Ordinary Irreducible Characters Research Articles

PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS

Abstract Let $\alpha $ be a complex-valued $2$ -cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $ -characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $ -regular are characterised by the values that the irreducible $\alpha $ -characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.

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Weights for ℓ-local compact groups

In this note, we initiate the study of F-weights for an ℓ-local compact group F over a discrete ℓ-toral group S with discrete torus T. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when T is the unique maximal abelian subgroup of S up to F-conjugacy and every element of S is F-fused into T, the number of weights of F is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of F with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when F is simple and |S:T|=ℓ. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.

Principal blocks with six ordinary irreducible characters

We classify Sylow p-subgroups of finite groups whose principal p-blocks have precisely six ordinary irreducible characters.

Navarro vertices and lifts in solvable groups

Let Q be a p-subgroup of a finite p-solvable group G, where p is a prime, and suppose that δ is a linear character of Q with the property that whenever are conjugate in G. In this situation, we show that restriction to p-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of G with Navarro vertex onto the set of irreducible Brauer characters of G with Green vertex Q. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups. Communicated by Mandi Schaeffer Fry

Degree divisibility in Alperin–McKay correspondences

Let p be a prime, B a p-block of a finite group G and b its Brauer correspondent. According to the Alperin–McKay Conjecture, there exists a bijection between the set of irreducible ordinary characters of height zero of B and those of b. In this paper, we show that whenever G is p-solvable such a bijection can be found, both for ordinary and Brauer characters, with the additional property of being compatible with divisibility of character degrees. In this case, we also show that the dimension of b divides the dimension of B.

On the Fischer matrices of a group of shape 21+2n + :G

In this paper, the Fischer matrices of the maximal subgroup G = 21+8+ : (U4(2):2) of U6(2):2 will be derived from the Fischer matrices of the quotient group Q = G/Z(21+8+) = 28 : (U4(2):2), where Z(21+8+) denotes the center of the extra-special 2-group 21+8+. Using this approach, the Fischer matrices and associated ordinary character table of G are computed in an elegantly simple manner. This approach can be used to compute the ordinary character table of any split extension group of the form 21+2n+ :G, n ∈ N, provided the ordinary irreducible characters of 21+2n+ extend to ordinary irreducible characters of its inertia subgroups in 21+2n+:G and also that the Fischer matrices M(gi) of the quotient group 21+2n+ :G/Z(21+2n+) = 22n:G are known for each class representative gi in G.

Height zero characters in principal blocks

We show that the principal p-block of a finite group of order divisible by p has at least 2p−1 height-zero characters. Along the way, we describe the p-local structure of finite groups whose principal p-blocks have at most five height-zero ordinary irreducible characters.

The blocks with four irreducible characters

AbstractSuppose that is a Brauer ‐block of a finite group with defect group . If exactly contains four ordinary irreducible characters, then we show that has order four or five, assuming the Alperin–McKay conjecture holds for .

Restrictions of irreducible characters of finite groups of Lie type to derived subgroups and regular semisimple elements

ABSTRACT In this note, we formulate an observation that “almost all” irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.

GALOIS AUTOMORPHISMS AND CLASSICAL GROUPS

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.

A 2-Compact Group as a Spets

In 1993, Broué, Malle and Michel initiated the study of spetses on the Greek island bearing the same name. These are mysterious objects attached to non-real Weyl groups. In algebraic topology, a p-compact group X is a space which is a homotopy-theoretic p-local analogue of a compact Lie group. A connected p-compact group X is determined by its root datum which in turn determines its Weyl group . In this article, we give strong numerical evidence for a connection between these two objects by considering the case when X is the exotic 2-compact group constructed by Dwyer–Wilkerson and is the complex reflection group . Inspired by results in Deligne–Lusztig theory for classical groups, if q is an odd prime power, then we propose a set of “ordinary irreducible characters” associated to the space of homotopy fixed points under the unstable Adams operation ψq. Notably, includes the set of unipotent characters associated to G24 constructed by Broué, Malle and Michel from the Hecke algebra of G24 using the theory of spetses. By regarding as the classifying space of a Benson–Solomon fusion system we formulate and prove an analogue of Robinson’s ordinary weight conjecture that the number of characters of defect d in can be counted locally.

Principal blocks with 5 irreducible characters

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

The principal p‐blocks with four irreducible characters

For a prime p, we determine a Sylow p-subgroup D of a finite group G such that the principal p-block B of G has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov 30 years ago. Our proof relies on the classification of finite simple groups.

Quasicharacters, recoupling calculus, and Hamiltonian lattice quantum gauge theory

We study the algebra R of G-invariant representative functions over the N-fold Cartesian product of copies of a compact Lie group G modulo the action of conjugation by the diagonal subgroup. Using the representation theory of G on the Hilbert space H=L2(GN)G, we construct a subset of G-invariant representative functions, which, by standard theorems, span H and thus generate R. The elements of this basis will be referred to as quasicharacters. For N = 1, they coincide with the ordinary irreducible group characters of G. The form of the quasicharacters depends on the choice of a certain unitary G-representation isomorphism, or reduction scheme, for every isomorphism class of irreps of G. We determine the multiplication law of R in terms of the quasicharacters with structure constants. Next, we use the one-to-one correspondence between complete bracketing schemes for the reduction of multiple tensor products of G-representations and rooted binary trees. This provides a link to the recoupling theory for G-representations. Using these tools, we prove that the structure constants of the algebra R are given by a certain type of recoupling coefficients of G-representations. For these recouplings, we derive a reduction law in terms of a product over primitive elements of 9j symbol type. The latter may be further expressed in terms of sums over products of Clebsch–Gordan coefficients of G. For G = SU(2), everything boils down to combinatorics of angular momentum theory. In the final part, we show that the above calculus enables us to calculate the matrix elements of bi-invariant operators occurring in quantum lattice gauge theory. In particular, both the quantum Hamiltonian and the orbit type relations may be dealt with in this way, thus reducing both the construction of the costratification and the study of the spectral problem to numerical problems in linear algebra. We spell out the spectral problem for G = SU(2), and we present sample calculations of matrix elements of orbit type relations for the gauge groups SU(2) and SU(3). The methods developed in this paper may be useful in the study of virtually any quantum model with polynomial constraints related to some symmetry.

On the sum of character degrees coprime to p and p-nilpotency of finite groups

The well-known Thompson theorem on character degrees states that if a prime p divides the degree of every nonlinear irreducible character of a finite group G , then G is p -nilpotent. In this paper, we give a strengthened version of Thompson's theorem in terms of ∑ χ ∈ Irr p ′ ( G ) χ ( 1 ) and ∑ χ ∈ Irr p ′ ( G ) χ ( 1 ) 2 , where Irr p ′ ( G ) denotes the set of all ordinary irreducible characters of G of degree coprime to p .

On a two-fold cover 2.(2⁶˙G₂(2)) of a maximal subgroup of Rudvalis group Ru

The Schur multiplier M(Ḡ1) ≅4 of the maximal subgroup Ḡ1 = 2⁶˙G₂(2)of the Rudvalis sporadic simple group Ru is a cyclic group of order 4. Hence a full representative group R of the type R = 4.(2⁶˙G₂(2)) exists for Ḡ1. Furthermore, Ḡ1 will have four sets IrrProj(Ḡ1;αi) of irreducible projective characters, where the associated factor sets α1, α2, α3 and α4, have orders of 1, 2, 4 and 4, respectively. In this paper, we will deal with a 2-fold cover 2. Ḡ1 of Ḡ1 which can be treated as a non-split extension of the form Ḡ = 27˙G2(2). The ordinary character table of Ḡ will be computed using the technique of the so-called Fischer matrices. Routines written in the computer algebra system GAP will be presented to compute the conjugacy classes and Fischer matrices of Ḡ and as well as the sizes of the sets |IrrProj(Hi; αi)| associated with each inertia factor Hi. From the ordinary irreducible characters Irr(Ḡ) of Ḡ, the set IrrProj(Ḡ1; α2) of irreducible projective characters of Ḡ1 with factor set α2 such that α22= 1, can be obtained.

The average character degree and an improvement of the Itô-Michler theorem

The classical Itô-Michler theorem states that the degree of every ordinary irreducible character of a finite group G is coprime to a prime p if and only if the Sylow p-subgroups of G are abelian and normal. In an earlier paper [8], we used the notion of average character degree to prove an improvement of this theorem for the prime p=2. In this follow-up paper, we obtain a full improvement for all primes.

Robinson’s conjecture for classical groups

Abstract Robinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks of finite quasi-simple classical groups.

2-blocks with abelian hyperfocal subgroup of rank 3

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G, whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has an abelian hyperfocal subgroup of rank 3.