On tensor induction for representations of finite groups

AbstractIt is well known that if D is an irreducible complex representation of a finite group G, then every direct summand of the restriction of D to a subgroup H must have degree at least as large as the degree of D divided by the index |G:H|; moreover, D is induced from H if and only if the restriction does have a direct summand whose dimension is equal to this quotient. This paper explores the possibility of an analogous result for tensor induction, under the additional assumption that D is faithful, quasi-primitive and not a tensor product (of projective representations of degree greater than 1), and that the Fitting subgroup F(G) is not in the centre Z(G). The main question is this: if the restriction has a (projective) tensor factor whose degree is the |G:H|th root of the degree of D, does it follow that D is tensor induced from H? Among other results, examples are given to show that the answer can be negative when the index is 2. An affirmative answer is proved for normal subgroups of odd index, and also for arbitrary subgroups of odd prime index. As might be expected, the key lies in the study of F(G)/Z(G) as a symplectic module over a finite prime field; in particular, in exploring the connection between (ordinary) induction and form-induction of such modules

Plasmonic Nanostructure Design for Efficient Light Coupling into Solar Cells

We demonstrate that subwavelength scatterers can couple sunlight into guided modes in thin film Si and GaAs plasmonic solar cells whose back interface is coated with a corrugated metal film. Using numerical simulations, we find that incoupling of sunlight is remarkably insensitive to incident angle, and that the spectral features of the coupling efficiency originate from several different resonant phenomena. The incoupling cross section can be spectrally tuned and enhanced through modification of the scatterer shape, semiconductor film thickness, and materials choice. We demonstrate that, for example, a single 100 nm wide groove under a 200 nm Si thin film can enhance absorption by a factor of 2.5 over a 10 μm area for the portion of the solar spectrum near the Si band gap. These findings show promise for the design of ultrathin solar cells that exhibit enhanced absorption

Zeros of Brauer characters and linear actions of finite groups: Small primes

We describe the finite groups whose p-Brauer character table, for p = 2 or p = 3, does not contain any zero. This completes the analysis in [6], where we considered the case p 65 5

A note on finite groups with few values in a column of the character table

Many structural properties of a finite group G are encoded in the set of irreducible character degrees of G. This is the set of (distinct) values appearing in the "first" column of the character table of G. In the current article, we study groups whose character table has a "non-first" column satisfying one particular condition. Namely, we describe groups having a nonidentity element on which all nonlinear characters take the same value

Groups whose degree graph has three independent vertices

Let G be a finite group, and let cd(G) denote the set of degrees of the irreducible complex characters of G. This paper is a contribution to the study of the degree graph of G, that is, the prime graph built on the set cd(G). Namely, we characterize finite groups whose degree graph has three independent vertices (i.e., three vertices that are pairwise non-adjacent). Our result turns out to be a generalization of several previously-known theorems concerning the structure of the degree graph

Finite groups with real conjugacy classes of prime size

We determine the structure of a finite group G whose noncentral real conjugacy classes have prime size. In particular, we show that G is solvable and that the set of the sizes of its real classes is one of the following: {1},{1, 2}, {1, p}, or {1, 2, p}, where p is an odd prime

Nonvanishing elements for Brauer characters

Let G be a finite group and p a prime. We say that a p-regular element g of G is p-nonvanishing if no irreducible p-Brauer character of G takes the value 0 on g. The main result of this paper shows that if G is solvable and g is a p-regular element which is p-nonvanishing, then g lies in a normal subgroup of G whose p-length and p'-length are both at most 2 (with possible exceptions for p\leq 7), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for p>7) some results in Dolfi and Pacifici [\u2018Zeros of Brauer characters and linear actions of finite groups\u2019, J. Algebra 340 (2011), 104\u2013113]

On tensor factorisation for representations of finite groups

We prove that, given a quasi-primitive complex representation D for a finite group G, the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G. More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G

On the vanishing prime graph of finite groups

Let G be a finite group. An element g 08 G is called a vanishing element of G if there exists an irreducible complex character \u3c7 of G such that \u3c7(g) = 0. In this paper we study the vanishing prime graph \u393(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. Among other things we prove that, similarly to what holds for the prime graph of G, the graph \u393(G) has at most six connected components

On the orders of zeros of irreducible characters

Let G be a finite group and p a prime number. We say that an element g in G is a vanishing element of G if there exists an irreducible character χ of G such that χ(g)=0. The main result of this paper shows that, if G does not have any vanishing element of p-power order, then G has a normal Sylow p-subgroup. Also, we prove that this result is a generalization of some classical theorems in Character Theory of finite groups