Irreducible Constituents Research Articles

On the minimal faithful degree of Rhodes semisimple semigroups

In this paper we compute the minimal degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all inverse semigroups, and hence our results generalize earlier results of Easdown and Schein on the minimal faithful degree of an inverse semigroup. It also includes well-studied monoids like full matrix monoids over finite fields and the monoid of binary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permutation representations of maximal subgroups that are faithful when restricted to distinguished normal subgroups. This is analogous to (and inspired by) recent results of the second author on the minimal number of irreducible constituents in a faithful completely reducible complex matrix representation of a finite semigroup that admits one. To illustrate what happens when a finite semigroup does not admit a faithful completely reducible representation, we compute the minimal faithful degree of the opposite monoid of the full transformation monoid.

Sylow Branching Coefficients and Hook Partitions

We give a description of the irreducible constituents of the restriction to Sylow 2-subgroups of irreducible characters of symmetric groups labelled by hook partitions.

Non-linear Sylow branching coefficients for symmetric groups

We study the restriction to Sylow subgroups of irreducible characters of symmetric groups. In particular, we give a precise description of the degrees of the irreducible constituents in terms of the shape of the partition that labels a given irreducible character. Our main result is a wide generalization of [Giannelli and Navarro (Proc Am Math Soc 146(5):1963–1976, 2018), Theorem 3.1].

On the Formal Degree Conjecture for Non-Singular Supercuspidal Representations

Abstract We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein’s work [16] proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein’s work is that in non-singular case, the Deligne–Lusztig representations can be reducible, and the $S$-groups are not necessarily abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne–Lusztig representations and the dimensions of irreducible representations of $S$-groups.

A question of Katz and Tiep on representations of finite general unitary groups

Katz and Tiep proved a characterization of the total Weil character of finite special unitary groups for any integer and any prime power q other than in terms of the character values and irreducible constituents. They asked whether a similar characterization can be done for finite general unitary groups In this article, we prove that such characterization for can be done up to endomorphisms and tensoring by real linear characters.

A conjecture of Cameron and Kiyota on sharp characters with prescribed values

Let χ be a virtual (generalized) character of a finite group G and be the image of χ on The pair is said to be sharp of type L or L-sharp if If the principal character of G is not an irreducible constituent of χ, the pair is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in 1988. This conjecture states that if is L-sharp and then the inner product is uniquely determined by L. We then prove that this conjecture is true in the case that is normalized, χ is a character of G, and L contains at least an irrational value.

Derived functor modules as irreducible constituents of degenerate principal series of the maximal Gelfand-Kirillov dimension

Derived functor modules as irreducible constituents of degenerate principal series of the maximal Gelfand-Kirillov dimension

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

AbstractFor a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

R-groups for unitary principal series of Spin groups

We study the unitary principal series of the split group Spinm(F), where F is a p-adic field. Let χ˜ be a unitary character of a maximal F-split torus T˜ of GSpinm(F), and let χ be its restriction to T=T˜∩Spinm(F). The R-groups Rχ˜ and Rχ of the corresponding principal series representations fit in the exact sequence 0→Rχ˜→Rχ→Rχ/Rχ˜→0. We give a complete answer to the question of splitting of this exact sequence. We also prove that the multiplicity is one when irreducible constituents in unitary principal series are restricted from GSpin(F) to Spin(F). Further, based on Keys' result, we prove Arthur's conjecture on R-groups for unitary principal series of Spin.

Some Structural Properties of Semidirect Sums of \(\mathfrak{so}(3)\) and Abelian Lie Algebras

Various structural properties of semidirect sums of the rotation Lie algebra of rank one and an Abelian algebra described in terms of real representations with at most two irreducible constituents are obtained. The stability properties of these semidirect sums are studied by means of the cohomological and the Jacobi scheme methods.

Local partial covering subgroups in finite groups

Let G be a finite group. Roughly speaking, a subgroup A of G is called a local partial covering subgroup if AG=AB for a suitable maximal G-invariant subgroup B of AG. Our main result is as follows: Let pd be a prime power with p≤pd≤|G|p, assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd=2 and a Sylow 2-subgroup of G is nonabelian, of G are local partial covering subgroups. Then G/Op′p(G) is p-supersolvable; furthermore if G is not p-supersolvable, then Op′p(G)/Φ is a homogeneous Fp[G]-module where Φ/Op′(G)=Φ(G/Op′(G)), and each irreducible constituent has dimension dividing d−logp⁡|Φ|p.

Geometry and topology of symmetric point arrangements

We investigate point arrangements vi∈Rd,i∈{1,...,n} with certain prescribed symmetries. The arrangement space of v is the column span of the matrix in which the vi are the rows. We characterize properties of v in terms of the arrangement space, e.g. we characterize whether an arrangement possesses certain symmetries or whether it can be continuously deformed into another arrangement while preserving symmetry in the process. We show that whether a symmetric arrangement can be continuously deformed into its mirror image depends non-trivially on several factors, e.g. the decomposition of its representation into irreducible constituents, and whether we are in even or odd dimensions.

𝒫-characters and the structure of finite solvable groups

Abstract Let G be a finite group. An irreducible character of G is called a 𝒫 {\mathcal{P}} -character if it is an irreducible constituent of ( 1 H ) G {(1_{H})^{G}} for some maximal subgroup H of G. In this paper, we obtain some conditions for a solvable group G to be p-nilpotent or p-closed in terms of 𝒫 {\mathcal{P}} -characters.

Automorphy lifting for residually reducible -adic Galois representations, II

We revisit the paper [Automorphy lifting for residually reducible$l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.

P-Blocks Relative to a Character of a Normal Subgroup II

Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let $$\theta $$ be a G-invariant irreducible character of N. In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set $$\mathrm{Irr}(G|\theta )$$ of irreducible constituents of the induced character $$\theta ^G$$ , relative to the prime p. We call the elements of this partition the $$\theta $$ -blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on $$\{ x \in G \, |\, x_p \in N\}$$ , which supersedes previous non-canonical constructions. This allows us to define $$\theta $$ -decomposition numbers in a natural way. We also prove that the elements of the partition of $${\text {Irr}}(G|\theta )$$ established by these $$\theta $$ -decomposition numbers are the $$\theta $$ -blocks.

Sylow branching coefficients for symmetric groups

Let $p\ge 5$ be a prime and let $n$ be a natural number. In this article we describe the irreducible constituents of the induced characters $\phi\big\uparrow^{\mathfrak{S}_n}$ for arbitrary linear characters $\phi$ of a Sylow $p$-subgroup of the symmetric group $\mathfrak{S}_n$, generalising earlier results of the authors. By doing so, we introduce Sylow branching coefficients for symmetric groups.

Hall subgroups and fully extendible characters

We say that an irreducible character ξ of a subgroup H of G is fully extendible to G if every irreducible constituent of the induced character ξG is actually an extension of ξ. Now let N◃G, where G/N is π-separable for some set π of primes, and let H/N be a Hall π-subgroup of G/N. If ξ∈Irr(H) is fully extendible to G, we show that a Hall π′-subgroup K/N of G/N must be abelian. Also, if λ is an irreducible constituent of ξN, we show that λ is K-invariant for some Hall π′-subgroup K/N of G/N, and furthermore, λ is fully extendible to K. Conversely, if λ∈Irr(N) is extendible to K, where K/N is a Hall π′ subgroup of G/N, we show that some irreducible character ξ of H lying over λ is extendible to G.

Character Table Groups and Extracted Simple and Cyclic Polygroups

Let G be a finite group and \hat{G} be the set of all irreducible complex characters of G. In this paper, we consider <\hat{G}, *> as a polygroup, where for each chi_i ,chi_j in \hat{G} the product \chi _{i} * \chi_{j} is the set of those irreducible constituents which appear in the element-wise product \chi_{i} \chi_{j}. We call that \hat{G} simple if it has no proper normal subpolygroup and show that if \hat{G} is a single power cyclic polygroup, then \hat{G} is a simple polygroup and hence \hat{S}_{n} and \hat{A}_{n} are simple polygroups. Also, we prove that if G is a non-abelian simple group, then \hat{G} is a single power cyclic polygroup. Moreover, we classify \hat{D}_{2n} for all n. Also, we prove that \hat{T}_{4n} and \hat{U}_{6n} are cyclic polygroups with finite period.

Kazhdan–Lusztig representations and Whittaker space of some genuine representations

We prove a formula for the dimension of Whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups. The formula explicitly relates such dimension to the Kazhdan-Lusztig representations associated with certain right cells of the Weyl group. We also state a refined version of the formula, which is proved under some natural assumption. The refined formula is also verified unconditionally in several important cases.

On the number of constituents of induced modules of Ariki-Koike algebras

We examine the crystal graph of the slˆe-module arising from an slˆe-categorification to study the defining endo-functors of the categorification. This yields lower bounds on the number of irreducible constituents of certain objects. We use Ariki's categorification result on Ariki-Koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules. In particular this will imply reducibility of every induced module.