcal A}[E]| E_0)\mid(\theta,\theta)=\mu\}\). Clearly \(\min\text{Ch}({\mathcal A}[E]| E_0)\) is \(\widehat F\)-stable. Two characters \(\phi,\psi\in\min\text{Ch}({\mathcal A}[E]| E_0)\) are “connected” if \((\phi,\lambda\psi)_{{\mathcal A}[E]}\neq 0\) for some \(\lambda\in\widehat F\).
The fundamental result (Theorem 4.7) of this paper proves that this connectivity is an equivalence relation and that its equivalence classes biject with \(\text{Irr}^E({\mathcal A}[G])\), the \(E\)-stable characters of \(\text{Irr}({\mathcal A}[G])\). Hence any isomorphism preserving the relevant structures induces a bijection of characters. Sections 6-10 of this paper develop applications of these results that include the Glauberman correspondence, Kawanaka’s bijection, the cases \(G=\text{SL}(2,2^n)\) with odd integer \(n\geq 1\), the Suzuki groups case and relative correspondences.

MSC:

20C15 Ordinary representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C20 Modular representations and characters
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dade, E., Counting characters in blocks, II, J. Reine Angew. Math., 448, 97-190 (1994) · Zbl 0790.20020
[2] Dade, E., Another way to count characters, J. Reine Angew. Math., 510, 1-55 (1999) · Zbl 0918.20007
[3] E. Dade, Relative Glauberman correspondences, preprint, 2000; E. Dade, Relative Glauberman correspondences, preprint, 2000
[4] Glauberman, G., Correspondences of characters for relatively prime operator groups, Canad. J. Math., 20, 1465-1488 (1968) · Zbl 0167.02602
[5] Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.07201
[6] Isaacs, I. M., Character Theory of Finite Groups (1976), Academic Press: Academic Press New York · Zbl 0337.20005
[7] Kawanaka, N., On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan, 29, 425-450 (1977) · Zbl 0353.20031
[8] Kawanaka, N., Liftings of irreducible characters of finite classical groups II, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 30, 499-516 (1984) · Zbl 0536.20023
[9] Shintani, T., Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Japan, 28, 396-414 (1976) · Zbl 0323.20041
[10] Suzuki, M., On a class of doubly transitive groups, Ann. of Math., 75, 105-145 (1962) · Zbl 0106.24702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.