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On the Heights of Group Characters
Abstract
For a finite p-soluble group G we derive a bound on the heights of the irreducible complex characters of G lying in a p-block B. This bound depends on the prime p and the exponent d of a defect group of B. We show by examples that this bound is of the right order of magnitude.
... This result is motivated by the case of blocks of finite groups of Lie type in characteristic p, where the hypotheses are satisfied, despite the presence of Qd(p). We remark that for p-solvable groups, it is known from the results of Haggarty [8] that the height of an irreducible character in a block of defect d is at most 3d−4 4 , a bound which can be (and, in [8], is) substantially improved for odd p. While it is an open question whether all p-blocks B of finite groups of Lie type in characteristic p satisfy OWC, it is of interest to note that it follows from a result of M. Geck [6] that for " good " characteristics p—in particular, for primes p > 5 in all types—defects and heights of irreducible characters of p-blocks of finite groups of Lie type in characteristic p do satisfy the bound given in Theorem 2. Geck's result verified directly that (for " good " primes p) defects of irreducible characters in p-blocks of finite groups of Lie type in characteristic p are defects of irreducible characters of unipotent radicals of parabolics, as the present author had previously observed should follow (for all primes p) from OWC. ...
... This result is motivated by the case of blocks of finite groups of Lie type in characteristic p, where the hypotheses are satisfied, despite the presence of Qd(p). We remark that for p-solvable groups, it is known from the results of Haggarty [8] that the height of an irreducible character in a block of defect d is at most 3d−4 4 , a bound which can be (and, in [8], is) substantially improved for odd p. While it is an open question whether all p-blocks B of finite groups of Lie type in characteristic p satisfy OWC, it is of interest to note that it follows from a result of M. Geck [6] that for " good " characteristics p—in particular, for primes p > 5 in all types—defects and heights of irreducible characters of p-blocks of finite groups of Lie type in characteristic p do satisfy the bound given in Theorem 2. Geck's result verified directly that (for " good " primes p) defects of irreducible characters in p-blocks of finite groups of Lie type in characteristic p are defects of irreducible characters of unipotent radicals of parabolics, as the present author had previously observed should follow (for all primes p) from OWC. ...
... We first require a crude approximation, though more precise formulae are well known. We next remind the reader that (though we modify notation of [8] here somewhat), OWC predicts that if B is p-block of positive defect of the finite group G, then for each non-negative integer e, we should have (the empty chain is excluded in this formulation). We do not need to insist that the subgroup V 1 is non-trivial, since if V 1 is trivial, then (as B has positive defect), chains beginning with (V 1 , b 1 ) will make zero contribution anyway. ...
We show that if the Ordinary Weight Conjecture is correct, then large character heights for a p-block B (p an odd prime) necessitate the involvement of Qd(p) in the normalizers of Alperin–Goldschmidt B-subpairs.
... This result is motivated by the case of blocks of finite groups of Lie type in characteristic p, where the hypotheses are satisfied, despite the presence of Qd(p). We remark that for p-solvable groups, it is known from the results of Haggarty [8] that the height of an irreducible character in a block of defect d is at most 3d−4 in Theorem 2. Geck's result verified directly that (for " good " primes p) defects of irreducible characters in p-blocks of finite groups of Lie type in characteristic p are defects of irreducible characters of unipotent radicals of parabolics, as the present author had previously observed should follow (for all primes p) from OWC. ...
We propose upper bounds for the number of modular constituents of the restriction modulo p of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.
We propose upper bounds for the number of modular constituents of the restriction modulo p of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.
Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften
- B Huppert
B. Huppert, Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin and New York, 1967. MR 37 #302.