The decomposition of the permutation character \(1^{\text{GL}(2n,q)}_{\text{GL}(n,q^2)}\). (English) Zbl 1031.20040
Let \(p\) be a prime and let \(q=p^a\), where \(a\) is a positive integer. Let \(\text{GL}_n(q)\) denote the general linear group of degree \(n\) over the field of \(q\) elements. The group \(H=\text{GL}_n(q^2)\) is a subgroup of \(G=\text{GL}_{2n}(q)\). N. F. J. Inglis, M. W. Liebeck and J. Saxl proved [in Math. Z. 192, 329-337 (1986; Zbl 0578.20006)] that the permutation character \(\pi\) of \(G\) acting on the cosets of \(H\) is multiplicity-free, meaning that each irreducible constituent of \(\pi\) has multiplicity one. This follows since each double coset \(HxH\) of \(H\) in \(G\) is equal to its inverse coset \(Hx^{-1}H\). The authors of the present paper identify the irreducible constituents of \(\pi\), using Green’s parametrization of the irreducible characters of \(G\). They show also that the number of irreducible constituents of \(\pi\) is \(\sum q^{l(\lambda)}\), the sum extending over all partitions \(\lambda\) of \(n\), where \(l(\lambda)\) equals the number of parts of \(\lambda\). The authors also note that A. Henderson has obtained their main theorem when \(q\) is odd [J. Algebra 261, No. 1, 102-144 (2003; Zbl 1020.20030)].
Reviewer: Roderick Gow (Dublin)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20G40 | Linear algebraic groups over finite fields |
| 20C33 | Representations of finite groups of Lie type |
Keywords:
general linear groups; multiplicity-free permutation characters; double cosets; irreducible constituents; irreducible charactersReferences:
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