Let G be a finite group, let N be a normal subgroup of G and let θ be an irreducible character of N. P.X. Gallagher showed that the number of irreducible characters of G lying θ equals the number of ...so-called θ-good conjugacy classes of Gθ.
Here we count the number of real irreducible characters of G lying over θ. To do so, we assign a new invariant, with value 0,+1 or −1, to each good conjugacy class. Then the desired number is the sum over these invariants.
We also compute the Frobenius-Schur indicator of the induced character θ↑G using a similar formula.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact ...groups with respect to the induction and restriction functors.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
We prove Clifford theoretic results which only hold in characteristic 2.
Let G be a finite group, let N be a normal subgroup of G and let φ be an irreducible 2-Brauer character of N. We show that φ ...occurs with odd multiplicity in the restriction of some self-dual irreducible Brauer character θ of G if and only if φ is G-conjugate to its dual. Moreover, if φ is self-dual then θ is unique and the multiplicity is 1.
Next suppose that θ is a self-dual irreducible 2-Brauer character of G which is not of quadratic type. We prove that the restriction of θ to N is a sum of distinct self-dual irreducible Brauer character of N, none of which have quadratic type. Moreover, G has no self-dual irreducible 2-Brauer character of non-quadratic type if and only if N and G/N satisfy the same property.
Finally, suppose that b is a real 2-block of N. We show that there is a unique real 2-block of G covering b which is weakly regular with respect to N.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider Noether's problem on the noncommutative rational functions invariant under a linear action of a finite group. For abelian groups the invariant skew-fields are always rational, for ...solvable group they are rational if the action is well-behaved - given by a so-called complete representation. We determine the groups that admit such representations and call them totally pseudo-unramified. We show that for a solvable group the invariant skew-field is finitely generated. Finally we study totally pseudo-unramified groups and classify totally pseudo-unramified p-groups of rank at most 5.
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Dade's Projective Conjecture is known to be true for finite p-solvable groups thanks to work of G.R. Robinson, but remains open in general. Work of Isaacs and Navarro suggested to Uno and Boltje ...refinements of this conjecture. These refinements were studied for finite p-solvable groups by Glesser. In the present paper, inspired by earlier work of Turull, we propose further refinements of the conjecture that take into account the Schur indices and the elements of the Brauer group. We prove that all these refinements of Dade's Projective Conjecture hold for all finite p-solvable groups. In particular, we obtain that the version of Dade's Projective Conjecture which involves character degree residues modulo p, fields of definition and Schur indices, as well as the full strength of Boltje's Conjecture both hold for all finite p-solvable groups. The proof develops a Clifford theory for normalizers of chains of p-subgroups which allows one to reduce the calculation of the relevant sums to simpler groups.
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In this paper it is shown that any irreducible representation of a Drinfeld double D(A) of a semisimple Hopf algebra A can be obtained as an induced representation from a certain Hopf subalgebra of ...D(A). This generalizes a well known result concerning the irreducible representations of Drinfeld doubles of finite groups 11. Using this description we also give a formula for the fusion rules of semisimple Drinfeld doubles. This shows that the Grothendieck rings of these Drinfeld doubles have a ring structure similar to the Grothendieck rings of Drinfeld doubles of finite groups.
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This paper concerns aspects of Clifford Theory of finite groups. In earlier papers, Turull proved that if two finite groups yielded the same element of the Brauer–Clifford group, then there was an ...endoisomorphism from one group to the other, and furthermore, that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other. The paper adapts the definition of Character Triple Isomorphism so that it involves ordinary and Brauer characters, and it preserves fields of definition, Schur indices, decomposition numbers, and blocks. It is proved that each endoisomorphism yields exactly one character triple isomorphism. Character triple isomorphisms can be composed, restricted, produced by direct sums, extension of fields, and these operations have their parallel for the endoisomorphisms. One goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications.
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We are interested in the McKay quiver Γ(
G
) and skew group rings
A
∗
G
, where
G
is a finite subgroup of GL(
V
), where
V
is a finite dimensional vector space over a field
K
, and
A
is a
K
−
G
...-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups
G
⊆
GL
(
V
)
and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups
G
(
r
,
p
,
n
). We first look at the case
G
(1,1,
n
), which is isomorphic to the symmetric group
S
n
, followed by
G
(
r
,1,
n
) for
r
> 1. Then, using Clifford theory, we can determine the McKay quiver for any
G
(
r
,
p
,
n
) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra
A
~
(
G
)
of a finite group
G
⊆
GL
(
V
)
, which is Morita equivalent to the skew group ring
A
∗
G
. This description gives us an embedding of the basic algebra Morita equivalent to
A
∗
G
into a matrix algebra over
A
.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this ...result to the situation of a Krull-Schmidt category on which a finite group acts as automorphisms. This then provides the orbit category introduced by Cibils and Marcos, and studied intensively by Keller in the context of cluster algebras, and by Asashiba in the context of Galois covering functors. We formulate and prove Clifford’s theorem for Krull-Schmidt orbit categories with respect to a finite group
Γ
of automorphisms, clarifying this way how the image of an indecomposable object in the original category decomposes in the orbit category. The pair of adjoint functors appears as the Kleisli category of the naturally appearing monad given by
Γ
.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We present a new criterion to predict if a character of a finite group extends. Let G be a finite group and p a prime. For N\vartriangleleft G, we consider p-blocks b and b' of N and \operatorname ...{N}_N(D), respectively, with (b')^N=b, where D is a defect group of b'. Under the assumption that G coincides with a normal subgroup Gb of G, which was introduced by Dade early in the 1970's, we give a character correspondence between the sets of all irreducible constituents of \phi ^G and those of (\phi ')^{\operatorname {N}_G(D)}, where \phi and \phi ' are irreducible Brauer characters in b and b', respectively. This implies a sort of generalization of the theorem of Harris-Knörr. An important tool is the existence of certain extensions that also help in checking the inductive Alperin-McKay and inductive Blockwise-Alperin-Weight conditions, due to the second author.
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