Representation Theory Of Finite Groups Research Articles

Computing character tables of symmetric groups using higher order permutation representations

In this paper, we will introduce the topic of representation theory by focusing on the techniques of computing character tables of finite groups, in particular, the symmetric group Sn . First, we will introduce some basic concepts of representation theory of finite groups over ℂ. Then we will compute the character tables for S 3 and S 4 using the classical methods introduced in Section 2. Also, we will construct a series of representations for Sn and compute their characters. Finally, we will compute the character tables for S 5 and S 6 using our method introduced in Section 3.

Some two-weight codes invariant under the 3-fold covers of the Mathieu groups M22 and Aut(M22)

Using an approach from finite group representation theory we construct quaternary non-projective codes with parameters [Formula: see text], quaternary projective codes with parameters [Formula: see text] and [Formula: see text] and binary projective codes with parameters [Formula: see text] as examples of two-weight codes on which a finite almost quasisimple group of sporadic type acts transitively as permutation groups of automorphisms. In particular, we show that these codes are invariant under the [Formula: see text]-fold covers [Formula: see text] and [Formula: see text], respectively, of the Mathieu groups [Formula: see text] and [Formula: see text]. Employing a known construction of strongly regular graphs from projective two-weight codes we obtain from the binary projective (respectively, quaternary projective) two-weight codes with parameters those given above, the strongly regular graphs with parameters [Formula: see text] and [Formula: see text] respectively. The latter graph can be viewed as a [Formula: see text]-[Formula: see text]-symmetric design with the symmetric difference property whose residual and derived designs with respect to a block give rise to binary self-complementary codes meeting the Grey–Rankin bound with equality.

Modular representations of finite groups and Lie theory

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent.The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character.We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.

Development of the Method of Averaging in Clifford Geometric Algebras

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli’s theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications.

Left regular bands of groups and the Mantaci–Reutenauer algebra

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci–Reutenauer algebra MRn[G] associated to any finite group G. When G is abelian, we give closed form expressions for these idempotents, and when G is the cyclic group of order two, we prove that they recover idempotents introduced by Vazirani.

Character Theory and Categorification

Over a hundred years after the work of Frobenius and Schur, the sheer enormity of what is not known about the character theory of symmetric and alternating groups continues to surprise and awe the uninitiated. How does one decompose the tensor product of a pair of complex characters? Or the restriction of a complex character to a Sylow or wreath product subgroup? Can we understand the vanishing sets of complex characters? What about the asymptotic behaviour of complex characters? What are the dimensions of the modular characters? These questions have been hailed as some of the definitive open problems in representation theory and algebraic combinatorics, they have deep connections with Lie theory, group theory, local-global conjectures in representation theory of finite groups, symplectic geometry, complexity theory, statistical mechanics and quantum information theory. The overarching theme of this proposal is the use of hidden, richer representation theoretic structures arising in modular, local-global, and categorical representation theory in order to prove and disprove conjectures concerning characters of symmetric and alternating groups.

Research in Representation Theory of Finite Groups

Research in Representation Theory of Finite Groups

A Tour of p-Permutation Modules and Related Classes of Modules

This survey provides an overview of numerous results on p-permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-p-permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the p-permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.

A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations

A hybrid classical-quantum approach for the solution of nonlinear ordinary differential equations using Walsh-Hadamard basis functions is proposed. Central to this hybrid approach is the computation of the Walsh-Hadamard transform of arbitrary vectors, which is enabled in our framework using quantum Hadamard gates along with state preparation, shifting, scaling, and measurement operations. It is estimated that the proposed hybrid classical-quantum approach for the Walsh-Hadamard transform of an input vector of size N results in a considerably lower computational complexity (O(N) operations) compared to the Fast Walsh-Hadamard transform (O(Nlog2(N)) operations). This benefit will also be relevant in the context of the proposed hybrid classical-quantum approach for the solution of nonlinear differential equations. Comparisons of results corresponding to the proposed hybrid classical-quantum approach and a purely classical approach for the solution of nonlinear differential equations (for cases involving one and two dependent variables) were found to be satisfactory. Some new perspectives relevant to the natural ordering of Walsh functions (in the context of both classical and hybrid approaches for the solution of nonlinear differential equations) and the representation theory of finite groups are also presented here.

Alperin weight conjecture and related developments

The Alperin weight conjecture is central to the modern representation theory of finite groups, and it is still open, despite many different approaches from different points of view. This paper surveys methods and results relating the Alperin weight conjecture, especially the recent developments in the inductive investigation.

Factoring the Dedekind-Frobenius determinant of a semigroup

The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutative chain ring. The former was motivated by combinatorics and the latter by coding theory over finite rings. We prove that the algebra of the multiplicative semigroup of a finite Frobenius ring is Frobenius over any field whose characteristic doesn't divide that of the ring. As a consequence we obtain an easier proof of Kovács's theorem that the algebra of the monoid of matrices over a finite field is a direct product of matrix algebras over group algebras of general linear groups (outside of the characteristic of the finite field).

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

Derived equivalence classification of Brauer graph algebras

We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider A∞-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences between Brauer graph algebras with the same derived invariants. As part of the proof we provide an interpretation of derived invariants of Brauer graph algebras as orbit invariants of line fields under the action of the mapping class group.

Quasi-isolated blocks and the Alperin–McKay conjecture

AbstractThe Alperin–McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin–McKay conjecture holds if the so-called inductive Alperin–McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper, we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this, we obtain a proof of the Alperin–McKay conjecture for$2$-blocks of finite groups with abelian defect.

Applying the symmetry groups to study the n body problem

We introduce an algebraic method to study local stability in the Newtonian n-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. Using representation theory of finite groups, we explicitly found the eigenvalues of certain 8×8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.

On the Galois symmetries for the character table of an integral fusion category

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. Based on these symmetries, we generalize a well-known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non-invertible object in the character table of a weakly integral fusion category contains a zero entry.

Jordan decomposition for the Alperin–McKay conjecture

Späth showed that the Alperin–McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin–McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnafé–Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.

The Dipper-Du conjecture revisited

We consider vertices, a notion originating in local representation theory of finite groups, for the categoryO\mathcal {O}of a rational Cherednik algebra and prove the analogue of the Dipper-Du Conjecture for Hecke algebras of symmetric groups in that setting. As a corollary we obtain a new proof of the Dipper-Du Conjecture overC\mathbb {C}.

A useful tool for constructing linear codes

We introduce and discuss an elementary tool from representation theory of finite groups for constructing linear codes invariant under a given permutation group G. The tool gives theoretical insight as well as a recipe for computations of generator matrices and weight distributions. In some interesting cases a classification of code vectors under the action of G can be obtained. As an explicit example a class of binary codes is studied extensively which is closely related to the class of binary codes associated to triangular graphs. A second explicit application is related to the action of the Mathieu simple group M24 on the set of octads giving many binary codes of length 759 with interesting properties. We also obtain new alternative proofs for several other theorems and construct several new codes invariant under various subgroups of the Conway simple group Co1.

Computing homological residue fields in algebra and topology

We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups.